# Fragment-wise differentiable structures

David Bate  Zeeman Building, University of Warwick, Coventry CV4 7AL, UK
<EMAIL_ADDRESS>,  Sylvester Eriksson-Bique  Department of
Mathematics and Statistics, University of Jyväskylä, Seminaarinkatu 15, PO Box
35, FI-40014 University of Jyväskylä, Finland  sylvester.d.eriksson-
<EMAIL_ADDRESS>and  Elefterios Soultanis  Department of Mathematics and
Statistics, University of Jyväskylä, Seminaarinkatu 15, PO Box 35, FI-40014
University of Jyväskylä, Finland<EMAIL_ADDRESS>
###### Abstract.

The $p$-modulus of curves, test plans, upper gradients, charts, differentials,
approximations in energy and density of directions are all concepts associated
to the theory of Sobolev functions in metric measure spaces. The purpose of
this paper is to give an analogous geometric and “fragment-wise” theory for
Lipschitz functions and Weaver derivations, where $\infty$-modulus of curve
fragments, $\ast$-upper gradients and Alberti representations play a central
role. We give a new definition of fragment-wise charts and prove that they
exists for spaces with finite Hausdorff dimension. We give a replacement for
$p$-duality in terms of Alberti representations and $\infty$-modulus and
present the theory of $\ast$-upper gradients. Further, we give new and sharper
results for approximations of Lipschitz functions, which yields the density of
directions. Our results are applicable to all complete and separable metric
measure spaces. In the process, we show that there are strong parallels
between the Sobolev and Lipschitz worlds.

###### 2020 Mathematics Subject Classification:

30L99, 51F30, 28A75

The first and third authors were supported by the European Union’s Horizon
2020 research and innovation programme grant number 948021. The second author
was supported by the Research Council of Finland grants 354241 and 356861. The
third author was also supported by the Research Council of Finland grant
355122. The research was started during a visit of the second named author to
University of Warwick. He thanks the institution for its hospitality. For the
purpose of open access, the authors have applied a Creative Commons
Attribution (CC BY) licence to any Author Accepted Manuscript version arising
from this submission.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Approximation
  4. 4 Representation of minimal $\ast$-upper gradients
  5. 5 Fragment-wise differentiable structure
  6. 6 Density of Directions
  7. 7 Comparisons

##  1\. Introduction

###  1.1. Background

Analysis on metric spaces, emerging at the turn of the millennium, is based on
the deep observation of the importance that collections of curves in the
metric space play in order to perform analysis. Upper gradients – introduced
and developed through the works of Heinonen–Koskela, Shanmugalingam, Cheeger
and others [27, 12, 38, 25] – control the oscillation of functions along
curves, and lead to a theory of Poincaré inequalities and Sobolev spaces in
metric spaces.

In his seminal work [12], Cheeger proved a Rademacher type theorem in this
context, giving rise to a fruitful first order calculus: doubling metric
measure spaces $(X,d,\mu)$ supporting a Poincaré inequality can be covered by
countably many _charts_ $(U,\varphi)$, where $U\subset X$ is Borel and
$\varphi:X\to\mathbb{R}^{n}$ is Lipschitz, such that every Lipschitz function
$f:X\to\mathbb{R}$ admits a unique differential
$\mathrm{d}_{x}f:\mathbb{R}^{n}\to\mathbb{R}$ with respect to $(U,\varphi)$,
characterized by

(1.1) $\displaystyle\limsup_{y\to
x}\frac{|f(y)-f(x)-\mathrm{d}_{x}f(\varphi(y)-\varphi(x))|}{d(x,y)}=0,$

for $\mu$-a.e. $x\in U$. Metric measure spaces satisfying the conclusion of
Cheeger’s differentiation theorem are now known as _Lipschitz
differentiability spaces_.

Lipschitz differentiability on metric spaces has since seen a flurry of
interest, and multiple research directions have developed, beginning with the
work of Keith [32] who gave a weaker sufficient condition, the Lip-lip
inequality, for the conclusion of Cheeger’s theorem to hold. The work of the
first named author [8] showed that Lipschitz differentiability spaces
necessarily possess a rich structure of 1-rectifiable curves (known as Alberti
representations, see below) which determine the differentiable structure.
Stronger variants which require (1.1) to hold for all Lipschitz functions
taking values in an RNP Banach space have been used to establish non-embedding
results [13]. It has been shown that such RNP differentiability spaces are
covered by countably many isometric images of spaces satisfying the Poincaré
inequality [10, 20]. It is known [36] that differentiability of _real_ valued
Lipschitz functions alone does not guarantee any Poincaré inequality. Notions
of _partial_ differentiability have been introduced in terms of Weaver
derivations [40, 26, 35].

On the other hand, Sobolev space theory can be developed in general metric
measure spaces [28, 23, 24], and many important properties of Sobolev spaces
(e.g. their reflexivity and the density of Lipschitz functions) hold without
assuming any structure on the metric space [3]. In this general context, in
[22], the second and third named authors introduced $p$-weak charts with
respect to which Sobolev functions admit differentials satisfying (1.1) along
curves in the space. The charts and the associated weak differentiable
structure in [22] exist whenever the space has finite Hausdorff dimension, and
give an explicit curvewise description of the differential, and minimal upper
gradient, of a Sobolev function.

In this paper, we combine techniques from Sobolev space theory and Lipschitz
differentiability spaces to obtain a weak notion of charts that generalizes
both (1.1) and $p$-weak charts in [22], and exists in spaces of finite
Hausdorff dimension (see Definition 1.1 and Theorem 5.5). Our results give a
geometric interpretation of the partial differentiability of Lipschitz
functions previously studied via Weaver derivations (see Theorem 1.4).

In order to study Lipschitz differentiability spaces, one must first note that
there need not be any curves in the space. Indeed, a Cantor set in the real
line of positive Lebesgue measure is easily observed to be a Lipschitz
differentiability space. Instead, one must consider curve _fragments_ : bi-
Lipschitz maps $\gamma:\operatorname{dom}(\gamma)\to X$ from a compact set
$\operatorname{dom}(\gamma)\subset\mathbb{R}$. An Alberti representation
$\mathcal{A}=\\{\mu_{\gamma},\mathbb{P}\\}$ of a measure $\mu$ on a metric
space $X$ is a disintegration of $\mu$ into 1-rectifiable measures
$\mu_{\gamma}\ll\mathcal{H}^{1}|_{\operatorname{im}(\gamma)}$:

$\displaystyle\mu(E)=\int_{\operatorname{Fr}(X)}\mu_{\gamma}(E)\mathrm{d}\mathbb{P}(\gamma),\quad
E\subset X\textrm{ Borel,}$

where $\operatorname{Fr}(X)$ is the space of curve fragments and $\mathbb{P}$
is a positive measure on $\operatorname{Fr}(X)$ (cf. Definition 2.16).
Independent collections of Alberti representations (see Definition 2.19) play
a central role in the theory: they formalize the notion of the “independent
directions” a measure sees, and act as an analogue of $p$-independence in
[22]. Moreover, admitting $n$ independent Alberti representations is a
necessary but not sufficient condition for the existence of an $n$-dimensional
chart [8, Theorem 6.6] in a Lipschitz differentiability space. (See Section 2
for the terminology and definitions employed in the introduction.)

######  Definition 1.1.

A fragment-wise chart $(U,\varphi)$ of dimension $n$ consists of a Borel set
$U\subset X$ with $\mu(U)>0$ and a Lipschitz function
$\varphi:X\to\mathbb{R}^{n}$ satisfying the following.

  * (i)

Independence: $\mu|_{U}$ admits $n$ $\varphi$-independent Alberti
representations;

  * (ii)

Maximality: If $V\subset U$ and $\mu|_{V}$ admits $k$ independent Alberti
representations, then $k\leq n$.

A result of Bate–Kangasniemi–Orponen [9] implies the Hausdorff dimension bound
$n\leq\dim_{H}U$ for any fragment-wise chart $(U,\varphi)$ of dimension $n$.
This bound also follows from a deep result of De Philippis and Rindler [16];
previous bounds in terms of the Assouad dimension can be found in Schioppa
[35] and [8]. From such a bound, standard arguments yield that _spaces of
finite Hausdorff dimension can be covered by a countable number of fragment-
wise charts_ , see Theorem 5.5. An alternative proof of this fact can be found
following [22].

In order to combine the Sobolev space theory with Alberti representations, we
use the notion of a $\ast$-upper gradient, introduced in [10] (see (1.2)), to
accommodate for the gaps in a curve fragment. This acts as a replacement for
upper gradients of Sobolev functions, which were introduced in [27]. The
proofs of our results employ elements of plan-modulus duality of [4] and the
close connection between 1-plans and Alberti representations. A novel aspect
of our work is connecting the defined notions and expressing many results in
terms of $\ast$-upper gradients and the $\infty$-modulus of curve fragments.
From a historical perspective this is quite natural. Indeed, in [38] Sobolev
spaces were introduced together with weak $p$-upper gradients, defined in
terms of $p$-modulus of curves. Thus, whereas weak $p$-upper gradients and
$p$-modulus of curves are the natural concepts for Sobolev functions,
$\ast$-upper gradients and $\infty$-modulus of curve fragments are the
corresponding natural notions for Lipschitz functions.

###  1.2. Fragment-wise differentials and density of directions

Throughout the paper, we consider a metric measure space $X=(X,d,\mu)$, i.e. a
complete separable metric space $(X,d)$ equipped with a Borel regular measure
$\mu$ that is finite on bounded sets. Given $f\in\operatorname{LIP}(X)$, we
say that a Borel function $\rho:X\to[0,\infty]$ is a $\ast$-upper gradient of
$f$, if

(1.2) $\displaystyle{\rm
osc}_{\gamma}f\leq\int_{\operatorname{im}(\gamma)}\rho\mathrm{d}\mathcal{H}^{1}+\operatorname{LIP}(f)\operatorname{gap}(\gamma)$

for every $\gamma\in\operatorname{Fr}(X)$. We refer to Section 2.3 for the
relevant definitions and the fact that, for every $f\in\operatorname{LIP}(X)$,
there exists a minimal $\ast$-upper gradient, denoted $|Df|_{\ast}$. (For
Sobolev functions, there exists a minimal _weak_ upper gradient [28, Section
6], but in our case, weak $\ast$-upper gradients are always a.e. equal to
$\ast$-upper gradients, see Remark 2.9.)

Let $U\subset X$ be a Borel set with $\mu(U)>0$ and
$\varphi\in\operatorname{LIP}(X;\mathbb{R}^{n})$. A Borel map
$\bm{\xi}:U\to(\mathbb{R}^{n})^{*}$ is a fragment-wise differential of $f$
with respect to $(U,\varphi)$, if

(1.3)
$\displaystyle(f\circ\gamma)_{t}^{\prime}=\bm{\xi}((\varphi\circ\gamma)^{\prime}_{t})\quad
a.e.\ t\in\gamma^{-1}(U)\quad\textrm{for }\operatorname{Mod}_{\infty}-a.e.\
\gamma\in\operatorname{Fr}(X).$

If any two fragment-wise differentials of $f$ agree $\mu$-a.e., we say that
$f$ admits a _unique_ fragment-wise differential, denoted
$\mathrm{d}f:U\to(\mathbb{R}^{n})^{*}$, with respect to $(U,\varphi)$.

Our first theorem establishes the existence of differentials and pointwise
norms associated to charts, which give rise to measurable cotangent bundles
discussed later on. It is a representation in the spirit of [14, Corollary
1.10].

######  Theorem 1.2.

Suppose $(U,\varphi)$ is a fragment-wise chart of dimension $n$. Then every
$f\in\operatorname{LIP}(X)$ admits a unique fragment-wise differential
$\mathrm{d}f:U\to(\mathbb{R}^{n})^{*}$. Moreover for $\mu$-a.e. $x\in U$ there
exists a norm $|\cdot|_{x}$ on $(\mathbb{R}^{n})^{*}$ such that
$x\mapsto|\xi|_{x}$ is Borel for every $\xi\in(\mathbb{R}^{n})^{*}$ and

(1.4) $\displaystyle|Df|_{\ast}(x)=|\mathrm{d}_{x}f|_{x}\quad\mu-a.e.\ x\in U$

for every $f\in\operatorname{LIP}(X)$.

The differential satisfies the usual calculus rules, see Section 5.2. As a
geometric consequence of the analytical structure provided by fragment-wise
charts we have the following theorem which states that, at a generic point in
the chart, there exist curve fragments arbitrarily close to any given
direction. This result generalizes [14, Theorem 5.3].

######  Theorem 1.3.

Let $(U,\varphi)$ be a fragment-wise chart of dimension $n\in\mathbb{N}$. For
$\mu$-a.e. $x\in U$, the set of directions

$\displaystyle
S_{x}(\varphi)=\bigg{\\{}\frac{(\varphi\circ\gamma)^{\prime}(t)}{|(\varphi\circ\gamma)^{\prime}(t)|}:\
\gamma_{t}=x\bigg{\\}}\subset S^{n-1}$

is dense in $S^{n-1}$.

This theorem is proven in a stronger form as Theorem 6.4.

###  1.3. Lipschitz differentiability spaces and Weaver derivations

A countable covering of $X$ by fragment-wise charts gives rise to a
differentiable structure as well as _tangent_ and _cotangent_ bundles $TX$ and
$T^{*}X$, respectively, see Section 5.4. The differential of
$f\in\operatorname{LIP}(X)$ given by Theorem 1.2 naturally defines an element
$\mathrm{d}f\in L^{\infty}(T^{*}X)$ in the space of essentially bounded
sections of $T^{*}X$.

An important property of the linear map
$f\mapsto\mathrm{d}f:\operatorname{LIP}(X)\to L^{\infty}(T^{*}X)$ is its
weak∗-continuity (cf. Theorem 5.8 and Remark 5.10), which hints at a
connection with _Weaver derivations_ (see Section 2.6). Introduced originally
by Weaver [40], Weaver derivations have inspired the theory of tangent modules
in [23], and turn out to be closely connected with Alberti representations and
metric 1-currents [37, 35]. The space $\mathcal{X}(\mu)$ of Weaver derivations
equipped with the norm
$\|\bm{b}\|_{\mathcal{X}(\mu)}=\||\bm{b}|_{\mathcal{X},\mathrm{loc}}\|_{L^{\infty}(\mu)}$
is a Banach space. See Section 2.6, and (2.3) for the definition of the
pointwise norm $|\bm{b}|_{\mathcal{X},\mathrm{loc}}$ of a Weaver derivation
$\bm{b}\in\mathcal{X}(\mu)$.

In Theorems 1.4 and 1.5 below, we identify Weaver derivations with _vector
fields_ on $X$, i.e. sections of the tangent bundle $TX$, and use this
identification to characterize Lipschitz differentiability spaces in the
spirit of [35] by using the minimal $\ast$-upper gradients, respectively.

######  Theorem 1.4.

Suppose $X=(X,d,\mu)$ admits a fragment-wise differentiable structure. Then
the map $\iota:L^{\infty}(TX)\to\mathcal{X}(\mu)$ defined by

$\displaystyle\iota(\bm{X})(f)\mathrel{\mathop{:}}=\mathrm{d}f(\bm{X}),\quad
f\in\operatorname{LIP}(X)$

is an isomorphism of normed $L^{\infty}(\mu)$-modules satisfying

(1.5)
$\displaystyle|\iota(\bm{X})|_{\mathcal{X},\mathrm{loc}}=|\bm{X}|\quad\mu-a.e.$

for every $\bm{X}\in L^{\infty}(TX)$.

In the claim below, a modulus of continuity is a non-decreasing continuous
function $\omega:[0,\infty)\to[0,\infty)$ with $\omega(0)=0$.

######  Theorem 1.5.

Let $(X,d,\mu)$ be a metric measure space. The following are equivalent.

  * (i)

$X$ is a Lipschitz differentiability space;

  * (ii)

$\operatorname{Lip}f=|Df|_{\ast}$ $\mu$-a.e. for every
$f\in\operatorname{LIP}(X)$;

  * (iii)

there is a collection $\\{\omega_{x}\\}_{x\in X}$ of moduli of continuities
such that

$\operatorname{Lip}f\leq\omega(|Df|_{\ast})\quad\mu-a.e.$

for every $f\in\operatorname{LIP}(X)$.

In closing, we remark that the equivalence (ii)$\iff$(iii) demonstrates a
self-improving property of the inequality
$\operatorname{Lip}f\leq\omega(|Df|_{\ast})$. In the case $\omega(t)=Ct$ it
follows from the self-improvement of Keith’s Lip–lip condition [35, 14].

###  1.4. Approximation

Both the theory of Sobolev spaces and that of Weaver differentials rests
heavily on finding good approximations of functions. Indeed, for us, the proof
of the density of directions, Theorem 6.4, is based on the following
approximation result.

######  Theorem 1.6.

Let $f\in\operatorname{LIP}_{bs}(X)$. Then there exists a sequence
$(f_{j})\subset\operatorname{LIP}_{bs}(X)$ with
$\operatorname{LIP}(f_{j})\leq\operatorname{LIP}(f)$ and $|f_{j}|\leq|f|$ for
each $j\in\mathbb{N}$, such that

  * (1)

$f_{j}\to f$ pointwise everywhere (and consequently uniformly on compact
sets);

  * (2)

$\operatorname{Lip}_{a}f_{j}\to|Df|_{\ast}$,
$\operatorname{Lip}f_{j}\to|Df|_{\ast}$, and $|Df_{j}|_{\ast}\to|Df|_{\ast}$
pointwise $\mu$-a.e.

This approximation result is of independent interest and brings together again
the two strands of research on Lipschitz algebras and Sobolev spaces. It has
many predecessors that have appeared in these different contexts.

In the case of differentiability spaces, such approximations were implicitly
used in [8, Lemma 6.3] and explicitly used in [35, Theorem 1.21], see also [2,
Theorem 2.1] for a recent application. In those cases, an embedding to
$\ell^{\infty}$ was used as an auxiliary tool. In our case, we use an
embedding into the free space over $X$, see Subsection 2.2 and specifically
Remark 2.1.

Independent from the differentiability space case, methods to approximate
Sobolev functions appeared earlier in Cheeger’s work [12, Sections 4 and 5],
where no embedding was employed. That work was very similar to a much older
argument of Ziemer, see [12, Remark 5.26], see also [27, Proposition 2.17].
Recently, these techniques were used in [21] to give a new proof of density of
energy of Lipschitz functions. This density in energy was earlier discovered
by entirely different means [6].

Theorem 1.6 is the Lipschitz analogue of the result in [21] and is based on
similar ideas. It gives a more direct way of obtaining the approximations from
[8, Lemma 6.3] and [35, Theorem 1.21] in the Lipschitz differentiability
category, while also giving slightly sharper conclusions. It also elucidates
how the different approximations appearing in the literature are really
modifications of the same one in disguise.

##  2\. Preliminaries

###  2.1. Notation and conventions

In this paper, we will only consider complete and separable metric measure
spaces $X$ that are equipped with a Radon measure $\mu$ that is finite on
bounded sets.

We say that a function $f:X\to Y$ is Lipschitz, if
$\operatorname{LIP}(f)=\sup_{x\neq y}\frac{d(f(x),f(y))}{d(x,y)}<\infty$. We
denote by $\operatorname{LIP}(X)$ the collection of Lipschitz functions
$f:X\to\mathbb{R}$, and treat it simply as a set without a topology. Further,
$\operatorname{LIP}(X,Y)$ denotes the space of Lipschitz functions $f:X\to Y$.
We denote by $\operatorname{LIP}_{\infty}(X)$ the collection of bounded
Lipschitz functions $f:X\to\mathbb{R}$, equipped with the norm
$\|f\|_{\operatorname{LIP}_{\infty}}=\max(\|f\|_{\infty},\operatorname{LIP}(f))$.
We let $\operatorname{LIP}_{bs}(X)$ be the space of Lipschitz functions with
bounded support, and we write

$\operatorname{Lip}_{a}f(x)=\lim_{r\to 0}\operatorname{LIP}(f|_{B(x,r)})\text{
and }\operatorname{Lip}f(x)=\lim_{r\to 0}\sup_{y\in
B(x,r)\setminus\\{x\\}}\frac{|f(x)-f(y)|}{d(x,y)}.$

For isolated points $x\in X$, we set
$\operatorname{Lip}_{a}f(x)=\operatorname{Lip}f(x)=0$.

If $\mu$ is a Radon measure on $X$, then the pushforward of $\mu$ under a
measurable map $f:X\to Y$ is denoted $f_{*}\mu$ and is given by
$f_{*}\mu(A)=\mu(f^{-1}(A))$.

