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#  On rectifiable measures in Carnot groups: representation

Gioacchino Antonelli* and Andrea Merlo**

#### Abstract

This paper deals with the theory of rectifiability in arbitrary Carnot groups,
and in particular with the study of the notion of $\mathscr{P}$-rectifiable
measure.

First, we show that in arbitrary Carnot groups the natural infinitesimal
definition of rectifiabile measure, i.e., the definition given in terms of the
existence of flat tangent measures, is equivalent to the global definition
given in terms of coverings with intrinsically differentiable graphs, i.e.,
graphs with flat Hausdorff tangents. In general we do not have the latter
equivalence if we ask the covering to be made of intrinsically Lipschitz
graphs.

Second, we show a geometric area formula for the centered Hausdorff measure
restricted to intrinsically differentiable graphs in arbitrary Carnot groups.
The latter formula extends and strengthens other area formulae obtained in the
literature in the context of Carnot groups.

As an application, our analysis allows us to prove the intrinsic
$C^{1}$-rectifiability of almost all the preimages of a large class of
Lipschitz functions between Carnot groups. In particular, from the latter
result, we obtain that any geodesic sphere in a Carnot group equipped with an
arbitrary left-invariant homogeneous distance is intrinsic
$C^{1}$-rectifiable. ††footnotetext: * Scuola Normale Superiore, Piazza dei
Cavalieri, 7, 56126 Pisa, Italy, ††footnotetext: ** Université Paris-Saclay,
307 Rue Michel Magat Bâtiment, 91400 Orsay, France.

#### Keywords

Carnot groups, Rectifiability, Rectifiable set, Rectifiable measure, Area
formula, Preiss’ tangent measure.

#### MSC (2010)

53C17, 22E25, 28A75, 49Q15, 26A16.

##  1 Introduction

In the Euclidean setting the notion of rectifiable set, and more in general
that of rectifiable measure, can be given in two equivalent ways. Either one
could prescribe the infinitesimal behaviour of the measure by saying that it
has flat tangent measures almost everywhere, i.e., Hausdorff measures
supported on vector subspaces of dimension $k$ or, following a global
approach, one could say that the measure is absolutely continuous with respect
to the Hausdorff $k$-dimensional measure, and that it is supported on a
countable union of $k$-dimensional Lipschitz graphs. In Euclidean spaces the
latter two notions are equivalent, pretty well-understood and thoroughly
studied, see [15, 42, 38, 10].

In the last two decades an increasing interest has grown towards the
understanding of rectifiability in some specific non-smooth contexts, such as
the context of Carnot groups, see Section 2 for details. A Carnot group
$\mathbb{G}$ is a simply connected nilpotent Lie group, whose Lie algebra is
stratified and generated by its first layer. Carnot groups are a
generalization of Euclidean spaces, and we remark that (quotients of) Carnot
groups arise as the infinitesimal models of sub-Riemannian manifolds. The
geometry of a Carnot group, even at an infinitesimal scale, might be very
different from the Euclidean one. On every Carnot group we have a natural
family of anisotropic dilations $\\{\delta_{\lambda}\\}_{\lambda>0}$. We
always endow $\mathbb{G}$ with an arbitrary left-invariant homogeneous (with
respect to $\\{\delta_{\lambda}\\}_{\lambda>0}$) distance $d$, and we recall
that any two of them are bi-Lipschitz equivalent. We denote $Q$ the Hausdorff
dimension of $\mathbb{G}$ with respect to any of such distances.

As shown in the fundamental papers [22, 21], in step-2 Carnot groups the
reduced boundary of a finite perimeter set can be covered up to
$\mathcal{H}^{Q-1}$-negligible sets by countably many intrinsic
$C^{1}$-regular hypersurfaces, $C^{1}_{\mathrm{H}}$ hypersurfaces from now on,
see 5.2. The positive De Giorgi’s rectifiability result in [22] has started an
effort to study Geometric Measure Theory in sub-Riemannian Carnot groups, and
in particular to study various notion(s) of rectifiability, mostly given
following the global approach described at the beginning of this paragraph.

One of the big efforts in this study is trying to understand what is the
correct class of building blocks to consider in order to give a satisfactory
global definition of rectifiable set, or measure, in the setting of Carnot
groups. The first notion of rectifiability that has been proposed and studied
is the one which considers as buliding blocks $C^{1}_{\mathrm{H}}$-surfaces,
as explained above, see [21, 23, 24, 35, 27]. Then a notion of intrinsic
Lipschitz graph (see 2.12) has been proposed and studied in [20, 17], and
relations between the notion of intrinsic Lipschitz rectifiability and the
notion of $C^{1}_{\mathrm{H}}$-rectifiability have been investigated in [19,
16]. The problem of linking the latter two definitions of rectifiability with
the infinitesimal viewpoint was raised in [39] in the setting of Heisenberg
groups $\mathbb{H}^{n}$. From the results in [39] one deduces that in
$\mathbb{H}^{n}$ the natural infinitesimal notion of rectifiable measure -
namely the one given in terms of the existence of flat tangent measures almost
everywhere - agrees with the one given in terms of intrinsic Lipschitz graphs
in low dimensions, and with the one given in terms of
$C^{1}_{\mathrm{H}}$-surfaces in low codimensions. Eventually, it took about
ten years to conclude that a Rademacher theorem for intrinsic Lipschitz
functions in low codimensions holds in $\mathbb{H}^{n}$, see [43]. As a
consequence, at least in $\mathbb{H}^{n}$, the natural infinitesimal
definition of rectifiability always agrees with the one given in terms of
coverings with intrinsic Lipschitz graphs. An analysis similar to the one of
[39] has been pursued in [26] in the setting of homogeneous groups and for
measures with horizontal tangents.

Other notions of rectifiability modelled on Lipschitz images of (homogeneous
subgroups of) Carnot groups have been proposed by Pauls and Cole-Pauls in [41,
8]. An interesting open question asks whether in $\mathbb{H}^{1}$ the notion
of rectifiability by means of $C^{1}_{\mathrm{H}}$-hypersurfaces is equivalent
to the one of Cole-Pauls given in [8], see [5, 11] for some partial results.
Nevertheless, in arbitrary Carnot groups, the two notions could be very
different, see the results in [2]. A weak notion of rectifiability in terms of
building blocks that satisfy some mild cone property has also been recently
investigated in [12].

On the other hand, from the infinitesimal viewpoint, a notion that makes sense
in arbitrary Carnot groups has been proposed in [40] by the second-named
author, namely the notion of $\mathscr{P}$-rectifiable measure, which we
recall here. We recall that a subgroup $\mathbb{V}$ of $\mathbb{G}$ is said to
be homogeneous if it is closed under the action of the dilations
$\\{\delta_{\lambda}\\}_{\lambda>0}$. Again we remark that $\mathbb{G}$ is
endowed with a left-invariant homogeneous distance $d$.

######  Definition 1.1 ($\mathscr{P}$-rectifiable measures).

Fix a natural number $1\leq h\leq Q$. A Radon measure $\phi$ on $\mathbb{G}$
is said to be $\mathscr{P}_{h}$-rectifiable if for $\phi$-almost every
$x\in\mathbb{G}$ we have

  * (i)

$0<\Theta^{h}_{*}(\phi,x)\leq\Theta^{h,*}(\phi,x)<+\infty$,

  * (ii)

$\mathrm{Tan}_{h}(\phi,x)\subseteq\\{\lambda\mathcal{S}^{h}\llcorner\mathbb{V}(x):\lambda\geq
0\\}$, where $\mathbb{V}(x)$ is a homogeneous subgroup of $\mathbb{G}$ of
Hausdorff dimension $h$,

where $\Theta^{h}_{*}(\phi,x)$ and $\Theta^{h,*}(\phi,x)$ are, respectively,
the lower and the upper $h$-density of $\phi$ at $x$, see 2.7,
$\mathrm{Tan}_{h}(\phi,x)$ is the set of $h$-tangent measures to $\phi$ at
$x$, see 2.6, and $\mathcal{S}^{h}$ is the spherical Hausdorff measure of
dimension $h$, see 2.4.

In [4] we started to study structure results for the class of
$\mathscr{P}$-rectifiable measures, proving a Marstrand-Mattila type
rectifiability criterion in the co-normal case [4, Theorem 1.3]. The latter
theorem directly leads to the proof of Preiss’s theorem in the first
Heisenberg group $\mathbb{H}^{1}$ equipped with the Koranyi norm, see [4,
Theorem 1.4]. In this paper we complete the study of the notion of
$\mathscr{P}$-rectifiable measure, when the tangents are complemented, showing
that the notion of $\mathscr{P}$-rectifiability, which is infinitesimal in
nature, is equivalent to the global one given in terms of intrinsic
differentiable graphs, see 1.1. We stress that our 1.1 extends to arbitrary
Carnot groups and to all the dimensions the results given in [39].

All in all we conclude that, in Carnot groups, the correct building blocks to
consider in order to give a global definition of rectifiability that agrees
with the infinitesimal one seem to be intrinsic differentiable graphs. We also
provide an area formula for such building blocks, see 1.3. We stress that, due
to the existence of intrinsic Lipschitz graphs that are nowhere intrinsically
differentiable, see [28], one cannot give a geometric area formula in the
spirit of 1.3 for arbitrary intrinisc Lipschitz graphs. Nevertheless the area
formulae in 1.2 and 1.3 extend the area formula given in [27, Theorem 1.1],
see the discussion after 1.3.

We stress that one of the main achievements of this paper is also the
rectifiability criterion in 3.9 which allows to prove the
$\mathscr{P}$-rectifiability of almost all the preimages of a large number of
Lipschitz functions, see 1.4, and 1.5.

Remark: Some of the results presented in this paper use results proven in [4,
Sections 2-3-4-6]. We recall the most important ones in the preliminary
section of this work, see Section 2. During the proofs we give precise
references to the results of [4] when we need them.

###  1.1 Main results

We discuss the main contributions of the present paper. We fix $\mathbb{G}$ a
Carnot group and we equip it with a left-invariant homogeneous distance. We
recall that when we say that a homogeneous subgroup $\mathbb{V}$ admits a
complementary subgroup, we mean that there exists a homogeneous subgroup
$\mathbb{L}$ such that $\mathbb{G}=\mathbb{V}\cdot\mathbb{L}$ and
$\mathbb{V}\cap\mathbb{L}=\\{0\\}$. The first result of this work is a
complete characterization of $\mathscr{P}_{h}$-rectifiable sets with
complemented tangents in arbitrary Carnot groups either in terms of the
existence of flat $h$-dimensional complemented Preiss’s tangents almost
everywhere or in terms of a covering property with $h$-dimensional
intrinsically differentiable graphs with complemented tangents. We recall
that, while $\mathrm{Tan}_{h}(\phi,x)$ captures the behaviour of tangent
measures obtained rescaling with the $h$-th power of the scale, see 2.6, the
Preiss’s tangent $\mathrm{Tan}(\phi,x)$, see 2.6, captures the behaviour of
all the possible tangent measures, namely

$\mathrm{Tan}(\phi,x):=\\{\nu:\,\,\text{there exist $\\{c_{i}\\}$, with
$c_{i}>0$, and $\\{r_{i}\\}$ with $r_{i}\to_{i}0$ such
that}\,\,c_{i}T_{x,r_{i}}\phi\rightharpoonup_{i}\nu\\},$

where the convergence of measures is meant in the duality with
$C_{c}(\mathbb{G})$, see 2.5. For the reader’s convenience we recall here that
an intrinsic graph with respect to a splitting
$\mathbb{G}=\mathbb{V}\cdot\mathbb{L}$ of the group is said to be
intrinsically differentiable at one of its points if the Hausdorff tangent at
that point is a homogeneous subgroup, see 3.1 for a precise definition. For
the proof of the next statement, see the end of Section 3.

######  Theorem 1.1.

Let $\mathbb{G}$ be a Carnot group and fix a natural number $1\leq h\leq Q$.
Let $\Gamma\subseteq\mathbb{G}$ be a Borel set such that
$\mathcal{S}^{h}(\Gamma)<+\infty$, where $\mathcal{S}^{h}$ is the
$h$-dimensional spherical Hausdorff measure. Then the following are equivalent

  1. 1.

$\mathcal{S}^{h}\llcorner\Gamma$ is a $\mathscr{P}_{h}$-rectifiable measure
with complemented tangents, i.e., a $\mathscr{P}_{h}^{c}$-rectifiable measure,
see 2.14,

  2. 2.

For $\mathcal{S}^{h}\llcorner\Gamma$-almost every $x\in\mathbb{G}$ we have

$\mathrm{Tan}(\mathcal{S}^{h}\llcorner\Gamma,x)=\\{\lambda\mathcal{S}^{h}\llcorner\mathbb{V}(x):\lambda>0,\mathbb{V}(x)\,\text{is
a {complemented} homogeneous subgroup of $\mathbb{G}$ with
$\mathrm{dim}_{H}\mathbb{V}(x)=h$}\\},$

  3. 3.

There exist countably many compact intrinsic Lipschitz graphs $\Gamma_{i}$
that are $h$-dimensional intrinsically differentiable graphs at
$\mathcal{S}^{h}$-almost every $x\in\Gamma_{i}$, that have complemented
Hausdorff tangents at $\mathcal{S}^{h}$-almost every $x\in\Gamma_{i}$, and
such that

$\mathcal{S}^{h}(\Gamma\setminus\cup_{i=1}^{+\infty}\Gamma_{i})=0.$

Moreover, denoting with $\mathcal{C}^{h}$ the centered Hausdorff measure of
dimension $h$, see 2.4, if any of the previous holds, then
$\Theta^{h}(\mathcal{C}^{h}\llcorner\Gamma,x)=1$ exists for
$\mathcal{C}^{h}\llcorner\Gamma$-almost every $x\in\mathbb{G}$ and

$r^{-h}(T_{x,r})_{*}(\mathcal{C}^{h}\llcorner\Gamma)\rightharpoonup\mathcal{C}^{h}\llcorner\mathbb{V}(x),\qquad\text{for
$\mathcal{C}^{h}\llcorner\Gamma$-almost every $x\in\mathbb{G}$,}$

where the convergence of measures is meant in the duality with
$C_{c}(\mathbb{G})$.

Let us observe that when a Rademacher theorem is available, we can
equivalently consider as the building blocks in item 3. of 1.1 the class of
$h$-dimensional intrinsically Lipschitz graphs, without asking anything a
priori on the differentiability. Let us recall that a Rademacher theorem is
proved in [16, 19] in the setting of Carnot groups $\mathbb{G}$ of type
$\star$, i.e., a class strictly larger than Carnot groups of step 2, and for
maps $\varphi:U\subseteq\mathbb{W}\to\mathbb{L}$, where $\mathbb{W}$ and
$\mathbb{L}$ are complementary subgroups of $\mathbb{G}$, with $\mathbb{L}$
horizontal and one-dimensional. Moreover, with the recent results of [32], the
latter codimension-one Rademacher theorem can be extended to the groups of
type diamond introduced in [32]. Recently, by making use of the theory of
currents, the author of [43] has proved the Rademacher theorem for
intrinsically Lipschitz maps between complementary subgroups of any dimension
in the Heisenberg groups $\mathbb{H}^{n}$, while in [3] we proved the validity
of a Rademacher theorem for co-normal intrinsically Lipschitz graphs.

Nevertheless, Rademacher theorem is now known to be false in arbitrary Carnot
groups in a very strict sense, i.e., there exists an $h$-dimensional
intrinsically Lipschitz graph in a Carnot group such that at every point of it
there exist infinitely many blow-ups and each of these blow-ups is not a
homogeneous subgroup, see [28, Theorem 1.1]. This latter result implies that
in general in item 3. of 1.1 one cannot equivalently consider as building
blocks of a locally well-behaved definition of rectifiable sets the family of
$h$-dimensional intrinsically Lipschitz graphs. So, in some sense, the result
of 1.1 is sharp also in view of the negative result of [28].

Let us further notice that we do not consider in this work the relations
between the three items in 1.1 and the existence of an approximate tangent in
the sense of [38, Definition 15.17] (cf. [39, Definition 3.7]). This relation
will be target of future investigations. All in all, taking into account that
$\mathcal{S}^{h}\llcorner\Gamma$ is $\mathscr{P}_{h}^{c}$-rectifiable with co-
horizontal tangents if and only if $\Gamma$ is
$C^{1}_{H}(\mathbb{G},\mathbb{R}^{Q-h})$-rectifiable, see 5.3 and 5.3, our
result in 1.1 extends and strengthens [39,
(i)$\Leftrightarrow$(ii)$\Leftrightarrow$(iv)$\Leftrightarrow$(v) of Theorem
3.15]. Notice also that in the previous chain of equivalences, we can also
drop the assumption on the lower density in [39, (iv),(v)]. Moreover, taking
into account the Rademacher theorem of [3] in the co-normal case, our result
in 1.1 extends [39,
(i)$\Leftrightarrow$(ii)$\Leftrightarrow$(iv)$\Leftrightarrow$(v) of Theorem
3.14] as well. Let us recall, for the reader’s convenience, that [39, Theorem
3.15] deals with the characterization of co-horizontal rectifiability in the
Heisenberg groups $\mathbb{H}^{n}$, while [39, Theorem 3.14] deals with the
characterization of horizontal rectifiability in the Heisenberg groups
$\mathbb{H}^{n}$.

Let us final notice that we stated our result in 1.1 for measures of the form
$\mathcal{S}^{h}\llcorner\Gamma$, but we could also give a version of it for
Radon measures with $\Theta^{h,*}(\phi,x)<+\infty$ for $\phi$-almost every
$x\in\mathbb{G}$, after having proven the analogue of 3.7 for measures.

The second result of the work is an area formula for intrinsic Lipschitz
graphs that are in addition $\mathscr{P}_{h}^{c}$-rectifiable. The proof of
the following statement is at the end of Section 4.

######  Theorem 1.2.

Let $\mathbb{V},\mathbb{L}$ be homogeneous complementary subgroups of a Carnot
group $\mathbb{G}$ such that $h:=\dim_{H}\mathbb{V}$. Let $\Gamma$ be the
graph of an intrinsic Lipschitz map
$\varphi:A\subseteq\mathbb{V}\to\mathbb{L}$ (see 2.12), with $A$ Borel, such
that $\mathcal{S}^{h}\llcorner\Gamma$ is $\mathscr{P}_{h}^{c}$-rectifiable
with tangent measures $\mathcal{S}^{h}\llcorner\Gamma$-almost everywhere
supported on homogeneous subgroups complemented by $\mathbb{L}$. Then, for
every Borel function $\psi:\Gamma\to[0,+\infty)$ the following area formula
holds

$\int_{\Gamma}\psi
d\mathcal{C}^{h}\llcorner\Gamma=\int_{A}\psi(a\cdot\varphi(a))\mathcal{A}(\mathbb{V}(a\cdot\varphi(a)))\mathcal{C}^{h}\llcorner\mathbb{V},$
(1)

where $\mathcal{C}^{h}$ is the centered Hausdorff measure, see 2.4,
$\mathbb{V}(a\cdot\varphi(a))$ is the tangent on which it is supported the
tangent measure of $\mathcal{S}^{h}\llcorner\Gamma$ at the point
$a\cdot\varphi(a)\in\Gamma$, and $\mathcal{A}(\cdot)$ is the centered area
factor defined with respect to the splitting
$\mathbb{G}=\mathbb{V}\cdot\mathbb{L}$, see 4.1.

A consequence of the previous result is the following one, which is an area
formula for intrinsic Lipschitz graphs that are also intrinsically
differentiable. The proof of the following statement can be found at the end
of Section 4.

######  Theorem 1.3.

Let $\mathbb{V},\mathbb{L}$ be homogeneous complementary subgroups of a Carnot
group $\mathbb{G}$ such that $h=\dim_{H}\mathbb{V}$. Let $\Gamma$ be the graph
of an intrinsic Lipschitz map $\varphi:A\subseteq\mathbb{V}\to\mathbb{L}$,
with $A$ Borel. Let us assume $\Gamma$ is an intrinsically differentiable
graph (see 3.1) at $\mathcal{S}^{h}$-almost every $x\in\Gamma$ and let us
assume that the Hausdorff tangent $\mathbb{V}(x)$ of $\Gamma$ at $x$ is
complemented by $\mathbb{L}$ at $\mathcal{S}^{h}$-almost every $x\in\Gamma$.
Then, for every Borel function $\psi:\Gamma\to[0,+\infty)$, the area formula
in (1) holds.

Let us remark that, taking into account that
$\Theta^{h}(\mathcal{C}^{h}\llcorner\Gamma,x)=1$ for
$\mathcal{C}^{h}\llcorner\Gamma$-almost every $x\in\mathbb{G}$, see 1.1, and
considering the result in [27, Corollary 3.6] one can show that (1) extends
and strengthens the area formula of [27, Theorem 1.1]. Indeed, the graph of a
$C^{1}_{\mathbb{W},\mathbb{V}}$ function as in the statement of [27, Theorem
1.1] is a $\mathscr{P}_{h}^{c}$-rectifiable set, see [4, Proposition 6.2].
Moreover, we stress that 1.2 strictly strengthens [27, Theorem 1.1] for two
reasons: there exists natural examples of graphs that are
$\mathscr{P}_{h}$-rectifiable but not $C^{1}_{\mathrm{H}}$-rectifiable, see
[4, Remark 6.3], and moreover in our result the map $\varphi$ does not need to
be defined on an open set but just on a Borel set.

Let us notice that the area formula in 1.3 is geometric. It roughly asserts
that when an intrinsic Lipschitz graph over the split
$\mathbb{V}\cdot\mathbb{L}$ has almost everywhere a flat Hausdorff tangent
complemented by $\mathbb{L}$, then the area of this graph can be obtained
integrating on $\mathbb{V}$ a geometric area factor. With geometric we mean
that the factor only depends on the tangent. Let us stress that when a
Rademacher theorem is available, one can remove the hypothesis about the
intrinsic differentiability in 1.3. Nevertheless, as discussed above, a
Rademacher theorem might not hold in arbitrary Carnot groups, see [28].

Let us point out that in the literature one can find many more analytic area
formulae in Carnot groups, i.e., in which the area element is expressed in
terms of properly defined intrinsic derivatives of the map $\varphi$. This is
the case of [9, Theorem 1.1 and Theorem 1.2] for low-codimensional
$C^{1}_{\mathrm{H}}$-surfaces in Heisenberg groups (cf. also [20, Theorem 2]),
which has been extended to intrinsic Lipschitz low-codimensional surfaces in
[43, Theorem 1.3] (cf. also [7, Theorem 1.6]); and of [1, Proposition 1.8] for
one-codimensional $C^{1}_{\mathrm{H}}$-graphs in arbitrary Carnot groups.
These formulas could be derived from 1.3 explicitly writing the area element
in terms of the intrinsic derivatives of the parametrisation map $\varphi$.
Other geometric area formulae for Euclidean $C^{1}$ or $C^{1,1}$-submanifolds
in Carnot groups have been investigated in [37, 36, 34]. Let us remark that
our point of view is intrinsic while on the contrary the works [37, 36, 34]
investigate Euclidean-regular manifolds. The results in [37, Theorem 1.1 and
Theorem 1.2], [36, Theorem 1.1], and [34, Theorem 1.1 and Theorem 1.2] roughly
assert that whenever a point of a Euclidean-regular submanifold is
sufficiently nice, then the intrinsic blow-up at that point exists and it is a
homogeneous subgroup; and as a consequence also the density - of the correct
dimension - of the (Euclidean) volume measure of the submanifold exists at
that point. Then what one notices is that in a lot of cases the nice points
are almost all - with respect to the intrinsic Hausdorff measure of the
correct dimension - the points of the submanifold, cf. [36, Theorem 1.2].
These latter results have to be compared with our 3.9 in which we prove that
having almost everywhere an intrinsic complemented blow-up implies the
existence of the density of the Hausdorff measure. Let us notice that when a
negligibility theorem, a blow-up theorem, and the existence of the density
hold in the sense of [37, 36, 34] discussed above for a Euclidean-regular
submanifold $\Sigma$, then one gets that
$\mathrm{Tan}_{h}(\mathcal{S}^{h}\llcorner\Sigma,x)\subseteq\\{\lambda\mathcal{S}^{h}\llcorner\mathbb{V}(x):\lambda>0\\}$
holds for $\mathcal{S}^{h}$-almost every $x\in\Sigma$, where $h$ is the
Hausdorff dimension of $\Sigma$, and where $\mathbb{V}(x)$ is a homogeneous
subgroup. This last observation easily follows arguing as in the last lines of
the proof of 3.9. As a result, when a negligibility theorem, a blow-up
theorem, and the existence of the density hold in the sense of [37, 36, 34]
for a Euclidean-regular submanifold $\Sigma$, one gets that
$\mathcal{S}^{h}\llcorner\Sigma$ is a $\mathscr{P}_{h}$-rectifiable measure.

Let us finally notice that the area formula in 1.3 is a formula for the
building blocks for the global notion of rectifiability in item 3. of 1.1.
Thus, by localization, one obtains from 1.3 and 1.1, an area formula for
arbitrary $\mathscr{P}_{h}^{c}$-rectifiable measures, see 4.7.

The third result of the work is a rectifiability result for the level sets of
Lipschitz functions between Carnot groups. For the proof of the following
statement, we refer the reader to 5.1.

######  Proposition 1.4.

Let $B$ be a Borel set in $\mathbb{G}$ and suppose $\mathbb{H}$ is a Carnot
group of homogeneous dimension $Q^{\prime}$ with $Q\geq Q^{\prime}$. Let
$f:B\subseteq\mathbb{G}\to\mathbb{H}$ be a Lipschitz map such that

$\text{$\mathrm{Ker}(df(x))$ is a {complemented} homogeneous sbugroup of
$\mathbb{G}$ for $\mathcal{S}^{Q}$-almost every $x\in\\{z\in
B:df(z)\,\text{exists surjective}\\}$},$

where $df(x)$ is the Pansu differential that exists for
$\mathcal{S}^{Q}$-almost every $x\in\mathbb{G}$, see 5.1. Then, for
$\mathcal{S}^{Q^{\prime}}$-almost every $y\in f(B)$, the measure
$\mathcal{S}^{Q-Q^{\prime}}\llcorner f^{-1}(y)$ is
$\mathscr{P}^{c}_{Q-Q^{\prime}}$-rectifiable in $\mathbb{G}$ and

$\mathrm{Tan}_{Q-Q^{\prime}}(\mathcal{S}^{Q-Q^{\prime}}\llcorner
f^{-1}(y),x)\subseteq\\{\lambda\mathcal{S}^{Q-Q^{\prime}}\llcorner\mathrm{Ker}(df(x)):\lambda>0\\},\quad\text{for
$\mathcal{S}^{Q-Q^{\prime}}$-almost every $x\in f^{-1}(y)$.}$

As an immediate consequence of the previous result we have the following
corollary, which is worth pointing out explicitly, since, even if simple to
state, it seems to be not present in the literature up to the authors’
knowledge. For a proof, see 5.4. We remark that a more general statement, in
the co-horizontal case, is still true and can be found in 5.2.

######  Corollary 1.5.

Suppose $f:B\subseteq\mathbb{G}\to\mathbb{R}$ is a Lipschitz map, where $B$ is
a Borel set. Then, for $\mathcal{S}^{1}$-almost every $y\in f(B)$ the set
$f^{-1}(y)$ is $C^{1}_{\mathrm{H}}$-rectifiable. In particular, for every
$r>0$ and $x\in\mathbb{G}$, every geodesic sphere $\partial B_{r}(x)$ is
$C^{1}_{\mathrm{H}}$-rectifiable.

Notice that the last part of the previous result comes from the first applied
to the distance function from $x$, and the observation that, by dilating, once
$\partial B_{r}(x)$ is $C^{1}_{\mathrm{H}}$-rectifiable for _one_ radius
$r>0$, then it is $C^{1}_{\mathrm{H}}$-rectifiable for _every_ radius $r>0$.
The previous corollary should be compared with [44, Theorem 3.2], where an
intrinisc Lipschitz rectifiability for Lipschitz surfaces is proved in CC-
spaces. Notice, however, that nowadayas it is not known wheter, in
codimension-one, intrinisc Lipschitz rectifiability and
$C^{1}_{\mathrm{H}}$-rectifiability are equivalent in arbitrary Carnot groups;
while being intrinisc Lipschitz rectifiable is weaker than being
$C^{1}_{\mathrm{H}}$-rectifiable. Nevertheless, our previous corollary
provides the $C^{1}_{\mathrm{H}}$-rectifiability of all the spheres in every
Carnot group. It is however interesting to point out how the Euclidean
rectifiability of the geodesic spheres in Carnot groups is still an intriguing
open question in general, and it is related to asymptotic volume expansion in
nilpotent Lie groups [6, 31].

Let us remark that the previous results in 1.4 and 1.5 follow from the
rectifiability criterion in 3.9. It is worth pointing out that, given a
Lipschitz function $f:B\subseteq\mathbb{G}\to\mathbb{R}$, for every $y\in
f(B)$, the set $\\{f\leq y\\}$ is of locally finite perimeter in $\mathbb{G}$,
see, e.g., [13, Theorem 2.40]. Hence, taking into account 1.5, we deduce the
following consequence: $\mathcal{S}^{1}$-almost all the sublevel sets of real-
valued Lipschitz functions defined on Borel subsets of Carnot groups are
examples of sets of locally finite perimeter whose boundary is
$C^{1}_{\mathrm{H}}$-rectifiable - namely De Giorgi’s rectifiability Theorem
holds for such sets.

Let us finally stress that the previous results in 1.4 and 1.5 open the way to
proving slicing theorems and coarea formulae for $\mathscr{P}$-rectifiable and
Lipschitz slicing functions. This will be target of future investigations.

Let us briefly comment on the proof of 1.1. For what concerns the implications
1. $\Rightarrow$ 2., and 1. $\Rightarrow$ 3., the first is just a matter of
routine argument, see [38, Remark 14.4(3)], and the second is a consequence of
[4, Theorem 1.8]. The main new contributions of this paper, which lead to the
equivalence in 1.1, are the implications 2. $\Rightarrow$ 1., and 3.
$\Rightarrow$ 1., both of them non-trivial.

For what concerns the implication 2. $\Rightarrow$ 1., we first use that the
hypothesis of flat Preiss’s tangents allows to conclude that $\Gamma$ is
$\mathcal{S}^{h}$-almost everywhere covered by countably many $h$-dimensional
graphs $\Gamma_{i}$ of intrinsically Lipschitz functions, namely
$\mathcal{S}^{h}(\Gamma\setminus\cup_{i=1}^{+\infty}\Gamma_{i})=0$, see 2.26.
Hence we exploit the general fact, that dates back to Preiss’s paper (cf. [42,
Corollary 2.7]), that a measure with a compact-based tangent at a point is
asymptotically doubling at that point. Joining the latter two observations, we
deduce that, for every $i$, the measure $\mathcal{S}^{h}\llcorner\Gamma_{i}$
is asymptotically doubling, and then this enables us to prove that
$\Gamma_{i}$ has big projections on the plane over which $\Gamma_{i}$ is a
graph, see 3.3. The latter proposition is just a straightforward empowerment
of our result already proved in [4, Proposition 4.6]. Finally, the big
projections property of 3.3 allows us to conclude that the $h$-lower density
$\Theta^{h}_{*}(\mathcal{S}^{h}\llcorner\Gamma_{i},\cdot)$ is positive
$\mathcal{S}^{h}\llcorner\Gamma_{i}$ almost everywhere, see 3.6. Hence, the
proof of the implication 2. $\Rightarrow$ 1\. is concluded since we can argue,
by exploiting Lebesgue differentiation theorem, that
$\Theta^{h}_{*}(\mathcal{S}^{h}\llcorner\Gamma,\cdot)$ is positive
$\mathcal{S}^{h}\llcorner\Gamma$-almost everywhere, which was the non-trivial
missing information to prove 1. Let us notice that in 2. we are not requiring
anything a priori on the positivity of the $h$-lower density of
$\mathcal{S}^{h}\llcorner\Gamma$, otherwise the implication 2. $\Rightarrow$
1\. would have been trivial. Nevertheless, we deduce the positivity of the
$h$-lower density from the fact that the tangents are flat and complemented as
we discussed above.

The proof of the implication 3. $\Rightarrow$ 1\. relies on the fact that an
arbitrary $h$-dimensional (almost everywhere) intrinsically differentiable
graph $\Gamma$ with complemented Hausdorff tangents has the property that
$\mathcal{S}^{h}\llcorner\Gamma$ is $\mathscr{P}_{h}^{c}$-rectifiable. This is
exactly the content of 3.9. In order to prove the latter proposition, we show
that when we have an arbitrary $h$-dimensional (almost everywhere)
intrinsically differentiable graph $\Gamma$ with complemented Hausdorff
tangents, at ($\mathcal{S}^{h}\llcorner\Gamma$-almost) every point we have
that the graph $\Gamma$ is, at arbitrarily small scales, contained in a cone
with arbitrarily small opening and with basis the Hausdorff tangent at that
point. This observation enables us to perform a covering argument and to show
directly that $\Theta^{h}(\mathcal{C}^{h}\llcorner\Gamma,\cdot)=1$ at
$\mathcal{C}^{h}\llcorner\Gamma$-almost every point. Then the fact that
$\mathcal{C}^{h}\llcorner\Gamma$, and hence $\mathcal{S}^{h}\llcorner\Gamma$,
is $\mathscr{P}_{h}^{c}$-rectifiable is reached by using a classical argument,
see 2.3 and 2.4. Let us notice that in 3.9 it is essential to work with the
centered Hausdorff measure $\mathcal{C}^{h}\llcorner\Gamma$, since we consider
coverings with balls centered on $\Gamma$. It is also worth noticing that
$\mathcal{S}^{h}\llcorner\Gamma$ and $\mathcal{C}^{h}\llcorner\Gamma$ are
mutually absolutely continuous so any rectifiability property for one measure
is transferred to the other by means of Lebesgue differentiation theorem and
the locality of tangents.

The final part of the statement in 1.1 is a consequence of the fact that the
$h$-density of $\mathcal{C}^{h}\llcorner\Gamma$ is 1 as a consequence of the
previous reasoning, and the fact that
$\mathcal{C}^{h}\llcorner\mathbb{V}(B(0,1))=1$ for every homogeneous subgroup
$\mathbb{V}$ of Hausdorff dimension $h$, see Lemma 2.1.

Let us briefly comment on the proof of 1.2. The strategy of the proof is
similar to the one in [27] and it is based on a continuity property of the
volumes through a blow-up procedure, see 4.4. Nevertheless, in order to prove
4.4, one needs to face some delicate technical problems due to the fact that
the map $\varphi$ is not necessarily defined on an open set, but just on a
Borel set. Hence, one needs to argue directly on the graph by using a Vitali-
type differentiation theorem, see 4.3, and a new delicate estimate on the
volumes of the projections of balls in $\Gamma$, see 3.8.

##  2 Preliminaries

###  2.1 Carnot Groups

In this subsection we briefly introduce some notations on Carnot groups that
we will extensively use throughout the paper. For a detailed account on Carnot
groups we refer to [29].

A Carnot group $\mathbb{G}$ of step $\kappa$ is a connected and simply
connected Lie group whose Lie algebra $\mathfrak{g}$ admits a stratification
$\mathfrak{g}=V_{1}\,\oplus\,V_{2}\,\oplus\dots\oplus\,V_{\kappa}$. We say
that $V_{1}\,\oplus\,V_{2}\,\oplus\dots\oplus\,V_{\kappa}$ is a stratification
of $\mathfrak{g}$ if
$\mathfrak{g}=V_{1}\,\oplus\,V_{2}\,\oplus\dots\oplus\,V_{\kappa}$,

$[V_{1},V_{i}]=V_{i+1},\quad\text{for any
$i=1,\dots,\kappa-1$},\quad\text{and}\quad[V_{1},V_{\kappa}]=\\{0\\},$

where $[A,B]:=\mathrm{span}\\{[a,b]:a\in A,b\in B\\}$. We call $V_{1}$ the
_horizontal layer_ of $\mathbb{G}$. We denote by $n$ the topological dimension
of $\mathfrak{g}$, by $n_{j}$ the dimension of $V_{j}$ for every
$j=1,\dots,\kappa$. Furthermore, we define $\pi_{i}:\mathbb{G}\to V_{i}$ to be
the projection maps on the $i$-th strata. We will often shorten the notation
to $v_{i}:=\pi_{i}v$.

The exponential map $\exp:\mathfrak{g}\to\mathbb{G}$ is a global
diffeomorphism from $\mathfrak{g}$ to $\mathbb{G}$. Hence, if we choose a
basis $\\{X_{1},\dots,X_{n}\\}$ of $\mathfrak{g}$, any $p\in\mathbb{G}$ can be
written in a unique way as $p=\exp(p_{1}X_{1}+\dots+p_{n}X_{n})$. This means
that we can identify $p\in\mathbb{G}$ with the $n$-tuple
$(p_{1},\dots,p_{n})\in\mathbb{R}^{n}$ and the group $\mathbb{G}$ itself with
$\mathbb{R}^{n}$ endowed with $\cdot$ the group operation determined by the
Baker-Campbell-Hausdorff formula. From now on, we will always assume that
$\mathbb{G}=(\mathbb{R}^{n},\cdot)$ and, as a consequence, that the
exponential map $\exp$ acts as the identity.

The stratificaton of $\mathfrak{g}$ carries with it a natural family of
dilations $\delta_{\lambda}:\mathfrak{g}\to\mathfrak{g}$, that are Lie algebra
automorphisms of $\mathfrak{g}$ and are defined by

$\delta_{\lambda}(v_{1},\dots,v_{\kappa}):=(\lambda
v_{1},\lambda^{2}v_{2},\dots,\lambda^{\kappa}v_{\kappa}),\quad\text{for any
$\lambda>0$},$

where $v_{i}\in V_{i}$. The stratification of the Lie algebra $\mathfrak{g}$
naturally induces a gradation on each of its homogeneous Lie sub-algebras
$\mathfrak{h}$, i.e., a sub-algebra that is $\delta_{\lambda}$-invariant for
any $\lambda>0$, that is

$\mathfrak{h}=V_{1}\cap\mathfrak{h}\oplus\ldots\oplus
V_{\kappa}\cap\mathfrak{h}.$ (2)

We say that $\mathfrak{h}=W_{1}\oplus\dots\oplus W_{\kappa}$ is a gradation of
$\mathfrak{h}$ if $[W_{i},W_{j}]\subseteq W_{i+j}$ for every $1\leq
i,j\leq\kappa$, where we mean that $W_{\ell}:=\\{0\\}$ for every
$\ell>\kappa$. Since the exponential map acts as the identity, the Lie algebra
automorphisms $\delta_{\lambda}$ are also group automorphisms of $\mathbb{G}$.

######  Definition 2.1 (Homogeneous subgroups).

A subgroup $\mathbb{V}$ of $\mathbb{G}$ is said to be _homogeneous_ if it is a
Lie subgroup of $\mathbb{G}$ that is invariant under the dilations
$\delta_{\lambda}$.

We recall the following basic terminology: a horizontal subgroup of a Carnot
group $\mathbb{G}$ is a homogeneous subgroup of it that is contained in
$\exp(V_{1})$; a Carnot subgroup $\mathbb{W}=\exp(\mathfrak{h})$ of a Carnot
group $\mathbb{G}$ is a homogeneous subgroup of it such that the first layer
$V_{1}\cap\mathfrak{h}$ of the grading of $\mathfrak{h}$ inherited from the
stratification of $\mathfrak{g}$ is the first layer of a stratification of
$\mathfrak{h}$.

Homogeneous Lie subgroups of $\mathbb{G}$ are in bijective correspondence
through $\exp$ with the Lie sub-algebras of $\mathfrak{g}$ that are invariant
under the dilations $\delta_{\lambda}$. For any Lie algebra $\mathfrak{h}$
with gradation $\mathfrak{h}=W_{1}\oplus\ldots\oplus W_{\kappa}$, we define
its _homogeneous dimension_ as

$\text{dim}_{\mathrm{hom}}(\mathfrak{h}):=\sum_{i=1}^{\kappa}i\cdot\text{dim}(W_{i}).$

Thanks to (2) we infer that, if $\mathfrak{h}$ is a homogeneous Lie sub-
algebra of $\mathfrak{g}$, we have
$\text{dim}_{\mathrm{hom}}(\mathfrak{h}):=\sum_{i=1}^{\kappa}i\cdot\text{dim}(\mathfrak{h}\cap
V_{i})$. It is well-known that the Hausdorff dimension (for a definition of
Hausdorff dimension see for instance [38, Definition 4.8]) of a graded Lie
group $\mathbb{G}$ with respect to a left-invariant homogeneous distance
coincides with the homogeneous dimension of its Lie algebra. For a reference
for the latter statement, see [30, Theorem 4.4]. From now on, if not otherwise
stated, $\mathbb{G}$ will be a fixed Carnot group.

For any $p\in\mathbb{G}$, we define the left translation
$\tau_{p}:\mathbb{G}\to\mathbb{G}$ as

$q\mapsto\tau_{p}q:=p\cdot q.$

As already remarked above, the group operation $\cdot$ is determined by the
Campbell-Hausdorff formula, and it has the form (see [21, Proposition 2.1])

$p\cdot q=p+q+\mathscr{Q}(p,q),\quad\mbox{for all }\,p,q\in\mathbb{R}^{n},$

where
$\mathscr{Q}=(\mathscr{Q}_{1},\dots,\mathscr{Q}_{\kappa}):\mathbb{R}^{n}\times\mathbb{R}^{n}\to
V_{1}\oplus\ldots\oplus V_{\kappa}$, and the $\mathscr{Q}_{i}$’s have the
following properties. For any $i=1,\ldots\kappa$ and any $p,q\in\mathbb{G}$ we
have

  * (i)

$\mathscr{Q}_{i}(\delta_{\lambda}p,\delta_{\lambda}q)=\lambda^{i}\mathscr{Q}_{i}(p,q)$,

  * (ii)

$\mathscr{Q}_{i}(p,q)=-\mathscr{Q}_{i}(-q,-p)$,

  * (iii)

$\mathscr{Q}_{1}=0$ and
$\mathscr{Q}_{i}(p,q)=\mathscr{Q}_{i}(p_{1},\ldots,p_{i-1},q_{1},\ldots,q_{i-1})$.

Thus, we can represent the product $\cdot$ as

$p\cdot
q=(p_{1}+q_{1},p_{2}+q_{2}+\mathscr{Q}_{2}(p_{1},q_{1}),\dots,p_{\kappa}+q_{\kappa}+\mathscr{Q}_{\kappa}(p_{1},\dots,p_{\kappa-1},q_{1},\dots,q_{\kappa-1})).$
(3)

######  Definition 2.2 (Homogeneous left-invariant distance and norm).

A metric $d:\mathbb{G}\times\mathbb{G}\to\mathbb{R}$ is said to be homogeneous
and left-invariant if for any $x,y\in\mathbb{G}$ we have, respectively

  * (i)

$d(\delta_{\lambda}x,\delta_{\lambda}y)=\lambda d(x,y)$ for any $\lambda>0$,

  * (ii)

$d(\tau_{z}x,\tau_{z}y)=d(x,y)$ for any $z\in\mathbb{G}$.

Given a homogeneous left-invariant distance, its associated norm is defined by
$\|g\|_{d}:=d(g,0)$, for every $g\in\mathbb{G}$, where $0$ is the identity
element of $\mathbb{G}$.