###  2.2. Lipschitz free space

Let $\operatorname{LIP}_{0}(X)$ be the space of Lipschitz functions vanishing
at a given base point $x_{0}\in X$, equipped with the Lipschitz norm
$\operatorname{LIP}(\cdot)$. For every $x\in X$ the associated evaluation map
$\iota_{x}(f):=f(x)$ belongs to $\operatorname{LIP}(X)^{*}$. Note that
$\iota_{x}$ is moreover continuous with respect to pointwise convergence. The
Lipschitz free space $\mathcal{F}(X)$ is defined as

$\mathcal{F}(X)=\overline{\operatorname{span}}\\{\iota_{x}:\ x\in
X\\}\subset\operatorname{LIP}(X)^{*}$

equipped with the dual norm. The canonical embedding
$\iota:X\hookrightarrow\mathcal{F}(X)$, given by $x\mapsto\iota_{x}$, is an
isometry. The Lipschitz free space $\mathcal{F}(X)$ over $X$ is a predual of
$\operatorname{LIP}_{0}(X)$ – i.e.
$\mathcal{F}(X)^{*}=\operatorname{LIP}_{0}(X)$; moreover on bounded sets, the
$w^{*}$-topology on $\operatorname{LIP}_{0}(X)$ induced by $\mathcal{F}(X)$
coincides with the topology of pointwise convergence, cf. [40, Theorem 2.2.2].
We refer to [40] for the more detailed definition of and thorough introduction
to $\mathcal{F}(X)$ and its properties.

We next define the _semihull_ $\operatorname{sh}(X)$ of $X$. Set

$\displaystyle\operatorname{sh}(X)=\bigcup\\{[x,y]:\ x,y\in X\\},$

where $[x,y]=\\{\iota_{x}+t(\iota_{y}-\iota_{x}):\ t\in[0,1]\\}$ is the line
segment in $\mathcal{F}(X)$ connecting the isometric images of $x$ and $y$.
Every element $v\in\operatorname{sh}(X)\setminus X$ has two representations
$v=\iota_{x}+t(\iota_{y}-\iota_{x})=\iota_{y}+(1-t)(\iota_{x}-\iota_{y})$ that
is, the representation is unique up to switching $x$ and $y$. We equip
$\operatorname{sh}(X)$ with the metric from $\mathcal{F}(X)$. It directly
follows that $X\subset\operatorname{sh}(X)$ isometrically.

The semihull provides a convenient technical tool which will be used in
Section 3 to obtain lower semicontinuity of suitable integrals over fragments
of curves. Its key property is a canonical extension of functions
$X\to\mathbb{R}$ to functions $\operatorname{sh}(X)\to\mathbb{R}$ which
preserves lower semicontinuity.

######  Remark 2.1.

The standard Kuratowski embedding $X\hookrightarrow\ell^{\infty}$ is not
appropriate for our canonical extension scheme. Indeed, the natural “convex”
extension we use (see Section 3.1) is not well-defined for the Kuratowski
embedding, and we do not know whether a canonical semicontinuity preserving
extension exists for it.

The following lemma gives a concrete description of convergence in
$\operatorname{sh}(X)$.

######  Lemma 2.2.

Let $v=\iota_{x}+t(\iota_{y}-\iota_{x})$ and suppose
$(v_{j})\subset\operatorname{sh}(X)$ converges to $v$. Then there exist
sequences $(x_{j}),(y_{j})\subset X$ and $(t_{j})\subset[0,1]$ with
$v_{j}=\iota_{x_{j}}+t_{j}(\iota_{y_{j}}-\iota_{x_{j}})$ such that the
following holds.

  * (1)

If $v\in\operatorname{sh}(X)\setminus\iota(X)$, then $t_{j}\to t$ and
$(x_{j},y_{j})\to(x,y)$;

  * (2)

If $v\in\iota(X)$, then up to a subsequence, we have one of the following
options: (a) $(x_{j},y_{j})\to(x,y)$, (b) $x_{j}\to x$ and $t_{j}\to 0$, or
(c) $y_{j}\to y$ and $t_{j}\to 1$.

###### Proof.

Assume first that $v=\iota_{x}+t(\iota_{y}-\iota_{x})\notin\iota(X)$, i.e.
$x\neq y$ and $t\in(0,1)$. Choose a representation
$v_{j}=\iota_{x_{j}}+t_{j}(\iota_{y_{j}}-\iota_{x_{j}})$ such that
$d(x,x_{j})+d(y,y_{j})$ is minimal. The functions
$f=\operatorname{dist}(\\{x,y\\},\cdot)$, $\tilde{f}=f-f(x_{0})$ are
1-Lipschitz, $\tilde{f}$ vanishes at $x_{0}$, and
$v_{j}(\tilde{f})-v(\tilde{f})=tf(x_{j})+(1-t)f(y_{j})$. Consequently

$\displaystyle 0\leq$
$\displaystyle\min\\{t,1-t\\}[\operatorname{dist}(\\{x,y\\},x_{j})+\operatorname{dist}(\\{x,y\\},y_{j})]$
$\displaystyle\leq$ $\displaystyle
tf(x_{j})+(1-t)f(y_{j})=|v_{j}(\tilde{f})-v(\tilde{f})|\leq\|v-v_{j}\|\to 0,$

i.e. the sequences $(x_{j})$ and $(y_{j})$ converge either to $x$ or $y$. By
the minimality of $d(x,x_{j})+d(y,y_{j})$ it follows that $x_{j}\to x$ and
$y_{j}\to y$. Similarly, if $g=d(x,\cdot)$ and $\tilde{g}=g-g(x_{0})$, then
$v_{j}(\tilde{g})-v(\tilde{g})=d(x_{j},x)+t_{j}(d(y_{j},x)-d(x_{j},x))-td(y,x)\to
0$, implying $t_{j}\to t$

Suppose next that $v\in\iota(X)$. Without loss of generality we may assume
that $v=\iota_{x}$ and $t\leq 1/2$. Considering the functions
$f=\operatorname{dist}(\\{x,y\\},\cdot)$ as above we obtain that $x_{j}\to x$
or $y_{j}\to x$. Up to relabeling $x_{j}$ and $y_{j}$ we may assume that
$x_{j}\to x$. If a subsequence of $(y_{j})$ converges to $x$ then (a) is
satisfied. If not, there exists $\delta>0$ such that $d(y_{j},x)\geq 2\delta$
for all $j$. Consequently $d(x_{j},y_{j})\geq\delta$ for all large $j$. It
follows that the functions $g_{j}=\frac{d(x_{j},\cdot)}{d(x_{j},y_{j})}$,
$\tilde{g}_{j}=g_{j}-g_{j}(x_{0})$ are uniformly Lipschitz and
$v_{j}(\tilde{g}_{j})-v(\tilde{g}_{j})=t_{j}-\frac{d(x,x_{j})}{d(y_{j},x_{j})}\to
0$. Since $\frac{d(x,x_{j})}{d(y_{j},x_{j})}\to 0$, it follows that $t_{j}\to
0$, completing the proof

∎

######  Remark 2.3.

Define the _edge-length function_
$l_{\operatorname{sh}}:\operatorname{sh}(X)\to[0,\infty)$ by

$\displaystyle
l_{\operatorname{sh}}(\iota_{x}+t(\iota_{y}-\iota_{x}))\mathrel{\mathop{:}}=d(x,y).$

In particular, $l_{\operatorname{sh}}(x)=0$ for $x\in
X\subset\operatorname{sh}(X)$. This is well-defined and
$\\{l_{\operatorname{sh}}>0\\}=\operatorname{sh}(X)\setminus X$. It follows
from Lemma 2.2 that $l_{\operatorname{sh}}$ is lower semicontinuous.

###  2.3. Curve fragments and $\ast$-upper gradients

A curve fragment in $X$ is a bi-Lipschitz map
$\gamma:\operatorname{dom}(\gamma)\to X$ from a compact set
$\operatorname{dom}(\gamma)\subset\mathbb{R}$, called the domain of $\gamma$.
The biLipschitz constant of $\gamma$ is denoted $\operatorname{biLip}(\gamma)$
and it is the smallest constant $L$ so that $L^{-1}|a-b|\leq
d(\gamma(a),\gamma(b))\leq L|a-b|$ holds for all
$a,b\in\operatorname{dom}(\gamma)$.

Let
$I_{\gamma}\mathrel{\mathop{:}}=\operatorname{conv}(\operatorname{dom}(\gamma))=[a,b]$
be the smallest interval containing $\operatorname{dom}(\gamma)$ and, writing
$\displaystyle[a,b]\setminus\operatorname{dom}(\gamma)=\bigcup_{i}(a_{i},b_{i})$,
the gap of the curve fragment $\gamma$ is defined as

$\operatorname{gap}(\gamma)=\sum_{i}d(\gamma_{b_{i}},\gamma_{a_{i}}).$

We also call the intervals $(a_{i},b_{i})$ appearing in this context the gaps
of the curve fragment $\gamma$. The gaps are indexed by a finite or countable
set. We define an extension $\bar{\gamma}:I_{\gamma}\to\operatorname{sh}(X)$
of $\gamma$:

$\displaystyle\bar{\gamma}_{t}\mathrel{\mathop{:}}=\left\\{\begin{array}[]{ll}\gamma_{t}\
,&t\in\operatorname{dom}(\gamma)\\\
\frac{t-a_{i}}{b_{i}-a_{i}}\gamma_{b_{i}}+\frac{b_{i}-t}{b_{i}-a_{i}}\gamma_{a_{i}},&t\in(a_{i},b_{i}).\end{array}\right.$

The _metric speed_ $|\bar{\gamma}_{t}^{\prime}|$ exists for
$\mathcal{L}^{1}$-a.e. $t\in I_{\gamma}$, see [7, Theorem 4.1.6]. In
particular, curve fragments also admit a metric speed: for
$\mathcal{L}^{1}$-a.e. $t\in\operatorname{dom}(\gamma)$ the limit

$|\gamma_{t}^{\prime}|:=\lim_{\operatorname{dom}(\gamma)\ni s\to
t}\frac{d(\gamma_{t},\gamma_{s})}{|t-s|}$

exists. At any non isolated point of $\operatorname{dom}(\gamma)$, this value
is independent of the chosen extension.

If $f:X\to\mathbb{R}$ is Lipschitz, then it can be extended to a Lipschitz
function $\bar{f}:\operatorname{sh}(X)\to\mathbb{R}$. Similarly, the
derivative $(\bar{f}\circ\bar{\gamma})^{\prime}(t)$ exists for
$\mathcal{L}^{1}$-a.e. $t\in I_{\gamma}$. Consequently, for
$\mathcal{L}^{1}$-a.e. $t\in\operatorname{dom}(\gamma)$ the limit

$(f\circ\gamma)^{\prime}(t):=\lim_{\operatorname{dom}(\gamma)\ni s\to
t}\frac{f(\gamma_{s})-f(\gamma_{t})}{s-t}$

exists. Again, the limit is independent of the choice of extension at non-
isolated points of the domain $\operatorname{dom}(\gamma)$.

The line integral of a Borel function $\rho:X\to[0,\infty]$ over $\gamma$
satisfies

$\int_{\gamma}\rho\mathrm{d}s:=\int_{\operatorname{dom}(\gamma)}\rho(\gamma_{t})|\gamma_{t}^{\prime}|\mathrm{d}t=\int_{\operatorname{im}(\gamma)}\rho\mathrm{d}\mathcal{H}^{1}.$

######  Remark 2.4.

Observe that

$\displaystyle\ell^{\ast}(\gamma)\mathrel{\mathop{:}}=\int_{\operatorname{dom}(\gamma)}|\gamma_{t}^{\prime}|\mathrm{d}t+\operatorname{gap}(\gamma)=\ell(\bar{\gamma}).$

The curve $\bar{\gamma}$ can be parametrized by constant speed, see e.g. [7,
Theorem 4.2.1]. Consequently also the curve fragment $\gamma$ has a bi-
Lipschitz reparametrization
$\tilde{\gamma}:\operatorname{dom}(\tilde{\gamma})\to X$ with
$[0,1]=I_{\tilde{\gamma}}$ such that
$|\tilde{\gamma}_{t}^{\prime}|=\ell^{*}(\gamma)$ for every
$t\in\operatorname{dom}(\tilde{\gamma})$.

The set of curve fragments on $X$ is denoted by $\operatorname{Fr}(X)$ and is
given a topology as follows. Using the injective map $\gamma\mapsto{\rm
graph}(\gamma)=\\{(t,\gamma_{t}):\ t\in\operatorname{dom}(\gamma)\\}$, we
identify $\operatorname{Fr}(X)$ with a subset of $\mathcal{H}(\mathbb{R}\times
X)$ (the space of closed bounded subsets of $\mathbb{R}\times X$) and equip it
with the Hausdorff metric. The following lemma follows by a straightforward
application of the definitions and Lemma 2.2 and we omit the proof.

######  Lemma 2.5.

We have that $\bar{\gamma}|_{\operatorname{dom}(\gamma)}=\gamma$ and
$\ell(\bar{\gamma})=\mathcal{H}^{1}(\operatorname{im}(\gamma))+\operatorname{gap}(\gamma)$
for $\gamma\in\operatorname{Fr}(X)$. Moreover, $\gamma_{j}\to\gamma$ in
$\operatorname{Fr}(X)$ if and only if $\bar{\gamma}_{j}\to\bar{\gamma}$ in
$\operatorname{Fr}(\operatorname{sh}(X))$.

Next we define curve fragment analogues of upper gradients, called
$\ast$-upper gradients.

######  Definition 2.6.

Let $f\in\operatorname{LIP}(X)$. A Borel function $g:X\to[0,\infty]$ is a
$\ast$-upper gradient of $f$ if

(2.1) ${\rm
osc}_{\gamma}f\leq\int_{\gamma}g\mathrm{d}s+\operatorname{LIP}(f){\rm
gap}(\gamma)$

for every $\gamma\in\operatorname{Fr}(X)$.

Here, we use the notation $\displaystyle{\rm
osc}_{\gamma}f=\sup_{x,y\in\operatorname{im}(\gamma)}|f(x)-f(y)|$. Note that
$\ast$-upper gradients are only defined for Lipschitz functions, and that
$\operatorname{LIP}(f)$ is always a $\ast$-upper gradient of $f$.

###  2.4. $\infty$-Modulus and minimal $\ast$-upper gradients

We relax the notion of $\ast$-upper gradients by introducing negligible
families of curve fragments in the sense of the _$\infty$ -modulus_
$\operatorname{Mod}_{\infty}$. The resulting class of _weak $\ast$-upper
gradients_ behaves better with respect to limits, and automatically contains
minimal elements, making it a suitable analogue of minimal $p$-weak upper
gradients in the context of curve fragments; compare [28, 11]. However, we
will also see that compared to the theory for finite $p$, the exponent
$p=\infty$ is special: a minimal $\ast$-upper gradient also exists.

The modulus $\operatorname{Mod}_{\infty}(\Gamma)$ of a family of curve
fragments $\Gamma\subset\operatorname{Fr}(X)$ is defined by

(2.2)
$\displaystyle\operatorname{Mod}_{\infty}(\Gamma)=\inf\left\\{\|\rho\|_{L^{\infty}(\mu)}:\
\int_{\gamma}\rho\mathrm{d}s\geq 1\textrm{ for all }\gamma\in\Gamma\right\\}.$

The $\infty$-modulus is an outer measure on $\operatorname{Fr}(X)$, and all
the basic properties of $\operatorname{Mod}_{p}$ for finite $p$ remain valid
for $\operatorname{Mod}_{\infty}$, see [19]. Families
$\Gamma_{0}\subset\operatorname{Fr}(X)$ of zero $\infty$-modulus have a
particularly simple description, given below; compare [19, Lemma 5.7].

######  Lemma 2.7.

A family $\Gamma_{0}\subset\operatorname{Fr}(X)$ satisfies
$\operatorname{Mod}_{\infty}(\Gamma_{0})=0$ if and only if there exists a
Borel set $N\subset X$ with $\mu(N)=0$ such that
$\Gamma_{0}\subset\Gamma_{N}^{+}$, where

$\Gamma_{N}^{+}:=\\{\gamma\in\operatorname{Fr}(X):\
\mathcal{H}^{1}(N\cap\operatorname{im}(\gamma))>0\\}.$

###### Proof.

The proof follows closely that of [19, Lemma 5.7]. If $\mu(N)=0$, let
$\rho=\infty\chi_{N}$, and we have
$\int_{\gamma}\chi_{N}ds=\infty\mathcal{H}^{1}(N\cap\operatorname{im}(\gamma))=\infty$.
Further, $\|\rho\|_{L^{\infty}(\mu)}=0$, and thus
$\operatorname{Mod}_{\infty}(\Gamma_{N}^{+})=0$. This gives the converse
direction of the lemma.

If on the other hand $\operatorname{Mod}_{\infty}(\Gamma_{0})=0$, then for
every $\epsilon>0$, there exists a $\rho_{\epsilon}$ with
$\|\rho_{\epsilon}\|_{L^{\infty}(\mu)}\leq\epsilon$ and
$\int_{\gamma}\rho_{\epsilon}ds\geq 1$ for all $\gamma\in\Gamma_{0}$. Let
$N=\bigcup_{n\in\mathbb{N}}\rho_{n^{-1}}^{-1}((n^{-1},\infty))$. From the
definition of the $L^{\infty}(\mu)$-norm it is direct to see $\mu(N)=0$. Now,
if $\gamma\in\Gamma_{0}$, we have some $n\in\mathbb{N}$ such that
$\mathcal{H}^{1}(\operatorname{im}(\gamma))<n$. Then,
$\mathcal{H}^{1}(\rho_{n^{-1}}^{-1}((n^{-1},\infty))\cap\operatorname{im}(\gamma))>0$
since otherwise
$\int_{\gamma}\rho_{n^{-1}}ds\leq\mathcal{H}^{1}(\operatorname{im}(\gamma))n^{-1}<1$.
Therefore, $\mathcal{H}^{1}(N\cap\operatorname{im}(\gamma))>0$ and
$\gamma\in\Gamma_{N}^{+}$. Since this holds for all $\gamma\in\Gamma_{0}$, the
claim follows. ∎

######  Definition 2.8.

A Borel function $\rho\colon X\to[0,\infty]$ is a weak $\ast$-upper gradient
of a function $f\in\operatorname{LIP}(X)$ if (2.1) holds for every curve
fragment outside a family $\Gamma_{0}\subset\operatorname{Fr}(X)$ with
$\operatorname{Mod}_{\infty}(\Gamma_{0})=0$.

######  Remark 2.9.

In contrast with finite exponents, a weak $\ast$-upper gradient $\rho$ of
$f\in\operatorname{LIP}(X)$ has a $\mu$-representative which is a genuine
upper gradient. Indeed, if $\Gamma_{0}\subset\operatorname{Fr}(X)$ is the set
where (2.1) fails for $\rho$ and $N\subset X$ is the Borel $\mu$-null set with
$\Gamma_{0}\subset\Gamma_{N}^{+}$, then

$\tilde{\rho}=\operatorname{LIP}(f)\chi_{N}+\rho\chi_{X\setminus N}$

is the required representative. Indeed, if $\gamma$ is any curve fragment,
then we can divide the domain as
$\operatorname{dom}(\gamma)=\bigcup_{i=1}^{\infty}K_{i}\cup B$, where $K_{i}$
are compact sets such that $\gamma(K_{i})\subset N$ or $\gamma(K_{i})\cap
N=\emptyset$, and $B$ is a null set. Then $\tilde{\rho}$ is an $\ast$-upper
gradient for each $\gamma|_{K_{i}}$. From this one can conclude that
$\tilde{\rho}$ is an $\ast$-upper gradient for $\gamma$.

######  Proposition 2.10.

Let $f\in\operatorname{LIP}(X)$. Then there exists a minimal $\ast$-upper
gradient $|Df|_{\ast}$ of $f$ in the following sense: $|Df|_{\ast}$ is a
$\ast$-upper gradient of $f$ and, if $\rho\in L^{\infty}(\mu)$ is a weak
$\ast$-upper gradient of $f$, then $|Df|_{\ast}\leq\rho$ $\mu$-a.e. Moreover,
any two minimal $\ast$-upper gradients of $f$ agree $\mu$-a.e.

In the proof below we use the observation that a Borel function $\rho\in
L^{\infty}(\mu)$ is a weak $\ast$-upper gradient of
$f\in\operatorname{LIP}(X)$ if and only if

$\displaystyle|(f\circ\gamma)^{\prime}(t)|\leq\rho(\gamma_{t})|\gamma^{\prime}_{t}|\quad
a.e.\ t\in\operatorname{dom}(\gamma)$

for $\operatorname{Mod}_{\infty}$-a.e. $\gamma\in\operatorname{Fr}(X)$. The
proof also uses the concept of a measure theoretic infimum, which is also
sometimes referred to as a lattice infimum.

######  Definition 2.11.

If $\mathcal{F}$ is any collection of measurable functions, then the measure
theoretic infimum ${\rm essinf}\mathcal{F}$ is a measurable function $\rho$
such that $\rho\leq f$ for all $f\in\mathcal{F}$ and such that if
$\rho^{\prime}$ is any other such function, then $\rho^{\prime}\leq\rho$
(a.e.).

Similarly, the measure theoretic supremum ${\rm esssup}\mathcal{F}$ is a
measurable function $\rho$ such that $\rho\geq f$ for all $f\in\mathcal{F}$
and such that if $\rho^{\prime}$ is any other such function, then
$\rho^{\prime}\geq\rho$ (a.e.).

It can be shown that the measure theoretic infimum exists for $\sigma$-finite
spaces, and that there exists always a sequence of $\rho_{n}\in\mathcal{F}$
such that $\inf_{n\in\mathbb{N}}\rho_{n}={\rm essinf}\mathcal{F}$. The proof
is quite standard, see e.g. [22, Appendix A.2, Essential supremum]. By
replacing functions by their negations, we can obtain the same results for
measure theoretic suprema.

###### Proof of Proposition 2.10.

Two minimal weak $\ast$-upper gradient must necessarily agree $\mu$-almost
everywhere: if $\rho_{1},\rho_{2}$ are both pointwise minimal, then
$\rho_{1}\leq\rho_{2}$ and $\rho_{2}\leq\rho_{1}$ $\mu$-a.e. Thus it suffices
to show its existence.

We show the following _lattice property_ of weak $\ast$-upper gradients: if
$\rho_{i}\in L^{\infty}(\mu)$, $i=1,2,\ldots$ are weak $\ast$-upper gradients
of $f$, then $\rho_{\infty}=\inf_{i}\rho_{i}$ is a weak $\ast$-upper gradient
of $f$. Indeed, if $\Gamma_{i}\subset\operatorname{Fr}(X)$ is the
$\operatorname{Mod}_{\infty}$-null set where (1.2) fails for $\rho_{i}$ and
$\Gamma_{0}=\Gamma_{1}\cup\Gamma_{2}\cup\cdots$, then for
$\gamma\notin\Gamma_{0}$ there exist null sets
$N_{i}\subset\operatorname{dom}(\gamma)$ such that
$|(f\circ\gamma)^{\prime}(t)|\leq\rho_{i}(\gamma_{t})|\gamma_{t}^{\prime}|$
for $t\in\operatorname{dom}(\gamma)\setminus N_{i}$. It follows that

$|(f\circ\gamma)^{\prime}(t)|\leq\rho_{\infty}(\gamma_{t})|\gamma_{t}^{\prime}|$

for $t\in\operatorname{dom}(\gamma)\setminus(N_{1}\cup N_{2}\cup\cdots)$. Thus
$\rho_{\infty}$ is a weak $\ast$-upper gradient of $f$ as claimed.

Now let $\rho$ be the measure theoretic infimum over all weak $\ast$-upper
gradients of $f$, and let $\rho_{i}\in L^{\infty}(\mu)$ be a sequence from
this family such that $\rho=\inf_{i}\rho_{i}$ $\mu$-a.e. By the preceding
argument $\rho$ is a weak $\ast$-upper gradient of $f$, and by Remark 2.9 we
may choose a representative of $\rho$ that is a genuine $\ast$-upper gradient.
The definition of measure theoretic infimum implies that $\rho\leq g$
$\mu$-a.e. for any weak $\ast$-upper gradient $g$ of $f$. ∎

The lattice property established in the proof above yields the basic locality
properties of the minimal $\ast$-upper gradient by standard arguments, see
[28, Chapter 6.3] or [11, Chapter 2.2]. We record below the basic properties
needed later on without proof, and refer the interested reader to the
references above.

######  Lemma 2.12.

Let $f,g\in\operatorname{LIP}(X)$. Then

  * (1)

$|D(f+g)|_{\ast}\leq|Df|_{\ast}+|Dg|_{\ast}$ $\mu$-a.e.;

  * (2)

$|Df|_{\ast}=|Dg|_{\ast}$ $\mu$-a.e. on $\\{f=g\\}$; and

  * (3)

$|Dh|_{\ast}=\chi_{\\{f<g\\}}|Df|_{\ast}+\chi_{\\{f\geq g\\}}|Dg|_{\ast}$
$\mu$-a.e. for $h=\min\\{f,g\\}$.

######  Lemma 2.13.

Suppose $(f_{j})\subset\operatorname{LIP}(X)$ converges pointwise to
$f\in\operatorname{LIP}(X)$ and $\sup_{j}\operatorname{LIP}(f_{j})<\infty$. If
$\rho_{j}\in L^{\infty}(\mu)$ is a weak $\ast$-upper gradient of $f_{j}$ for
each $j$ and $(\rho_{j})$ converges pointwise $\mu$-a.e. to $\rho\in
L^{\infty}$ with $\sup_{j}\|\rho_{j}\|_{L^{\infty}(\mu)}<\infty$, then $\rho$
is a weak $\ast$-upper gradient of $f$.

###### Proof.

Let $N=\bigcup_{i=0}^{\infty}N_{i}\subset X$ be the Borel set, where

$N_{0}:=\\{x\in X:\lim_{j\to\infty}\rho_{j}(x)\neq\rho(x)\text{ or
}\rho(x)>\sup_{j}\|\rho_{j}\|_{L^{\infty}(\mu)}\\}$

and $N_{j}$ is a Borel set so that for $\gamma\not\in\Gamma_{N_{j}}^{*}$ the
function $\rho_{j}$ is a $\ast$-upper gradient of $f_{j}$. The latter is
guaranteed by Lemma 2.7.

Then, whenever $\gamma\notin\Gamma_{N}^{+}$, we have that (a) $\rho_{j}$ is a
$\ast$-upper gradient of $f_{j}$ along $\gamma$, and (b)
$\rho_{j}\circ\gamma\to\rho\circ\gamma$ pointwise a.e. with
$\sup_{j}\|\rho_{j}\circ\gamma\|_{L^{\infty}(\operatorname{dom}(\gamma))}<\infty$.
By the dominated convergence theorem we have

$\displaystyle|f(\gamma_{t})-f(\gamma_{s})|=$
$\displaystyle\lim_{j\to\infty}|f_{j}(\gamma_{t})-f_{j}(\gamma_{s})|\leq\liminf_{j\to\infty}\bigg{(}\int_{\gamma}\rho_{j}\mathrm{d}s+\operatorname{LIP}(f_{j})\operatorname{gap}(\gamma)\bigg{)}$
$\displaystyle\leq$
$\displaystyle\int_{\gamma}\rho\mathrm{d}s+C\operatorname{gap}(\gamma)$

for all $t,s\in\operatorname{dom}(\gamma)$, where
$C=\sup_{j}\operatorname{LIP}(f_{j})$. From this and the Lebesgue
differentiation theorem it follows that
$|(f\circ\gamma)_{t}|\leq\rho(\gamma_{t})|\gamma_{t}^{\prime}|$ a.e.
$t\in\operatorname{dom}(\gamma)$ for $\gamma\notin\Gamma_{N}^{+}$. Thus $\rho$
is a weak $\ast$-upper gradient of $f$. ∎

The same conclusion remains true when the $\mu$-a.e. convergence
$\rho_{j}\to\rho$ is weakened to $w^{\ast}$-convergence in
$L^{\infty}(\mu)=L^{1}(\mu)^{\ast}$. The argument uses a slightly non-standard
version of Mazur’s lemma.

######  Lemma 2.14.

Suppose $(f_{j})\subset\operatorname{LIP}(X)$ converges pointwise to
$f\in\operatorname{LIP}(X)$ and $\sup_{j}\operatorname{LIP}(f_{j})<\infty$. If
$\rho_{j}\in L^{\infty}(\mu)$ is a weak $\ast$-upper gradient of $f_{j}$ for
each $j$ and $(\rho_{j})$ $w^{\ast}$-converges in $L^{\infty}(\mu)$ to $\rho$,
then $\rho$ is a weak $\ast$-upper gradient of $f$.

###### Proof.

First, by Banach-Steinhausen, we get
$\sup_{j}\|\rho_{j}\|_{L^{\infty}(\mu)}<\infty$. We will show that there
exists non-negative numbers $\alpha_{k}(n),\dots\alpha_{N_{n}}(n)$,
$N_{n}\in\mathbb{N}$ (with $n\leq N_{n})$ and with
$\sum_{j=n}^{N_{n}}\alpha_{j}(n)=1$, so that

$\tilde{\rho}_{n}=\sum_{j=n}^{N_{n}}\alpha_{j}(n)\rho_{j}\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}\rho$

converges pointwise almost everywhere and with
$\sup_{j}\|\tilde{\rho}_{j}\|_{L^{\infty}(\mu)}<\infty$. If this is true, then
it is direct to verify that $\tilde{\rho}_{k}$ is a weak $\ast$-upper gradient
for $\tilde{f}_{n}=\sum_{j=n}^{N_{n}}\alpha_{j}(n)f_{j}$. Indeed, for
$\operatorname{Mod}_{\infty}$-a.e. curve fragment $\gamma$, and any
$s,t\in\operatorname{dom}(\gamma)$, we have

$\displaystyle|\tilde{f}_{n}(\gamma_{t})-\tilde{f}_{n}(\gamma_{s})|$
$\displaystyle\leq\sum_{j=n}^{N_{n}}\alpha_{j}(n)|f_{j}(\gamma_{t})-f_{j}(\gamma_{s})|$
$\displaystyle\leq\sum_{j=n}^{N_{n}}\alpha_{j}(n)\bigg{(}\int_{\gamma}\rho_{j}\mathrm{d}s+\operatorname{LIP}(f_{j})\operatorname{gap}(\gamma)\bigg{)}$
$\displaystyle\leq$
$\displaystyle\int_{\gamma}\tilde{\rho}_{n}\mathrm{d}s+C\operatorname{gap}(\gamma),$

where $C=\sup_{j}\operatorname{LIP}(f_{j})$. From this, we get that
$\tilde{\rho}_{n}$ is a weak $\ast$-upper gradient for $\tilde{f}_{n}$. Now,
the claim follows from Lemma 2.13. We are left to prove the existence of
$\alpha_{j}(n)$ and $N_{n}$ such that the previous claims hold.

Fix $n\in\mathbb{N}$ and fix $x_{0}\in X$. Since
$L^{\infty}(B(x_{0},n),\mu)\subset L^{2}(B(x_{0},n),\mu)\newline \subset
L^{1}(B(x_{0},n),\mu)$, we have that the sequence
$\rho_{j}\chi_{R}\to\rho\chi_{R}$ weakly in $L^{2}(B(x_{0},n),\mu)$. Thus, by
Mazur’s lemma, together with passing to a subsequence, we can find non-
negative values $\beta^{n}_{k}(k),\dots\beta^{n}_{M^{n}_{k}}(k)$, $k\leq
M^{n}_{k}\in\mathbb{N}$, with $\sum_{j=k}^{M^{n}_{k}}\beta_{j}(k)=1$, so that

$\tilde{\rho}^{n}_{k}=\sum_{j=k}^{M^{n}_{k}}\beta_{j}(k)\rho_{j}\stackrel{{\scriptstyle
k\to\infty}}{{\longrightarrow}}\rho$

converges almost everywhere in $B(x_{0},n)$. So, for some index
$m_{n}\in\mathbb{N}$ we have $m_{n}\geq n$ and $\mu(\\{x\in
B(x_{0},n):|\tilde{\rho}^{n}_{m_{n}}(x)-\rho(x)|\geq\frac{1}{n}\\})\leq
2^{-n}$. Set $N_{n}=M_{m_{n}}$, and $\alpha_{j}(n)=\beta_{j}(m)$ for
$j\in[m,M_{m}]$, and otherwise $\alpha_{j}(n)=0$ for $j\in[n,m]$.

By the following Borel-Cantelli-type argument we have that almost everywhere

$\tilde{\rho}_{n}=\sum_{j=n}^{N_{n}}\alpha_{j}(n)\rho_{j}\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}\rho.$

Indeed, let $A_{n}=\\{x\in
B(x_{0},n):|\tilde{\rho}^{n}_{m_{n}}(x)-\rho(x)|\geq\frac{1}{n}\\}=\\{\\{x\in
B(x_{0},n):|\tilde{\rho}_{n}(x)-\rho(x)|\geq\frac{1}{n}\\}.$ Then for the
limit superior set
$\displaystyle\limsup_{n\to\infty}A_{n}=\bigcap_{m=1}^{\infty}\bigcup_{n\geq
m}A_{n}$ we have

$\mu\left(\bigcap_{m=1}^{\infty}\bigcup_{n\geq
m}\limsup_{n\to\infty}A_{n}\right)=0,$

and for every $x\not\in\limsup_{n\to\infty}A_{n}$, we get pointwise
convergence $\lim_{n\to\infty}\tilde{\rho}_{n}(x)=\rho(x)$. ∎

###  2.5. Plans and Alberti representations

Let $\mathbb{P}$ be a finite measure on $\operatorname{Fr}(X)$. The
_barycenter_ $\mathbb{P}^{\\#}$ of $\mathbb{P}$ is the Borel regular measure
on $X$ given by

$\displaystyle\mathbb{P}^{\\#}(E):=\int\int_{\gamma}\chi_{E}\mathrm{d}s\mathrm{d}\mathbb{P}(\gamma),\quad
E\subset X\textrm{ Borel.}$

######  Definition 2.15.

A finite measure $\mathbb{P}$ on $\operatorname{Fr}(X)$ is called a 1-plan, if
$\mathbb{P}^{\\#}=\rho\mu$ for some $\rho\in L^{1}(\mu)$.

These 1-plans are closely related to Alberti representations, defined next.

######  Definition 2.16.

Let $\mu$ be a Radon measure. An Alberti representation
$\mathcal{A}=\\{\mu_{\gamma},\mathbb{P}\\}$ of $\mu$ consists of a finite
measure $\mathbb{P}$ on $\operatorname{Fr}(X)$ and, for $\mathbb{P}$-a.e.
$\gamma$, a finite measure
$\mu_{\gamma}\ll\mathcal{H}^{1}|_{\operatorname{im}(\gamma)}$ such that the
following hold. For every Borel set $E\subset X$ we have that

  * (a)

$\displaystyle\gamma\mapsto\mu_{\gamma}(E)$ is Borel, and

  * (b)

$\displaystyle\mu(E)=\int_{\operatorname{Fr}(X)}\mu_{\gamma}(E)\mathrm{d}\mathbb{P}(\gamma)$.

At this juncture, we remark, that there are subtle differences in the
definitions and conventions regarding Alberti representations. Our definition
follows that of [8], which is essentially equivalent to [35]. For other
definitions see e.g. [13, 1, 21]. We record the close relationship between
1-plans and Alberti representations in the next remark.

######  Remark 2.17.

If $\mathbb{P}$ is a 1-plan on $X$ and $\rho$ is a Borel representative of
$\frac{\mathrm{d}\mathbb{P}^{\\#}}{\mathrm{d}\mu}$, then $\mu|_{\\{\rho>0\\}}$
has an Alberti representation $\mathcal{A}=\\{\mu_{\gamma},\mathbb{P}\\}$,
where
$\mu_{\gamma}=\frac{\chi_{\\{\rho>0\\}}}{\rho}\mathcal{H}^{1}|_{\operatorname{im}(\gamma)}$.

Conversely, if $\mathcal{A}=\\{\mu_{\gamma},\mathbb{P}\\}$ is an Alberti
representation of $\mu$ with

$\int\mathcal{H}^{1}(\operatorname{spt}(\mu_{\gamma}))\mathrm{d}\mathbb{P}(\gamma)<\infty,$

there exists a compact $K\subset X$ such that, if
$R:\operatorname{Fr}(X)\to\operatorname{Fr}(X)$ is the map
$\gamma\mapsto\gamma|_{\gamma^{-1}(K)}$, then
$\mathbb{P}^{\prime}:=R_{\ast}\mathbb{P}$ is a 1-plan. Such a $K$ can be
obtained from the Lebesgue decomposition of $\mu$ with respect to
$\mathbb{P}^{\\#}$. Note that $\mathbb{P}^{\prime}$ is concentrated on
subcurves of $\operatorname{spt}\mathbb{P}$.

Depending on the application, there are advantages to both Alberti
representations and plans. We highlight some crucial features to pay attention
to. For example, Alberti representations are easier to restrict to subsets (we
can just replace $\mu_{\gamma}$ with $\mu_{\gamma}|_{A}$ when $A\subset X$,
while $\gamma|_{\gamma^{-1}(A)}$ may not be a curve fragment), whereas it is
easier to take superpositions of plans (we can add
$\mathbb{P}^{1}+\mathbb{P}^{2}$, but it is more difficult to combine
$\mu_{\gamma}^{1}$ and $\mu_{\gamma}^{2}$ in order to preseve b) in Definition
2.16). Plans are a bit more flexible, since they do not impose (b) in
Definition 2.16, however, a plan may not “see” all of the measure $\mu$.
Moreover, for Alberti representations, a crucial technical issue is that
$\mu_{\gamma}$ only “sees” a part of the curve fragment $\gamma$. These formal
details are reflected in the details of the arguments employing the different
notions, and have some subtle consequences for the formulations of different
results. A reader interested in these subtleties can see the arguments in [35,
8].

Denote by $C(v,\vartheta)\mathrel{\mathop{:}}=\\{w\in\mathbb{R}^{n}:\ v\cdot
w>\cos\vartheta|w|\\}$ the (open) cone of width $\vartheta\in(0,\infty)$ in
the direction $v\in S^{n-1}$. We say that a $k$-tuple
$C_{1},\ldots,C_{k}\subset\mathbb{R}^{n}$ of cones is _independent_ , if
$\\{w_{1},\ldots,w_{k}\\}$ are linearly independent whenever $w_{j}\in C_{j}$,
$j=1,\ldots,k$ and _$\varepsilon$ -separated_ ($\varepsilon>0$), if

$\displaystyle\bigg{\|}\sum_{j}^{k}\lambda_{j}w_{j}\bigg{\|}>\varepsilon\max_{1\leq
j\leq k}\|\lambda_{j}w_{j}\|$

whenever $w_{j}\in C_{j}$, $\lambda_{j}\in\mathbb{R}$, $j=1,\ldots,k$.

######  Remark 2.18.

If $\bm{v}:X\to S^{n-1}$, $\bm{\vartheta},\bm{\varepsilon}:X\to(0,\infty)$ are
Borel functions, we may similarly define linearity and
$\bm{\varepsilon}$-separatedness pointwise for _cone fields_. Here, following
the terminology of [35], a cone field $\bm{C}=C(\bm{v},\bm{\vartheta})$ is the
map $x\mapsto\bm{C}_{x}\mathrel{\mathop{:}}=C(\bm{v}_{x},\bm{\vartheta}_{x})$.

Observe that $\bm{\varepsilon}$-separated cone fields are always independent.
Conversely, independent (closed) cones are $\bm{\varepsilon}$-separated for
some Borel $\bm{\varepsilon}:X\to(0,\infty)$.

######  Definition 2.19.

Suppose $\varphi\in\operatorname{LIP}(X,\mathbb{R}^{n})$.

  * (a)

An Alberti representation $\mathcal{A}=\\{\mu_{\gamma},\mathbb{P}\\}$ of $\mu$
is in the $\varphi$-direction of a cone field $\bm{C}$, if
$(\varphi\circ\gamma)^{\prime}(t)\in\bm{C}_{\gamma_{t}}$ a.e.
$t\in\operatorname{dom}(\gamma)$ for $\mathbb{P}$-a.e.
$\gamma\in\operatorname{Fr}(X)$.

  * (b)

A collection $\mathcal{A}_{1},\ldots,\mathcal{A}_{k}$ of Alberti
representations is $\varphi$-independent (resp.
$(\varphi,\bm{\varepsilon})$-separated) if they are in the $\varphi$-direction
of independent (resp. $\bm{\varepsilon}$-separated) cone fields.

###  2.6. Weaver derivations

A Weaver derivation is a bounded linear map

$\bm{b}:\operatorname{LIP}_{\infty}(X)\to L^{\infty}(\mu)$

that (a) satisfies the Leibniz rule $\bm{b}(fg)=f\bm{b}(g)+g\bm{b}(f)$
$\mu$-a.e. for every $f,g\in\operatorname{LIP}_{\infty}(X)$, and (b) is
weak∗-continuous in the following sense: if a sequence
$(f_{j})\subset\operatorname{LIP}_{\infty}(X)$ with
$\sup_{j}\|f_{j}\|_{\operatorname{LIP}_{\infty}}<\infty$ converges to
$f\in\operatorname{LIP}_{\infty}(X)$ pointwise, then
$\bm{b}(f_{j})\stackrel{{\scriptstyle w^{*}}}{{\rightharpoonup}}\bm{b}(f)$ in
$L^{\infty}(\mu)$. The collection $\mathcal{X}(\mu)$ of Weaver derivations on
$X$ is a Banach space when equipped with the norm
$\|\bm{b}\|_{\mathcal{X}(\mu)}=\||\bm{b}|_{\mathcal{X},\mathrm{loc}}\|_{L^{\infty}(\mu)}$,
where

(2.3)
$\displaystyle|\bm{b}|_{\mathcal{X},\mathrm{loc}}\mathrel{\mathop{:}}=\operatorname{esssup}\\{|\bm{b}(f)|:\
\operatorname{LIP}(f)\leq 1\\}\in L^{\infty}(\mu)$

is the pointwise norm defined as a measure theoretic supremum, see Definition
2.11.