Given a homogeneous left-invariant distance $d$ on $\mathbb{G}$, for every
$x\in\mathbb{G}$ and every $E\subseteq\mathbb{G}$ we define
$\mathrm{dist}(x,E):=\inf\\{d(x,y):y\in E\\}$.

Throughout the paper we will always endow, if not otherwise stated, the group
$\mathbb{G}$ with an arbitrary homogeneous and left-invariant metric. We will
denote such a distance with $d$. We remark that two homogeneous left-invariant
distances on a Carnot group are always bi-Lipschitz equivalent.

######  Definition 2.3 (Metric balls and tubular neighbourhoods).

Suppose a homogeneous and left-invariant metric $d$ has been fixed on
$\mathbb{G}$. Then, we define $U_{d}(x,r):=\\{z\in\mathbb{G}:d(x,z)<r\\}$ to
be the open metric ball relative to the distance $d$ centred at $x$ at radius
$r>0$. The closed ball will be denoted with
$B_{d}(x,r):=\\{z\in\mathbb{G}:d(x,z)\leq r\\}$. Moreover, for a subset
$E\subseteq\mathbb{G}$ and $r>0$, we denote with
$B_{d}(E,r):=\\{z\in\mathbb{G}:\operatorname*{dist}(z,E)\leq r\\}$ the closed
$r$-tubular neighborhood of $E$ and with
$U_{d}(E,r):=\\{z\in\mathbb{G}:\operatorname*{dist}(z,E)<r\\}$ the open
$r$-tubular neighborhood of $E$. When the metric $d$ is understood, we will
tacitly drop the dependence on the metric in the notation.

######  Definition 2.4 (Hausdorff Measures).

Let $d$ be a homogeneous and left-invariant metric on $\mathbb{G}$ and
$A\subseteq\mathbb{G}$ be a Borel set. For any $0\leq h\leq\mathcal{Q}$ and
$\delta>0$, define

$\begin{split}&\mathscr{C}^{h}_{d,\delta}(A):=\inf\Bigg{\\{}\sum_{j=1}^{\infty}r_{j}^{h}:A\subseteq\bigcup_{j=1}^{\infty}B_{d}(x_{j},r_{j}),\text{
}r_{j}\leq\delta\text{ and }x_{j}\in A\Bigg{\\}},\\\
&\mathscr{S}^{h}_{d,\delta}(A):=\inf\Bigg{\\{}\sum_{j=1}^{\infty}r_{j}^{h}:A\subseteq\bigcup_{j=1}^{\infty}B_{d}(x_{j},r_{j}),\text{
}r_{j}\leq\delta\Bigg{\\}},\end{split}$

and
$\mathscr{S}^{h}_{\delta,E}(\emptyset):=0=:\mathscr{C}^{h}_{\delta}(\emptyset)$.
Eventually, we let

$\begin{split}\mathcal{C}^{h}_{d}(A):=\sup_{B\subseteq
A}\sup_{\delta>0}\mathscr{C}^{h}_{d,\delta}(B)=\sup_{B\subseteq
A}\mathcal{C}_{d,0}^{h}(B)&\qquad\text{be the {\em centred spherical Hausdorff
measure}},\\\
\mathcal{S}^{h}_{d}(A):=\sup_{\delta>0}\mathscr{S}^{h}_{d,\delta}(A)&\qquad\text{be
the {\em spherical Hausdorff measure}}.\end{split}$

We stress that $\mathcal{C}^{h}_{d}$ is an outer measure, and thus it defines
a Borel regular measure, see [14, Proposition 4.1], and that the measures
$\mathcal{S}^{h}_{d},\mathcal{H}^{h}_{d},\mathcal{C}^{h}_{d}$ are all
equivalent measures, see [15, Section 2.10.2] and [14, Proposition 4.2]. When
the metric $d$ is understood, we will tacitly drop the dependence on the
metric in the notation.

We recall here the following result that has been proved in [4, item (iii) of
Proposition 2.11], and that will often be used in the paper.

######  Lemma 2.1.

Let $\mathbb{V}$ be a homogeneous subgroup of a Carnot group $\mathbb{G}$
endowed with a left-invariant homogeneous distance $d$. Let us call $h$ the
homogeneous dimension of $\mathbb{V}$. Hence

$\mathcal{C}^{h}_{d}(B_{d}(x,r)\cap\mathbb{V})=r^{h},$

for every $x\in\mathbb{V}$ and any $r>0$.

###  2.2 Densities and tangents of Radon measures

Throughout the rest of the paper we will always assume that $\mathbb{G}$ is a
fixed Carnot group endowed with a left-invariant homogeneous distance $d$. The
homogeneous, and thus Hausdorff, dimension with respect to $d$ will be denoted
with $Q$. Furthermore, as discussed in the previous subsection, we will assume
without loss of generality that $\mathbb{G}$ coincides with $\mathbb{R}^{n}$
endowed with the product induced by the Baker-Campbell-Hausdorff formula.

######  Definition 2.5 (Weak convergence of measures).

Given a family $\\{\phi_{i}\\}_{i\in\mathbb{N}}$ of Radon measures on
$\mathbb{G}$ we say that the sequence $\phi_{i}$ weakly* converges to a Radon
measure $\phi$, and we write $\phi_{i}\rightharpoonup\phi$, if

$\int fd\phi_{i}\to\int fd\phi\qquad\text{for any }f\in C_{c}(\mathbb{G}),$

where $C_{c}(\mathbb{G})$ is the space of compactly supported functions on
$\mathbb{G}$.

######  Definition 2.6 (Tangent measures).

Let $\phi$ be a Radon measure on $\mathbb{G}$. For any $x\in\mathbb{G}$ and
any $r>0$ we define the measure

$T_{x,r}\phi(E):=\phi(x\cdot\delta_{r}(E)),\qquad\text{for any Borel set }E.$

Furthermore, we define $\mathrm{Tan}(\phi,x)$, the tangent measures to $\phi$
at $x$, to be the collection of the _non-null_ Radon measures $\nu$ for which
there is a sequence $\\{r_{i}\\}_{i\in\mathbb{N}}$, with $r_{i}\to 0$, and a
sequence $\\{c_{i}\\}_{i\in\mathbb{N}}$, with $c_{i}>0$, such that

$c_{i}T_{x,r_{i}}\phi\rightharpoonup\nu.$

Moreover, we define $\mathrm{Tan}_{h}(\phi,x)$, the $h$-tangent measures to
$\phi$ at $x$, to be the collection of Radon measures $\nu$ for which there is
a sequence $\\{r_{i}\\}_{i\in\mathbb{N}}$, with $r_{i}\to 0$, such that

$r_{i}^{-h}T_{x,r_{i}}\phi\rightharpoonup\nu.$

######  Definition 2.7 (Lower and upper densities).

Suppose $d$ is a fixed homogeneous left-invariant metric on $\mathbb{G}$. If
$\phi$ is a Radon measure on $\mathbb{G}$, and $h>0$, we define

$\Theta_{d,*}^{h}(\phi,x):=\liminf_{r\to
0}\frac{\phi(B_{d}(x,r))}{r^{h}},\qquad\text{and}\qquad\Theta^{h,*}_{d}(\phi,x):=\limsup_{r\to
0}\frac{\phi(B_{d}(x,r))}{r^{h}},$

and we say that $\Theta_{d,*}^{h}(\phi,x)$ and $\Theta^{h,*}_{d}(\phi,x)$ are
the lower and upper $h$-density of $\phi$ at the point $x\in\mathbb{G}$,
respectively. Furthermore, we say that measure $\phi$ has $h$-density if

$0<\Theta_{d,*}^{h}(\phi,x)=\Theta^{h,*}_{d}(\phi,x)<\infty,\qquad\text{for
}\phi\text{-almost any }x\in\mathbb{G}.$

When the metric $d$ is understood, we will tacitly drop the dependence on the
metric in the notation.

######  Proposition 2.2.

Assume $\phi$ is a Radon measure on $\mathbb{G}$ and suppose that
$r_{i}^{-h}T_{x,r_{i}}\phi\rightharpoonup\nu$. Then, for any
$z\in\operatorname*{supp}(\nu)$ there exists a sequence
$y_{i}\in\operatorname*{supp}(\phi)$ such that
$\delta_{1/r_{i}}(x^{-1}y_{i})\to z$.

###### Proof.

A simple argument by contradiction yields the claim. The proof follows
verbatim its Euclidean analogue, see for instance the proof of [10,
Proposition 3.4]. ∎

######  Definition 2.8 (Definition of $E(\vartheta,\gamma)$).

Let $\phi$ be a Radon measure on $\mathbb{G}$ that is supported on the compact
set $K$, i.e., such that $\phi(\mathbb{G}\setminus K)=0$. For any
$\vartheta,\gamma\in\mathbb{N}$ we define

$E(\vartheta,\gamma):=\big{\\{}x\in
K:\vartheta^{-1}r^{h}\leq\phi(B(x,r))\leq\vartheta r^{h}\text{ for any
}0<r<1/\gamma\big{\\}}.$ (4)

The following two propositions can be found in [4, Proposition 2.4 and
Proposition 2.5].

######  Proposition 2.3 ([4, Proposition 2.4 and Proposition 2.5]).

Assume $\phi$ is a Radon measure supported on the compact set $K$ such that
$0<\Theta^{h}_{*}(\phi,x)\leq\Theta^{h,*}(\phi,x)<\infty$ for $\phi$-almost
every $x\in\mathbb{G}$. Then, for every $\vartheta,\gamma\in\mathbb{N}$ the
set $E(\vartheta,\gamma)$ is compact and
$\phi(\mathbb{G}\setminus\bigcup_{\vartheta,\gamma\in\mathbb{N}}E(\vartheta,\gamma))=0$.

######  Proposition 2.4.

Suppose $\phi$ is a Radon measure with $h$-density. Then for $\phi$-almost
every $x\in\mathbb{G}$ we have that $\mathrm{Tan}_{h}(\phi,x)$ is not empty
and for any $\nu\in\mathrm{Tan}_{h}(\phi,x)$ we have
$0\in\operatorname*{supp}(\nu)$, and $\nu(B(y,s))=\Theta^{h}(\phi,x)s^{h}$ for
any $y\in\operatorname*{supp}(\nu)$ and any $s>0$.

###### Proof.

The proof follows verbatim its Euclidean counterpart, see for instance [10,
Proposition 3.4]. ∎

######  Proposition 2.5.

Suppose that $\mu$ is a Borel regular measure on $\mathbb{G}$ supported on a
homogeneous subgroup $\mathbb{V}\in Gr(h)$, such that
$0\in\operatorname*{supp}(\mu)$ and assume that there exists a constant $C>0$
such that for any $z\in\operatorname*{supp}(\mu)$ and any $s>0$ we have

$\mu(B(z,s))=Cs^{h}.$

Then $\mu$ is a Haar measure of the subgroup $\mathbb{V}$.

###### Proof.

Without loss of generality we can assume $C=1$. Thanks to [18, Theorem 3.1] we
thus infer that $\mu=\mathcal{C}^{h}\llcorner\text{supp}(\mu)$. Moreover, for
any $x\in\operatorname*{supp}(\mu)$, thanks to Lemma 2.1, we have that
$\mu(B(x,r))=\mathcal{C}^{h}\llcorner\mathbb{V}(B(x,r))$ for every $r>0$. If
by contradiction $\text{supp}(\mu)\neq\mathbb{V}$, then there would exist a
$p\in\mathbb{V}$ and a $r_{0}>0$ such that
$B(p,r_{0})\cap\text{supp}(\mu)=\emptyset$. This however is impossible since
we would have

$\mathcal{C}^{h}\llcorner\mathbb{V}(B(0,2(\|p\|+r_{0})))\geq\mathcal{C}^{h}(B(0,2(\|p\|+r_{0}))\cap\text{supp}(\mu))+\mathcal{C}^{h}(B(p,r_{0})\cap\mathbb{V})>\mu(B(0,2(\|p\|+r_{0})),$

and this is a contradiction with what we found above, since by assumption
$0\in\text{supp}(\mu)$. ∎

A very useful property of locally asymptotically doubling measures is that
Lebesgue theorem holds and thus local properties are stable under restriction
to Borel subsets. The forhcoming result is a direct consequence of [25,
Theorem 3.4.3] and the Lebesgue differentiation Theorem in [25, page 77].

######  Proposition 2.6.

Suppose $d$ is a fixed homogeneous left-invariant metric on $\mathbb{G}$ and
that $\phi$ is a Radon measure on $\mathbb{G}$ such that for $\phi$-almost
every $x\in\mathbb{G}$ we have

$\limsup_{r\to 0}\frac{\phi(B_{d}(x,2r))}{\phi(B_{d}(x,r))}<\infty.$ (5)

Then

  * (i)

for any Borel set $B\subseteq\mathbb{G}$ the measure $\phi\llcorner B$ is a
locally asymptotically doubling measure, and we have that the following
equalities hold for $\phi$-almost every $x\in B$

$\Theta^{h}_{d,*}(\phi\llcorner
B,x)=\Theta^{h}_{d,*}(\phi,x),\qquad\text{and}\qquad\Theta^{h,*}_{d}(\phi\llcorner
B,x)=\Theta^{h,*}_{d}(\phi,x),$

  * (ii)

for every non-negative $\rho\in L^{1}(\phi)$, and for $\phi$-almost every
$x\in\mathbb{G}$ we have
$\mathrm{Tan}(\rho\phi,x)=\rho(x)\mathrm{Tan}(\phi,x)$. More precisely, for
$\phi$-almost every $x\in\mathbb{G}$ the following holds

$\begin{split}\text{if $r_{i}\to 0$ is such that}\quad
c_{i}T_{x,r_{i}}\phi\rightharpoonup\nu\quad\text{then}\quad
c_{i}T_{x,r_{i}}(\rho\phi)\rightharpoonup\rho(x)\nu.\\\ \end{split}$ (6)

We recall that any Radon measure $\phi$ on $(\mathbb{G},d)$ that satisfies (5)
for $\phi$-almost every $x\in\mathbb{G}$ is said to be locally asymptotically
doubling, or simply asymptotically doubling.

###  2.3 Intrinsic Grassmannian in Carnot groups

We recall in this section some useful properties about homogeneous subgroups
in Carnot groups. We equip $\mathbb{G}$ with a fixed left-invariant
homogeneous distance $d$ that will sometimes be understood.

######  Definition 2.9 (Intrinsic Grassmanian on Carnot groups).

For any $1\leq h\leq Q$, we define $Gr(h)$ to be the family of homogeneous
subgroups $\mathbb{V}$ of $\mathbb{G}$ that have Hausdorff dimension $h$. Let
us recall that if $\mathbb{V}$ is a homogeneous subgroup of $\mathbb{G}$, any
other homogeneous subgroup $\mathbb{L}$ such that

$\mathbb{V}\cdot\mathbb{L}=\mathbb{G},\qquad\text{and}\qquad\mathbb{V}\cap\mathbb{L}=\\{0\\},$

is said to be a _complement_ of $\mathbb{G}$. We let $Gr_{c}(h)$ to be the
subfamily of those $\mathbb{V}\in Gr(h)$ that have a complement and we will
refer to $Gr_{c}(h)$ as the $h$-dimensional _complemented_ Grassmanian.
Finally, for any $1\leq h\leq Q$ we endow $Gr(h)$ with the metric

$d_{\mathbb{G}}(\mathbb{W}_{1},\mathbb{W}_{2}):=d_{H,\mathbb{G}}(\mathbb{W}_{1}\cap
B(0,1),\mathbb{W}_{2}\cap B(0,1)),$

where $d_{H,\mathbb{G}}$ is the Hausdorff distance of sets induced by the
distance $d$. For more details we refer to [4].

We recall here the following proposition from [4, Proposition 2.7] that will
be used several times.

######  Proposition 2.7 (Compactness of the Grassmannian).

For any $1\leq h\leq Q$ $(Gr(h),d_{\mathbb{G}})$ is a compact metric space.

######  Definition 2.10 (Projections related to a splitting).

For any $\mathbb{V}\in Gr_{c}(h)$, if we choose a complement $\mathbb{L}$ of
$\mathbb{V}$, we can find two unique elements
$g_{\mathbb{V}}:=P_{\mathbb{V}}g\in\mathbb{V}$ and
$g_{\mathbb{L}}:=P_{\mathbb{L}}g\in\mathbb{L}$ such that

$g=P_{\mathbb{V}}(g)\cdot P_{\mathbb{L}}(g)=g_{\mathbb{V}}\cdot
g_{\mathbb{L}}.$

We will refer to $P_{\mathbb{V}}(g)$ and $P_{\mathbb{L}}(g)$ as the _splitting
projections_ , or simply projections, of $g$ onto $\mathbb{V}$ and
$\mathbb{L}$, respectively.

######  Proposition 2.8 ([17, Proposition 2.12 and Corollary 2.15], [4,
Proposition 2.14]).

Suppose $d$ is a fixed homogeneous left-invariant metric on $\mathbb{G}$ and
let $\lVert\cdot\rVert_{d}$ be the associated homogeneous norm. Then, for any
$\mathbb{V}\in Gr_{c}(h)$ with complement $\mathbb{L}$ there is a constant
$C_{1}(d,\mathbb{V},\mathbb{L})>0$ such that for any $p\in\mathbb{G}$ we have

$\begin{split}C_{\ref{ProjC}}(d,\mathbb{V},\mathbb{L})\lVert
P_{\mathbb{L}}(p)\rVert_{d}\leq\mathrm{dist}&(p,\mathbb{V})\leq\lVert
P_{\mathbb{L}}(p)\rVert_{d},\\\
C_{\ref{ProjC}}(d,\mathbb{V},\mathbb{L})(\lVert
P_{\mathbb{L}}(p)\rVert_{d}+\lVert P_{\mathbb{V}}(p)\rVert_{d})\leq&\lVert
p\rVert_{d}\leq\lVert P_{\mathbb{L}}(p)\rVert_{d}+\lVert
P_{\mathbb{V}}(p)\rVert_{d}.\end{split}$ (7)

Furthermore, for any $r>0$, there exists a constant $C_{2}$ we have
$\mathcal{S}^{h}_{d}\llcorner\mathbb{V}\big{(}P_{\mathbb{V}}(B_{d}(p,r))\big{)}=C_{\ref{BallC}}(d,\mathbb{V},\mathbb{L})r^{h}$
and, for any Borel set $A\subseteq\mathbb{G}$ for which
$\mathcal{S}^{h}_{d}(A)<\infty$, we have

$\begin{split}\mathcal{S}^{h}_{d}\llcorner\mathbb{V}(P_{\mathbb{V}}(A))\leq
2C_{\ref{BallC}}(d,\mathbb{V},\mathbb{L})\mathcal{S}^{h}_{d}(A).\end{split}$
(8)

When the metric $d$ is understood, we will tacitly drop the dependence on the
metric in the notation.

######  Remark 2.1 (About the definition of $C_{\ref{ProjC}}$).

We stress that we fix $C_{\ref{ProjC}}$ to be the supremum of all the
constants for which the inequalities in (7) hold.

We recall here the following proposition that will be useful later on.

######  Proposition 2.9 ([4, Proposition 2.10], [17, Proof of Lemma 2.20]).

Suppose $d$ is a fixed homogeneous left-invariant metric on $\mathbb{G}$. Let
us fix $\mathbb{V}\in Gr_{c}(h)$ and $\mathbb{L}$ two complementary
homogeneous subgroups of $\mathbb{G}$. Then, for any $x\in\mathbb{G}$ the map
$\Psi:\mathbb{V}\to\mathbb{V}$ defined as $\Psi(z):=P_{\mathbb{V}}(xz)$ is
invertible and it has unitary Jacobian. As a consequence
$\mathcal{S}^{h}_{d}(P_{\mathbb{V}}(\mathcal{E}))=\mathcal{S}^{h}_{d}(P_{\mathbb{V}}(xP_{\mathbb{V}}(\mathcal{E})))=\mathcal{S}^{h}_{d}(P_{\mathbb{V}}(x\mathcal{E}))$
for every $x\in\mathbb{G}$ and $\mathcal{E}\subseteq\mathbb{G}$ Borel.

The following proposition will be useful in the proof of 5.2. We omit the
proof. The first part of the statement, i.e., the one about the homeomorphism,
can be reached by using elementary Linear Algebra; while the second part of
the statement follows from the first. We stress that in the following
statement we are endowing $\mathfrak{g}$, that is identified with
$\mathbb{R}^{n}$ through a choice of a basis of left-invariant vector fields
$X_{1},\dots,X_{n}$, with a scalar product $\langle\cdot,\cdot\rangle$ that
makes $X_{1},\dots,X_{n}$ orthonormal.

######  Proposition 2.10.

Let $\mathcal{L}(\mathfrak{g},\mathfrak{g})$ be the set of linear maps from
the Lie algebra $\mathfrak{g}$ of $\mathbb{G}$ into itself, endowed with the
operator norm $\rho$. Then, being $Gr(\mathfrak{g})$ the Grassmannian of the
vector space $\mathfrak{g}$, the map
$\mathfrak{P}:Gr(\mathfrak{g})\to\mathcal{L}(\mathfrak{g},\mathfrak{g})$
defined as $\mathfrak{P}(V):=\Pi_{V^{\perp}}$, where $\Pi_{V^{\perp}}$ is the
orthogonal projection onto $V^{\perp}$, is a homeomorphism onto its image.