It is a standard fact that the estimate

(2.4)
$\displaystyle|\bm{b}(f)|\leq|\bm{b}|_{\mathcal{X},\mathrm{loc}}\operatorname{Lip}_{a}f\quad\mu-a.e.$

holds for every $f\in\operatorname{LIP}_{\infty}(X)$. Indeed, in order to
introduce some of the common techniques involved in the study of Weaver
derivations, we will sketch this argument.

The Leibniz rule together with weak∗-continuity implies locality: If $A$ is a
positive measure subset and $f|_{A}=g|_{A}$, then
$\bm{b}(f)|_{A}=\bm{b}(g)|_{A}$ almost everywhere, see e.g. [26, Lemma 13.4].
Now, for every ball $B(x,r)$, we can take $f_{x,r}$ to be the McShane
extension of $f|_{B(x,r)}$, which is
$\operatorname{LIP}(f|_{B(x,r)})$-Lipschitz. Then, locality implies that
$\bm{b}(f_{x,r})|_{B(x,r)}=\bm{b}(f)|_{B(x,r)}$ almost everywhere. From (2.3)
we thus get

$|\bm{b}(f)|_{B(x,r)}|\leq|\bm{b}|_{\mathcal{X},\mathrm{loc}}\operatorname{LIP}(f|_{B(x,r)}).$

Then, taking a countable basis of the topology given by balls
$\\{B(x_{i},r_{i}):i\in\mathbb{N}\\}$, we see that for a.e. $x\in X$ we have

$|\bm{b}(f)|(x)\leq|\bm{b}|_{\mathcal{X},\mathrm{loc}}(x)\inf\\{\operatorname{LIP}(f|_{B(x_{i},r_{i})}):x\in
B(x_{i},r_{i})\\}.$

Now, the final part is to observe that
$\inf\\{\operatorname{LIP}(f|_{B(x_{i},r_{i})}):x\in
B(x_{i},r_{i})\\}=\operatorname{Lip}_{a}(f)(x)$, whenever
$\\{B(x_{i},r_{i}):i\in\mathbb{N}\\}$ forms a basis for the topology.

######  Remark 2.20.

The locality of Weaver derivation can also be used to define $\bm{b}(f)$ for
any $f\in\operatorname{LIP}(X)$ by
$\bm{b}(f):=\lim_{j\to\infty}\bm{b}(f_{j})$, where
$f_{j}=(1-\operatorname{dist}(\cdot,B(x_{0},j))_{+}f$ for some arbitrary
$x_{0}\in X$. The pointwise norm bound (2.4) remains valid for
$f\in\operatorname{LIP}(X)$ and we will tacitly use this extension in the
sequel.

##  3\. Approximation

Before turning to the representation of minimal $\ast$-upper gradients, in
this section, we prove the main approximation result, Theorem 1.6. This
theorem will be used later in Sections 6 and 7 in connection with the density
of directions and the characterization of Lipschitz differentiability spaces
(cf. Theorems 6.4 and 1.5).

###  3.1. Curve fragments, line integrals, and lower semicontinuity

In the proof of Theorem 1.6, but also of the representation Theorem 4.1, we
will need semicontinuity properties of the line integral. However, in contrast
to the situation with curves, the line integral of a continuous function need
not be lower semicontinuous with respect to convergence of curve fragments,
compare [31, Proposition 4]. Nevertheless here we obtain the semicontinuity of
a closely related quantity, called the $\ast$-integral in [10], defined by

$\displaystyle\int^{\ast}_{\gamma}\rho\mathrm{d}s:=\int_{\gamma}\rho\mathrm{d}s+\operatorname{gap}(\rho,\gamma)$

for Borel functions $\rho:X\to[0,\infty]$ and $\gamma\in\operatorname{Fr}(X)$.
Here the weighted gap-functional $\operatorname{gap}(\rho,\gamma)$ is defined
by

$\displaystyle\operatorname{gap}(\rho,\gamma)\mathrel{\mathop{:}}=\sum_{i}\frac{\rho(\gamma_{a_{i}})+\rho(\gamma_{b_{i}})}{2}d(\gamma_{a_{i}},\gamma_{b_{i}}),$

where
$\bigcup_{i}(a_{i},b_{i})=I_{\gamma}\setminus\operatorname{dom}(\gamma)$. We
also recall the edge length functional $l_{\operatorname{sh}}$ and the
extension $\bar{\gamma}:I_{\gamma}\to\operatorname{sh}(X)$ of $\gamma$ (cf.
Remark 2.3 and Section 2.2, respectively), and define
$\operatorname{gap}_{\delta}$ for $\delta\geq 0$ by

$\displaystyle\operatorname{gap}_{\delta}(\gamma)\mathrel{\mathop{:}}=\int_{\bar{\gamma}}\chi_{\\{l_{\operatorname{sh}}>\delta\\}}\mathrm{d}s=\sum_{d(\gamma_{a_{i}},\gamma_{b_{i}})>\delta}d(\gamma_{a_{i}},\gamma_{b_{i}}),$

where $(a_{i},b_{i})$ are the gaps of the curve fragment $\gamma$; recall the
terminology in Subsection 2.3. Observe that
$\operatorname{gap}(1,\gamma)=\operatorname{gap}(\gamma)=\operatorname{gap}_{0}(\gamma)$.

######  Proposition 3.1.

Suppose $(\gamma_{j})\subset\operatorname{Fr}(X)$ converges to $\gamma$ in
$\operatorname{Fr}(X)$, and $(\rho_{j})$ is a pointwise non-decreasing
sequence of lower semicontinuous functions with (pointwise) limit $\rho$. Then

$\displaystyle\int_{\gamma}\rho\mathrm{d}s+\operatorname{gap}(\rho,\gamma)\leq\liminf_{j\to\infty}\left(\int_{\gamma_{j}}\rho_{j}\mathrm{d}s+\operatorname{gap}(\rho_{j},\gamma_{j})\right)$

and

$\displaystyle\operatorname{gap}_{\delta}(\gamma)\leq\liminf_{j\to\infty}\operatorname{gap}_{\delta}(\gamma_{j}).$

We will prove this proposition after a bit of preliminary work. Given any
function $\rho:X\to\mathbb{R}\cup\\{\infty\\}$ we may define the extension
$\bar{\rho}:\operatorname{sh}(X)\to\mathbb{R}\cup\\{\infty\\}$ by

$\displaystyle\bar{\rho}(\iota_{x}+t(\iota_{y}-\iota_{x}))\mathrel{\mathop{:}}=t\rho(y)+(1-t)\rho(x).$

Using the properties of $\mathcal{F}(X)$ it is not difficult to see that the
extension is well-defined. The extensions $\bar{\gamma}$ and $\bar{\rho}$ can
be used to identify the $\ast$-integral with a line-integral over the extended
curve in $\operatorname{sh}(X)$.

######  Lemma 3.2.

Let $\gamma:\operatorname{dom}(\gamma)\to X$ be a curve fragment and
$\bar{\gamma}$ the associated extension. Let $\rho:X\to[0,\infty]$ be a Borel
function and let $\bar{\rho}$ be it’s extension. Then

$\displaystyle\int_{\gamma}\rho\mathrm{d}s+\operatorname{gap}(\rho,\gamma)=\int_{\bar{\gamma}}\bar{\rho}\mathrm{d}s.$

###### Proof.

Given $\gamma$ with the decomposition using gaps,
$I_{\gamma}\setminus\operatorname{dom}(\gamma)=\bigcup_{i}(a_{i},b_{i})$,
consider the extended curve $\bar{\gamma}$ and note that, for each $i$,

$\int_{\bar{\gamma}|_{[a_{i},b_{i}]}}\bar{\rho}\mathrm{d}s=d(\gamma_{b_{i}},\gamma_{a_{i}})\int_{0}^{1}[t\rho(\gamma_{b_{i}})+(1-t)\rho(\gamma_{a_{i}})]\mathrm{d}t=\frac{\rho(\gamma_{a_{i}})+\rho(\gamma_{b_{i}})}{2}d(\gamma_{b_{i}},\gamma_{a_{i}})$

so that

$\displaystyle\int_{\bar{\gamma}}\bar{\rho}\mathrm{d}s=\int_{\operatorname{dom}(\gamma)}\rho\mathrm{d}s+\int_{[a,b]\setminus\operatorname{dom}(\gamma)}\bar{\rho}\mathrm{d}s=\int_{\operatorname{dom}(\gamma)}\rho\mathrm{d}s+\operatorname{gap}(\rho,\gamma),$

as claimed. ∎

######  Lemma 3.3.

If $\rho:X\to[0,\infty]$ is lower semicontinuous, then
$\bar{\rho}:\operatorname{sh}(X)\to[0,\infty]$ is lower semicontinuous.

###### Proof of Lemma 3.3.

Let $v=\iota_{x}+t(\iota_{y}-\iota_{x})$ with $t\leq 1/2$ and $v_{j}\to v$ in
$sh(X)$. Let $(x_{j},y_{j})$ and $(t_{j})$ be as in Lemma 2.2. If $x\neq y$
and $t\in(0,1)$, then

$\displaystyle\bar{\rho}(v)=$ $\displaystyle
t\rho(y)+(1-t)\rho(x)\leq\liminf_{j\to\infty}t_{j}\rho(y_{j})+\liminf_{j\to\infty}(1-t_{j})\rho(x_{j})$
$\displaystyle\leq$
$\displaystyle\liminf_{j\to\infty}[t_{j}\rho(y_{j})+(1-t_{j})\rho(x_{j})]=\liminf_{j\to\infty}\bar{\rho}(v_{j}).$

In the second case we have that $v=\iota_{x}$ and $t=0$, Thus $t_{j}\to 0$ and
$x_{j}\to x$. In this case

$\bar{\rho}(v)=\rho(x)\leq\liminf_{j\to\infty}(1-t_{j})\rho(x_{j})\leq\liminf_{j\to\infty}[t_{j}\rho(x_{j})+(1-t_{j})\rho(y_{j})]=\liminf_{j\to\infty}\bar{\rho}(v_{j}).$

This completes the proof. ∎

We now provide the proof of the main result of this subsection.

###### Proof of Proposition 3.1.

The first estimate follows from [15, Lemma 4.2] and Lemmas 2.5 and 3.2. The
second similarly follows from [15, Lemma 4.2] since
$\chi_{\\{l_{\operatorname{sh}}>\delta\\}}$ is lower semicontinuous by Remark
2.3. ∎

###  3.2. Construction of an approximating sequence

Suppose $f\in\operatorname{LIP}_{bs}(X)$. By Remark 2.9, $|Df|_{\ast}$ has a
Borel representative $\rho$ that is a genuine $\ast$-upper gradient of $f$.
Since $f$ has bounded support, $\rho$ vanishes outside some bounded set. We
fix $\rho$ for this construction.

If $j\in\mathbb{N}$, we set

$E_{j,l}\mathrel{\mathop{:}}=\rho^{-1}\Big{(}\frac{l-1}{2^{j}},\frac{l}{2^{j}}\Big{]},$

for $l=1,\ldots,N_{j}$, where $N_{j}$ is the unique integer in
$(2^{j}\|\rho\|_{L^{\infty}(\mu)},2^{j}\|\rho\|_{L^{\infty}(\mu)}+1]$. Given
$j,l$, let $K_{j,l}$ be a compact subsets of $E_{j,l}$, such that

$\mu(E_{j,l}\setminus K_{j,l})\leq 2^{-j^{2}}.$

Set $\displaystyle K_{j}:=\bigcup_{l=1}^{N_{j}}K_{j,l}$. For any compact set
$A\subset X$ and $r>0$, let $B(A,r)=\\{x\in X:d(x,A)<r\\}$. Let $k_{j}$ be
such that

$2^{-k_{j}}<\frac{\min_{l\neq\tilde{l}}\operatorname{dist}(K_{j,l},K_{j,\tilde{l}})}{2}.$

Then, for each $k\geq k_{j}$ define the functions $\rho_{j,l}^{k}:X\to[0,1]$,

$\rho_{j,l}^{k}:=\frac{\operatorname{dist}(x,X\setminus
B(K_{j},2^{-k}))}{\operatorname{dist}(x,X\setminus
B(K_{j},2^{-k}))+\operatorname{dist}(x,K_{j,l})},$

which vanish outside $B(K_{j},2^{-k})$ and equal $1$ on $K_{j,l}$. Finally set

$\displaystyle\rho_{j}^{k}:=\operatorname{LIP}(f)-\sum_{l=1}^{N_{j}}\Big{(}\operatorname{LIP}(f)-\frac{l}{2^{j}}\Big{)}_{+}\rho_{j,l}^{k}.$

######  Remark 3.4.

We have for every $k\geq k_{j}$ that
$\rho_{j}^{k}=\min\\{l/2^{j},\operatorname{LIP}(f)\\}$ on $K_{j,l}$,
$\rho_{j}^{k}=\operatorname{LIP}(f)$ outside $B(K_{j},2^{-k})$ and
$\rho_{j}^{k}\leq\operatorname{LIP}(f)$ everywhere. Furthermore,
$(\rho_{j}^{k})_{k}$ is a non-decreasing sequence of continuous functions
converging pointwise to the function

$\rho_{j}\mathrel{\mathop{:}}=\operatorname{LIP}(f)\chi_{X\setminus
K_{j}}+\sum_{l=1}^{N_{j}}\frac{l}{2^{j}}\chi_{K_{j,l}},$

which satisfies $\rho\leq\rho_{j}$ and is thus a $\ast$-upper gradient of $f$.
Further, $\rho_{j}\geq\rho_{j}^{k}\geq 2^{-j}$.

######  Lemma 3.5.

Given $j\in\mathbb{N}$, and $\varepsilon>0$ there exists
$k=k(j,\varepsilon)\in\mathbb{N}$ such that

$\displaystyle|f(y)-f(x)|\leq$
$\displaystyle\int_{\gamma}\rho_{j}^{k}\mathrm{d}s+\sum_{d(\gamma_{a_{i}},\gamma_{b_{i}})\leq
2^{-k}}\frac{\rho(\gamma_{a_{i}})+\rho(\gamma_{b_{i}})}{2}d(\gamma_{a_{i}},\gamma_{b_{i}})$
$\displaystyle+\operatorname{LIP}(f)\operatorname{gap}_{2^{-k}}(\gamma)+\varepsilon$

for all $\gamma\in\operatorname{Fr}(X)$ with start and endpoints $x$ and $y$,
respectively, and

$\displaystyle\bigcup_{i}(a_{i},b_{i})=\operatorname{conv}{\operatorname{dom}(\gamma)}\setminus\operatorname{dom}(\gamma).$

The proof scheme is based on a contradiction argument that has appeared much
earlier. Indeed, Cheeger attributes it to Ziemer in [12, Remark 5.22]. For the
proof we define the function
$\widehat{\rho}_{j}^{k}:\operatorname{sh}(X)\to[0,\infty)$ by

$\widehat{\rho}_{j}^{k}:=\max\\{\overline{\rho_{j}^{k}},\operatorname{LIP}(f)\chi_{\\{l_{\operatorname{sh}(X)}>2^{-k}\\}}\\}$

for each $j,k\in\mathbb{N}$, where $\overline{\rho_{j}^{k}}$ denotes the
extension of $\rho_{j}^{k}$ into $\operatorname{sh}(X)$, cf. Section 3.1, and
note that

$\displaystyle\int_{\bar{\gamma}}\widehat{\rho}_{j}^{k}\mathrm{d}s=\int_{\gamma}\rho_{j}^{k}\mathrm{d}s$
$\displaystyle+\sum_{d(\gamma_{a_{i}},\gamma_{b_{i}})\leq
2^{-k}}\frac{\rho(\gamma_{a_{i}})+\rho(\gamma_{b_{i}})}{2}d(\gamma_{a_{i}},\gamma_{b_{i}})+\operatorname{LIP}(f)\operatorname{gap}_{2^{-k}}(\gamma).$

######  Remark 3.6.

The line-integral above has the following geometric interpretation: from the
perspective of the curve fragment $\gamma$, the line integral of
$\widehat{\rho}_{j}^{k}$ over $\bar{\gamma}$ penalizes gaps in $\gamma$ with
length above a certain threshold, while short gaps only contribute an amount
determined by $\rho_{j}^{k}$.

###### Proof of Lemma 3.5.

Observe that, as the maximum of two lower semicontinuous functions,
$\widehat{\rho}_{j}^{k}$ is lower semicontinuous, cf. Lemma 3.3 and Remark
2.3. Moreover the sequence $(\widehat{\rho}_{j}^{k})_{k}$ increases pointwise
to
$\widehat{\rho}_{j}:=\chi_{X}\rho_{j}+\operatorname{LIP}(f)\chi_{\operatorname{sh}(X)\setminus
X}$ as $k\to\infty$, see Remark 3.4. Note that

$\displaystyle\int_{\bar{\gamma}}\widehat{\rho}_{j}\mathrm{d}s=\int_{\gamma}\rho_{j}\mathrm{d}s+\operatorname{LIP}(f)\operatorname{gap}(\gamma),\quad\gamma\in\operatorname{Fr}(X).$

Fix $\varepsilon>0$ and suppose by contradiction that, for every $k\geq
k_{j}$, there exists $\gamma_{k}\in\operatorname{Fr}(X)$ with start and
endpoints $x_{k},y_{k}\in X$ respectively, such that

(3.1)
$|f(y_{k})-f(x_{k})|\geq\int_{\bar{\gamma}_{k}}\widehat{\rho}_{j}^{k}\mathrm{d}s+\varepsilon.$

Choose $\gamma_{k}$ to be a minimal curve-fragment such that (3.1) holds,
where curve fragments are ordered according to the inclusions of their graphs.
By Zorn’s lemma, since the intersection of any chain of curve fragments
satisfying the previous inequality will also satisfy that same inequality (by
lower-semicontinuity of the integrand in (3.1)), such a minimal curve fragment
exists. Denote $C_{k}=\operatorname{dom}(\gamma_{k})$ and
$\operatorname{conv}C_{k}=[a_{k},b_{k}]$.

We claim that
$\operatorname{im}(\gamma_{k})\subset\overline{B}(K_{j},2^{1-k})$. The idea
here is that $\widehat{\rho}_{j}^{k}=\operatorname{LIP}(f)$ on the parts of
the fragment lying outside $\overline{B}(K_{j},2^{1-k})$ and thus they can be
removed while preserving (3.1).

Suppose that
$F_{k}:=\gamma_{k}^{-1}(X\setminus\overline{B}(K_{j},2^{1-k}))\neq\varnothing$.
If $t_{0}\in F_{k}$ is isolated in $C_{k}$, let
$t_{0}^{\pm}\in(C_{k}\setminus\\{t_{0}\\})\cup\\{a_{k},b_{k}\\}$,
$t_{0}^{-}\leq t_{0}\leq t_{0}^{+}$, be its nearest points. Consider two cases
$\sigma\in\\{-,+\\}.$ Either
$\gamma_{k}(t_{0}^{\sigma})\in\overline{B}(K_{j},2^{-k})$, in which case
$d(\gamma_{k}(t_{0}),\gamma_{k}(t_{0}^{\sigma}))>2^{-k}$, or
$\gamma_{k}(t_{0}^{\sigma})\notin\overline{B}(K_{j},2^{-k})$, in which case
$\rho_{j}^{k}(t_{0}^{\sigma})=\operatorname{LIP}(f)$. In both cases we have
$\widehat{\rho}_{j}^{k}(\overline{\gamma}_{k}(t)))=\operatorname{LIP}(f)$ for
all $t\in(t_{0}^{\pm},t_{0}]\cup[t_{0},t_{0}^{\pm})$. Consequently
$\widehat{\rho}_{j}^{k}(\overline{\gamma}_{k}(t))=\operatorname{LIP}(f)$ for
$t\in\\{t_{0}\\}\cup(t_{0}^{-},t_{0}^{+})$ and
$|f(\gamma_{k}(t_{0}^{+}))-f(\gamma_{k}(t_{0}^{-}))|\leq\int_{\overline{\gamma}_{k}|_{[t_{0}^{-},t_{0}^{+}]}}\widehat{\rho}_{j}^{k}\mathrm{d}s$,
in particular

$\displaystyle\varepsilon+\int_{\overline{\gamma}_{k}}\widehat{\rho}_{j}^{k}\mathrm{d}s\leq$
$\displaystyle|f(y_{k})-f(x_{k})|$ $\displaystyle\leq$
$\displaystyle|f(y_{k})-f(\gamma_{k}(t_{0}^{+}))|+\int_{\overline{\gamma}_{k}|_{[t_{0}^{-},t_{0}^{+}]}}\widehat{\rho}_{j}^{k}\mathrm{d}s+|f(\gamma_{k}(t_{0}^{-}))-f(x_{k})|.$

It follows that the proper sub-fragment
$\gamma_{k}^{\prime}:=\gamma_{k}|_{C_{k}\setminus\\{t_{0}\\}}$ also satisfies
(3.1), contradicting the minimality of $\gamma_{k}$.