Then, a map $V:\mathfrak{g}\to\exp^{-1}(Gr(h))$ is Borel measurable if and
only if the projection map
$\pi_{V^{\perp}}:\mathfrak{g}\to\mathcal{L}(\mathfrak{g},\mathfrak{g})$
defined as

$\pi_{V^{\perp}}(x):=\Pi_{V(x)^{\perp}},$

where $\Pi_{V(x)^{\perp}}$ is the orthogonal projection onto $V(x)^{\perp}$,
is Borel measurable. In addition to this, the Borelianity of $\pi_{V^{\perp}}$
is also equivalent to saying that for any fixed $v,w\in\mathfrak{g}$, the map
$x\mapsto\langle v,\Pi_{V(x)^{\perp}}[w]\rangle$ is Borel.

###  2.4 Cones over homogeneous subgroups and intrinisc Lipschitz functions

In this section we recall some basic definitions about intrinsic cones in
Carnot groups.

######  Definition 2.11 (Intrinsic cone).

Suppose $d$ is an homogeneous left-invariant distance on $\mathbb{G}$. For any
$\alpha>0$ and $\mathbb{V}\in Gr(h)$, we define the cone
$C_{\mathbb{V},d}(\alpha)$ as:

$C_{\mathbb{V},d}(\alpha):=\\{w\in\mathbb{G}:\mathrm{dist}_{d}(w,\mathbb{V})\leq\alpha\|w\|_{d}\\}.$

Furthermore, given $\mathbb{V}\in Gr(h)$ and an $\alpha>0$, we say that a
Borel set $E\subseteq\mathbb{G}$ is a $C_{\mathbb{V},d}(\alpha)$-set if

$E\subseteq pC_{\mathbb{V},d}(\alpha)\qquad\text{for any }p\in E.$

When the metric $d$ is understood, we will tacitly drop the dependence on the
metric in the notation.

######  Remark 2.2 (Equivalent intrinsic cones).

Suppose $d$ is a homogeneous left-invariant distance on $\mathbb{G}$. For some
benefits toward the rest of the paper, let us prove that if $\mathbb{V}\in
Gr_{c}(h)$, $\mathbb{L}$ is a complementary subgroup of $\mathbb{V}$, and
$\alpha<C_{\ref{ProjC}}(d,\mathbb{V},\mathbb{L})$, then

$C_{\mathbb{V},d}(\alpha)\subseteq\left\\{w\in\mathbb{G}:\|P_{\mathbb{L}}w\|_{d}\leq\frac{\alpha}{C_{\ref{ProjC}}(d,\mathbb{V},\mathbb{L})-\alpha}\|P_{\mathbb{V}}w\|_{d}\right\\}.$
(9)

Indeed, let us take an element $w$ in the complement of the set in the right-
hand-side above. Thanks to the fact that
$\|w\|_{d}\leq\|P_{\mathbb{L}}w\|_{d}+\|P_{\mathbb{V}}w\|_{d}<C_{\ref{ProjC}}(d,\mathbb{V},\mathbb{L})\alpha^{-1}\|P_{\mathbb{L}}w\|_{d}$
and to 2.8 we have

$\begin{split}\mathrm{dist}_{d}(w,\mathbb{V})\geq
C_{\ref{ProjC}}(d,\mathbb{V},\mathbb{L})\|P_{\mathbb{L}}(w)\|_{d}>\alpha\|w\|_{d}.\end{split}$
(10)

Therefore, any such $w$ is contained in the complement of the left-hand-side
of (9), and thus we get the sought conclusion. Moreover, for any
$\mathbb{V}\in Gr_{c}(h)$ and any of its complementary subgroup $\mathbb{L}$,
let us show that for any $\alpha>0$

$C_{\mathbb{V},\mathbb{L},d}(\alpha):=\\{w\in\mathbb{G}:\|P_{\mathbb{L}}w\|_{d}\leq\alpha\|P_{\mathbb{V}}w\|_{d}\\}\subseteq
C_{\mathbb{V},d}(C_{\ref{ProjC}}(d,\mathbb{V},\mathbb{L})^{-1}\alpha).$

Indeed, if $w$ is an element in the left-hand-side above, we can readily see
thanks to 2.8 that

$\mathrm{dist}_{d}(w,\mathbb{V})\leq\|P_{\mathbb{L}}w\|_{d}\leq\alpha\|P_{\mathbb{V}}w\|_{d}\leq\alpha
C_{\ref{ProjC}}({d},\mathbb{V},\mathbb{L})^{-1}\|w\|_{d}.$

All in all we have proved that if $\mathbb{V}\in Gr_{c}(h)$, $\mathbb{L}$ is
one of its complementary subgroups and
$\alpha<C_{\ref{ProjC}}(d,\mathbb{V},\mathbb{L})$ we have

$C_{\mathbb{V},\mathbb{L},d}(C_{\ref{ProjC}}(d,\mathbb{V},\mathbb{L})\alpha)\subseteq
C_{\mathbb{V},d}(\alpha)\subseteq
C_{\mathbb{V},\mathbb{L},d}(\alpha/(C_{\ref{ProjC}}(d,\mathbb{V},\mathbb{L})-\alpha)),$

thus showing that, below some threshold on the opening, the cones
$C_{\mathbb{V},d}$ and $C_{\mathbb{V},\mathbb{L},d}$ are equivalent.

######  Remark 2.3 (Equivalent distances and cones).

Let $d_{1},d_{2}$ be two homogeneous and left-invariant metrics on
$\mathbb{G}$. Since they are bi-Lipschitz equivalent, we can find a constant
$0<\mathfrak{c}<1$ such that $\mathfrak{c}d_{1}(x,y)\leq
d_{2}(x,y)\leq\mathfrak{c}^{-1}d_{1}(x,y)$ for any $x,y\in\mathbb{G}$. Then
for any $\alpha>0$ and any $\mathbb{V}\in Gr(h)$ we have

$C_{\mathbb{V},d_{1}}(\mathfrak{c}^{2}\alpha)\subseteq
C_{\mathbb{V},d_{2}}(\alpha)\subseteq
C_{\mathbb{V},d_{1}}(\mathfrak{c}^{-2}\alpha).$

We now recall two results that were already proven in [4]. We refer the reader
to the reference [4] for the simple proofs.

######  Lemma 2.11 ([4, Lemma 2.16]).

Suppose $d$ is a homogeneous left-invariant distance on $\mathbb{G}$. For any
$\mathbb{V}\in Gr_{c}(h)$, given $\mathbb{L}$ to be a complementary subgroup
of $\mathbb{V}$, there exists
$0<\varepsilon_{1}:=\varepsilon_{\ref{ep:Cool}}(d,\mathbb{V},\mathbb{L})<1$
such that

$\mathbb{L}\cap C_{\mathbb{V},d}(\varepsilon_{\ref{ep:Cool}})=\\{0\\}.$

and for the aims of this paper, we can and will fix
$\varepsilon_{\ref{ep:Cool}}(d,\mathbb{V},\mathbb{L}):=C_{\ref{ProjC}}(d,\mathbb{V},\mathbb{L})/2$.

######  Proposition 2.12.

Suppose $d$ is a homogeneous left-invariant distance on $\mathbb{G}$. The
function $\mathfrak{e}:Gr_{c}(h)\to\mathbb{R}$ defined as

$\mathfrak{e}(\mathbb{V}):=\sup\\{\varepsilon_{\ref{ep:Cool}}(d,\mathbb{V},\mathbb{L}):\mathbb{L}\text{
is a complement of }\mathbb{V}\\},$ (11)

is lower semicontinuous. In particular the following conclusion holds

  * if $\mathscr{G}\subseteq Gr_{c}(h)$ is compact with respect to $d_{\mathbb{G}}$, then there exists a $\mathfrak{e}_{\mathscr{G}}>0$ such that $\mathfrak{e}(\mathbb{V})\geq\mathfrak{e}_{\mathscr{G}}$ for any $\mathbb{V}\in{\mathscr{G}}$.

###### Proof.

It follows verbatim from the proof of [4, Proposition 2.22], taking [4, Remark
2.5] into account. ∎

Let us recall the classical definition of intrinisc Lipschits function, see
[17, Definition 11].

######  Definition 2.12 (Intrinsic Lipschitz function).

Let $\mathbb{W}\in Gr_{c}(h)$, assume $\mathbb{L}$ is a complement of
$\mathbb{W}$, and let $E\subseteq\mathbb{W}$. A function $f:E\to\mathbb{L}$ is
said to be an _intrinsic Lipschitz function_ if there exists an $\alpha>0$
such that for every $p\in\text{graph}(f):=\\{v\cdot f(v):v\in E\\}$ we have

$\text{graph}(f)\cap pC_{\mathbb{W},\mathbb{L}}(\alpha)=\text{graph}(f),$

where the cones $C_{\mathbb{W},\mathbb{L}}(\alpha)$ have been defined in 2.2.

We finally state two properties of intrinisc Lipschitz graphs that will be
useful later on, and whose simple proofs can be found in [4].

######  Proposition 2.13 ([4, Proposition 2.19]).

Suppose $d$ is a homogeneous left-invariant distance on $\mathbb{G}$ and let
us fix $\mathbb{V}\in Gr_{c}(h)$ with complement $\mathbb{L}$. If
$\Gamma\subset\mathbb{G}$ is a $C_{\mathbb{V}}(\alpha)$-set for some
$\alpha\leq\varepsilon_{\ref{ep:Cool}}(d,\mathbb{V},\mathbb{L})$, then the map
$P_{\mathbb{V}}:\Gamma\to\mathbb{V}$ is injective. As a consequence $\Gamma$
is the intrinsic graph of an intrinsically Lipschitz map defined on
$P_{\mathbb{V}}(\Gamma)$.

######  Lemma 2.14 ([4, Lemma 2.21]).

Let $\mathbb{L}$ and $\mathbb{V}$ be homogeneous complementary subgroups of
$\mathbb{G}$ endowed with a left-invariant homogeneous distance $d$. Suppose
$\Gamma$ is a $C_{\mathbb{V},d}(\alpha)$-set with
$\alpha\leq\varepsilon_{\ref{ep:Cool}}(d,\mathbb{V},\mathbb{L})$. Then there
exists a constant
$\mathfrak{C}(\alpha)=\mathfrak{C}(\alpha,d,\mathbb{V},\mathbb{L})>0$ such
that

$\mathcal{S}^{h}_{d}(P_{\mathbb{V}}(B_{d}(x,r)\cap\Gamma))\geq\mathcal{S}^{h}_{d}\Big{(}P_{\mathbb{V}}\big{(}B_{d}(x,\mathfrak{C}(\alpha)r)\cap
xC_{\mathbb{V},d}(\alpha)\big{)}\cap P_{\mathbb{V}}(\Gamma)\Big{)},$

for any $x\in\Gamma$, and any $r>0$.

###  2.5 Rectifiable measures in Carnot groups

In what follows we are going to define the class of $h$-flat measures on a
Carnot group and then we will give proper definitions of rectifiable measures
on Carnot groups.

######  Definition 2.13 (Flat measures).

For any $h\in\\{1,\ldots,Q\\}$ we let $\mathfrak{M}(h)$ to be the family of
flat $h$-dimensional measures in $\mathbb{G}$, i.e.,

$\mathfrak{M}(h):=\\{\lambda\mathcal{S}^{h}\llcorner\mathbb{W}:\text{ for some
}\lambda>0\text{ and }\mathbb{W}\in Gr(h)\\}.$

Furthermore, if $G$ is a subset of the $h$-dimensional Grassmanian $Gr(h)$, we
let $\mathfrak{M}(h,G)$ to be the set

$\mathfrak{M}(h,G):=\\{\lambda\mathcal{S}^{h}\llcorner\mathbb{W}:\text{ for
some }\lambda>0\text{ and }\mathbb{W}\in G\\}.$ (12)

######  Definition 2.14 ($\mathscr{P}_{h}$ and
$\mathscr{P}_{h}^{c}$-rectifiable measures).

Let $h\in\\{1,\ldots,Q\\}$. A Radon measure $\phi$ on $\mathbb{G}$ is said to
be a $\mathscr{P}_{h}$-rectifiable measure if for $\phi$-almost every
$x\in\mathbb{G}$ we have

  * (i)

$0<\Theta^{h}_{*}(\phi,x)\leq\Theta^{h,*}(\phi,x)<+\infty$,

  * (ii)

there exists a $\mathbb{V}(x)\in Gr(h)$ such that
$\mathrm{Tan}_{h}(\phi,x)\subseteq\\{\lambda\mathcal{S}^{h}\llcorner\mathbb{V}(x):\lambda\geq
0\\}$.

Furthermore, we say that $\phi$ is $\mathscr{P}_{h}^{c}$-rectifiable if (ii)
is replaced with the weaker

  * (ii)*

there exists a $\mathbb{V}(x)\in Gr_{c}(h)$ such that
$\mathrm{Tan}_{h}(\phi,x)\subseteq\\{\lambda\mathcal{S}^{h}\llcorner\mathbb{V}(x):\lambda\geq
0\\}$.

######  Remark 2.4.

(About $\lambda=0$ in 2.14) It is readily noticed that, since in 2.14 we are
asking $\Theta^{h}_{*}(\phi,x)>0$ for $\phi$-almost every $x$, we can not have
the zero measure as an element of $\mathrm{Tan}_{h}(\phi,x)$ thanks to [4,
Proposition 2.26]. As a consequence, a posteriori, we have that in item (ii)
and item (ii)* we can restrict to $\lambda>0$. We will tacitly work in this
restriction from now on, see [4, Remark 2.7].

######  Remark 2.5 (About the rectifiability of Hausdorff measures).

We observe that if $\Gamma$ is a Borel set in $\mathbb{G}$,
$\mathcal{S}^{h}\llcorner\Gamma$ is $\mathscr{P}_{h}$-rectifiable if and only
if $\mathcal{C}^{h}\llcorner\Gamma$ (or also $\mathcal{H}^{h}\llcorner\Gamma)$
is $\mathscr{P}_{h}$-rectifiable. This is because
$\mathcal{S}^{h},\mathcal{H}^{h},\mathcal{C}^{h}$ are equivalent measures (see
2.4), the $\mathscr{P}_{h}$-rectifiability implies being locally
asymptotically doubling, and then we can transfer the property of being
$\mathscr{P}_{h}$-rectifiable from one measure to the other by using
Lebesgue–Radon–Nikodym theorem (see [25, page 82]) and the locality of
tangents in 2.6.

We introduce now a way to estimate how far two measures are.

######  Definition 2.15 (Definition of $F_{K}$).

Given $\phi$ and $\psi$ two Radon measures on $\mathbb{G}$, and given
$K\subseteq\mathbb{G}$ a compact set, we define

$F_{K}(\phi,\psi):=\sup\left\\{\left|\int fd\phi-\int
fd\psi\right|:f\in\mathrm{Lip}_{1}^{+}(K)\right\\},$ (13)

where $\mathrm{Lip}_{1}^{+}(K)$ denotes the class of $1$-Lipschitz nonnegative
function with compact support contained in $K$. We also write $F_{x,r}$ for
$F_{B(x,r)}$.

######  Remark 2.6 (Properties of $F_{K}$).

With few computations that we omit, it is easy to see that
$F_{x,r}(\phi,\psi)=rF_{0,1}(T_{x,r}\phi,T_{x,r}\psi)$. Furthermore, $F_{K}$
enjoys the triangular inequality. Indeed, if $\phi_{1},\phi_{2},\phi_{3}$ are
Radon measures and $f\in\operatorname*{Lip_{1}^{+}}(K)$, then

$\Big{\lvert}\int fd\phi_{1}-\int fd\phi_{2}\Big{\rvert}\leq\Big{\lvert}\int
fd\phi_{1}-\int fd\phi_{3}\Big{\rvert}+\Big{\lvert}\int fd\phi_{3}-\int
fd\phi_{2}\Big{\rvert}\leq F_{K}(\phi_{1},\phi_{2})+F_{K}(\phi_{2},\phi_{3}).$

The arbitrariness of $f$ concludes that $F_{K}(\phi_{1},\phi_{2})\leq
F_{K}(\phi_{1},\phi_{3})+F_{K}(\phi_{3},\phi_{2})$.

######  Definition 2.16 (Definition of $F_{r}$).

For a given Radon measure $\phi$ on $\mathbb{G}$ and for $r>0$, let us define
$F_{r}(\phi):=\int\text{dist}(z,U(0,r)^{c})d\phi(z)$.

######  Lemma 2.15.

For any Radon measure $\phi$ on $\mathbb{G}$ and any $r>0$ we have that
$F_{r}(\phi)=F_{0,r}(\phi,0)$.

###### Proof.

It is immediate to see that $F_{0,r}(\phi,0)\geq F_{r}(\phi)$ for any $r>0$.
In order to prove the viceversa, note that for any
$f\in\operatorname*{Lip_{1}^{+}}(B(0,r))$ we have that $f\rvert_{\partial
B(0,r)}=0$. Thanks to this observation, for any $y\in B(0,r)$ if we let
$x\in\partial B(0,r)$ be a point of minimal distance of $y$ from $U(0,r)^{c}$
we have

$f(y)=\lvert f(y)-f(x)\rvert\leq d(y,x)=\operatorname*{dist}(y,U(0,r)^{c}),$

and this finally shows that $F_{0,r}(\phi,0)=F_{r}(\phi)$, concluding the
proof of the lemma. ∎

######  Proposition 2.16.

The function defined on $\mathcal{M}\times\mathcal{M}$ as

$d(\phi,\psi):=\sum_{p=0}^{\infty}2^{-p}\min\\{1,F_{0,p}(\phi,\psi)\\},$

is a distance and $(\mathcal{M},d)$ is a separable metric space. The topology
induced by $d$ on $\mathcal{M}$ coincides with the one given by the weak*
topology.

Moreover, let us assume $\\{\phi_{i}\\}_{i\in\mathbb{N}}$ is a sequence of
Radon measures such that $\limsup_{i\to\infty}\phi_{i}(B(0,r))<\infty$ for
every $r>0$. Then $\\{\phi_{i}\\}_{i\in\mathbb{N}}$ has a converging
subsequence with respect to the weak* topology.

###### Proof.

The result is stated in [42, Proposition 1.12] in the Euclidean case, but the
proof works verbatim for Radon measures on Carnot groups. ∎

######  Proposition 2.17.

The function $F_{0,1}(\cdot,\cdot)$ is a metric on
$\mathfrak{B}(h):=\\{\psi\in\mathfrak{M}(h):F_{1}(\psi)=1\\}$ and
$(\mathfrak{B}(h),F_{0,1})$ is a compact metric space.

###### Proof.

First of all, we note that for any $\mu,\nu\in\mathfrak{B}(h)$ we have that
$F_{0,1}(\mu,\nu)=0$ if and only if $\mu=\nu$ and this is an immediate
consequence of the fact that $\mu$ and $\nu$ are cones. Symmetry follows
directly form the definition and the triangular inequality follows from2.6.

We are left to show that $(\mathfrak{B}(h),F_{0,1})$ is a compact metric
space. Let $\Psi_{i}$ be a sequence in $\mathfrak{B}(h)$ and note that since
$\mathcal{C}^{h}\llcorner\mathbb{V}(B(0,1))=1$ for every $\mathbb{V}\in
Gr(h)$, because of Lemma 2.1, we deduce that
$\Psi_{i}=(h+1)\mathcal{C}^{h}\llcorner\mathbb{V}_{i}$ for some
$\mathbb{V}_{i}\in Gr(h)$. Thus, we can find a (non-relabeled) subsequence of
the planes $\mathbb{V}_{i}$ that converges to some $\mathbb{V}\in Gr(h)$ in
the Hausdorff metric thanks to the compactness of the Grassmanian $Gr(h)$, see
2.7. Hence, by [4, Proposition 2.29] we infer that
$\Psi_{i}\rightharpoonup(h+1)\mathcal{C}^{h}\llcorner\mathbb{V}\in\mathfrak{B}(h)$
and therefore the compactness follows. ∎

Now we are going to define some functionals that quantifies how far is a
measure from being flat around a point $x\in\mathbb{G}$ and at a certain scale
$r>0$.

######  Definition 2.17 (Definition of $d_{x,r}$).

For any $x\in\mathbb{G}$, any $h\in\\{1,\ldots,Q\\}$ and any $r>0$ we define
the functional

$d_{x,r}(\phi,\mathfrak{M}(h)):=\inf_{\begin{subarray}{c}\mathbb{V}\in
Gr(h)\end{subarray}}F_{0,1}(T_{x,r}\phi/F_{1}(T_{x,r}\phi),(h+1)\mathcal{C}^{h}\llcorner\mathbb{V}).$
(14)

Furthermore, if $G$ is a subset of the $h$-dimensional Grassmanian $Gr(h)$, we
also define

$d_{x,r}(\phi,\mathfrak{M}(h,G)):=\inf_{\begin{subarray}{c}\mathbb{V}\in
G\end{subarray}}F_{0,1}(T_{x,r}\phi/F_{1}(T_{x,r}\phi),(h+1)\mathcal{C}^{h}\llcorner\mathbb{V}).$
(15)

######  Remark 2.7 (About the definition of $d_{x,r}$).

For any Radon measure $\phi$ on $\mathbb{G}$ and any $r>0$ it is immediate to
see that $F_{1}(T_{0,r}\phi)=r^{-1}F_{r}(\phi)$. Moreover, thanks to the first
part of 2.6, we get by few simple computations

$F_{0,1}\left(\frac{T_{x,r}\phi}{F_{1}(T_{x,r}\phi)},(h+1)\mathcal{C}^{h}\llcorner\mathbb{V}\right)=r^{-(h+1)}F_{0,r}\left(\frac{T_{x,1}\phi}{r^{-(h+1)}F_{r}(T_{x,1}\phi)},(h+1)\mathcal{C}^{h}\llcorner\mathbb{V}\right),$
(16)

for all $r>0$ and $\mathbb{V}\in Gr(h)$. Hence, since
$F_{1}((h+1)\mathcal{C}^{h}\llcorner\mathbb{V})=1$ as a consequence of [4,
Proposition 2.12] and Lemma 2.1, we notice that the definition in (15) agrees
with the definition given in [42, §2.1(3)]. Namely,
$d_{x,r}(\phi,\mathfrak{M}(h,G))=d_{r}(T_{x,1}\phi,\mathfrak{M}(h,G))=d_{1}(T_{x,r}\phi,\mathfrak{M}(h,G))$,
where $d_{r}$ is the one defined in [42, §2.1(3)].

For the sake of completeness, and for some benefits toward subsequent
calculations, let us recall here the precise definition of the function $d$
Preiss gave in his setting. Let $\mathscr{C}$ be an arbitrary cone of measures
without the origin, that means $0\not\in\mathscr{C}$ and $\mu\in\mathscr{C}$
implies $\lambda T_{0,\nu}\mu\in\mathscr{C}$ for every $\lambda,\nu>0$. Then,
for every $r>0$ and $\phi$ a Radon measure we define

$d_{r}(\phi,\mathscr{C}):=\inf\left\\{F_{r}\left(\frac{\phi}{F_{r}(\phi)},\psi\right):\psi\in\mathscr{C},\,\,F_{r}(\psi)=1\right\\}.$
(17)

By the explicit expression and the continuity of $F_{r}(\cdot)$ with respect
to the weak* convergence, one easily verifies that, for every $r>0$

$\phi_{k}\rightharpoonup_{k}\phi,\,\,F_{r}(\phi)>0\,\,\Rightarrow
d_{r}(\phi_{k},\mathscr{C})\to_{k}d_{r}(\phi,\mathscr{C}),$ (18)

compare [42, 2.1(6)]. Moreover, due to a slight modification of (16), we have,
for every $r>0$ and every Radon measure $\phi$,

$d_{r}(\phi,\mathscr{C})=d_{1}(T_{0,r}\phi,\mathscr{C}).$ (19)

We now adapt some classical results contained in [42] to our context. The aim
will be to prove that when a Radon measure on $\mathbb{G}$ has a tangent at a
point that is a cone (of measures) with compact basis, then the measure is
locally asymptotically doubling. The following proposition is the analogue of
[42, Propostion 2.2].

######  Proposition 2.18.

Assume that $\mathcal{T}$ is a non-empty cone of Radon measures, i.e., for any
$\nu\in\mathcal{T}$ and any $\lambda,\eta>0$ we have $\eta
T_{0,\lambda}\nu\in\mathcal{T}$, and moreover $0\not\in\mathcal{T}$. Then, the
following are equivalent

  1. (i)

the set $\mathfrak{B}(\mathcal{T}):=\\{\nu\in\mathcal{T}:F_{1}(\nu)=1\\}$ is
weak* compact,

  2. (ii)

for any sequence $\\{\nu_{i}\\}_{i\in\mathbb{N}}\subseteq\mathcal{T}$ such
that $\lim_{i\to\infty}F_{1}(\nu_{i})=0$, we have $\nu_{i}\rightharpoonup 0$,

  3. (iii)

there is a $q\in(0,\infty)$ such that $\nu(B(0,2r))\leq q\nu(B(0,r))$ for
every $r>0$ and any $\nu\in\mathcal{T}$.

###### Proof.

Let us first prove that (i)$\Rightarrow$(ii). Let $\nu_{i}$ be a sequence in
$\mathcal{T}$ and let us assume that $\lim_{i\to\infty}F_{1}(\nu_{i})=0$. We
note that $\nu_{i}\rightharpoonup 0$ if and only if
$F_{0,t}(\nu_{i},0)=F_{t}(\nu_{i})\to_{i}0$ for any $t>0$. This means that if
$\nu_{i}$ does not converge to $0$, we infer that there are a $t>0$ and an
$\varepsilon>0$ such that, up to passing to subsequences, we have
$F_{t}(\nu_{i})>\varepsilon$ for any $i\in\mathbb{N}$. Let us define

$r_{i}:=\sup\\{r\in[1,t]:F_{r}(\nu_{i})\leq F_{1}(\nu_{i})+1/i\\}.$

It is immediate to see that up to further subsequences
$F_{1}(T_{0,r_{i}}\nu_{i})=r_{i}^{-1}F_{r_{i}}(\nu_{i})>0$ and that

$\lim_{i\to\infty}\frac{F_{t/r_{i}}(T_{0,r_{i}}\nu_{i})}{F_{1}(T_{0,r_{i}}\nu_{i})}=\lim_{i\to\infty}\frac{F_{t}(\nu_{i})}{F_{r_{i}}(\nu_{i})}>\varepsilon\lim_{i\to\infty}(F_{1}(\nu_{i})+1/i)^{-1}=\infty.$

Thanks to the fact that $\mathcal{T}$ is a cone, we know that
$F_{1}(T_{0,r_{i}}\nu_{i})^{-1}T_{0,r_{i}}\nu_{i}\in\mathfrak{B}(\mathcal{T})$
and thus there must exists a converging (non-relabeled) subsequence of $r_{i}$
and a $\nu\in\mathfrak{B}(\mathcal{T})$ such that
$F_{1}(T_{0,r_{i}}\nu_{i})^{-1}T_{0,r_{i}}\nu_{i}\rightharpoonup\nu$. This
however implies that

$\infty=\lim_{i\to\infty}\frac{F_{t/r_{i}}(T_{0,r_{i}}\nu_{i})}{F_{1}(T_{0,r_{i}}\nu_{i})}\leq\lim_{i\to\infty}\frac{F_{t}(T_{0,r_{i}}\nu_{i})}{F_{1}(T_{0,r_{i}}\nu_{i})}=\lim_{i\to\infty}F_{t}(F_{1}(T_{0,r_{i}}\nu_{i})^{-1}T_{0,r_{i}}\nu_{i})=F_{t}(\nu),$

that is a contradiction with the fact that $\nu$ is a Radon measure.

Secondly, let us show that (ii)$\Rightarrow$(iii). Since $\mathcal{T}$ is a
cone, it suffices to prove that there exists $q\in(0,+\infty)$ such that
$\nu(B(0,2))\leq q\nu(B(0,1))$ for every $\nu\in\mathcal{T}$. Indeed, we thus
would get that for every $\nu\in\mathcal{T}$ and $r>0$ we have
$\nu(B(0,2r))=T_{0,r}\nu(B(0,2))\leq qT_{0,r}\nu(B(0,1))=q\nu(B(0,r))$.
Suppose by contradiction that there exists a sequence of measures
$\nu_{i}\in\mathcal{T}$ such that $\nu_{i}(B(0,2))>i\nu_{i}(B(0,1))$. Note now
that since $\mathcal{T}$ is a cone, the measures $\nu_{i}(B(0,2))^{-1}\nu_{i}$
are still in $\mathcal{T}$ and
$\lim_{i\to\infty}F_{1}(\nu_{i}(B(0,2))^{-1}\nu_{i})=0$. Thanks to (ii) this
shows in particular that

$\nu_{i}(B(0,2))^{-1}\nu_{i}\rightharpoonup 0$ (20)

However, since $F_{3}(\nu_{i}(B(0,2))^{-1}\nu_{i})\geq 1$ for any
$i\in\mathbb{N}$, this is a contradiction with (20), according to which one
should have

$\lim_{i\to\infty}F_{3}(\nu_{i}(B(0,2))^{-1}\nu_{i})=F_{3}(0)=0,$

as clearly $F_{3}$ is a weak* continuous operator on Radon measures.

Finally, let us prove the implication (iii)$\Rightarrow$(i). Let
$\\{\nu_{i}\\}_{i\in\mathbb{N}}$ be a sequence in $\mathfrak{B}(\mathcal{T})$
and note that for any $i\in\mathbb{N}$ we have

$\nu_{i}(B(0,1/2))\leq 2F_{1}(\nu_{i})=2,$

and thus thanks to (iii) we infer that for any $r>0$ we have
$\nu_{i}(B(0,r))\leq 4\max\\{1,q^{\log_{2}(r)+1}\\}$ for any $i\in\mathbb{N}$.
2.16 and the weak* continuity of $F_{1}$ conclude the proof. ∎

######  Remark 2.8.

For some benefit towards the remaining part of this section, let us notice
that if $\mathcal{T}$ is a non-empty cone of Radon measure such that
$\mathfrak{B}(\mathcal{T})$ is weak* compact, for every $\lambda>1$ there is
$\tau>1$ such that $F_{\tau r}(\psi)\leq\lambda F_{r}(\psi)$ for every $r>0$
and $\psi\in\mathcal{T}$. The proof follows verbatim from the five lines in
[42, (1)$\Rightarrow$(5) of Proposition 2.2].

######  Proposition 2.19.

For any Radon measure $\phi$ on $\mathbb{G}$ and $\phi$-almost every
$x\in\mathbb{G}$ the set $\mathrm{Tan}(\phi,x)$ is either empty or a cone.
Suppose $\phi$ is a Radon measure on $\mathbb{G}$ such that the set
$\mathfrak{B}(\phi,x):=\\{\nu\in\mathrm{Tan}(\phi,x):F_{1}(\nu)=1\\}$ is a
non-empty weak* compact for $\phi$-almost every $x\in\mathbb{G}$. Then $\phi$
is locally asymptotically doubling.

###### Proof.

In order to prove the first part of the statement, let
$x\in\operatorname*{supp}(\phi)$ be a point where $\mathrm{Tan}(\phi,x)$ is
non-empty, choose a $\nu\in\mathrm{Tan}(\phi,x)$ and assume that $r_{i}$ and
$c_{i}$ are two sequences such that

$c_{i}T_{x,r_{i}}\phi\rightharpoonup\nu.$

To conclude the proof of the claim we need to show that for any
$\eta,\lambda>0$ we have $\eta T_{0,\lambda}\nu\in\mathrm{Tan}(\phi,x)$ and to
do this, we just note that

$\eta c_{i}T_{x,\lambda r_{i}}\phi=\eta
T_{0,\lambda}(c_{i}T_{x,r_{i}}\phi)\rightharpoonup\eta T_{0,\lambda}\nu.$

This shows that $\eta T_{0,\lambda}\nu\in\mathrm{Tan}(\phi,x)$ and thus
$\mathrm{Tan}(\phi,x)$ is a cone.

Fix a point in $\mathbb{G}$ where the set $\mathfrak{B}(\phi,x)$ is a compact
cone and thanks to 2.18(iii) we infer there exists a $q>0$ such that
$\nu(B(0,2r))\leq q\nu(B(0,r))$ for any $\nu\in\mathrm{Tan}(\phi,x)$ and every
$r>0$. Let $\mathfrak{d}:=\inf\\{\operatorname*{dist}(z,U(0,1/2)^{c}):z\in
B(0,1/4)\\}>0$. We now prove that

$\limsup_{r\to 0}F_{1}(T_{x,2r}\phi)/F_{1}(T_{x,r}\phi)\leq 2d^{-1}q^{2}.$
(21)

Indeed, if by contradiction $r_{i}$ is an infinitesimal sequence such that
$F_{1}(T_{x,2r_{i}}\phi)>2\mathfrak{d}^{-1}q^{2}F_{1}(T_{x,r_{i}}\phi)$, then
for any $\nu\in\mathfrak{B}(\phi,x)$ we have

$F_{1}(T_{x,2r_{i}}\phi/F_{1}(T_{x,2r_{i}}\phi),\nu)\geq
F_{1/2}(T_{x,2r_{i}}\phi/F_{1}(T_{x,2r_{i}}\phi),\nu)\geq
F_{1/2}(\nu)-F_{1/2}(T_{x,2r_{i}}\phi)/F_{1}(T_{x,2r_{i}}\phi),$ (22)

where the last inequality comes from 2.6 and Lemma 2.15. Furthermore, we also
have for any $\nu\in\mathfrak{B}(\phi,x)$ that

$F_{1/2}(\nu)=\frac{F_{1/2}(\nu)}{F_{1}(\nu)}\geq\frac{\mathfrak{d}\nu(B(0,1/4))}{2\nu(B(0,1))}\geq\frac{\mathfrak{d}}{2q^{2}}.$
(23)

Thanks to the absurd hypothesis and the fact that for any $s>0$ we have
$F_{s}(T_{x,r}\phi)=sF_{1}(T_{x,rs}\phi)$, we infer that

$F_{1/2}(T_{x,2r_{i}}\phi)/F_{1}(T_{x,2r_{i}}\phi)=F_{1}(T_{x,r_{i}}\phi)/2F_{1}(T_{x,2r_{i}}\phi)\leq\mathfrak{d}/4q^{2}.$
(24)

Putting (22), (23) and (24) together, we conclude that

$F_{1}(T_{x,2r_{i}}\phi/F_{1}(T_{x,2r_{i}}\phi),\nu)\geq\mathfrak{d}/4q^{2}\geq\min\\{\mathfrak{d}/4q^{2},1/2\\}=:\varepsilon,$
(25)

for any $\nu\in\mathfrak{B}(\phi,x)$. Let us now denote, for simplicity,
$\mathscr{T}:=\mathrm{Tan}(\phi,x)$. By taking into account the definition of
$d_{1}$ in (17), we get from the previous computations that
$d_{1}(T_{x,2r_{i}}\phi,\mathscr{T})\geq\varepsilon$ for every $i$. Let us fix
$\nu\in\mathrm{Tan}(\phi,x)$ such that
$c_{i}T_{x,s_{i}}\phi\rightharpoonup\nu$ and let us note that (17) and (18)
imply that

$\lim_{i\to
0}d_{1}(T_{x,s_{i}}\phi,\mathscr{T})=\lim_{i\to\infty}d_{1}(c_{i}T_{x,s_{i}}\phi,\mathscr{T})=d_{1}(\nu,\mathscr{T})=0.$

Thanks to the above chain of identities, for $i$ sufficiently large, we denote
by $\ell_{i}$ the smallest number among those $\ell\in[0,s_{i}]$ with the
property that $d_{1}(T_{x,\eta},\mathscr{T})<\varepsilon$ for every
$\ell<\eta\leq s_{i}$. Since
$d_{1}(T_{x,2r_{i}}\phi,\mathscr{T})\geq\varepsilon$ we conclude that
$\ell_{i}>0$ for $i$ sufficiently large and
$d_{1}(T_{x,\ell_{i}}\phi,\mathscr{T})=\varepsilon$ by the minimality of
$\ell_{i}$ and the continuity of the map $\eta\mapsto
d_{1}(T_{x,\eta}\phi,\mathscr{T})$.

If, up to subsequences, $\ell_{i}/s_{i}\to_{i}t>0$, we conclude that, thanks
to (18),

$d_{1}(T_{0,t}\nu,\mathscr{T})=\lim_{i\to+\infty}d_{1}(T_{x,ts_{i}}\phi,\mathscr{T})\geq\varepsilon,$

where the last inequality is true since $ts_{i}$ is arbitrarily near to
$\ell_{i}$ for $i$ large enough, and
$d_{1}(T_{x,\ell_{i}}\phi,\mathscr{T})\geq\varepsilon$. The previous
inequality gives a contradiction since $T_{0,t}\nu\in\mathscr{T}$ and hence we
should have $d_{1}(T_{0,t}\nu,\mathscr{T})=0$. Thus, $\ell_{i}/s_{i}\to 0$.
This means that for every $r\geq 1$, taking into account (19), we have

$\limsup_{i\to+\infty}d_{r}(T_{x,\ell_{i}}\phi,\mathscr{T})=\limsup_{i\to+\infty}d_{1}(T_{x,r\ell_{i}}\phi,\mathscr{T})\leq\varepsilon,$
(26)

since $\ell_{i}\leq r\ell_{i}\leq s_{i}$ for $i$ sufficiently large. Since
$\varepsilon<1$, we have that $\lambda:=2/(1+\varepsilon)>1$, and hence, by
2.8, there exists $\tau>1$ such that $F_{\tau r}(\psi)\leq\lambda F_{r}(\psi)$
for every $\psi\in\mathscr{C}$ and for every $r>0$, since $\mathscr{C}$ has a
compact basis. Hence, taking (26) into account with $\tau r$ instead of $r$,
we get that, whenever $r\geq 1$ and $i$ is sufficiently big, there exists
$\psi\in\mathscr{T}$ with $F_{\tau r}(\psi)=1$ and

$F_{\tau r}\left(\frac{T_{x,\ell_{i}}\phi}{F_{\tau
r}(T_{x,\ell_{i}}\phi)},\psi\right)\leq\varepsilon/2.$

As a consequence, whenever $r\geq 1$ and $i$ is sufficiently big, by the
triangle inequality for $F$ (cf. 2.17) and by the fact that $F_{\tau
r}(\cdot)\geq F_{r}(\cdot)$, we get that

$\frac{F_{r}(T_{x,\ell_{i}}\phi)}{F_{\tau r}(T_{x,\ell_{i}}\phi)}\geq
F_{r}(\psi)-\varepsilon/2\geq\lambda^{-1}F_{\tau r}(\psi)-\varepsilon\geq
1/2.$

Hence, iterating, we have shown that there exists $\tau>1$ such that that for
every $r\geq 1$ and every $p\in\mathbb{N}$,

$\limsup_{i\to+\infty}\frac{F_{\tau^{p}r}(T_{x,\ell_{i}}\phi)}{F_{r}(T_{x,\ell_{i}}\phi)}<+\infty.$

By the arbitrariness of $p\in\mathbb{N}$ and $r\geq 1$, this implies that we
are in a position to apply 2.16 to the sequence
$\frac{T_{x,\ell_{i}}\phi}{F_{1}(T_{x,\ell_{i}}\phi)}$, which then converges,
up to subsequences, to $\widetilde{\nu}\in\mathscr{T}$ with
$F_{1}(\widetilde{\nu})=1$. But then, by (18),

$d_{1}(\widetilde{\nu},\mathscr{T})=\lim_{i\to+\infty}d_{1}(T_{x,\ell_{i}}\phi,\mathscr{T})\geq\varepsilon,$

that is a contradiction since $d_{1}(\widetilde{\nu},\mathscr{T})=0$. Hence we
finally have proven (21).

Hence, from (21), we deduce

$\limsup_{r\to 0}\frac{\phi(B(x,2r))}{\phi(B(x,r))}\leq\limsup_{r\to
0}\frac{2F_{1}(T_{x,4r}\phi)}{2^{-1}F_{1}(T_{x,r}\phi)}\leq
16\mathfrak{d}^{-2}q^{4},$

whence the conclusion. ∎

Let us now prove a simple consequence of the previous Proposition.

######  Proposition 2.20.

Let $\phi$ be a Radon measure on $\mathbb{G}$ such that for $\phi$-almost
every $x\in\mathbb{G}$ we have
$\mathrm{Tan}(\phi,x)=\\{\lambda\mathcal{S}^{h}\llcorner\mathbb{V}(x),\lambda>0\\}$
for some homogeneous subgroup $\mathbb{V}(x)$ of Hausdorff dimension
$h\in\mathbb{N}$. Then, for $\phi$-almost every $x\in\mathbb{G}$, the measure
$T_{x,r}\phi/F_{1}(T_{x,r}\phi)$ weak* converges to
$(h+1)\mathcal{C}^{h}\llcorner\mathbb{V}(x)$.

###### Proof.

For $\phi$-almost every $x\in\mathbb{G}$ we have that
$\mathfrak{B}(\phi,x)=\\{(h+1)\mathcal{C}^{h}\llcorner\mathbb{V}(x)\\}$,
taking into account [4, Proposition 2.12] and Lemma 2.1. Hence
$\mathfrak{B}(\phi,x)$ is clearly compact for $\phi$-almost every
$x\in\mathbb{G}$, and then $\phi$ is locally asymptotically doubling, due to
2.19. Hence for every sequence $r_{i}\to 0$ we can extract a subsequence in
$i$ such that $T_{x,r_{i}}\phi/F_{1}(T_{x,r_{i}}\phi)$ weak* converges to some
$\nu\in\mathrm{Tan}(\phi,x)$, due to the fact that $\phi$ is locally
asymptotically doubling and thus the hypothesis of 2.16 is verified. Since
$F_{1}(\nu)=1$ by continuity of $F_{1}$, we conclude that
$\nu=(h+1)\mathcal{C}^{h}\llcorner\mathbb{V}(x)$. Thus, being the sequence
$r_{i}$ arbitrary, we obtain the thesis. ∎

The following proposition, which is inspired by [42, 4.4(4)], will be of
crucial importance in the proof of the two fundamental results of this
section, namely 2.25, and 2.26.

######  Proposition 2.21.

Let $0<\sigma<1/5$, $\phi$ be a Radon measure on $\mathbb{G}$,
$h\in\\{1,\ldots,Q\\}$, and
$d_{z,t}(\phi,\mathfrak{M}(h,\\{\mathbb{V}\\}))\leq\sigma^{h+4}$, then

$\phi(B(y,s)\cap
B(y\mathbb{V},\sigma^{2}t/(h+1)))\geq(1-5\sigma)(s/r)^{h}\phi(B(x,r)),$

whenever $x,y\in z\mathbb{V}\cap B(z,(1-\sigma)t)$, $\sigma t\leq
r\leq(1-\sigma)t-\lVert z^{-1}x\rVert$, and $\sigma t\leq
s\leq(1-\sigma)t-\lVert z^{-1}y\rVert$.