Therefore, we have that every point in $F_{k}$ is an accumulation point of of
$C_{k}$ and $F_{k}$ is relatively open. Thus, there exist
$a_{k}^{\prime},b_{k}^{\prime}\in F_{k}$ such that
$\varnothing\neq[a_{k}^{\prime},b_{k}^{\prime}]\cap C_{k}\subset F_{k}$. In
particular
$\widehat{\rho}_{j}^{k}(\overline{\gamma}_{k}(t))=\operatorname{LIP}(f)$ for
all $t\in[a_{k}^{\prime},b_{k}^{\prime}]$. As above we obtain that
$|f(\gamma_{k}(b_{k}^{\prime}))-f(\gamma_{k}(a_{k}^{\prime}))|\leq\int_{\overline{\gamma}_{k}|_{[a_{k}^{\prime},b_{k}^{\prime}]}}\widehat{\rho}_{j}^{k}\mathrm{d}s$,
and that
$\gamma_{k}^{\prime}:=\gamma_{k}|_{C_{k}\setminus(a_{k}^{\prime},b_{k}^{\prime})}$
satisfies (3.1), contradicting the minimality of $\gamma_{k}$.

Thus $\operatorname{im}(\gamma_{k})\subset\overline{B}(K_{j},2^{1-k})$ for all
$k$ and furthermore

$2^{-j}\ell^{\ast}(\gamma_{k})\leq\varepsilon+\int_{\bar{\gamma}_{k}}\hat{\rho}_{j}^{k}\mathrm{d}s\leq|f(y_{k})-f(x_{k})|\leq\operatorname{LIP}(f)\operatorname{diam}(\operatorname{spt}(f))$

so that $\sup_{k\geq
k_{j}}\ell^{\ast}(\gamma_{k})=\mathrel{\mathop{:}}L<\infty$. By Remark 2.4, we
can parametrize all curves $\bar{\gamma}_{k}$ to be $L-$Lipschitz with the
parameter interval $[0,1]$. Then the sequence
$\bar{\gamma}_{k}:[0,1]\to\operatorname{sh}(X)$ of uniformly Lipschitz curves
satisfies
$\bar{\gamma}_{k}([0,1])\subset\operatorname{sh}(\overline{B(K_{j},2^{-k})})$
and thus, it is not hard to see that for each $t\in[0,1]$, the sequence
$(\bar{\gamma}_{k}(t))\subset\operatorname{sh}(X)$ is precompact. By the
Arzelá–Ascoli Theorem [25, Theorem 10.28] a subsequence of
$(\bar{\gamma}_{k})$ converges to a Lipschitz curve
$\bar{\gamma}:[0,1]\to\operatorname{sh}(X)$, and thus the corresponding
subsequence of $(\gamma_{k})\subset\operatorname{Fr}(X)$ converges to the
curve fragment
$\gamma\mathrel{\mathop{:}}=\bar{\gamma}|_{\bar{\gamma}^{-1}(X)}$, cf. Lemma
2.5. By Remark 3.4 and Proposition 3.1 we have that

$\displaystyle\varepsilon+\int_{\gamma}\rho_{j}\mathrm{d}s+\operatorname{LIP}(f)\operatorname{gap}(\gamma)$
$\displaystyle=\varepsilon+\int_{\bar{\gamma}}\hat{\rho}_{j}\mathrm{d}s\leq\varepsilon+\liminf_{k\to\infty}\int_{\bar{\gamma}_{k}}\hat{\rho}_{j}^{k}\mathrm{d}s$
$\displaystyle\leq\liminf_{k\to\infty}|f(y_{k})-f(x_{k})|=|f(y)-f(x)|,$

which contradicts the fact that $\rho_{j}$ is a $\ast$-upper gradient of $f$,
cf. Remark 3.4. This completes the proof. ∎

By Lemma 3.5 there exists $k=k(j)>j^{2}$ so that
$\hat{\rho}_{j}:=\widehat{\rho}_{j}^{k(j)}:\operatorname{sh}(X)\to(0,\infty)$
satisfies

(3.2)
$\displaystyle|f(x)-f(y)|\leq\frac{\operatorname{LIP}(f)}{2^{j+1}}+\int_{\overline{\gamma}}\hat{\rho}_{j}\mathrm{d}s,\quad\gamma\in\operatorname{Fr}(X).$

We define

$\displaystyle
f_{j}(x)=\inf\left\\{f(y)+\int_{\overline{\gamma}}\hat{\rho}_{j}\mathrm{d}s\Big{|}\Big{.}\
\gamma\in\operatorname{Fr}(X),\ \gamma:y\curvearrowright x\right\\},$

where we write $\gamma:y\curvearrowright x$ to denote that the curve fragment
starts at $y$ and ends at $x$, that is
$\gamma(\min(\operatorname{dom}(\gamma)))=y,\gamma(\max(\operatorname{dom}(\gamma)))=x$.

In the next lemma and proposition we collect some key properties of the
functions $\hat{\rho}_{j}$ and $f_{j}$ defined above.

######  Lemma 3.7.

For each $j\in\mathbb{N}$ we have that

  * (a)

$\hat{\rho}_{j}|_{X}$ is continuous and
$\min\\{2^{-j},\operatorname{LIP}(f)\\}\leq\hat{\rho}_{j}|_{X}\leq\operatorname{LIP}(f)$;

  * (b)

$|\hat{\rho}_{j}-\rho|\leq 2^{-j+1}$ on $K_{j}$; and

  * (c)

$\mu(\\{\rho>0\\}\setminus K_{j})\leq
2^{-j^{2}}(2^{j}\|\rho\|_{L^{\infty}(\mu)}+1)$.

###### Proof.

The claims are immediate from the construction, see also Remark 3.4. ∎

######  Proposition 3.8.

Suppose $f\in\operatorname{LIP}_{bs}(X)$ is non-negative. For each
$j\in\mathbb{N}$,

  * (a)

the function $f_{j}$ satisfies $0\leq f_{j}(x)\leq f(x)$ for every $x\in X$;

  * (b)

there exists $\delta=\delta(j)>0$ so that, for each $x,y\in X$ with
$d(x,y)\leq\delta$, we have

$\displaystyle|f_{j}(x)-f_{j}(y)|\leq\frac{\hat{\rho}_{j}(x)+\hat{\rho}_{j}(y)}{2}d(x,y),$

while $|f_{j}(x)-f_{j}(y)|\leq\operatorname{LIP}(f)d(x,y)$ if $d(x,y)>\delta$;
and

  * (c)

we have that $|f(x)-f_{j}(x)|\leq 2^{-j}\operatorname{LIP}(f)$ for all $x\in
X$.

###### Proof.

The statement in (a) is immediate from the construction. To see this, it is
useful to observe that $f,\hat{\rho}_{j}\geq 0$ and that a curve fragment
$\gamma$ can be chosen to consist of only one point.

To see (b), let $k=k(j)\in\mathbb{N}$ be as in (3.2) and the definition of
$f_{j}$. Suppose $x,y\in X$ and assume without loss of generality that
$f_{j}(x)\leq f_{j}(y)$. Given $\varepsilon>0$, let
$\gamma:y_{\gamma}\curvearrowright x$ be a curve fragment such that
$f_{j}(x)+\varepsilon>f(y_{\gamma})+\int_{\overline{\gamma}}\hat{\rho}_{j}\mathrm{d}s$.
Set $t_{y}=\max\operatorname{dom}(\gamma)+d(x,y)$,
$\operatorname{dom}(\tilde{\gamma}):=\operatorname{dom}(\gamma)\cup\\{t_{y}\\}$,
and define $\tilde{\gamma}:\operatorname{dom}(\tilde{\gamma})\to X$ by
$\tilde{\gamma}|_{\operatorname{dom}(\gamma)}=\gamma$ and
$\tilde{\gamma}(t_{y})=y$. If $d(x,y)\leq 2^{-k}=\mathrel{\mathop{:}}\delta$,
then

$\int_{\overline{\tilde{\gamma}}}\hat{\rho}_{j}\mathrm{d}s=\int_{\overline{\gamma}}\hat{\rho}_{j}\mathrm{d}s+\frac{\hat{\rho}_{j}(x)+\hat{\rho}_{j}(y)}{2}d(x,y)$

Consequently

$\displaystyle f_{j}(y)\leq
f(y_{\gamma})+\int_{\overline{\tilde{\gamma}}}\hat{\rho}_{j}\mathrm{d}s$
$\displaystyle=f(y_{\gamma})+\int_{\overline{\gamma}}\hat{\rho}_{j}\mathrm{d}s+\frac{\hat{\rho}_{j}(x)+\hat{\rho}_{j}(y)}{2}d(x,y)$
$\displaystyle<f_{j}(x)+\varepsilon+\frac{\hat{\rho}_{j}(x)+\hat{\rho}_{j}(y)}{2}d(x,y)$

and thus
$f_{j}(y)-f_{j}(x)\leq\frac{\hat{\rho}_{j}(x)+\hat{\rho}_{j}(y)}{2}d(x,y)+\varepsilon$.
If $d(x,y)>\delta$ we have

$\int_{\overline{\tilde{\gamma}}}\hat{\rho}_{j}\mathrm{d}s=\int_{\overline{\gamma}}\hat{\rho}_{j}\mathrm{d}s+\operatorname{LIP}(f)d(x,y)$

and the argument above yields
$f_{j}(y)-f_{j}(x)\leq\operatorname{LIP}(f)d(x,y)+\varepsilon$. Since
$\varepsilon$ is arbitrary, (b) follows.

It remains to show (c). Let $x\in X$ and, for each $j$, let
$\gamma:y_{\gamma}\curvearrowright x$ be such that

$f(y_{\gamma})+\int_{\overline{\gamma}}\hat{\rho}_{j}\mathrm{d}s<f_{j}(x)+\frac{\operatorname{LIP}(f)}{2^{j+1}}.$

By (3.2) we have that

$f(x)\leq
f(y_{\gamma})+\int_{\overline{\gamma}}\hat{\rho}_{j}\mathrm{d}s+\frac{\operatorname{LIP}(f)}{2^{j+1}}\leq
f_{j}(x)+\frac{\operatorname{LIP}(f)}{2^{j}}.$

Since $f_{j}\leq f$ it follows that $|f(x)-f_{j}(x)|\leq
2^{-j}\operatorname{LIP}(f)$, completing the proof. ∎

We now prove Theorem 1.6.

###### Proof of Theorem 1.6.

Let $f^{+}=\max\\{f,0\\}$ and $f^{-}=-\min\\{f,0\\}$, so that
$f^{\pm}\in\operatorname{LIP}_{bs}(X)$ is non-negative and $f=f^{+}-f^{-}$.
Let $(f_{j}^{\pm})$ be the sequence constructed above for $f^{\pm}$, with
minimal $\ast$-upper gradient $\rho^{\pm}$, and $\tilde{\rho}_{j}^{\pm}$
satisfying (3.2). Set $f_{j}\mathrel{\mathop{:}}=f_{j}^{+}-f_{j}^{-}$. Then
$\operatorname{LIP}(f_{j})\leq\operatorname{LIP}(f)$, $|f_{j}|\leq|f|$, and
$f_{j}\to f$ pointwise by Proposition 3.8.

It remains to show (2). Set
$\tilde{\rho}_{j}\mathrel{\mathop{:}}=\tilde{\rho}_{j}^{+}+\tilde{\rho}_{j}^{-}$
and $\rho\mathrel{\mathop{:}}=\rho^{+}+\rho^{-}$. We observe that

(3.3) $\displaystyle\operatorname{Lip}_{a}f_{j}\leq\tilde{\rho}_{j}\ \textrm{
on }X,\quad\tilde{\rho}_{j}\to\rho\ \mu-a.e.\textrm{ on }X$

by Proposition 3.8(b) (with the continuity of $\tilde{\rho}_{j}$) and Lemma
3.7(b),(c), respectively. By Lemma 2.12 we have that $|Df|_{\ast}=\rho$
$\mu$-a.e..

Let $A\subset X$ be a Borel set with $\mu(A)<\infty$. We claim that
$\operatorname{Lip}_{a}f_{j}\rightharpoonup\rho$ weakly in $L^{2}(A)$. Indeed,
since the sequence $(\operatorname{Lip}_{a}f_{j})$ is bounded in $L^{2}(A)$,
given any subsequence, a further subsequence converges weakly to some $g\in
L^{2}(A)$. By Mazur’s Lemma a sequence of convex combination’s of the tail of
this subsequence converges to $g$ in $L^{2}(A)$ and pointwise. Since
$\operatorname{Lip}_{a}f_{j}\leq\rho_{j}\to\rho$ pointwise, we have that
$g\leq\rho$ on $A$. On the other hand by Lemma 2.13 $g$ is a weak $\ast$-upper
gradient of $f$ (which equals the limit of the corresponding convex
combination of $(f_{j})$), implying $\rho\leq g$. We have proven that any
subsequence of $(\operatorname{Lip}_{a}f_{j})$ has a further subsequence
converging weakly in $L^{2}(A)$ to $\rho$, establishing the claim.

For every $\varepsilon>0$ we have the estimate

$\displaystyle\mu(A\cap\\{|\rho-\operatorname{Lip}_{a}f_{j}|\geq\varepsilon\\})\leq$
$\displaystyle\frac{1}{\varepsilon}\int_{A}|\rho-\operatorname{Lip}_{a}f_{j}|\mathrm{d}\mu$
$\displaystyle\leq$
$\displaystyle\frac{1}{\varepsilon}\int_{A}|\rho-\rho_{j}|\mathrm{d}\mu+\frac{1}{\varepsilon}\int_{A}(\rho_{j}-\operatorname{Lip}_{a}f_{j})\mathrm{d}\mu.$

The $\mu$-a.e. pointwise convergence $\rho_{j}\to\rho$, the weak
$L^{2}(A)$-convergence $\rho_{j}-\operatorname{Lip}_{a}f_{j}\rightharpoonup
0$, and the dominated convergence theorem imply that

$\limsup_{j\to\infty}\mu(A\cap\\{|\rho-\operatorname{Lip}_{a}f_{j}|\geq\varepsilon\\})=0,$

i.e. $\operatorname{Lip}_{a}f_{j}\to\rho$ in measure (in $A$). Together with
the weak $L^{2}(A)$-convergence this implies pointwise convergence $\mu$-a.e.
in $A$ after passing to a subsequence. Since $A\subset X$ is arbitrary this
shows that, after passing to a subsequence,
$\operatorname{Lip}_{a}f_{j}\to\rho$ $\mu$-a.e. on $X$.

Since $|Dh|_{\ast}\leq\operatorname{Lip}h\leq\operatorname{Lip}_{a}h$ for
$h\in\operatorname{LIP}(X)$, (3.3) holds for $|Df_{j}|_{\ast}$ and
$\operatorname{Lip}f_{j}$ in place of $\operatorname{Lip}_{a}f_{j}$.
Therefore, the same argument yields the $\mu$-a.e. convergence
$|Df_{j}|_{\ast}\to\rho$ and $\operatorname{Lip}f_{j}\to\rho$, finishing the
proof.

∎

##  4\. Representation of minimal $\ast$-upper gradients

In this section we establish the following representation of the minimal weak
$\ast$-upper gradient as a maximal directional derivative along curve
fragments, analogous to [14, Corollary 1.10], [22, Theorem 1.1] and [13,
Theorem 4.2], as a tool towards constructing the pointwise norm and
differential of functions in Theorem 1.2.

######  Theorem 4.1.

Let $f\in\operatorname{LIP}(X)$. There exists a 1-plan $\mathbb{P}$ with
$\mu|_{\\{|Df|>0\\}}\ll\mathbb{P}^{\\#}$ and disintegration
$\\{\bar{\mathbb{P}}_{x}\\}$, such that

$\displaystyle|Df|_{\ast}(x):=\chi_{\\{|Df|>0\\}}(x)\left\|\frac{(f\circ\gamma)^{\prime}(t)}{|\gamma_{t}^{\prime}|}\right\|_{L^{\infty}(\bar{\mathbb{P}}_{x})}$

$\mu$-a.e.

The representation is based on the disintegration of a measure with respect to
a mapping [5, Theorem 5.3.1]: if $\Phi:X\to Y$ is a Borel measurable map
between two separable metric spaces, and $\mu$ is a finite measure on $X$,
then the disintegration of $\mu$ with respect to $\Phi$ is a collection
$\\{\mu_{y}\\}\subset\mathcal{M}(X)$ for $\Phi_{\ast}\mu$-a.e. $y\in Y$ such
that

  1. (1)

for $\Phi_{\ast}\mu$-a.e. $y\in Y$, $\mu_{y}$ is concentrated on $f^{-1}(y)$;

  2. (2)

for every Borel set $B$ in $X$, the map $y\mapsto\mu_{y}(B)$ is Borel; and

  3. (3)

for every measurable set $A\subset X$ we have

$\int_{X}\chi_{A}(x)\mathrm{d}\mu(x)=\int_{Y}\int_{X}\chi_{A}(x)d\mu_{y}(x)\mathrm{d}\Phi_{\ast}\mu(y).$

Moreover, if $\\{\tilde{\mu}_{y}\\}$ is another collection satisfying (1) and
(2) above, then $\mu_{y}=\tilde{\mu}_{y}$ for $\Phi_{\ast}\mu$-a.e. $y\in Y$.
See also [17, III-70].

Set

$\overline{\operatorname{Fr}}(X)\mathrel{\mathop{:}}=\\{(\gamma,t)\in\operatorname{Fr}(X)\times\mathbb{R}:\
\gamma\in\operatorname{Fr}(X),\ t\in\operatorname{dom}(\gamma)\\},$

and equip it with the subspace topology of
$\operatorname{Fr}(X)\times\mathbb{R}$. Define the evaluation map
$e:\overline{\operatorname{Fr}}(X)\to X$, $(\gamma,t)\mapsto\gamma_{t}$. Given
a finite measure $\mathbb{P}\in\mathcal{M}(\operatorname{Fr}(X))$, the measure
$\mathrm{d}\bar{\mathbb{P}}\mathrel{\mathop{:}}=\mathrm{d}s\mathrm{d}\mathbb{P}$
on $\overline{\operatorname{Fr}}(X)$ is given by

$\displaystyle\bar{\mathbb{P}}(B)=\int_{\operatorname{Fr}(X)}\int_{\gamma}\chi_{B}(\gamma,\cdot)\mathrm{d}s\mathrm{d}\mathbb{P}(\gamma),\quad
B\subset\overline{\operatorname{Fr}}(X).$

Observe that $e_{\ast}\bar{\mathbb{P}}=\mathbb{P}^{\\#}$ so that, if
$\mathbb{P}$ is a 1-plan, then $\mathbb{P}^{\\#}=\rho_{\mathbb{P}}\mu$, where
$\rho_{\mathbb{P}}\in L^{1}(\mu)$ is the Radon–Nikodym derivative of
$\mathbb{P}^{\\#}$ with respect to $\mu$. Consequently, if $\mathbb{P}$ is a
1-plan, then the disintegration $\\{\bar{\mathbb{P}}_{x}\\}$ of
$\bar{\mathbb{P}}$ with respect to $e$ is defined for $\mu$-a.e.
$x\in\\{\rho_{\mathbb{P}}>0\\}$.

###  4.1. Plan-modulus duality

The proof of Theorem 4.1 is based on duality, and is modeled on the argument
in [22]. However, compared to the setting in [22], we face the central
difficulty that the duality theory of modulus of curve families from [4] is
only applicable for a finite exponent $p\in(1,\infty)$. (See, however, [29,
18, 15] for duality results that are applicable for $p=1$.) This duality
theory does not extend directly to the case of $p=\infty$. As a substitute,
for the exponent $p=\infty$, we prove a partial duality result (Proposition
4.3) which is based on the use of Rainwater’s lemma. Historically, this is
quite natural, since this lemma was used in [8] and [35] to obtain results on
the geometry of Lipschitz differentiability spaces. Thus, Rainwater’s Lemma
really should be thought of as the $p=\infty$ substitute for the duality
result in [4]. Indeed, a close examination of the proofs in [4] and the
following shows that these are related.

######  Lemma 4.2.

Let $\mu$ be a finite measure on a metric space $X$ and $\Gamma$ a compact
family of curve fragments. Then one of the following two conditions holds.

  * (1)

There exists a $\mu$-null Borel set $N\subset X$ such that
$\mathcal{H}^{1}|_{\operatorname{im}(\gamma)}$ is concentrated on $N$ for
every $\gamma\in\Gamma$; or

  * (2)

there exists a compact set $K\subset X$ with $\mu(K)>0$ and a 1-plan
$\mathbb{P}\neq 0$ concentrated on the curve fragment family $\Gamma|_{K}$.

In the proof we use a variant of the barycenter called the _parametrized
barycenter_. Given a measure $\mathbb{P}\in\mathcal{M}(\operatorname{Fr}(X))$,
define a measure $\mathbb{P}_{b}$ on $X$ by

$\mathbb{P}_{b}(E)=\int_{\Gamma}\int_{\operatorname{dom}(\gamma)}\chi_{E}(\gamma_{t})\mathrm{d}t\mathrm{d}\mathbb{P}(\gamma),\quad
E\subset X\textrm{ Borel}.$

The reason for this is that the integration over curve fragments with respect
to $\mathrm{d}t$ instead of $\mathrm{d}s$ ensures suitable upper
semicontinuity properties for the functional $\Phi$ defined in the proof
below. Note moreover that
$\mathbb{P}^{\\#}\ll\mathbb{P}_{b}\ll\mathbb{P}^{\\#}$.

###### Proof.

Suppose (2) does not hold. Then for each $\mathbb{P}\in\mathcal{P}(\Gamma)$
the barycenter $\mathbb{P}^{\\#}$ of $\mathbb{P}$ is singular with respect to
$\mu$. Indeed, if there existed a set $K\subset X$ with $\mu(K)>0$ (which
could be taken compact by the inner regularity of $\mu$) and
$\mathbb{P}\in\mathcal{P}(\Gamma)$ such that $0\neq\mathbb{P}^{\\#}|_{K}\in
L^{1}(\mu)$, then $\mathbb{P}|_{K}\in\mathcal{P}(\Gamma|_{K})$ given by

$\mathbb{P}|_{K}=(R_{K})_{\ast}\mathbb{P},\text{ where
}R_{K}(\gamma)=\gamma|_{\gamma^{-1}(K)},$

would yield a 1-plan, because

$\displaystyle(\mathbb{P}|_{K})^{\\#}(E)=\int\int_{\gamma}\chi_{E}\mathrm{d}s\mathrm{d}\mathbb{P}|_{K}=\int\int_{\gamma|_{\gamma^{-1}(K)}}\chi_{E}\mathrm{d}s\mathrm{d}\mathbb{P}=\mathbb{P}^{\\#}|_{K}(E)$

for any Borel set $E\subset X$.

To complete the proof we follow the argument in [34, Lemma 9.4.3] to show that
if $\mathbb{P}^{\\#}\perp\mu$ for all $\mathbb{P}\in\mathcal{P}(\Gamma)$, then
(1) holds. Since $\mathbb{P}_{b}$ and $\mathbb{P}^{\\#}$ are mutually
absolutely continuous, it is equivalent to show that $\mathbb{P}_{b}\perp\mu$
for all $\mathbb{P}\in\mathcal{P}(\Gamma)$ implies (1). Let
$\Gamma_{L}=\\{\gamma\in\Gamma:\operatorname{biLip}(\gamma)\leq L\\}$ for
$L\in\mathbb{N}$. It suffices to show that there exists a $\mu$-null set
$N_{L}\subset X$ so that $\mathcal{H}^{1}|_{\operatorname{im}(\gamma)}$ is
concentrated on $N_{L}$ for every $\Gamma_{L}$, because then
$N=\bigcap_{L=1}^{\infty}N_{L}$ is the set we search for.

Let $L\in\mathbb{N}$. Equip $\mathcal{P}(\Gamma_{L})$ with the
$w^{\ast}$-topology and $G=C(X;[0,1])$ with the topology of uniform
convergence, and define the functional $\Phi:\mathcal{P}(\Gamma_{L})\times
G\to\mathbb{R}$ by

$\displaystyle\Phi(\mathbb{P},g):=\int(1-g)\mathrm{d}\mu+\int
g\mathrm{d}\mathbb{P}_{b}.$

$\Phi$ is bi-linear and continuous in $g$ for any fixed
$\mathbb{P}\in\mathcal{P}(\Gamma_{L})$. Since
$\gamma\mapsto\int_{\operatorname{dom}(\gamma)}g(\gamma_{t})\mathrm{d}t$ is
upper semicontinuous for any $g\in G$, cf. [35, Proof of Lemma 2.28], it
follows that $\Phi(\cdot,g)$ is upper semicontinuous for any fixed $g\in G$.
As $\mathcal{P}(\Gamma_{L})$ is compact, we may apply Sion’s minimax theorem
(see [39] and [15, Theorem 4.7]) to obtain

(4.1)
$\sup_{\mathbb{P}}\inf_{g}\Phi(\mathbb{P},g)=\inf_{g}\sup_{\mathbb{P}}\Phi(\mathbb{P},g).$

For every $\mathbb{P}\in\mathcal{P}(\Gamma_{L})$ the fact that
$\mathbb{P}_{b}\perp\mu$ implies $\displaystyle\inf_{g}\Phi(\mathbb{P},g)=0$
by an application of Urysohn’s lemma. By (4.1) there are functions $g_{j}\in
G$, for $j\in\mathbb{N}$, such that
$\displaystyle\sup_{\mathbb{P}}\Phi(\mathbb{P},g_{j})<2^{-j}$. Set
$N_{L}=\\{x\in X:\ \sum_{j\in\mathbb{N}}g_{j}(x)<\infty\\}$. For each
$\mathbb{P}\in\mathcal{P}(\Gamma_{L})$

(4.2)
$0\leq\sum_{j\in\mathbb{N}}\Phi(\mathbb{P},g_{j})=\int\bigg{(}\sum_{j\in\mathbb{N}}[1-g_{j}]\bigg{)}\mathrm{d}\mu+\int\bigg{(}\sum_{j\in\mathbb{N}}g_{j}\bigg{)}\mathrm{d}\mathbb{P}_{b}\leq
1.$

Since $\sum_{j\in\mathbb{N}}(1-g_{j})=\infty$ on $N_{L}$, (4.2) implies that
$\mu(N_{L})=0$ and moreover that $\mathbb{P}_{b}$ is concentrated on $N_{L}$.
The set $N_{L}$ satisfies the claim in (1) since for any $\gamma\in\Gamma_{L}$
we may choose $\mathbb{P}=\delta_{\gamma}$ to obtain that
$\mathcal{H}^{1}|_{\operatorname{im}(\gamma)}\simeq_{L}\mathbb{P}_{b}$ is
concentrated on $N_{L}$. ∎

Recall that a Borel function $\rho:X\to[0,\infty)$ is a weak $\ast$-upper
gradient of $f\in\operatorname{LIP}(X)$ if and only if, for
$\operatorname{Mod}_{\infty}$-a.e. $\gamma\in\operatorname{Fr}(X)$, we have
that $(f\circ\gamma)^{\prime}(t)\leq\rho(\gamma_{t})|\gamma_{t}^{\prime}|$
a.e. $t\in\operatorname{dom}(\gamma)$, cf. the discussion before the proof of
Proposition 2.10. In the next proposition we apply Lemma 4.2 to the family of
curve-fragments failing this condition. Namely, we define

$\Gamma_{\rho}:=\\{\gamma\in\operatorname{Fr}(X):\
(f\circ\gamma)^{\prime}(t)>\rho(\gamma_{t})|\gamma_{t}^{\prime}|\textrm{ on a
positive measure subset of }\operatorname{dom}(\gamma)\\}.$

Notice that $\rho$ fails to be a weak $\ast$-upper gradient of $f$ if and only
if $\operatorname{Mod}_{\infty}\Gamma_{\rho}>0$.

######  Proposition 4.3.

Let $f\in\operatorname{LIP}(X)$, $\rho\in L^{\infty}(\mu)$, and suppose that
$\rho$ is not a weak $\ast$-upper gradient of $f$, i.e. that we have
$\mu(\\{\rho<|Df|_{\ast}\\})>0$, or that, equivalently,
$\operatorname{Mod}_{\infty}\Gamma_{\rho}>0$. Then there exists a 1-plan
concentrated on $\Gamma_{\rho}$.

###### Proof.

Let $(K_{j})$ be a sequence of compact sets in $X$ such that $K_{j}\subset
K_{j+1}$, $N=X\setminus\bigcup_{j}K_{j}$ is $\mu$-null, and moreover
$\rho|_{K_{j}}$ is continuous for each $j$. Given $j,n\in\mathbb{N}$, we
define $\Gamma_{n,j}$ as the family of curve fragments
$\gamma:\operatorname{dom}(\gamma)\to K_{j}$ for which

$|f(\gamma_{s})-f(\gamma_{t})|\geq(1+1/n)\int_{\gamma|_{\operatorname{dom}(\gamma)\cap[t,s]}}^{*}\rho\mathrm{d}s=(1+1/n)\int_{\bar{\gamma}|_{[t,s]}}\bar{\rho}\mathrm{d}s$

for all $t,s\in\operatorname{dom}(\gamma)$. Note that for each
$n,j\in\mathbb{N}$, the family $\Gamma_{n,j}$ is compact by the lower
semicontinuity of the $\ast$-integral, cf. Proposition 3.1. We first claim
that every fragment in $\Gamma_{\rho}\setminus\Gamma_{N}^{+}$ contains a sub-
fragment in $\Gamma_{n,j}$ for some $j,n\in\mathbb{N}$.

Let $\gamma\in\Gamma_{\rho}\setminus\Gamma_{N}^{+}$. By a standard exhaustion
argument there exists $\lambda>1$ and a (compact)
$A\subset\operatorname{dom}(\gamma)$ with $|A|>0$ such that

$(f\circ\gamma)^{\prime}_{t}\geq\lambda\rho(\gamma_{t})|\gamma_{t}^{\prime}|$

for all $t\in A$. Since
$0=|A\cap\gamma^{-1}(N)|=\lim_{j\to\infty}|A\setminus\gamma^{-1}(K_{j})|$, we
may assume that $\operatorname{im}(\gamma)\subset K_{j}$ for some suitably
large $j$ by considering the restriction $\gamma|_{\gamma^{-1}(K_{j})}$. If
$\rho(\gamma_{\tau})|\gamma_{\tau}^{\prime}|=0$ for a.e. $t\in A$ then
$\gamma\in\Gamma_{n,j}$ for all $n$. Otherwise we may pass to a compact subset
$C\subset A$ with $|C|>0$ such that

  * •

$\tau\mapsto\rho(\gamma_{\tau})|\gamma_{\tau}^{\prime}|$ and
$\tau\mapsto(f\circ\gamma)_{\tau}^{\prime}$ restricted to $C$ are continuous
and bounded below away from zero;

  * •

$\tau\mapsto\frac{1}{h}\int_{\bar{\gamma}|_{[\tau,\tau+h]}}\bar{\rho}\mathrm{d}s$
and
$\tau\mapsto\frac{\bar{f}(\bar{\gamma}_{t+h})-\bar{f}(\bar{\gamma}_{t})}{h}$
converge uniformly in $C$ to $\rho(\gamma_{\tau})|\gamma_{\tau}^{\prime}|$ and
$(f\circ\gamma)_{\tau}^{\prime}$, respectively, as $h\to 0$.

From these it follows that there exists $h_{0}>0$ such that, for some
$0<\lambda^{\prime}<\lambda$, we have

$\lambda^{\prime}\frac{1}{h}\int_{\bar{\gamma}|_{[t,t+h]}}\bar{\rho}\mathrm{d}s\leq\frac{\bar{f}(\bar{\gamma}_{t+h})-\bar{f}(\bar{\gamma}_{t})}{h}$

whenever $t\in C$ and $h\in(0,h_{0}]$. Choosing $t_{0}\in C$ such that
$|C\cap[t_{0},t_{0}+h_{0}]|>0$ and $n\in\mathbb{N}$ such that
$1+1/n<\lambda^{\prime}$ we obtain that the restriction of $\gamma$ to
$C\cap[t_{0},t_{0}+h_{0}]$ belongs to $\Gamma_{n,j}$, proving the first claim.

Since every fragment in $\Gamma_{\rho}\setminus\Gamma_{N}^{+}$ contains a
subfragment in $\displaystyle\bigcup_{n,j}\Gamma_{n,j}$, we have that

$\operatorname{Mod}_{\infty}(\Gamma_{\rho})=\operatorname{Mod}_{\infty}(\Gamma_{\rho}\setminus\Gamma_{N}^{+})\leq\operatorname{Mod}_{\infty}(\bigcup_{n,j}\Gamma_{n,j})\leq\sum_{n,j}\operatorname{Mod}_{\infty}(\Gamma_{n,j}).$

Thus $\operatorname{Mod}_{\infty}(\Gamma_{n,j})>0$ for some
$n,j\in\mathbb{N}$. In particular the measures
$\mathcal{H}^{1}|_{\operatorname{im}(\gamma)}$, $\gamma\in\Gamma_{n,j}$,
cannot be all concentrated on a $\mu$-null set. Since $\Gamma_{n,j}$ is
compact, Rainwater’s lemma (Lemma 4.2) yields the existence of a compact set
$K\subset K_{j}$ and a 1-plan $\mathbb{P}$ concentrated on
$\Gamma_{n,j}|_{K}$. This implies the claim since
$\Gamma_{n,j}|_{K}\subset\Gamma_{n,j}\subset\Gamma_{\rho}$. ∎

###  4.2. Proof of Theorem 4.1

In order to prove Theorem 4.1 recall that, given a 1-plan $\mathbb{P}$, we set
$\bar{\mathbb{P}}=\mathrm{d}s\mathrm{d}\mathbb{P}=|\gamma_{t}^{\prime}|\mathrm{d}t\mathrm{d}\mathbb{P}$.
Note that $\mathbb{P}^{\\#}=e_{\ast}\bar{\mathbb{P}}$. We define

$\displaystyle|Df|_{\mathbb{P}}(x)=\chi_{\\{\mathbb{P}^{\\#}>0\\}}(x)\left\|\frac{(f\circ\gamma)_{t}^{\prime}}{|\gamma_{t}^{\prime}|}\right\|_{L^{\infty}(\bar{\mathbb{P}}_{x})},$

where $\\{\bar{\mathbb{P}}_{x}\\}_{x}$ denotes the disintegration of
$\bar{\mathbb{P}}$ with respect to $e$. The measurability, and a.e.
definedness of the expression $(\varphi\circ\gamma)_{t}^{\prime}$ follows from
the disintegration theorem and standard arguments, see e.g. [22, Appendix A].
These, and similar measurability arguments, are also used later in this paper
without mention. Since $\mathbb{P}^{\\#}\in L^{1}(\mu)$,
$\bar{\mathbb{P}}_{x}$ is defined for $\mu$-a.e.
$x\in\\{\mathbb{P}^{\\#}>0\\}$, and consequently $|Df|_{\mathbb{P}}$ is well-
defined $\mu$-a.e. on $X$. We record the following properties, whose proofs
are similar to [22, Lemma 3.2] and are left to the interested reader.

######  Lemma 4.4.

Let $f\in\operatorname{LIP}(X)$ and $\mathbb{P}$ a 1-plan. Then
$|Df|_{\mathbb{P}}$ has the following properties.

  * (1)

$|(f\circ\gamma)_{t}^{\prime}|\leq|Df|_{\mathbb{P}}(\gamma_{t})|\gamma_{t}^{\prime}|$
a.e. $t\in\operatorname{dom}(\gamma)$ for $\mathbb{P}$-a.e. $\gamma$;

  * (2)

$|Df|_{\mathbb{P}}\leq|Df|_{\ast}$ $\mu$-a.e.

  * (3)

If $\tilde{\mathbb{P}}$ is a 1-plan with $\tilde{\mathbb{P}}\ll\mathbb{P}$,
then $|Df|_{\tilde{\mathbb{P}}}\leq|Df|_{\mathbb{P}}$.

###### Proof of Theorem 4.1.

Define the function
$\rho=\operatorname{esssup}_{\mathbb{P}}|Df|_{\mathbb{P}}$, where the measure
theoretic supremum is taken over all 1-plans $\mathbb{P}$. We have
$\rho\leq|Df|_{\ast}$ $\mu$-a.e by Lemma 4.4(2). We claim that
$\rho=|Df|_{\ast}$. Suppose to the contrary that the set
$\\{\rho<|Df|_{\ast}\\}$ has positive $\mu$-measure.

By Proposition 4.3 there exists a 1-plan $\tilde{\mathbb{P}}$ concentrated on
$\Gamma_{\rho}$. By the definition of $\Gamma_{\rho}$, (1) in Lemma 4.4 and
the properties of disintegration we have that $|Df|_{\tilde{\mathbb{P}}}>\rho$
on a set of positive measure, contradicting the definition of $\rho$. Arguing
as in [22, Lemma 3.2] we obtain the existence of a 1-plan $\mathbb{P}$ such
that $\rho=|Df|_{\mathbb{P}}$. This completes the proof of Theorem 4.1.

∎

We finish this section by recording the following vector valued analogue of
Theorem 4.1. The two propositions below are in the spirit of [22, Propositions
4.1–4.3] and the proofs therein establish their claims, except for (iii),
which is special for the $p=\infty$ case. We therefore omit the details except
for this claim.

######  Proposition 4.5.

Let $\varphi:X\to\mathbb{R}^{n}$ be a Lipschitz map. Then there exists a Borel
set $D\subset X$ and a 1-plan $\mathbb{P}$ with $\mu|_{D}\ll\mathbb{P}^{\\#}$
and disintegration $\\{\bar{\mathbb{P}}_{x}\\}$ so that the map
$\Phi:(\mathbb{R}^{n})^{*}\times X\to[0,\infty)$ given by

$\Phi(\lambda,x)=\chi_{D}(x)\left\|\frac{\lambda((\varphi\circ\gamma)_{t}^{\prime})}{|\gamma_{t}^{\prime}|}\right\|_{L^{\infty}(\bar{\mathbb{P}}_{x})}$

is Borel and has the following properties.

  * (i)

For each $\lambda\in(\mathbb{R}^{n})^{*}$,
$\Phi_{\lambda}:=\Phi(\lambda,\cdot)$ is a Borel representative of
$|D(\lambda\circ\varphi)|_{\ast}$;

  * (ii)

For $\mu$-a.e. $x\in X$, $\Phi^{x}:=\Phi(\cdot,x)$ is a seminorm on
$(\mathbb{R}^{n})^{*}$;

  * (iii)

there exists a null set $E$ so that
$\Phi(\lambda,\cdot)+\operatorname{LIP}(\varphi)\chi_{E}$ is a $\ast$-upper
gradient for $\lambda\circ\varphi$ for every $\lambda\in(\mathbb{R}^{n})^{*}$
and any curve fragment $\gamma$.

###### Proof.

We only prove (iii) as its proof is distinct from those in [22]. First, let
$S\subset(\mathbb{R}^{n})^{*}$ be a countable dense subset. Then by (i) and
Remark 2.9 there exists for every $s\in S$ a set $E_{s}$ so that
$\Phi(s,\cdot)+\operatorname{LIP}(\varphi)\chi_{E_{s}}$ is a $\ast$-upper
gradient for $s\circ\varphi$. If we set $E=\bigcup_{s\in S}E_{s}$, we observe
that $\Phi(s,\cdot)+\operatorname{LIP}(\varphi)\chi_{E}$ is a $\ast$-upper
gradient for $s\circ\varphi$ for all $s\in S$. Since
$\lambda\mapsto\Phi(\lambda,x)$ is Lipschitz continuous, and since
$\|s-s^{\prime}\|\operatorname{LIP}(\varphi)$ is a $\ast$-upper gradient for
$(s-s^{\prime})\circ\varphi$, we obtain directly that
$\Phi(\lambda,\cdot)+\operatorname{LIP}(\varphi)\chi_{E}$ is an $\ast$-upper
gradient for all $\lambda\circ\varphi$. ∎

Let $\lambda\mapsto\|\lambda\|_{*}$ denote the dual norm to the standard
Euclidean norm on $\mathbb{R}^{n}$. In the next statement, we define

$\displaystyle
I(\varphi,\cdot):=\underset{\|\lambda\|_{*}=1}{\operatorname{essinf}}\,|D(\lambda\circ\varphi)|.$

This is the measure theoretic essential infimum of $|D(\lambda\circ\varphi)|$.

######  Proposition 4.6.

$\Phi$ has the following useful properties.

  * (i)

For any Borel map $\bm{\lambda}:X\to(\mathbb{R}^{n})^{*}$ we have that
$\Phi(\bm{\lambda}_{x},x)=0$ $\mu$-a.e. $x\in X$ if and only if
$\bm{\lambda}_{\gamma_{t}}((\varphi\circ\gamma)_{t}^{\prime})=0$ a.e.
$t\in\operatorname{dom}(\gamma)$ for $\infty$-a.e. $\gamma$;

  * (ii)

$\displaystyle
I(\varphi,x)=\underset{\|\lambda\|_{*}=1}{\inf}\,\Phi(\lambda,x)$ for
$\mu$-a.e. $x\in X$.

##  5\. Fragment-wise differentiable structure

Throughout this section we fix a Borel set $U\subset X$ with $\mu(U)>0$ and
$\varphi\in\operatorname{LIP}(X,\mathbb{R}^{n})$, $n\geq 0$, and let $\Phi$ be
as in Proposition 4.5.

###  5.1. Charts and differentials

For what follows we define the Borel map ${\rm
Dir}_{\varphi}:\overline{\operatorname{Fr}}(X)\to S^{n-1}$ by ${\rm
Dir}_{\varphi}(\gamma,t)=\frac{(\varphi\circ\gamma)^{\prime}(t)}{|(\varphi\circ\gamma)^{\prime}(t)|}$
whenever $(\varphi\circ\gamma)^{\prime}(t)$ exists and is non-zero, and ${\rm
Dir}_{\varphi}(\gamma,t)=e_{1}$ otherwise.