###### Proof.

The definition of $d_{z,t}(\cdot,\mathfrak{M}(h,\\{\mathbb{V}\\}))$ implies
that

$F_{0,1}(T_{z,t}\phi/F_{1}(T_{z,t}\phi),(h+1)\mathcal{C}^{h}\llcorner\mathbb{V})\leq\sigma^{h+4},$

and thus up to redefining $\phi$ we can assume without loss of generality that
$z=0$, $t=1$ and that $F_{1}(\phi)=1$. Thus, let $q:=\sigma^{2}/(h+1)$,
$x\in\mathbb{V}$ and $r>0$ as in the hypothesis of the proposition. Define

$g(w):=\min\\{1,\operatorname*{dist}(w,\mathbb{G}\setminus B(x,r+q))/q\\}.$

Notice that $B(x,r)\Subset B(0,1)$, and thanks to the assumptions on $\phi$ we
infer that, calling $\mathrm{Lip}(g)$ the Lipschitz constant of the function
$g$,

$\begin{split}\phi(B(x,r))&\leq\int g(w)d\phi(w)\leq(h+1)\int
g(w)d\mathcal{C}^{h}\llcorner\mathbb{V}(w)+\text{Lip}(g)F_{0,1}(\phi,(h+1)\mathcal{C}^{h}\llcorner\mathbb{V}(w))\\\
&\leq(h+1)\mathcal{C}^{h}\llcorner\mathbb{V}(B(x,r+q))+\sigma^{h+4}/q=(h+1)(r+q)^{h}+\sigma^{h+4}/q.\end{split}$
(27)

With the same argument used above,see [4, Equation (37)], for any $y$ and
$s>0$ as in the hypothesis of the proposition one can also show that

$(h+1)(s-q)^{h}=(h+1)\mathcal{C}^{h}\llcorner\mathbb{V}(B(y,s-q))\leq\phi(B(y,s)\cap
B(\mathbb{V},q))+\sigma^{h+4}/q.$ (28)

Thus, putting together (27) and (28) we infer that

$\begin{split}\frac{\phi(B(y,s)\cap
B(\mathbb{V},q))}{\phi(B(x,r))}&\geq\frac{(h+1)(s-q)^{h}-\sigma^{h+4}/q}{(h+1)(r+q)^{h}+\sigma^{h+4}/q}=\left(\frac{s}{r}\right)^{h}\frac{\left(1-\frac{\sigma^{2}}{s(h+1)}\right)^{h}-\frac{\sigma^{h+2}}{s^{h}}}{\left(1+\frac{\sigma^{2}}{r(h+1)}\right)^{h}+\frac{\sigma^{h+2}}{r^{h}}}\\\
&\geq\left(\frac{s}{r}\right)^{h}\frac{\left(1-\frac{\sigma}{h+1}\right)^{h}-\sigma^{2}}{\left(1+\frac{\sigma}{h+1}\right)^{h}+\sigma^{2}}\geq\left(\frac{s}{r}\right)^{h}\frac{1-h/(h+1)\sigma-\sigma^{2}}{1+2h/(h+1)\sigma+\sigma^{2}}\geq\left(\frac{s}{r}\right)^{h}\frac{1-2\sigma}{1+3\sigma}\geq(1-5\sigma)\left(\frac{s}{r}\right)^{h},\end{split}$

where in the third inequality above we are using that $\sigma\leq r$ and
$\sigma\leq s$; in the fourth inequality we are using that
$(1-\sigma/(h+1))^{h}\geq 1-h/(h+1)\sigma$ by Bernoulli inequality, and
$(1+\sigma/(h+1))^{h}\leq 1+2h/(h+1)\sigma$, which can be easily verified by
induction since $2h\sigma/(h+1)\leq 1$. ∎

Before proving the main results of this section, namely 2.25, and 2.26, we now
state and prove three measurability results that will play a crucial role in
the rest of the paper. Roughly speaking, we prove that when a measure has
unique tangents (or unique approximate tangents), the map that associates a
point $x\in\mathbb{G}$ to its tangent (or approximate tangent) is measurable.

######  Lemma 2.22.

Let $\phi$ be a Radon measure such that, for $\phi$-almost every
$x\in\mathbb{G}$, there exists $\tau(\phi,x)\in Gr(h)$ such that

$\mathrm{Tan}(\phi,x)=\\{\lambda\mathcal{C}^{h}\llcorner\tau(\phi,x):\lambda>0\\}.$

Then the map $x\mapsto\tau(\phi,x)$ is $\phi$-measurable as a map from
$\mathbb{G}$ to $Gr(h)$.

###### Proof.

First of all, from 2.19 we get that $\phi$ is locally asymptotically doubling.
We let $\\{\mathbb{V}_{\ell}\\}_{\ell\in\mathbb{N}}$ be a countable dense set
in $Gr(h)$ that exists thanks to the compactness of the Grassmanian, see 2.7.
Furthermore, for any $r\in(0,1)\cap\mathbb{Q}$ any $\varepsilon>0$, and any
$\ell\in\mathbb{N}$ we define the function

$f_{r,\ell,\varepsilon}(x):=\phi(B(x,r))^{-1}\phi(\\{w\in
B(x,r):\operatorname*{dist}(x^{-1}w,\mathbb{V}_{\ell})\geq\varepsilon\lVert
x^{-1}w\rVert\\})=:\phi(B(x,r))^{-1}\phi(I(x,r,\ell,\varepsilon)),$

when $\phi(B(x,r))>0$ and we set it to be $+\infty$ if $\phi(B(x,r))=0$. We
claim that the functions $f_{r,\ell,\varepsilon}$ are upper semicontinuous.
Let $\\{x_{i}\\}_{i\in\mathbb{N}}$ be a sequence of points converging to some
$x\in\mathbb{G}$. If $\phi(B(x,r))=0$ the upper semicontinuity on the sequence
$\\{x_{i}\\}_{i\in\mathbb{N}}$ is trivially verified by definition of
$f_{r,\ell,\varepsilon}$. So let us assume without loss of generality that
$\phi(B(x,r))>0$. Since $x_{i}\to x$ and $\phi$ is a Radon measure we have
$\phi(B(x,r))=\lim_{i}\phi(B(x_{i},r))$, and then we can assume withouot loss
of generality that $\phi(B(x_{i},r))>0$ for every $i$.

Since the sets $I(x_{i},r,\ell,\varepsilon)$ are contained in $B(x,2)$
provided $i$ is sufficiently big, we infer thanks to Fatou’s Lemma that

$\limsup_{i\to+\infty}f_{r,\ell,\varepsilon}(x_{i})=\limsup_{i\to+\infty}\phi(B(x_{i},r))^{-1}\int\chi_{I(x_{i},r,\ell,\varepsilon)}(z)d\phi(z)\leq\phi(B(x,r))^{-1}\int\limsup_{i\to+\infty}\chi_{I(x_{i},r,\ell,\varepsilon)}(z)d\phi(z).$
(29)

Furthermore, since $x_{i}\to x$ and the sets $I(x_{i},r,\ell,\varepsilon)$ and
$I(x,r,\ell,\varepsilon)$ are closed, we have

$\limsup_{i\to+\infty}\chi_{I(x_{i},r,\ell,\varepsilon)}=\chi_{\limsup_{i\to+\infty}I(x_{i},r,\ell,\varepsilon)}\leq\chi_{I(x,r,\ell,\varepsilon)},$

where the first equality is true in general. Then, from (29), we infer that

$\limsup_{i\to\infty}f_{r,\ell,\varepsilon}(x_{i})\leq\phi(B(x,r))^{-1}\int\limsup_{i\to+\infty}\chi_{I(x_{i},r,\ell,\varepsilon)}(z)d\phi(z)\leq\phi(B(x,r))^{-1}\int\chi_{I(x,r,\ell,\varepsilon)}(z)d\phi(z)=f_{r,\ell,\varepsilon}(x),$

and this concludes the proof that $f_{r,\ell,\varepsilon}$ is upper
semicontinuous. This shows that for every $\ell\in\mathbb{N}$ and
$\varepsilon>0$, the function

$f_{\ell,\varepsilon}:=\liminf_{r\in\mathbb{Q}\cap(0,1),r\to
0}f_{r,\ell,\varepsilon},$

is $\phi$-measurable. Hence also
$\widetilde{f}_{\ell,\varepsilon}:=\sup_{\widetilde{\varepsilon}\in\mathbb{Q},\widetilde{\varepsilon}>\varepsilon}f_{\ell,\varepsilon}$
is $\phi$-measurable. As a consequence, since
$\mathrm{Tan}(\phi,x)=\\{\lambda\mathcal{C}^{h}\llcorner\tau(\phi,x):\lambda>0\\}$
for $\phi$-almost every $x\in\mathbb{G}$, we infer that the set

$\begin{split}B_{\ell,\varepsilon}&:=\big{\\{}x\in\mathbb{G}:\widetilde{f}_{\ell,\varepsilon}(x)=0\\}\cap\\{x\in\mathbb{G}:\text{there
exists $\tau(\phi,x)$}\big{\\}}=\\{x\in\mathbb{G}:\tau(\phi,x)\subseteq
C_{\mathbb{V}_{\ell}}(\varepsilon)\\},\end{split}$ (30)

is $\phi$-measurable as well. Let us justify the last equality in the previous
line. If $\widetilde{f}_{\ell,\varepsilon}(x)=0$, then
$f_{\ell,\widetilde{\varepsilon}}(x)=0$ for every
$\widetilde{\varepsilon}>\varepsilon$, $\widetilde{\varepsilon}\in\mathbb{Q}$.
Hence, arguing as in [4, Proof of Proposition 5.5], in particular as in the
lines slightly above [4, Equation (109)], we get $\tau(\phi,x)\subseteq
C_{\mathbb{V}_{\ell}}(\widetilde{\varepsilon})$ for every
$\widetilde{\varepsilon}\in\mathbb{Q}$ and
$\widetilde{\varepsilon}>\varepsilon$. Let us explain this with further
details. We first get that there exist $r_{i}\to 0$ such that
$f_{r_{i},\ell,\widetilde{\varepsilon}}(x)\to_{i}0$. Since $\phi$ is locally
asymptotically doubling, thanks to 2.16 we deduce that
$\phi(B(x,r_{i}))^{-1}T_{x,r_{i}}\phi$ converges, up to subsequences, to some
tangent measure $\nu\in\mathrm{Tan}(\phi,x)$, and then from the hypothesis we
have $\nu=\lambda\mathcal{C}^{h}\llcorner\tau(\phi,x)$, for some $\lambda>0$.
Then the same computations in the two displayed equations before [4, Equation
(109)] give the sought conclusion. Taking
$\widetilde{\varepsilon}\to\varepsilon$ we get the first inclusion of (30). On
the other hand, if $\tau(\phi,x)\subseteq C_{\mathbb{V}_{\ell}}(\varepsilon)$,
we get that
$\tau(\phi,x)\subseteq\\{w\in\mathbb{G}:\operatorname*{dist}(w,\mathbb{V}_{\ell})<\widetilde{\varepsilon}\|w\|\\}$
for every $\widetilde{\varepsilon}>\varepsilon$. Hence we can argue as in [4,
Equation (111)] in order to obtain that
$f_{\ell,\widetilde{\varepsilon}}(x)=0$ for every
$\widetilde{\varepsilon}>\varepsilon$ and then passing to the limit as
$\widetilde{\varepsilon}\to\varepsilon$ we get the sought conclusion.

In order to prove that the map $x\mapsto\tau(\phi,x)$ is $\phi$-measurable, it
suffices to check that the for any open $\Omega\subseteq Gr(h)$ we have that
$\tau^{-1}(\Omega)$ is $\phi$-measurable. To show this we note that, thanks to
[4, Lemma 2.15], there is a sequence of radii $r_{k}>0$ such that

$\Omega=\bigcup_{\begin{subarray}{c}k\in\mathbb{N}\\\
\mathbb{V}_{k}\in\Omega\end{subarray}}\\{\mathbb{W}\in
Gr(h):\mathbb{W}\subseteq C_{\mathbb{V}_{k}}(r_{k})\\}.$

This implies that, up to $\phi$-null sets,
$\tau^{-1}(\Omega)=\bigcup_{k\in\mathbb{N}}B_{k,r_{k}}$, which thanks to the
above discussion is a $\phi$-measurable set. ∎

######  Lemma 2.23.

Let $\phi$ be a $\mathscr{P}_{h}$-rectifiable measure. Denote $\tau(\phi,x)$
to be the unique element of $Gr(h)$, that exists $\phi$-almost everywhere by
definition, for which

$\mathrm{Tan}_{h}(\phi,x)\subseteq\\{\lambda\mathcal{C}^{h}\llcorner\tau(\phi,x):\lambda>0\\}.$

Then the map $x\mapsto\tau(\phi,x)$ is $\phi$-measurable as a map from
$\mathbb{G}$ to $Gr(h)$.

###### Proof.

From a routine argument (cf. [38, Remark 14.4(3)]), we get that
$\mathrm{Tan}(\phi,x)=\\{\lambda\mathcal{C}^{h}\llcorner\tau(\phi,x)\\}$ for
$\phi$-almost every $x\in\mathbb{G}$. Hence we can apply Lemma 2.22 ∎

The proof of the following lemma follows as the ones above. We omit the
details.

######  Lemma 2.24.

Suppose $d$ is a homogeneous left-invariant metric on $\mathbb{G}$, let $E$ be
a Borel set of finite $\mathcal{S}^{h}_{d}$-measure, and suppose that for
$\mathcal{S}^{h}_{d}$-almost every $x\in E$ there exists $\mathbb{V}(x)\in
Gr(h)$ for which for any $0<\varepsilon<1$ and any $0<\beta<1$ there exist a
$\rho(x,\varepsilon,\beta)>0$ such that

$\mathcal{S}^{h}_{d}\llcorner E(B_{d}(x,r)\setminus
xC_{\mathbb{V}(x),{d}}(\beta))\leq\varepsilon\mathcal{S}^{h}_{d}\llcorner
E(B_{d}(x,r)),$ (31)

for any $0<r<\rho(x,\varepsilon,\beta)$. Then the map $x\mapsto\mathbb{V}(x)$
from $E$ to $Gr(h)$ is $\mathcal{S}^{h}_{d}\llcorner E$-measurable.

Notice that the previous statement could be also obtained arguing as in [39,
Proposition 3.9], after having noticed that, since
$2^{-h}\leq\Theta^{h,*}_{d}(\mathcal{S}^{h}_{d}\llcorner E,x)\leq 1$ for
$\mathcal{S}^{h}_{d}\llcorner E$-almost every $x\in\mathbb{G}$ due to [15,
2.10.19(1) and 2.10.19(5)], the condition (31) is equivalent to asking that
$\mathbb{V}(x)$ is an approximate tangent plane to $E$ at $x$ in the sense of
[39, Equation (3.2)].

######  Remark 2.9.

The results in Lemma 2.22, Lemma 2.23, and Lemma 2.24 are readily true also
when we allow $\tau(\phi,x)$ (or $\mathbb{V}(x)$) to be in some Borel subset
of $Gr(h)$.

######  Proposition 2.25.

Suppose $\phi$ is a Radon measure on $\mathbb{G}$ such that, for $\phi$-almost
every $x\in\mathbb{G}$, we have
$\mathrm{Tan}(\phi,x)=\\{\lambda\mathcal{S}^{h}\llcorner\mathbb{V}(x):\lambda>0\\}$
for some $\mathbb{V}(x)\in Gr(h)$. Then, for every $\alpha\in(0,1)$ there
exist $\\{\mathbb{V}_{i}\\}_{i\in\mathbb{N}}\subseteq Gr(h)$, and a family of
compact $C_{\mathbb{V}_{i}}(\alpha)$-sets $\\{\Gamma_{i}\\}_{i\in\mathbb{N}}$
such that

$\phi(\mathbb{G}\setminus\cup_{i\in\mathbb{N}}\Gamma_{i})=0.$

###### Proof.

First of all, by 2.19, the measure $\phi$ is locally asymptotically doubling.
Up to restricting $\phi$ to closed balls and by using the locality of tangents
in 2.6 and Lebesgue Theorem in 2.6, we may assume that $\phi$ is supported on
a compact set $K$ and that it is still locally asymptotically doubling. Let
$S$ be dense countable subset of $(Gr(h),d_{\mathbb{G}})$, that exists thanks
to 2.7. Thanks to [4, Proposition 2.29], we infer that also the countable set
$\\{(h+1)\mathcal{C}^{h}\llcorner\mathbb{W}:\mathbb{W}\in S\\}$ is dense in
the metric space $(\\{(h+1)\mathcal{C}^{h}\llcorner\mathbb{V}:\mathbb{V}\in
Gr(h)\\},F_{0,1})$.

Let us now fix $\mathfrak{e}<1/10$,
$\sigma<1/100(\mathfrak{e}/(3(1+\mathfrak{e})))^{h}$,$\mathbb{V}\in S$ and let
us denote

$K_{\mathbb{V}}:=\\{x\in
K:F_{0,1}((h+1)\mathcal{C}^{h}\llcorner\mathbb{V},(h+1)\mathcal{C}^{h}\llcorner\mathbb{V}(x))<\sigma^{h+4}\\},$

where $\mathbb{V}(x)\in Gr(h)$ is such that
$\mathrm{Tan}(\phi,x)=\\{\lambda\mathcal{S}^{h}\llcorner\mathbb{V}(x),\lambda>0\\}$.
Since $\\{(h+1)\mathcal{C}^{h}\llcorner\mathbb{W}:\mathbb{W}\in S\\}$ is dense
in the metric space $(\\{(h+1)\mathcal{C}^{h}\llcorner\mathbb{V}:\mathbb{V}\in
Gr(h)\\},F_{0,1})$ we conclude that $K=\cup_{\mathbb{V}\in S}K_{\mathbb{V}}$.
By Lemma 2.22, one gets that $K_{\mathbb{V}}$ is $\phi$-measurable for every
$\mathbb{V}\in S$. Thus by 2.6, we can assume without loss of generality that
$\phi$ is locally asymptotically doubling and supported on $K_{\mathbb{V}}$
for some $\mathbb{V}\in S$, which from now on we fix.

We now claim that for $\phi$-almost every $x\in\mathbb{G}$ the following holds

$\lim_{r\to
0^{+}}d_{x,r}(\phi,\mathfrak{M}(h,\\{\mathbb{V}\\}))=F_{0,1}((h+1)\mathcal{C}^{h}\llcorner\mathbb{V},(h+1)\mathcal{C}^{h}\llcorner\mathbb{V}(x)).$
(32)

Indeed, for $\phi$-almost every $x\in\mathbb{G}$ the measure
$T_{x,r}\phi/F_{1}(T_{x,r}\phi)$ converges to
$(h+1)\mathcal{C}^{h}\llcorner\mathbb{V}(x)$ as $r\to 0^{+}$, see 2.20 and
thus, from the definition of $d_{x,r}$, we get that

$d_{x,r}(\phi,\mathfrak{M}(h,\\{\mathbb{V}\\}))=F_{0,1}(T_{x,r}\phi/F_{1}(T_{x,r}\phi),(h+1)\mathcal{C}^{h}\llcorner\mathbb{V}),$
(33)

from which we deduce the claim (32) by using the previous convergence and the
continuity of $F_{0,1}$, see, e.g., [40, Proposition 1.10] or 2.16. Moreover,
the function $x\to d_{x,r}(\phi,\mathfrak{M}(h,\\{\mathbb{V}\\}))$ is
continuous in $x$ for every $r>0$. Indeed, by (33) and the continuity of
$F_{0,1}$, it is sufficent to see that, for every $r>0$, the map $x\to
T_{x,r}\phi/F_{1}(T_{x,r}\phi)$ is continuous from $\mathbb{G}$ to the space
of Radon measures equipped with the weak* convergence, which is clear again by
the continuity of $F_{1}(\cdot)$ and by the continuity of the map $x\to
T_{x,r}\phi$, which is readily verified (see, e.g., the computations at the
end of [40, page 22]). Hence, by using Severini-Egoroff Theorem, we can assume
without loss of generality that $\phi$ is supported on a compact set $E$ such
that $\operatorname*{diam}(E)<s$ and such that
$d_{x,r}(\phi,\mathfrak{M}(h,\\{\mathbb{V}\\}))<\sigma^{h+4}$ whenever $x\in
E$ and $r\in(0,400(h+1)s)$. Let us now fix $\widetilde{x},\widetilde{y}\in E$
and denote $a:=d(\widetilde{x},\widetilde{y})$,
$\widetilde{t}:=2a(1+\mathfrak{e})$, $\widetilde{r}:=a(1+\mathfrak{e})$ and
$\widetilde{s}:=a\mathfrak{e}$.

Let us apply 2.21 first with the choices $x=y=z=\widetilde{y}$,
$s=r=\widetilde{r}$, $t=\widetilde{t}$ and $\sigma$ as above, that yields

$\phi(B(\widetilde{y},\widetilde{r})\cap
B(\widetilde{y}\mathbb{V},\sigma^{2}\widetilde{t}/(h+1)))\geq(1-5\sigma)\phi(B(\widetilde{y},\widetilde{r})),$
(34)

and secondly with $x=y=z=\widetilde{x}$, $r=\widetilde{r}+a$,
$s=\widetilde{s}$, $t=3a(1+\mathfrak{e})$ and $\sigma$, we get

$\phi(B(\widetilde{x},\widetilde{s})\cap
B(\widetilde{x}\mathbb{V},\sigma^{2}\cdot
3a(1+\mathfrak{e})/(h+1)))\geq(1-5\sigma)(\widetilde{s}/(\widetilde{r}+a))^{h}\phi(B(\widetilde{x},\widetilde{r}+a)).$
(35)

Putting together (34) and (35), we conclude that

$\begin{split}\phi(B(\widetilde{y},\widetilde{r})\setminus
B(\widetilde{y}\mathbb{V},2a\sigma^{2}(1+\mathfrak{e})/(h+1)))&=\phi(B(\widetilde{y},\widetilde{r}))-\phi(B(\widetilde{y},\widetilde{r})\cap
B(\widetilde{y}\mathbb{V},2a\sigma^{2}(1+\mathfrak{e})/(h+1)))\\\ \leq
5\sigma\phi(B(\widetilde{y},\widetilde{r}))&\leq
5\sigma\phi(B(\widetilde{x},\widetilde{r}+a))\leq\frac{5\sigma}{1-5\sigma}\left(\frac{2(1+\mathfrak{e})}{\mathfrak{e}}\right)^{h}\phi(B(\widetilde{x},\widetilde{s}))\end{split}$
(36)

If by contradiction $B(\widetilde{x},\mathfrak{e}a)\cap
B(\widetilde{y}\mathbb{V},2a\sigma^{2}(1+\mathfrak{e})/(h+1)))=\emptyset$ then
from (36) and the fact that $B(\widetilde{x},\widetilde{s})\subseteq
B(\widetilde{y},\widetilde{r})$, we infer

$\begin{split}\phi(B(\widetilde{x},\widetilde{s}))\leq\frac{5\sigma}{1-5\sigma}\left(\frac{2(1+\mathfrak{e})}{\mathfrak{e}}\right)^{h}\phi(B(\widetilde{x},\widetilde{s})),\end{split}$

that is in contradiction thanks with the choice of $\sigma$. Hence, for every
$\widetilde{x},\widetilde{y}\in E$ we have that
$B(\widetilde{x},\mathfrak{e}a)\cap
B(\widetilde{y}\mathbb{V},2a\sigma^{2}(1+\mathfrak{e})/(h+1)))\neq\emptyset$
and thus $d(\widetilde{x},\widetilde{y}\mathbb{V})\leq
a(\mathfrak{e}+2\sigma^{2}(1+\mathfrak{e})/(h+1))=d(\widetilde{x},\widetilde{y})(\mathfrak{e}+2\sigma^{2}(1+\mathfrak{e})/(h+1))$.
Hence, the compact set $E$ is a
$C_{\mathbb{V}}(\mathfrak{e}+2\sigma^{2}(1+\mathfrak{e})/(h+1))$-set. Since it
is clear that, for any given $\alpha>0$, $\sigma$ and $\mathfrak{e}$ can be
chosen small enough in order to have
$\mathfrak{e}+2\sigma^{2}(1+\mathfrak{e})/(h+1)<\alpha$, the proof is thus
concluded. ∎

In the case the tangents are complemented we can give the following
improvement of the latter Proposition.