######  Lemma 5.1.

Let $\mathbb{P}^{\prime}$ be a 1-plan with
$U\subset\\{(\mathbb{P}^{\prime})^{\\#}>0\\}$ and set

$\Phi^{\prime}(\lambda,x)=\left\|\frac{\lambda((\varphi\circ\gamma)^{\prime}(t))}{|\gamma_{t}^{\prime}|}\right\|_{L^{\infty}(\bar{\mathbb{P}}^{\prime}_{x})},\quad
I^{\prime}(\varphi,x)=\inf_{\|\lambda\|_{*}=1}\Phi^{\prime}(\lambda,x)$

for $\mu$-a.e. $x\in U$. Let

$G=\Big{\\{}(\gamma,t):\ \gamma_{t}\in U,\
\frac{|(\varphi\circ\gamma)^{\prime}(t)|}{|\gamma_{t}^{\prime}|}\geq\frac{I^{\prime}(\varphi,\gamma_{t})}{2}\Big{\\}}\subset\overline{\operatorname{Fr}}(X).$

Then for $\mu$-a.e. $x\in U$ we have that $I^{\prime}(\varphi,x)>0$ if and
only if ${\rm
Dir}_{\varphi\ast}(\chi_{G}\bar{\mathbb{P}}^{\prime}_{x})\in\mathcal{P}(S^{n-1})$
is supported on a set spanning $\mathbb{R}^{n}$.

In particular if $I^{\prime}(\varphi,\cdot)>0$ $\mu$-a.e. on $U$, then
$\operatorname{spt}({\rm Dir}_{\varphi\ast}(\chi_{G}\bar{\mathbb{P}}_{x}))$
spans $\mathbb{R}^{n}$ for $\mu$-a.e. $x\in U$.

###### Proof.

Let $x\in U$ be such that $\bar{\mathbb{P}}^{\prime}_{x}$ exists and
$\Phi^{\prime}(\cdot,x)$ is a seminorm. Since $\varphi\circ\gamma$ is
differentiable for a.e. $t\in{\rm dom}(\gamma)$, by disintegration, this
occurs for a.e. $x\in U$. Let $V_{x}\subset\mathbb{R}^{n}$ be the subspace
spanned by $\operatorname{spt}({\rm
Dir}_{\varphi\ast}\chi_{G}\bar{\mathbb{P}}^{\prime}_{x})$. We have that
$(\varphi\circ\gamma)_{t}^{\prime}\in V_{x}$ for
$\bar{\mathbb{P}}^{\prime}_{x}$-a.e.
$(\gamma,t)\in\overline{\operatorname{Fr}}(X)\cap G$, since
$\operatorname{spt}({\rm
Dir}_{\varphi\ast}\chi_{G}\bar{\mathbb{P}}^{\prime}_{x})$ equals the
$\bar{\mathbb{P}}^{\prime}_{x}$-essential image of ${\rm Dir}_{\varphi}(G)$.
If $V_{x}\neq\mathbb{R}^{n}$, there exists $\lambda\in(\mathbb{R}^{n})^{*}$
such that $|\lambda|^{*}=1$ and $\lambda|_{V_{x}}=0$ implying that either
$|\lambda(\varphi\circ\gamma)^{\prime}(t)|=0$ or $(\gamma,t)\not\in G$ for
$\mathbb{P}^{\prime}_{x}$-a.e. $(\gamma,t)\in\overline{\operatorname{Fr}}(X)$.
In either case, we have
$|\lambda((\varphi\circ\gamma)^{\prime}(t))|\leq|(\varphi\circ\gamma)^{\prime}(t)|\leq\frac{I^{\prime}(\varphi,\gamma_{t})}{2}|\gamma_{t}^{\prime}|$
(or the derivative is not defined). Thus

$\left\|\frac{\lambda((\varphi\circ\gamma)^{\prime}(t))}{|\gamma_{t}^{\prime}|}\right\|_{L^{\infty}(\bar{\mathbb{P}}^{\prime}_{x})}\leq
I^{\prime}(\varphi,x)/2,$

which implies $I^{\prime}(\varphi,x)\leq I^{\prime}(\varphi,x)/2$, i.e.
$I^{\prime}(\varphi,x)=0$.

Conversely, if $V_{x}=\mathbb{R}^{n}$ then for each
$\lambda\in(\mathbb{R}^{n})^{*}\setminus\\{0\\}$,
$|\lambda((\varphi\circ\gamma)^{\prime}(t))|>0$ for a set of $(\gamma,t)\in
G$’s of positive $\bar{\mathbb{P}}^{\prime}_{x}$-measure, and thus
$\Phi^{\prime}(\lambda,x)>0$. Since $\lambda\mapsto\Phi^{\prime}(\lambda,x)$
is continuous it follows that $I^{\prime}(\varphi,x)>0$. ∎

In the next statement we define $\displaystyle
I(\varphi,x):=\inf_{\|\lambda\|_{\ast}=1}\Phi(\lambda,x)$.

######  Proposition 5.2.

We have that $I(\varphi,\cdot)>0$ $\mu$-a.e. on $U$ if and only if there
exists a Borel function $\bm{\delta}:U\to(0,\infty)$ and $\varphi$-independent
Alberti representations
$\mathcal{A}_{j}=\\{\mu_{\gamma}^{j},\mathbb{P}^{j}\\}$ of $\mu|_{U}$ such
that

(5.1)
$\displaystyle|(\varphi\circ\gamma)^{\prime}(t)|\geq\bm{\delta}_{\gamma_{t}}|\gamma_{t}^{\prime}|\quad
a.e.\ t\in\operatorname{dom}(\gamma)$

$\mathbb{P}^{j}$-a.e. $\gamma$, for each $j=1,\ldots,n$.

###### Proof.

Let $\mathbb{P}$ be the 1-plan in Proposition 4.5. We assume that
$I(\varphi,\cdot)>0$ $\mu$-a.e. on $U$, and proceed to prove the existence of
the Alberti representations. By Lemma 5.1, when

$G=\Big{\\{}(\gamma,t):\ \gamma_{t}\in U,\
\frac{|(\varphi\circ\gamma)^{\prime}(t)|}{|\gamma_{t}^{\prime}|}\geq\frac{I(\varphi,\gamma_{t})}{2}\Big{\\}}\subset\overline{\operatorname{Fr}}(X).$

we have that $\operatorname{spt}({\rm
Dir}_{\varphi\ast}(\chi_{G}\bar{\mathbb{P}}_{x}))=\mathbb{R}^{n}$ for
$\mu$-a.e. $x\in U$. We also have

(5.2)
$\displaystyle|(\varphi\circ\gamma)^{\prime}(t)|\geq\bm{\delta}_{\gamma_{t}}|\gamma_{t}^{\prime}|\
a.e.\ t\in\operatorname{dom}(\gamma)\ \textrm{ for every }(\gamma,t)\in G.$

where $\bm{\delta}=I(\varphi,\cdot)/2$.

Now, let $S$ be a countable dense set in $S^{n-1}$ and define

$\displaystyle\mathcal{C}=\\{((\xi_{1},\dots,\xi_{n}),\vartheta)\ :\ $
$\displaystyle\xi_{i}\in S,\vartheta\in\mathbb{Q}\cap(0,\pi/2)$ (5.3)
$\displaystyle\text{ and the cones }C(\xi_{i},\vartheta)\text{ are independent
for }i=1,\dots,n\\}.$

The set $\mathcal{C}$ is countable and we can enumerate its elements as
$\mathcal{C}=\\{((\xi_{1}^{j},\dots,\xi_{n}^{j}),\vartheta^{j}):j\in\mathbb{N}\\}$.
For each $j\in\mathbb{N}$, let $C_{j}$ be the set of $x\in X$ so that
$\bar{\mathbb{P}}_{x}(\\{(\gamma,t)\in G:(\phi\circ\gamma)^{\prime}(t)\in
C(\xi_{i}^{j},\vartheta^{j}))\\}$ for every $i=1,\dots,n$. The sets $C_{j}$
are Borel, by the measurability condition in the definition of the
disintegration; recall the definition at the beginning of Section 4.

Next, we will perform a measurable selection argument. This, and the similar
arguments below, could alternatively be accomplished by quoting an appropriate
measurable selection theorem, see [30, Theorem 18.1]. Instead, we will be
explicit with the rather simple choices that we need. Since
$\operatorname{spt}({\rm Dir}_{\varphi\ast}(\chi_{G}\bar{\mathbb{P}}_{x}))$
spans $\mathbb{R}^{n}$ for $\mu$-a.e. $x\in X$, we have that for $\mu$-a.e.
$x\in U$, there exists a $j\in\mathbb{N}$ so that $x\in C_{j}$. Now, let
$\tilde{C}_{j}=C_{j}\setminus\bigcup_{i<j}C_{i}$. Then,
$\mu(A\setminus\bigcup_{j\in\mathbb{N}}\tilde{C}_{j})=0$. Define the following
Borel maps $\bm{v}_{1},\ldots,\bm{v}_{n}:U\to S^{n-1}$ and
$\bm{\vartheta}:U\to(0,\pi/2)$: $\bm{v}_{i}(x)=\xi_{i}^{j}$ and
$\bm{\vartheta}(x)=\vartheta^{j}$ for $x\in\tilde{C}_{j}$. For $x\in
U\setminus\bigcup_{j\in\mathbb{N}}\tilde{C}_{j}$ one can define the maps
arbitrarily. Now, by construction the cone fields
$\bm{C}_{j}=C(\bm{v}_{j},\bm{\vartheta})$ are $\varphi$-independent and
$\bar{\mathbb{P}}_{x}({\rm Dir}_{\varphi}^{-1}\bm{C}_{j}(x)\cap G)>0$
$\mu$-a.e. $x\in U$ for each $j=1,\ldots,n$.

We will conclude the proof by using [8, Corollary 5.4]. By this result, for
every $j=1,\dots,n$ we can decompose $U$ as $U=A_{j}\cup S_{j}$, with
$\mu|_{A_{j}}$ admitting an Alberti representation $\mathcal{A}_{j}$ in the
cone direction $\bm{C}_{j}$ which satisfies (5.1), and $S_{j}$ satisfies the
following. For every $\gamma\in\operatorname{Fr}(X)$ which is in
$\varphi$-direction $\bm{C}_{j}$ and which satisfies (5.1) we have
$\mathcal{H}^{1}(\gamma\cap S_{j})=0$.

Denote

$G_{j}={\rm Dir}_{\varphi}^{-1}\bm{C}_{j}\cap G=\\{(\gamma,t)\in G:\ {\rm
Dir}_{\varphi}(\gamma,t)\in\bm{C}_{j}(\gamma_{t})\\}\subset\overline{\operatorname{Fr}}(X).$

We have $\bar{\mathbb{P}}_{x}(G_{j})>0$ for a.e. $x\in S_{j}$, since
$S_{j}\subset U$. If $\mu(S_{j})>0$, then from Proposition 4.5 and the
disintegration identity we get

(5.4)
$0<\int_{S_{j}}\bar{\mathbb{P}}_{x}(G_{j})\mathrm{d}\mathbb{P}^{\\#}=\int\int_{\gamma}\chi_{G_{j}}(\gamma,t)\chi_{S_{j}}(\gamma_{t})ds\mathrm{d}\mathbb{P}.$

Now, if $\gamma$ is any curve fragment, we claim that for a.e.
$t\in\operatorname{dom}(\gamma)$ we have $(\gamma,t)\not\in G_{j}$ or
$\gamma_{t}\not\in S_{j}$. First, let $K\subset\operatorname{dom}(\gamma)$ be
a compact set, where $(\gamma,t)\in G_{j}$ and $\gamma_{t}\in S_{j}$. Since
$(\gamma,t)\in G_{j}$ for all $t\in K$, we have that $\gamma|_{K}$ is in
$\varphi$-direction of $\bm{C}_{j}$ and it satisfies (5.1). Thus,
$\mathcal{H}^{1}(\gamma\cap S_{j})=0$ by the previous paragraph, and since
$\gamma$ is biLipschtiz we get $\lambda(K)=0$. Thus, for a.e.
$t\in\operatorname{dom}(\gamma)$ we have $(\gamma,t)\not\in G_{j}$ or
$\gamma_{t}\not\in S_{j}$. This contradicts (5.4), since the previous implies
that the integrand in the right hand side vanishes and thus

$\int\int_{\gamma}\chi_{G_{j}}(\gamma,t)\chi_{S_{j}}(\gamma_{t})ds\mathrm{d}\mathbb{P}=0.$

This concludes the first part of the proof.

Next assume that $\mathcal{A}_{j}=\\{\mu_{\gamma}^{j},\mathbb{P}^{j}\\}$,
$j=1,\ldots,n$ are $\varphi$-independent and satisfy (5.1) for some Borel
function $\bm{\delta}:U\to(0,\infty)$. Let $N=\\{x\in U:I(\varphi,x)=0\\}$. We
will show that $\mu(N)=0$.

By definition, there exists independent cone fields
$\bm{C}_{j}=C(\bm{v}_{j},\bm{\vartheta})$, so that
$(\varphi\circ\gamma)_{t}^{\prime}\in\bm{C}_{j,\gamma_{t}}$ for a.e.
$t\in\operatorname{dom}(\gamma)$ and $\mathbb{P}^{i}$-a.e.
$\gamma\in\operatorname{Fr}(X)$ for $j=1,\dots,n$. Since the cone fields
$\bm{C}_{j}$ are independent, there exists a positive function
$\bm{\varepsilon}$ so that for a.e. $x\in U$ we have

(5.5)
$\bm{\varepsilon}_{x}\|\lambda\|_{*}\leq\sum_{j=1}^{n}\inf_{w\in\bm{C}_{j,x}\cap
S^{n-1}}|\lambda(w)|$

for every $\lambda\in(\mathbb{R}^{n})^{*}$. Choose a Borel function
$\bm{\lambda}:N\to(\mathbb{R}^{n})^{*}$ so that $\|\bm{\lambda}\|_{*}=1$, and
$\Phi(\bm{\lambda},x)<\bm{\delta}\bm{\varepsilon}/(2n)$ for a.e. $x\in N$. The
existence of such a choice follows from the the separability of the unit
sphere in $(\mathbb{R}^{n})^{*}$ and by an argument similar to above.

We obtain from (5.5) that for a.e. $x$ there exists a $j(x)$ so that

(5.6) $\bm{\varepsilon}/n|w|\leq|\bm{\lambda}(w)|$

for all unit vectors $w\in\bm{C}_{j}$. Then, by scale invariance this holds
for all $w\in\bm{C}_{j}$. Now let $N_{j}=\\{x\in N:j(x)=j\\}$, and observe
that $\mu(N)\leq\sum_{j=1}^{n}\mu(N_{j})$. We claim that $\mu(N_{j})=0$ for
all $j=1,\dots,n$. First, by (iii) of Lemma 4.5 there exists a null set $E$ so
that $\Phi(\lambda,x)+\operatorname{LIP}(\varphi)\chi_{E}$ is an $\ast$-upper
gradient for $\varphi$. Thus for every $\gamma\in\operatorname{Fr}(X)$ we have
for a.e. $t\in\gamma^{-1}(N_{j}\setminus E)$

(5.7) $\displaystyle|\bm{\lambda}(\varphi\circ\gamma)^{\prime}(t)|\leq$
$\displaystyle(\Phi(\bm{\lambda},\gamma_{t})+\chi_{E}(\gamma_{t}))|\gamma_{t}^{\prime}|\leq\bm{\delta}^{-1}\bm{\delta}\bm{\varepsilon}/(2n)|(\varphi\circ\gamma)^{\prime}(t)|$
$\displaystyle=$
$\displaystyle\bm{\varepsilon}/(2n)|(\varphi\circ\gamma)^{\prime}(t)|.$

Since $\mu(E)=0$, for $\mathbb{P}^{j}$-a.e. $\gamma$ we have
$\mu_{\gamma,j}(E)=0$. However, for $\mathbb{P}^{j}$ a.e. $\gamma$ and a.e.
$t\in\operatorname{dom}(\gamma)$, we have
$(\varphi\circ\gamma)_{t}^{\prime}\in\bm{C}_{j}$ and thus by (5.6) we get for
a.e. $t\in\gamma^{-1}(N_{j})$ that

$|\bm{\lambda}(\varphi\circ\gamma)^{\prime}(t)|\geq\bm{\varepsilon}/n|(\varphi\circ\gamma)^{\prime}(t)|.$

This estimate can not hold simultaneously with (5.7). Thus, for a.e.
$t\in\gamma^{-1}(N_{j})$ we have $\gamma(t)\in E$ for $\mathbb{P}^{j}$-a.e.
$\gamma$. Therefore, $\mu_{\gamma}(N_{j})=\mu_{\gamma}(E)$ for a.e.
$\gamma\in\mathbb{P}^{j}$. since $\mathcal{A}_{j}$ is an Alberti
representation, we conclude $\mu(N_{j})=\mu(E)=0$. Since this holds for all
$j=1,\dots,n$, we conclude that $\mu(N)=0$.

∎

We now establish the existence of differentials and pointwise norms associated
to fragment-wise charts.

###### Proof of Theorem 1.2.

Suppose $(U,\varphi)$ is a fragment-wise chart, and define
$|\cdot|_{x}=\Phi(\cdot,x)$ for $\mu$-a.e. $x\in U$. By Proposition 5.2 we
have that $I(\varphi,\cdot)>0$ $\mu$-a.e. on $U$. By Proposition 4.6(i),
$|\cdot|_{x}$ is thus a norm on $(\mathbb{R}^{n})^{*}$ for $\mu$-a.e. $x\in
U$. Moreover $x\mapsto|\lambda|_{x}$ is a Borel representative of
$|D(\lambda\circ\varphi)|_{\ast}$ for each $\lambda\in(\mathbb{R}^{n})^{*}$.

Given $f\in\operatorname{LIP}(X)$, let $\Psi:(\mathbb{R}^{n+1})^{*}\times
X\to[0,\infty)$ be the map from Proposition 4.5 associated to
$\psi:=(\varphi,f)\in\operatorname{LIP}(X,\mathbb{R}^{n+1})$. In what follows,
given $\lambda\in(\mathbb{R}^{n})^{*}$ and $a\in\mathbb{R}$, we use the
notation $(\lambda,a)\in(\mathbb{R}^{n+1})^{*}$ for the linear map
$\mathbb{R}^{n+1}\ni(x,x_{n+1})\mapsto\lambda(x)+ax_{n+1}$. It is
straightforward to check, by applying Propositions 4.5 and 4.6 that

$\displaystyle\Psi((\lambda,0),x)=\Phi(\lambda,x),\quad\lambda\in(\mathbb{R}^{n})^{*}$

for $\mu$-a.e. $x\in U$. In particular, if $I(\varphi,x)>0$, the seminorm
$\Psi^{x}$ restricts to a norm on the hyperplane
$(\mathbb{R}^{n})^{*}\times\\{0\\}\subset(\mathbb{R}^{n+1})^{*}$. On the other
hand, since $\mu|_{V}$ does not admit $n+1$ $\psi$-independent Alberti
representations for any positive measure $V\subset U$, by Proposition 5.2 we
have that $I(\psi,x)=0$ $\mu$-a.e. $x\in U$. Thus $\Psi^{x}$ is degenerate for
$\mu$-a.e. $x\in U$. It follows that for $\mu$-a.e. $x\in U$, there exists a
unique $\mathrm{d}_{x}f\in(\mathbb{R}^{n})^{*}$ such that
$\Psi((\mathrm{d}_{x}f,-1),x)=0$, and the map $x\mapsto\mathrm{d}_{x}f$ is
Borel. To see measurability, observe that whenever $A$ is a closed set, we
have $\\{x\in U:\mathrm{d}_{x}f\in A\\}=\\{x\in U:\inf_{a\in
A}\Psi(a,-1),x)=0\\}$. From this together with the continuity of
$a\mapsto\Psi(a,-1)$, measurability follows. By Proposition 4.6(i) this
implies that
$\mathrm{d}_{x}f((\varphi\circ\gamma)_{t}^{\prime})-(f\circ\gamma)_{t}^{\prime}=0$
a.e. $t\in\gamma^{-1}(U)$ for $\operatorname{Mod}_{\infty}$-a.e. $\gamma$,
i.e. $\mathrm{d}f$ is a fragment-wise differential of $f$ with respect to
$(U,\varphi)$. The uniqueness follows, since if $\bm{\xi}_{x}$ was another
differential, then
$(\mathrm{d}_{x}f-\bm{\xi}_{x})(\varphi\circ\gamma)_{t}^{\prime}=0$ for a.e.
$t\in\gamma^{-1}(U)$ for $\operatorname{Mod}_{\infty}$-a.e. $\gamma$. Then,
$\Phi(\mathrm{d}_{x}f-\bm{\xi}_{x},x)=0$ for $\mu$-a.e. $x\in U$ by (i) of
Proposition 4.6. This, together with $I(\varphi,x)>0$ implies
$\mathrm{d}_{x}f=\bm{\xi}_{x}$ for a.e. $x\in U$. Moreover we have

$\displaystyle||\mathrm{d}_{x}f|_{x}-|Df|(x)||=|\Psi((\mathrm{d}_{x}f,0),x)-\Psi((\bar{0}_{(\mathbb{R}^{n})^{*}},1),x)|\leq\Psi((\mathrm{d}_{x}f,-1),x)=0$

$\mu$-a.e. $x\in U$, proving (1.4). This completes the proof of the theorem. ∎

###  5.2. Basic properties and existence

In the following two lemmas we collect some basic properties of fragment-wise
charts and differentials, such as the usual calculus rules for the
differential.