######  Proposition 2.26.

Let $1\leq h\leq Q$ be a natural number. There exist
$\\{\mathbb{V}_{i}\\}_{i\in\mathbb{N}}\subseteq Gr_{c}(h)$ and
$\mathbb{L}_{i}$ complementary subgroups of $\mathbb{V}_{i}$ such that the
following holds.

Suppose $\phi$ is a Radon measure on $\mathbb{G}$ such that, for $\phi$-almost
every $x\in\mathbb{G}$, we have
$\mathrm{Tan}(\phi,x)=\\{\lambda\mathcal{S}^{h}\llcorner\mathbb{V}(x):\lambda>0\\}$
for some $\mathbb{V}(x)\in Gr_{c}(h)$. Then, for every $\alpha\in(0,1)$ there
exists a family of compact sets $\\{\Gamma_{i}\\}_{i\in\mathbb{N}}$ such that

$\phi(\mathbb{G}\setminus\cup_{i\in\mathbb{N}}\Gamma_{i})=0,$

and, for every $i\in\mathbb{N}$, $\Gamma_{i}$ is a compact intrinsic Lipschitz
graph, which is also a $C_{\mathbb{V}_{i}}(\alpha)$-set, of a map
$\varphi_{i}:A_{i}\subseteq\mathbb{V}_{i}\to\mathbb{L}_{i}$, where $A_{i}$ is
compact.

###### Proof.

The proof follows exactly the same lines as the proof of 2.25, so we just
sketch it underlying the main differences. For every $\ell\in\mathbb{N}$, with
$\ell\geq 2$, let us define

$Gr_{c}(h,\ell):=\\{\mathbb{V}\in
Gr_{c}(h):\exists\,\mathbb{L}\,\,\text{complement
of}\,\,\mathbb{V}\,\,\text{s.t.}\,\,1/\ell<\varepsilon_{\ref{ep:Cool}}(\mathbb{V},\mathbb{L})\leq
1/(\ell-1)\\}.$

Observe that 2.7 implies that $Gr_{c}(h,\ell)$ is separable for any
$\ell\in\mathbb{N}$, since $Gr_{c}(h,\ell)\subseteq Gr(h)$ and
$(Gr(h),d_{\mathbb{G}})$ is a compact metric space. Let

$\mathscr{D}_{\ell}:=\\{\mathbb{V}_{i,\ell}\\}_{i\in\mathbb{N}},$ (37)

be a countable dense subset of $Gr_{c}(h,\ell)$ and

$\text{for all $i\in\mathbb{N}$, choose a complement $\mathbb{L}_{i,\ell}$ of
$\mathbb{V}_{i,\ell}$ s.t.
$1/\ell<\varepsilon_{\ref{ep:Cool}}(\mathbb{V}_{i,\ell},\mathbb{L}_{i,\ell})\leq
1/(\ell-1)$}.$

Now, let $S:=\\{\mathbb{V}_{i,\ell}\\}_{i,\ell\in\mathbb{N}}$, which is a
dense countable subset of $(Gr_{c}(h),d_{\mathbb{G}})$ thanks to the
definition given above. As in the above proposition we infer that also the
countable set $\\{(h+1)\mathcal{C}^{h}\llcorner\mathbb{W}:\mathbb{W}\in S\\}$
is dense in the metric space
$(\\{(h+1)\mathcal{C}^{h}\llcorner\mathbb{V}:\mathbb{V}\in
Gr_{c}(h)\\},F_{0,1})$. Let us now fix, for every $\ell\in\mathbb{N}$,
$\mathfrak{e}_{\ell}<\min\\{1/10,1/(2\ell),\alpha/2\\}$, where $\alpha$ is as
in the statement, and
$\sigma_{\ell}<\min\\{1/100(\mathfrak{e}_{\ell}/(3(1+\mathfrak{e}_{\ell})))^{h},\sigma^{\prime}_{\ell}\\}$,
where $\sigma^{\prime}_{\ell}$ is chosen small enough such that
$\mathfrak{e}_{\ell}+2(\sigma_{\ell}^{\prime})^{2}(1+\mathfrak{e}_{\ell})/(h+1)<\min\\{\alpha,1/\ell\\}$.
Moreover, for every $\mathbb{V}_{i,\ell}\in\mathscr{D}_{\ell}$, let us denote

$K_{\mathbb{V}_{i,\ell}}:=\\{x\in
K:F_{0,1}((h+1)\mathcal{C}^{h}\llcorner\mathbb{V}_{i,\ell},(h+1)\mathcal{C}^{h}\llcorner\mathbb{V}(x))<\sigma_{\ell}^{h+4}\\},$

where $\mathbb{V}(x)$ is the element of $Gr(h)$ for which
$\mathrm{Tan}(\phi,x)=\\{\lambda\mathcal{S}^{h}\llcorner\mathbb{V}(x),\lambda>0\\}$.
Arguing as in the above proposition, being $K$ the compact set on which we can
assume $\phi$ is supported without loss of generality, we have
$K=\cup_{\ell\in\mathbb{N}}\cup_{\mathbb{V}_{i,\ell}\in\mathscr{D}_{\ell}}K_{\mathbb{V}_{i,\ell}}$.
Hence, we can assume without loss of generality that $\phi$ is supported on
$K_{\mathbb{V}_{i,\ell}}$ for some $\mathbb{V}_{i,\ell}$. The computations in
the proof of the above proposition can be repeated substituting
$\sigma_{\ell}$ with $\sigma$ accordingly, allowing us to conclude that
$\phi$-almost every $K_{\mathbb{V}_{i,\ell}}$ can be covered by compact sets
that are
$C_{\mathbb{V}_{i,\ell}}(\mathfrak{e}_{\ell}+2(\sigma_{\ell})^{2}(1+\mathfrak{e}_{\ell})/(h+1))$.
By the very choice of $\sigma^{\prime}_{\ell}$ this implies that the latter
compact sets are $C_{\mathbb{V}_{i,\ell}}(\min\\{\alpha,1/\ell\\})$-sets, and
since
$1/\ell<\varepsilon_{\ref{ep:Cool}}(\mathbb{V}_{i,\ell},\mathbb{L}_{i,\ell})$,
we also conclude that they are graphs according to the splitting
$\mathbb{G}=\mathbb{V}_{i,\ell}\cdot\mathbb{L}_{i,\ell}$, see 2.13. ∎

##  3 From flat tangents to $\mathscr{P}$-rectifiability

In this section we first prove that, in an arbitrary Carnot group, having flat
(complemented) tangent measures à la Preiss implies being
$\mathscr{P}$-rectifiable, see 3.7. Then we will prove a rectifiable
criterion, see 3.9, which will allow us to complete the proof of 1.1.

In this section a Carnot group $\mathbb{G}$ will be fixed, along with a left-
invariant homogeneous distance on it, that sometimes will be understood.
Throughout this section we assume that $\mathbb{V}\in Gr_{c}(h)$ and that
$\mathbb{V}\mathbb{L}=\mathbb{G}$. In this chapter whenever we deal with
$C_{\mathbb{V}}(\alpha)$-sets we are tacitly assuming that
$\alpha\leq\varepsilon_{\ref{ep:Cool}}(\mathbb{V},\mathbb{L})$.

###  3.1 From flat Preiss’s tangents to $\mathscr{P}$-rectifiability

We are going to prove that whenever a measure has flat (complemented) tangents
á la Preiss, then it is $\mathscr{P}$-rectifiable. Throughout this section we
assume that $\mathbb{V}\in Gr_{c}(h)$ and that $\mathbb{L}$ is a complementary
subgroup of $\mathbb{V}$. Let us begin with a proposition that roughly tells
us the following. If $\Gamma$ is a compact $C_{\mathbb{V}}(\alpha)$-set with
$\alpha\leq\varepsilon_{\ref{ep:Cool}}(\mathbb{V},\mathbb{L})$, and moreover
we know that the measure $\mathcal{S}^{h}\llcorner\Gamma$ is locally
asymptotically doubling, hence $\Gamma$ has big projections on $\mathbb{V}$.
This will allow us to prove that the lower $h$-density of
$\mathcal{S}^{h}\llcorner\Gamma$ is positive almost everywhere, see 3.6. The
latter conclusion eventually leads to the following result: if a set has flat
(complemented) Preiss’s tangents, then it is $\mathscr{P}^{c}$-rectifiable,
see 3.7.

Let us start by recalling two results from [4] and proving an adaption of [4,
Proposition 4.6] to our context.

######  Lemma 3.1 ([4, Lemma 4.2]).

Suppose $d$ is a homogeneous left-invariant metric on $\mathbb{G}$. Then,
there exists an $A:=A(d,\mathbb{V},\mathbb{L})>1$ such that for any $w\in
B_{d}(0,1/5A)$, any $y\in\partial B_{d}(0,1)\cap
C_{\mathbb{V},d}(\varepsilon_{\ref{ep:Cool}}(d,\mathbb{V},\mathbb{L}))$ and
any $z\in B_{d}(y,1/5A)$, we have $w^{-1}z\not\in\mathbb{L}$.

######  Proposition 3.2 ([4, Proposition 4.3]).

Suppose $d$ is a homogeneous left-invariant metric on $\mathbb{G}$. Let us fix
$\alpha<\varepsilon_{\ref{ep:Cool}}(d,\mathbb{V},\mathbb{L})$ and suppose
$\Gamma$ is a $C_{\mathbb{V},d}(\alpha)$-set. For any $x\in\Gamma$ let
$\rho(x)$ to be the biggest number satisfying the following condition. For any
$0<r<\rho(x)$ and any $y\in B(x,r)\cap\Gamma$ we have

$P_{\mathbb{V}}(B_{d}(x,r))\cap P_{\mathbb{V}}(B_{d}(y,s))=\emptyset\,\,\text{
for any }r,s<d(x,y)/5A,$

where $A=A(d,\mathbb{V},\mathbb{L})$ is the constant yielded by Lemma 3.1.
Then, the function $x\mapsto\rho(x)$ is positive everywhere on $\Gamma$ and
upper semicontinuous.

######  Proposition 3.3.

Let $\alpha\leq\varepsilon_{\ref{ep:Cool}}(\mathbb{V},\mathbb{L})$ and suppose
$\Gamma$ is a compact $C_{\mathbb{V}}(\alpha)$-set of $\mathcal{S}^{h}$-finite
measure such that for $\mathcal{S}^{h}\llcorner\Gamma$-almost every
$x\in\mathbb{G}$ we have

$\limsup_{r\to
0}\frac{\mathcal{S}^{h}\llcorner\Gamma(B(x,2r))}{\mathcal{S}^{h}\llcorner\Gamma(B(x,r))}<\infty.$

Then, there exists a constant $C_{3}:=C_{\ref{C:proj}}(h,A)>0$ such that for
$\mathcal{S}^{h}$-almost every $x\in\Gamma$ there exists an infinitesimal
sequence $\\{\ell_{i}(x)\\}_{i\in\mathbb{N}}$ such that for any
$i\in\mathbb{N}$ we have

$\mathcal{S}^{h}(P_{\mathbb{V}}(\Gamma\cap
B(x,\ell_{i}(x))))>C_{\ref{C:proj}}\ell_{i}(x)^{h}$ (38)

###### Proof.

We will just sketch the proof, that is an adaptation of [4, Proposition 4.6]
and we refer the reader to [4, Proposition 4.6] for the missing details. Let
$N\in\mathbb{N}$ be the unique natural number for which $5^{N-2}\leq
A<5^{N-1}$, where $A$ is as in Lemma 3.1. Notice that, since
$2^{-h}\leq\Theta^{h,*}(\mathcal{S}^{h}\llcorner\Gamma,x)\leq 1$ for
$\mathcal{S}^{h}\llcorner\Gamma$-almost every $x\in\mathbb{G}$ (cf. [15,
2.10.19(1) and 2.10.19(5)]), hence, for
$\mathcal{S}^{h}\llcorner\Gamma$-almost every $x\in\mathbb{G}$ there exists an
infinitesimal sequence $\\{\ell_{i}(x)\\}_{i\in\mathbb{N}}$ such that

$\frac{1}{2^{h+1}}\leq\frac{\mathcal{S}^{h}(\Gamma\cap
B(x,\ell_{i}(x)))}{\ell_{i}(x)^{h}}\leq 2.$ (39)

Thus, for any $k\in\mathbb{N}$ and $0<\delta<1/2$ we define the following sets

$\begin{split}A(k):=&\\{x\in\Gamma:\rho(x)>1/k\\},\\\
\mathscr{D}(k):=&\bigg{\\{}x\in A(k):\lim_{r\to
0}\frac{\mathcal{S}^{h}(B(x,r)\cap
A(k))}{\mathcal{S}^{h}(B(x,r)\cap\Gamma)}=1\bigg{\\}},\\\
\mathscr{F}_{\delta}(k):=&\bigg{\\{}B(x,r):x\in\mathscr{D}(k)\text{,
}r\leq\frac{\min\\{k^{-1},\delta\\}}{1000A}\text{ and
}\frac{1}{2^{h+1}}\leq\frac{\mathcal{S}^{h}\llcorner\Gamma(B(x,5^{N+1}r))}{(5^{N+1}r)^{h}}\leq
2\bigg{\\}},\end{split}$ (40)

where $\rho(x)$ is the number defined in 3.2. First of all notice that, thanks
to (39), $\mathscr{F}_{\delta}(k)$ is a fine covering of
$\mathcal{S}^{h}\llcorner\Gamma$-almost all $\mathscr{D}(k)$. Furthermore, for
any $k$ the sets $A(k)$ are Borel since thanks to 3.2, the function $\rho$ is
upper semicontinuous and, since by assumption $\mathcal{S}^{h}\llcorner\Gamma$
is locally asymptotically doubling, we also know that
$\mathcal{S}^{h}\llcorner\Gamma(A(k)\setminus\mathscr{D}(k))=0$. Finally, from
3.2 we infer that
$\mathcal{S}^{h}(\Gamma\setminus\cup_{k=1}^{+\infty}A(k))=0$. Let us apply [4,
Lemma 4.5] to $N$ and $\mathscr{F}_{\delta}(k)$ and we obtain the disjoint
subfamily $\mathscr{G}_{\delta}(k)$ of $\mathscr{F}_{\delta}(k)$ such that

  * ($\alpha$)

for any $B,B^{\prime}\in\mathscr{G}_{\delta}(k)$ we have that $5^{N}B\cap
5^{N}B^{\prime}=\emptyset$,

  * ($\beta$)

$\bigcup_{B\in\mathscr{F}_{\delta}(k)}B\subseteq\bigcup_{B\in\mathscr{G}_{\delta}(k)}5^{N+1}B$.

Throughout the rest of the proof we fix a $w\in\mathscr{D}(k)$ such that there
exists a sequence $\\{\ell_{i}(w)\\}_{i\in\mathbb{N}}$ satisfying (39),
$\ell_{i}(w)\leq\min\\{k^{-1},\delta\\}/8$, and

$\frac{\mathcal{S}^{h}\llcorner\mathscr{D}(k)(B(w,\ell_{i}(w)))}{\mathcal{S}^{h}\llcorner\Gamma(B(w,\ell_{i}(w)))}\geq\frac{1}{2}\qquad\text{for
any }i\in\mathbb{N},$ (41)

where the inequality follows from the fact that
$\mathcal{S}^{h}\llcorner\Gamma$-almost every point of $\mathscr{D}(k)$ has
density one with respect to the locally asymptotically doubling measure
$\mathcal{S}^{h}\llcorner\Gamma$. Notice that, according to the previous
discussion, the previous choice on $w$ is made in a set of full
$\mathcal{S}^{h}\llcorner\Gamma$-measure, so that if we prove the estimate
(38) with such a $w$ we are done. For the ease of notation we continue the
proof fixing the radius $\ell_{i}(w)=R$. We stress that the forthcoming
estimates are verified also for any $\ell_{i}(w)$. As in [4, Proposition 4.6],
one can prove that for any couple of closed balls
$B(x,r),B(y,s)\in\mathscr{G}_{\delta}(k)$ such that $B(w,R)$ intersects both
$B(x,5^{N+1}r)$ and $B(y,5^{N+1}s)$, we have

$P_{\mathbb{V}}(B(x,r))\cap P_{\mathbb{V}}(B(y,s))=\emptyset.$ (42)

In order to proceed with the conclusion of the proof, let us define

$\begin{split}\mathscr{F}_{\delta}(w,R):=&\\{B\in\mathscr{F}_{\delta}(k):5^{N+1}B\cap
B(w,R)\cap\mathscr{D}(k)\neq\emptyset\\},\\\
\mathscr{G}_{\delta}(w,R):=&\\{B\in\mathscr{G}_{\delta}(k):5^{N+1}B\cap
B(w,R)\cap\mathscr{D}(k)\neq\emptyset\\},\end{split}$

Thanks to our choice of $R$, see (41), and the definition of
$\mathscr{G}_{\delta}(w,R)$ we have

$\frac{R^{h}}{2^{h+1}}\leq\mathcal{S}^{h}\llcorner\Gamma(B(w,R))\leq
2\mathcal{S}^{h}\llcorner\mathscr{D}(k)(B(w,R))\leq
2\mathcal{S}^{h}\llcorner\mathscr{D}(k)\bigg{(}\bigcup_{B\in\mathscr{G}_{\delta}(w,R)}5^{N+1}B\bigg{)}.$

Let $\mathscr{G}_{\delta}(w,R):=\\{B(x_{i},r_{i})\\}_{i\in\mathbb{N}}$ and
recall that $x_{i}\in\mathscr{D}(k)$. This implies, thanks to 2.8, that

$\begin{split}\mathcal{S}^{h}\llcorner\mathscr{D}(k)\bigg{(}\bigcup_{B\in\mathscr{G}_{\delta}(w,R)}5^{N+1}B\bigg{)}\leq
2\cdot 5^{(N+1)h}\sum_{i\in\mathbb{N}}r_{i}^{h}&=2\cdot
5^{(N+1)h}C_{\ref{BallC}}(\mathbb{V},\mathbb{L})^{-1}\sum_{i\in\mathbb{N}}\mathcal{S}^{h}(P_{\mathbb{V}}(B(x_{i},r_{i})))\\\
=2\cdot
5^{(N+1)h}C_{\ref{BallC}}(\mathbb{V},\mathbb{L})^{-1}\mathcal{S}^{h}\bigg{(}P_{\mathbb{V}}\bigg{(}\bigcup_{i\in\mathbb{N}}B(x_{i},r_{i})\bigg{)}\bigg{)}&\leq
2\cdot
5^{(N+1)h}C_{\ref{BallC}}(\mathbb{V},\mathbb{L})^{-1}\mathcal{S}^{h}\bigg{(}P_{\mathbb{V}}\bigg{(}\bigcup_{B\in\mathscr{F}_{\delta}(w,R)}B\bigg{)}\bigg{)},\end{split}$

where the first inequality comes from the subadditivity of the measure and the
upper estimate that we have in the definition of $\mathscr{F}_{\delta}(k)$;
while the first identity of the second line above comes from (42). Summing up,
for any $\delta>0$ we have

$\frac{C_{\ref{BallC}}(\mathbb{V},\mathbb{L})R^{h}}{5^{(N+1)h}2^{h+3}}\leq\mathcal{S}^{h}\bigg{(}P_{\mathbb{V}}\bigg{(}\bigcup_{B\in\mathscr{F}_{\delta}(w,R)}B\bigg{)}\bigg{)}.$

Arguing as in the end of the proof of [4, Proposition 4.6], we get the
Hausdorff convergence

$P_{\mathbb{V}}\bigg{(}\bigcup_{B\in\mathscr{F}_{\delta}(w,R)}B\bigg{)}\underset{H,\delta\to
0}{\longrightarrow}P_{\mathbb{V}}\bigg{(}\overline{\mathscr{D}(k)}\cap
B(w,R)\bigg{)}.$

Thanks to the upper semicontinuity of the Lebesgue measure with respect to the
Hausdorff convergence we eventually infer that

$\frac{C_{\ref{BallC}}(\mathbb{V},\mathbb{L})R^{h}}{5^{(N+1)h}2^{h+3}}\leq\limsup_{\delta\to
0}\mathcal{S}^{h}\bigg{(}P_{\mathbb{V}}\bigg{(}\bigcup_{B\in\mathscr{F}_{\delta}(w,R)}B\bigg{)}\bigg{)}\leq\mathcal{S}^{h}(P_{\mathbb{V}}(\overline{\mathscr{D}(k)}\cap
B(w,R)))\leq\mathcal{S}^{h}(P_{\mathbb{V}}(\Gamma\cap B(w,R))),$

where the last inequality above comes from the compactness of $\Gamma$ and the
fact that $\mathscr{D}(k)\subseteq\Gamma$. ∎

######  Proposition 3.4.

Let us fix $\alpha\leq\varepsilon_{\ref{ep:Cool}}(\mathbb{V},\mathbb{L})$ and
suppose $\Gamma$ is a compact $C_{\mathbb{V}}(\alpha)$-set of finite
$\mathcal{S}^{h}$-measure such that

$\limsup_{r\to
0}\frac{\mathcal{S}^{h}\llcorner\Gamma(B(x,2r))}{\mathcal{S}^{h}\llcorner\Gamma(B(x,r))}<\infty,$

for $\mathcal{S}^{h}$-almost every $x\in\Gamma$. Let us set
$\varphi:P_{\mathbb{V}}(\Gamma)\to\mathbb{L}$ the map whose graph is $\Gamma$,
see 2.13, and set $\Phi:P_{\mathbb{V}}(\Gamma)\to\mathbb{G}$ to be the graph
map of $\varphi$. Let us define $\Phi_{*}\mathcal{S}^{h}\llcorner\mathbb{V}$
to be the measure on $\Gamma$ such that for every measurable
$A\subseteq\Gamma$ we have
$\Phi_{*}\mathcal{S}^{h}\llcorner\mathbb{V}(A):=\mathcal{S}^{h}\llcorner\mathbb{V}(\Phi^{-1}(A))=\mathcal{S}^{h}\llcorner\mathbb{V}(P_{\mathbb{V}}(A))$.
Then $\Phi_{*}\mathcal{S}^{h}\llcorner\mathbb{V}$ is mutually absolutely
continuous with respect to $\mathcal{S}^{h}\llcorner\Gamma$.

###### Proof.

The fact that $\Phi_{*}\mathcal{S}^{h}\llcorner\mathbb{V}$ is absolutely
continuous with respect to $\mathcal{S}^{h}\llcorner\Gamma$ is an immediate
consequence of the second part of 2.8. Viceversa, suppose by contradiction
that there exists a compact subset $C$ of $\Gamma$ of positive
$\mathcal{S}^{h}$-measure such that

$0=\Phi_{*}\mathcal{S}^{h}\llcorner\mathbb{V}(C)=\mathcal{S}^{h}(P_{\mathbb{V}}(C)).$
(43)

Since $\mathcal{S}^{h}\llcorner C$ is locally asymptotically doubling by 2.6
and $C$ has positive and finite $\mathcal{S}^{h}$-measure, we infer thanks to
3.3 that the set $C$ must have a projection of positive
$\mathcal{S}^{h}$-measure. This however comes in contradiction with (43). ∎

In order to prove the forthcoming 3.6 we need the following result, which is
precisely [4, Proposition 4.10].

######  Proposition 3.5.

Suppose $d$ is a homogeneous left-invariant distance on $\mathbb{G}$, let
$\mathbb{V},\mathbb{L}$ be complementary subgroups of $\mathbb{G}$ such that
$\mathbb{V}\in Gr_{c}(h)$, and let us fix
$\alpha\leq\varepsilon_{\ref{ep:Cool}}(\mathbb{V},\mathbb{L})$. Suppose that
$\Gamma$ is a compact $C_{\mathbb{V},d}(\alpha)$-set of finite
$\mathcal{S}^{h}$-measure. As in 3.4, let us denote with
$\Phi:P_{\mathbb{V}}(\Gamma)\to\mathbb{G}$ the graph map of
$\varphi:P_{\mathbb{V}}(\Gamma)\to\mathbb{L}$ whose intrinsic graph is
$\Gamma$. Then, for $\mathcal{S}^{h}_{d}$-almost every $w\in
P_{\mathbb{V}}(\Gamma)$ we have

$\lim_{r\to
0}\frac{\mathcal{S}^{h}_{d}\big{(}P_{\mathbb{V}}\big{(}B_{d}(\Phi(w),r)\cap\Phi(w)C_{\mathbb{V},d}(\alpha)\big{)}\cap
P_{\mathbb{V}}(\Gamma)\big{)}}{\mathcal{S}_{d}^{h}\big{(}P_{\mathbb{V}}\big{(}B_{d}(\Phi(w),r)\cap\Phi(w)C_{\mathbb{V},d}(\alpha)\big{)}\big{)}}=1.$
(44)

######  Proposition 3.6.

Let $\alpha\leq\varepsilon_{\ref{ep:Cool}}(\mathbb{V},\mathbb{L})$ and suppose
$\Gamma$ is a $C_{\mathbb{V}}(\alpha)$-set such that
$\mathcal{S}^{h}\llcorner\Gamma$ is locally asymptotically doubling. Then,
$\Theta^{h}_{*}(\mathcal{S}^{h}\llcorner\Gamma,x)>0$ for
$\mathcal{S}^{h}$-almost every $x\in\Gamma$.

###### Proof.

Assume by contradiction that there exists a compact set $C\subseteq\Gamma$ of
positive $\mathcal{S}^{h}$-measure such that
$\Theta^{h}_{*}(\mathcal{S}^{h}\llcorner\Gamma,x)=0$ for every $x\in C$. Since
by 3.4 the measures $\mathcal{S}^{h}\llcorner\Gamma$ and
$\Phi_{*}\mathcal{S}^{h}\llcorner\mathbb{V}$ are mutually absolutely
continuous, the set $P_{\mathbb{V}}(C)$ must have positive
$\mathcal{S}^{h}$-measure as well. In particular we have thanks to 3.5, Lemma
2.14, 2.8, and 2.9 that for $\mathcal{S}^{h}$-almost every $x\in C$ we have

$\begin{split}&\mathcal{S}^{h}\big{(}P_{\mathbb{V}}\big{(}B(0,1)\cap
C_{\mathbb{V}}(\alpha)\big{)}\big{)}\\\ =&\liminf_{r\to
0}\frac{\mathcal{S}^{h}\big{(}P_{\mathbb{V}}\big{(}B(x,\mathfrak{C}(\alpha)r)\cap
xC_{\mathbb{V}}(\alpha)\big{)}\cap
P_{\mathbb{V}}(\Gamma)\big{)}}{\mathcal{S}^{h}\big{(}P_{\mathbb{V}}\big{(}B(\Phi(w),\mathfrak{C}(\alpha)r)\cap\Phi(w)C_{\mathbb{V}}(\alpha)\big{)}\big{)}}\frac{\mathcal{S}^{h}\big{(}P_{\mathbb{V}}\big{(}B(\Phi(w),\mathfrak{C}(\alpha)r)\cap\Phi(w)C_{\mathbb{V}}(\alpha)\big{)}\big{)}}{(\mathfrak{C}(\alpha)r)^{h}}\\\
=&\liminf_{r\to
0}\frac{\mathcal{S}^{h}\big{(}P_{\mathbb{V}}\big{(}B(x,\mathfrak{C}(\alpha)r)\cap
xC_{\mathbb{V}}(\alpha)\big{)}\cap
P_{\mathbb{V}}(\Gamma)\big{)}}{(\mathfrak{C}(\alpha)r)^{h}}\\\
\leq&\liminf_{r\to
0}\frac{\mathcal{S}^{h}\llcorner\mathbb{V}(P_{\mathbb{V}}(B(x,r)\cap\Gamma))}{(\mathfrak{C}(\alpha)r)^{h}}\leq
2C_{\ref{ProjC}}(\mathbb{V},\mathbb{L})\liminf_{r\to
0}\frac{\mathcal{S}^{h}\llcorner\Gamma(B(x,r))}{(\mathfrak{C}(\alpha)r)^{h}}=0,\end{split}$

where $\mathfrak{C}(\alpha)$ is the constant introduced in Lemma 2.14. The
above computation is in contradiction with the fact that
$\mathcal{S}^{h}\big{(}P_{\mathbb{V}}\big{(}B(0,1)\cap
C_{\mathbb{V}}(\alpha)\big{)}\big{)}$ is positive thus concluding the proof of
the proposition. ∎

We are now in a position to state the main result of this subsection.

######  Theorem 3.7.

Let $\Gamma\subseteq\mathbb{G}$ be compact such that
$\mathcal{S}^{h}(\Gamma)<+\infty$. Assume that for
$\mathcal{S}^{h}\llcorner\Gamma$-almost every $x\in\mathbb{G}$ we have
$\mathrm{Tan}(\mathcal{S}^{h}\llcorner\Gamma,x)=\\{\lambda\mathcal{S}^{h}\llcorner\mathbb{V}(x):\lambda>0,\\}$,
where $\mathbb{V}(x)\in Gr_{c}(h)$. Then, $\mathcal{S}^{h}\llcorner\Gamma$ is
$\mathscr{P}_{h}^{c}$-rectifiable.

###### Proof.

We have that $\mathcal{S}^{h}\llcorner\Gamma$ is locally asymptotically
doubling, see 2.19. Moreover, from 2.26, there exist
$\\{\mathbb{V}_{i}\\}_{i\in\mathbb{N}}\subseteq Gr_{c}(h)$, and
$\\{\mathbb{L}_{i}\\}_{i\in\mathbb{N}}$, such that $\mathbb{L}_{i}$ and
$\mathbb{V}_{i}$ are homogeneous complementary subgroups, with the property
that for every $\alpha>0$ there exists a family of compact sets
$\\{\Gamma_{i}\\}$ such that $\Gamma_{i}$ is a
$C_{\mathbb{V}_{i}}(\min\\{\alpha,\varepsilon_{\ref{ep:Cool}}(\mathbb{V}_{i},\mathbb{L}_{i})\\})$-set,
and

$\mathcal{S}^{h}(\Gamma\setminus\cup_{i\in\mathbb{N}}\Gamma_{i})=0.$ (45)

Since $\mathcal{S}^{h}\llcorner\Gamma$ is locally asymptotically doubling,
then $\mathcal{S}^{h}\llcorner\Gamma_{i}$ is locally asymptotically doubling
for every $i\in\mathbb{N}$, see 2.6. Hence, we can apply 3.6 to conclude that
$\Theta^{h}_{*}(\mathcal{S}^{h}\llcorner\Gamma_{i},x)>0$ for every
$i\in\mathbb{N}$ and for $\mathcal{S}^{h}$-almost every $x\in\Gamma_{i}$. In
addition, from the previous inequality and [15, 2.10.19(5)] for every
$i\in\mathbb{N}$, we get that

$0<\Theta^{h}_{*}(\mathcal{S}^{h}\llcorner\Gamma_{i},x)\leq\Theta^{h,*}(\mathcal{S}^{h}\llcorner\Gamma_{i},x)<+\infty,\quad\text{for
$\mathcal{S}^{h}$-almost every $x\in\Gamma_{i}$.}$ (46)

Moreover, since for $\mathcal{S}^{h}$-almost every $x\in\Gamma$ we have
$\mathrm{Tan}(\mathcal{S}^{h}\llcorner\Gamma,x)=\\{\lambda\mathcal{S}^{h}\llcorner\mathbb{V}(x):\lambda>0\\}$
with $\mathbb{V}(x)\in Gr_{c}(h)$, we deduce that, for every $i\in\mathbb{N}$,
the locality of tangents in 2.6 ensures that for $\mathcal{S}^{h}$-almost
every $x\in\Gamma_{i}$ we have
$\mathrm{Tan}(\mathcal{S}^{h}\llcorner\Gamma_{i},x)=\\{\lambda\mathcal{S}^{h}\llcorner\mathbb{V}(x):\lambda>0\\}$.
From the previous equality, we conclude that for every $i\in\mathbb{N}$ we
have
$\mathrm{Tan}_{h}(\phi,x)\subseteq\\{\lambda\mathcal{S}^{h}\llcorner\mathbb{V}(x):\lambda>0\\}$.
Hence, from the latter conclusion and (46) we get that
$\mathcal{S}^{h}\llcorner\Gamma_{i}$ is a $\mathscr{P}_{h}^{c}$-rectifiable
measure for every $i\in\mathbb{N}$. Finally, from (45) and 2.6 we conclude
that $\mathcal{S}^{h}\llcorner\Gamma$ is a $\mathscr{P}^{c}_{h}$-rectifiable
measure. ∎

###  3.2 From approximate tangent planes to $\mathscr{P}$-rectifiability

In this section we aim at proving that whenever an approximate (complemented)
$h$-dimensional tangent plane to a set $\Gamma$ exists almost everywhere in
the sense of 3.9, then the measure $\mathcal{S}^{h}\llcorner\Gamma$ is
$\mathscr{P}^{c}_{h}$-rectifiable. First, we need a crucial estimate on
projections that will be useful also later on. Since none of the main results
of this subsection depends on the choice of the metric, in the following $d$
will be an arbitrary homogeneous left-invariant metric and $\lVert\rVert_{d}$
its associated homogeneous norm.

######  Proposition 3.8.

Suppose $d$ is a homogeneous left-invariant distance on $\mathbb{G}$ and let
$\mathbb{V},\mathbb{W}\in Gr_{c}(h)$ be complemented by the same subgroup
$\mathbb{L}$. Then, there exists an increasing function
$\Delta:(0,C_{\ref{ProjC}}(d,\mathbb{W},\mathbb{L})]\to(0,+\infty)$, depending
only on $\mathbb{V}$, $\mathbb{W}$, $\mathbb{L}$ and $d$, such that
$\lim_{\beta\to 0}\Delta(\beta)=0$ and satisfying the following condition.

For any $\alpha\leq\varepsilon_{\ref{ep:Cool}}(\mathbb{V},\mathbb{L})$ and any
$C_{\mathbb{V},d}(\alpha)$-set $\Gamma$ of finite $\mathcal{S}^{h}$-measure if
there are an $x\in\Gamma$, a $\beta\leq
C_{\ref{ProjC}}(d,\mathbb{W},\mathbb{L})$ and a $\rho>0$ such that

$\Gamma\cap B_{d}(x,r)\subseteq xC_{\mathbb{W},d}(\beta),\qquad\text{for all
$0<r<\rho$.}$ (47)

then

$\Big{\lvert}\frac{\mathcal{S}_{d}^{h}(P_{\mathbb{V}}(B_{d}(x,r)\cap
xC_{\mathbb{W},d}(\beta))\cap
P_{\mathbb{V}}(\Gamma))}{r^{h}}-\frac{\mathcal{S}^{h}(P_{\mathbb{V}}(B_{d}(x,r)\cap
xC_{\mathbb{W},d}(\beta)\cap\Gamma))}{r^{h}}\Big{\rvert}\leq\Delta(\beta),$

for any
$0<r<(1+\alpha(C_{\ref{ProjC}}(d,\mathbb{V},\mathbb{L})-\alpha)^{-1})^{-1}C_{\ref{ProjC}}(d,\mathbb{V},\mathbb{L})\rho=:\varrho(\rho,\alpha)$.

###### Proof.

In order to simplify notation throughout the proof of the proposition, we will
drop everywhere the dependence on $d$.

Let us fix an $x\in\Gamma$, a $0<\beta\leq
C_{\ref{ProjC}}(\mathbb{W},\mathbb{L})$ and a $\rho$ where (47) holds. We
denote with $P_{\mathbb{V}},P_{\mathbb{L}}^{\mathbb{V}}$, respectively, the
projections associated to the splitting
$\mathbb{G}=\mathbb{V}\cdot\mathbb{L}$, and analogously for the splitting
$\mathbb{G}=\mathbb{W}\cdot\mathbb{L}$. For the sake of notation, for any
fixed
$0<r<(1+\alpha(C_{\ref{ProjC}}(\mathbb{V},\mathbb{L})-\alpha)^{-1})^{-1}C_{\ref{ProjC}}(\mathbb{V},\mathbb{L})\rho$
we let

$\begin{split}A_{r}:=P_{\mathbb{V}}(B(x,r)\cap xC_{\mathbb{W}}(\beta))\cap
P_{\mathbb{V}}(\Gamma)\qquad\text{and}\qquad B_{r}:=P_{\mathbb{V}}(B(x,r)\cap
xC_{\mathbb{W}}(\beta)\cap\Gamma).\end{split}$

Since the inclusion $B_{r}\subseteq A_{r}$ is always verified, we want to
estimate the measure of those $w$ contained in $A_{r}\setminus B_{r}$. If
$y\in A_{r}$, there are $w\in\Gamma$ such that $P_{\mathbb{V}}(w)=y$, and an
$\ell\in\mathbb{L}$ such that $y\ell\in B(x,r)\cap xC_{\mathbb{W}}(\beta)$.
Let us notice that 2.8 implies that
$\|P_{\mathbb{V}}(x^{-1}y)\|=\|P_{\mathbb{V}}(x^{-1}y\ell)\|\leq
C_{\ref{ProjC}}(\mathbb{V},\mathbb{L})^{-1}r$. Moreover, since $\Gamma$ is a
$C_{\mathbb{V}}(\alpha)$-set, we even get that, by exploiting 2.2,

$\|P_{\mathbb{L}}^{\mathbb{V}}(x^{-1}w)\|\leq\alpha(C_{\ref{ProjC}}(\mathbb{V},\mathbb{L})-\alpha)^{-1}\|P_{\mathbb{V}}(x^{-1}w)\|=\alpha(C_{\ref{ProjC}}(\mathbb{V},\mathbb{L})-\alpha)^{-1}\|P_{\mathbb{V}}(x^{-1}y)\|\leq\alpha(C_{\ref{ProjC}}(\mathbb{V},\mathbb{L})-\alpha)^{-1}C_{\ref{ProjC}}(\mathbb{V},\mathbb{L})^{-1}r.$

This implies in particular that

$\|x^{-1}w\|\leq\|P_{\mathbb{V}}(x^{-1}w)\|+\|P_{\mathbb{L}}^{\mathbb{V}}(x^{-1}w)\|=\|P_{\mathbb{V}}(x^{-1}y)\|+\|P_{\mathbb{L}}^{\mathbb{V}}(x^{-1}w)\|\leq(1+\alpha(C_{\ref{ProjC}}(\mathbb{V},\mathbb{L})-\alpha)^{-1})C_{\ref{ProjC}}^{-1}(\mathbb{V},\mathbb{L})r.$

Hence, from the choice of $r$, we infer that
$(1+\alpha(C_{\ref{ProjC}}(\mathbb{V},\mathbb{L})-\alpha)^{-1})C_{\ref{ProjC}}(\mathbb{V},\mathbb{L})^{-1}r<\rho$
and thus we can use the hypothesis in (47) applied to $w$ to obtain that
$x^{-1}w\in C_{\mathbb{W}}(\beta)$. Thus, by also exploiting 2.2 and the fact
that $x^{-1}y\ell\in C_{\mathbb{W}}(\beta)$ we get that

$\displaystyle\lVert
P_{\mathbb{L}}^{\mathbb{W}}(x^{-1}y)\ell\rVert\leq\beta(C_{\ref{ProjC}}(\mathbb{W},\mathbb{L})-\beta)^{-1}\lVert
P_{\mathbb{W}}(x^{-1}y)\rVert\quad$ $\displaystyle\text{and}\quad\lVert
P_{\mathbb{L}}^{\mathbb{W}}(x^{-1}y)P_{\mathbb{L}}^{\mathbb{V}}(w)\rVert\leq\beta(C_{\ref{ProjC}}(\mathbb{W},\mathbb{L})-\beta)^{-1}\lVert
P_{\mathbb{W}}(x^{-1}y)\rVert,$ (48)

where the last inequality comes from the fact that
$P_{\mathbb{L}}^{\mathbb{W}}(x^{-1}w)=P_{\mathbb{L}}^{\mathbb{W}}(x^{-1}yP_{\mathbb{L}}^{\mathbb{V}}(w))=P_{\mathbb{L}}^{\mathbb{W}}(x^{-1}y)P_{\mathbb{L}}^{\mathbb{V}}(w)$.
Thanks to (48) we deduce that

$\begin{split}\lVert\ell^{-1}P_{\mathbb{L}}^{\mathbb{V}}(w)\rVert&\leq\lVert
P_{\mathbb{L}}^{\mathbb{W}}(x^{-1}y)\ell\rVert+\lVert
P_{\mathbb{L}}^{\mathbb{W}}(x^{-1}y)P_{\mathbb{L}}^{\mathbb{V}}(w)\rVert\leq
2\beta(C_{\ref{ProjC}}(\mathbb{W},\mathbb{L})-\beta)^{-1}\lVert
P_{\mathbb{W}}(x^{-1}y)\rVert\\\
&=2\beta(C_{\ref{ProjC}}(\mathbb{W},\mathbb{L})-\beta)^{-1}\lVert
P_{\mathbb{W}}(x^{-1}y\ell)\rVert\\\ &\leq
2\beta(C_{\ref{ProjC}}(\mathbb{W},\mathbb{L})-\beta)^{-1}C_{\ref{ProjC}}(\mathbb{W},\mathbb{L})^{-1}\lVert
x^{-1}y\ell\rVert\\\ &\leq
2\beta(C_{\ref{ProjC}}(\mathbb{W},\mathbb{L})-\beta)^{-1}C_{\ref{ProjC}}(\mathbb{W},\mathbb{L})^{-1}r.\end{split}$
(49)

This in particular implies that

$\lVert x^{-1}w\rVert=\lVert x^{-1}yP_{\mathbb{L}}^{\mathbb{V}}(w)\rVert\leq