######  Lemma 5.3.

Suppose $(U,\varphi)$ is a fragment-wise chart. If $V\subset U$ is Borel and
$\mu(V)>0$, then $(V,\varphi)$ is a fragment-wise chart. Likewise if $N\subset
X$ $\mu$-null, then $(U\cup N,\varphi)$ is a fragment-wise chart.

In the proof, given an Alberti representation
$\mathcal{A}=\\{\mu_{\gamma},\mathbb{P}\\}$, its restriction to a (Borel) set
$E\subset X$ is defined as
$\mathcal{A}|_{E}\mathrel{\mathop{:}}=\\{\mu_{\gamma}|_{E},\mathbb{P}|_{\\{\gamma:\
\mu_{\gamma}(E)>0\\}}\\}$.

###### Proof.

Any Alberti representation of $\mu|_{U}$ is an Alberti representation of
$\mu|_{U\cup N}$ since $\mu|_{U}=\mu|_{U\cup N}$, implying the second claim.
To see the first, note that the maximality in Definition 1.1 holds for
$(V,\varphi)$ and, if $\mathcal{A}_{1},\ldots,\mathcal{A}_{n}$ are
$\varphi$-independent Alberti representations of $\mu|_{U}$, then the
restrictions $\mathcal{A}_{j}|_{V}$ form $n$ $\varphi$-independent Alberti
representations of $\mu|_{V}$, proving independence. ∎

######  Lemma 5.4.

Suppose $(U,\varphi)$ is a fragment-wise chart. Then

  * (1)

$\mathrm{d}(af+bg)=a\mathrm{d}f+b\mathrm{d}g$ for every
$f,g\in\operatorname{LIP}(X)$ and $a,b\in\mathbb{R}$ (linearity);

  * (2)

$\mathrm{d}(fg)=f\mathrm{d}g+g\mathrm{d}f$ $\mu$-a.e. on $U$ for every
$f,g\in\operatorname{LIP}(X)$ (product rule);

  * (3)

$\mathrm{d}(h\circ f)=(h^{\prime}\circ f)\mathrm{d}f$ for every
$f\in\operatorname{LIP}(X)$, $h\in C^{1}(\mathbb{R})$ (chain rule); and
finally

  * (4)

$\mathrm{d}f=\mathrm{d}g$ $\mu$-a.e. on $\\{f=g\\}\cap U$ for every
$f,g\in\operatorname{LIP}(X)$ (locality).

###### Proof.

Since the differential of a function is uniquely determined by (1.3), we
obtain the following fragment-wise identities (which are obtained by
differentiation of single variable Lipschitz functions in the parameter $t$)

$\displaystyle((af+bg)\circ\gamma)^{\prime}(t)$
$\displaystyle=a(f\circ\gamma)^{\prime}(t)+b(g\circ\gamma)^{\prime}(t)\ a.e.\
t\in\operatorname{dom}(\gamma)$ $\displaystyle(fg\circ\gamma)^{\prime}(t)$
$\displaystyle=f(\gamma_{t})(g\circ\gamma)^{\prime}(t)+g(\gamma_{t})(f\circ\gamma)^{\prime}(t)\
a.e.\ t\in\operatorname{dom}(\gamma)$ $\displaystyle(h\circ
f\circ\gamma)^{\prime}(t)$
$\displaystyle=h^{\prime}(f(\gamma_{t}))(f\circ\gamma)^{\prime}(t)\ a.e.\
t\in\operatorname{dom}(\gamma).$

These combined with uniqueness of the differential from Theorem 1.2 imply
(1)–(3). Similarly, since $((f-g)\circ\gamma)^{\prime}(t)=0$ for a.e.
$t\in\operatorname{dom}(\gamma)\cap\gamma^{-1}(\\{f=g\\})$, we have that
$\mathrm{d}(f-g)((\varphi\circ\gamma)^{\prime}(t))=0$ for a.e.
$t\in\gamma^{-1}(U\cap\\{f=g\\})$. Propositition 4.6(i) with
$\bm{\lambda}\mathrel{\mathop{:}}=\chi_{\\{f=g\\}}\mathrm{d}(f-g)$ now implies
(4).

∎

######  Theorem 5.5.

Let $X=(X,d,\mu)$ be a metric measure space and suppose $X$ can be covered up
to a $\mu$-null set by countably many Borel sets $X_{i}$ with finite Hausdorff
dimension. Then there is a countable collection of fragment-wise charts
$(U_{j},\varphi_{j})$ of dimension $n_{j}$ such that $X=\bigcup_{j}U_{j}$, and
moreover $n_{j}\leq\dim_{H}X_{i}$ if $\mu(U_{j}\cap X_{i})>0$.

###### Proof of Theorem 5.5.

Let $V\subset X$ be a Borel set with $\mu(V)>0$. There exists $i\in\mathbb{N}$
such that $\mu(V\cap X_{i})>0$. Consider the family $\mathcal{C}$ of pairs
$(U,\varphi)$ where $U\subset V\cap X_{i}$, $\mu(U)>0$,
$\varphi\in\operatorname{LIP}(X,\mathbb{R}^{n_{\varphi}})$ for some
$n_{\varphi}\in\mathbb{N}$ and $\mu|_{U}$ admits $n_{\varphi}$
$\varphi$-independent Alberti representations. By [9, Theorem 5.3] we have
$n_{\varphi}\leq\dim_{H}X_{i}<\infty$ for every $(U,\varphi)\in\mathcal{C}$,
and thus there exists a pair $(U,\varphi)\in\mathcal{C}$ attaining the maximum
$N:=\sup_{(U,\varphi)\in\mathcal{C}}n_{\varphi}\leq\dim_{H}X_{i}$. Thus
$\mu|_{U}$ admits $N$ $\varphi$-independent Alberti representations and, since
$N$ is maximal, $\mu|_{W}$ cannot admit more than $N$ independent Alberti
representations for any $W\subset U$. That is, $(U,\varphi)$ is a fragment-
wise chart.

We have shown that, given any $V\subset X$ with $\mu(V)>0$, there exists a
Borel set $U\subset V$ with $\mu(U)>0$, and a map
$\varphi\in\operatorname{LIP}(X,\mathbb{R}^{n})$ such that $(U,\varphi)$ is a
fragment-wise chart with dimension $n\leq\dim_{H}X_{i}$ whenever $\mu(U\cap
X_{i})>0$. By [33, Proposition 3.1.1] there exists a countable decomposition
$X=N\cup\bigcup_{j}U_{j}$ and maps
$\varphi_{j}\in\operatorname{LIP}(X,\mathbb{R}^{n_{j}})$ such that $\mu(N)=0$
and $(U_{j},\varphi_{j})$ are fragment-wise charts. This readily implies the
claim. ∎

###  5.3. Weak∗ continuity of differentials

Assume that $(U,\varphi)$ is a fragment-wise chart. Given $p\in[1,\infty]$, we
denote by $L^{p}(T^{*}(U,\varphi))$ the space of Borel functions
$\bm{\xi}:U\to(\mathbb{R}^{n})^{*}$ up to $\mu$-a.e. equivalence for which
$\|\bm{\xi}\|_{p}<\infty$, where

(5.8)
$\displaystyle\|\bm{\xi}\|_{p}\mathrel{\mathop{:}}=\|x\mapsto|\bm{\xi}_{x}|_{x}\|_{L^{p}(\mu|_{U})}.$

The space $L^{p}(T^{*}(U,\varphi))$ equipped with the norm (5.8) is a Banach
space. Denoting by $|\cdot|_{x}^{*}$ the dual norm of $|\cdot|_{x}$ on
$\mathbb{R}^{n}$, we similarly define the space $L^{p}(T(U,\varphi))$ of
($\mu$-a.e. equivalence classes of) Borel functions
$\bm{X}:U\to\mathbb{R}^{n}$ with finite norm

(5.9)
$\displaystyle\|\bm{X}\|_{p}\mathrel{\mathop{:}}=\|x\mapsto|\bm{X}(x)|^{*}_{x}\|_{L^{p}(\mu|_{U})}.$

Note that if $\\{W_{i}\\}$ is a countable Borel partition of $U$ then we have
an isometric identification

$\displaystyle\Gamma_{1}(T(U,\varphi))=\bigoplus_{l^{1}}\Gamma_{1}(T(W_{i},\varphi))\textrm{
and thus
}\Gamma_{1}(T(U,\varphi))^{*}=\bigoplus_{l^{\infty}}\Gamma_{1}(T(W_{i},\varphi))^{*},$

where $\bigoplus_{l^{\infty}}B_{i}$ consists of sequences
$\bar{v}=(v_{i})_{i}$ such that $v_{i}\in B_{i}$ and
$\|\bar{v}\|=\sup_{i}\|v_{i}\|_{B_{i}}<\infty$. Using a partition $W_{i}$ of
$U$ such that
$\delta_{i}|\cdot|_{Eucl}\leq|\cdot|_{x}^{*}\leq\delta_{i}^{-1}|\cdot|_{Eucl}$
$\mu$-a.e. on $W_{i}$ for some $\delta_{i}>0$, we obtain the isomorphism
$\Gamma_{1}(T(W_{i},\varphi))\simeq L^{1}(W_{i},\mathbb{R}^{n})$. By standard
arguments we deduce from this the following lemma. We omit the detailed proof.

######  Lemma 5.6.

We have the isometric identification
$L^{\infty}(T^{*}(U,\varphi))=L^{1}(T(U,\varphi))^{*}$.

Likewise a simple change of variables reduces the following lemma to the well-
known weak∗-continuity of derivatives of functions of one variable, see [40,
Chapter 7, Example 7.1.5] for the statement. For the sake of completeness, we
sketch a proof.

######  Lemma 5.7.

Suppose $(f_{j})\subset\operatorname{LIP}(X)$ converges pointwise to
$f\in\operatorname{LIP}(X)$ with $\sup_{j}\operatorname{LIP}(f_{j})<\infty$.
Let $\gamma\in\operatorname{Fr}(X)$, and $h\in
L^{1}(\operatorname{dom}(\gamma))$. Then

$\displaystyle\lim_{j\to\infty}\int_{\operatorname{dom}(\gamma)}h(t)(f_{j}\circ\gamma)^{\prime}(t)\mathrm{d}t=\int_{\operatorname{dom}(\gamma)}h(t)(f\circ\gamma)^{\prime}(t)\mathrm{d}t$

for all $h\in L^{1}(\nu)$.

###### Proof.

Let $\overline{\gamma}$ be the extension of $\gamma$ to a curve in
$\operatorname{sh}(X)$. Extend $f_{j}$ and $f$ to Lipschitz functions on
$\operatorname{sh}(X)$. When $h=\chi_{[a,b]}$ is a characteristic function of
an interval, with $a,b\in\operatorname{dom}(\gamma)$, we have

$\displaystyle\lim_{j\to\infty}\int_{\operatorname{dom}(\overline{\gamma})}h(t)(f_{j}\circ\overline{\gamma})^{\prime}(t)\mathrm{d}t$
$\displaystyle=\lim_{j\to\infty}f_{j}(\gamma(b))-f_{j}(\gamma(a))=f(\gamma(b))-f(\gamma(a))$
$\displaystyle=\int_{\operatorname{dom}(\overline{\gamma})}h(t)(f\circ\gamma)^{\prime}(t)\mathrm{d}t.$

If $h\in L^{1}(\operatorname{dom}(\gamma))$, then we can approximate $h$ by a
linear combination of functions of the previous type. This yields the claim. ∎

With these auxiliary results we are ready to prove the weak∗-continuity of
differentials with respect to $(U,\varphi)$. This statement, and its proof,
are quite similar to Lemma 2.14.

######  Theorem 5.8.

Let $(U,\varphi)$ be a fragment-wise chart. If
$(f_{j})\subset\operatorname{LIP}(X)$ converges pointwise to
$f\in\operatorname{LIP}(X)$ and $\sup_{j}\operatorname{LIP}(f_{j})<\infty$,
then $\mathrm{d}f_{j}\to\mathrm{d}f$ weak-$\ast$ in
$L^{\infty}(T^{*}(U,\varphi))=L^{1}(T(U,\varphi))^{*}$. In other words:

$\displaystyle\lim_{j\to\infty}\int_{U}\mathrm{d}f_{j}(\bm{X})\mathrm{d}\mu=\int_{U}\mathrm{d}f(\bm{X})\mathrm{d}\mu,\quad\bm{X}\in
L^{1}(T(U,\varphi)).$

###### Proof.

Since we have $\sup_{j}\operatorname{LIP}(f_{j})<\infty$, then
$\mathrm{d}f_{j}$ belong to some bounded subset of
$L^{\infty}(T^{*}(U,\varphi))$. By Banach-Alaoglu, since $L^{1}(T(U,\varphi)$
is separable, a subsequence $\mathrm{d}f_{j}$ converges weakly to some
$\bm{\xi}$. We will show that this subsequential weak-$\ast$ limit is the
differential of $f$, that is $\bm{\xi}=\mathrm{d}f$. Therefore, the weak limit
is unique, and thus the full sequence $\mathrm{d}f_{j}$ converges weakly to
$\bm{\xi}$. For the rest of the proof, suppose that we have passed to the
subsequence so that $\mathrm{d}f_{j}$ converges weak-$\ast$ to $\bm{\xi}$.

We first do a standard argument to convert the weak convergence to pointwise
almost everywhere convergence. This employs Mazur’s lemma applied for a
reflexive $L^{2}$-space. Let $U_{n}\subset U$ is an increasing sequence of
sets, $n\in\mathbb{N}$, with $\mu(U_{n})<\infty$, with
$\mu(U\setminus\bigcup_{n\in\mathbb{N}}U_{n})=0$. we have
$\mathrm{d}f_{j}|_{U_{n}}\in L^{2}(T^{*}(U_{n},\varphi))$, and
$\mathrm{d}f_{j}|_{U_{n}}\to\bm{\xi}\chi_{V}$ weakly in
$L^{2}(T^{*}(U_{n},\varphi))$. Therefore, by Mazur’s lemma, there exist values
$\alpha_{k,n,m}\geq 0$ and $\sum_{k=m}^{N_{n,m}}\alpha_{k,n,m}=1$ and convex
combinations
$\mathrm{d}g_{n,m}=\sum_{k=m}^{N_{n,m}}\alpha_{k,n,m}\mathrm{d}f_{k}$, which
converge strongly to $\bm{\xi}|_{U_{n}}$ in $L^{2}(T^{*}(U_{n},\varphi))$.
Moreover, by Lemma 5.4 we have that $\mathrm{d}g_{n}$ is the differential of
$g_{n,m}=\sum_{k=n}^{N_{n}}\alpha_{k,n}f_{k}$. Choose now $M_{n}$ so large
that

$\mu(\\{x\in U_{n}:|\bm{\xi}-\mathrm{d}g_{n,M_{n}}|_{x}\geq
2^{-n}\\})<2^{-n}.$

We let $g_{n}=g_{n,M_{n}}$. We have $\mathrm{d}g_{n}\to\bm{\xi}$ a.e. (by a
standard Borel-Cantelli argument), that $g_{n}\to f$ pointwise and
$\sup_{n}\operatorname{LIP}(g_{n})\leq\sup_{j}\operatorname{LIP}(f_{j})<\infty$.

Since $\mathrm{d}g_{n}\to\bm{\xi}$ a.e., there exists a null-set $E$ so that
for all $x\not\in E$ we have $\mathrm{d}g_{n,x}\to\bm{\xi}_{x}$. For
$\operatorname{Mod}_{\infty}$-a.e. $\gamma\in\operatorname{Fr}(X)$ we have
$(g_{n}\circ\gamma)^{\prime}(t)=\mathrm{d}g_{n}(\varphi\circ\gamma)^{\prime}_{t}$
and
$(f\circ\gamma)^{\prime}(t)\mathrm{d}t=\mathrm{d}f(\varphi\circ\gamma)^{\prime}_{t}$
for a.e. $t$. Let $\Gamma_{B}$ be the set of curve fragments where this fails.

Now, if
$\gamma\in\operatorname{Fr}(X)\setminus(\Gamma_{E}^{+}\cup\Gamma_{B})$, we
have by dominated convergence and Lemma 5.7 that for all $h\in L^{1}({\rm
dom(\gamma)})$

$\displaystyle\int_{\operatorname{dom}(\gamma)}h(t)\chi_{U}(\gamma_{t})\bm{\xi}(\varphi\circ\gamma)^{\prime}_{t}\mathrm{d}t$
$\displaystyle=\lim_{j\to\infty}\int_{\operatorname{dom}(\gamma)}h(t)\chi_{U}(\gamma_{t})\mathrm{d}g_{n}(\varphi\circ\gamma)^{\prime}_{t}\mathrm{d}t$
$\displaystyle=\lim_{j\to\infty}\int_{\operatorname{dom}(\gamma)}h(t)\chi_{U}(\gamma_{t})(g_{n}\circ\gamma)^{\prime}(t)\mathrm{d}t$
$\displaystyle=\int_{\operatorname{dom}(\gamma)}h(t)\chi_{U}(\gamma_{t})(f\circ\gamma)^{\prime}(t)\mathrm{d}t$
$\displaystyle=\int_{\operatorname{dom}(\gamma)}h(t)\chi_{U}(\gamma_{t})\mathrm{d}f(\varphi\circ\gamma)^{\prime}_{t}\mathrm{d}t$

Since this holds for all $h\in L^{1}(\operatorname{dom}(\gamma))$, we conclude
$\bm{\xi}(\varphi\circ\gamma)^{\prime}_{t}=\mathrm{d}f(\varphi\circ\gamma)^{\prime}_{t}$
for a.e. $t\in\gamma^{-1}(U)$. Thus, by $\bm{\xi}$ is a differential for $f$
and by uniqueness from Theorem 1.2 we have $\bm{\xi}=\mathrm{d}f$. This
concludes the proof. ∎

###  5.4. Differentiable structure

A _measurable $L^{\infty}$-bundle_ $\mathcal{T}$ over $X=(X,d,\mu)$ consists
of collections $(\\{U_{i},V_{i,x}\\})_{i\in I}$ and $(\\{\phi_{i,j,x}\\})$,
where $I$ is countable and

  1. (1)

$U_{i}\subset X$ are Borel for each $i\in I$, and cover $X$ up to a $\mu$-null
set;

  2. (2)

for any $i\in I$ and $\mu$-a.e. $x\in U_{i}$, $V_{i,x}=(V_{i},|\cdot|_{i,x})$
is a finite dimensional normed space so that $x\mapsto|v|_{i,x}$ is Borel for
any $v\in V_{i}$;

  3. (3)

for any $i,j\in I$ and $\mu$-a.e. $x\in U_{i}\cap U_{j}$, $\phi_{i,j,x}\colon
V_{i,x}\to V_{j,x}$ is an isometric bijective linear map satisfying the
_cocycle condition_ : for any $i,j,k\in I$ and $\mu$-a.e. $x\in U_{i}\cap
U_{j}\cap U_{k}$, we have $\phi_{j,k,x}\circ\phi_{i,j,x}=\phi_{i,k,x}$.

A _section_ of $\mathcal{T}$ is a collection $\omega=\\{\omega_{i}:U_{i}\to
V_{i}\\}$ of Borel measurable maps with
$\pi\circ\omega_{i}=\mathrm{id}_{U_{i}}$ and
$\omega_{j}=\phi_{i,j,x}\circ\omega_{i}$ $\mu$-a.e. for all $i,j\in I$. Here
$\pi:\mathcal{T}\to X$ is the projection map $(x,v)\mapsto x$. We remark that
the map $x\mapsto|\omega(x)|_{x}$, given by

(5.10) $\displaystyle|\omega(x)|_{x}\mathrel{\mathop{:}}=|\omega(x)|_{i,x}\
\mu-a.e.\ x\in U_{i},$

is well-defined up to $\mu$-negligible sets as a consequence of the cocycle
condition and the isometry of $\phi_{i,j,x}$. We denote by
$\Gamma(\mathcal{T})$ the vector space of sections of $\mathcal{T}$, and by
$\Gamma_{p}(\mathcal{T})$ the elements of $\Gamma(\mathcal{T})$ such that
$x\mapsto|\omega(x)|_{x}\in L^{p}(\mu)$.

A countable collection $(U_{i},\varphi_{i})$ of fragment-wise charts (of
dimension $n_{i}$) covering $X$ up to a null set gives rise to _measurable
tangent and cotangent bundles_ , denoted
$TX=(\\{U_{i},(\mathbb{R}^{n_{i}},|\cdot|_{x}^{*})\\})$ and
$T^{*}X=(\\{U_{i},((\mathbb{R}^{n_{i}})^{*},|\cdot|_{x})\\})$, respectively.
Indeed, (1) holds by assumption, while (2) follows by Theorem 1.2, and (3)
follows from the following lemma, which can be proven exactly as [22,
Proposition 5.1 and Lemma 5.2] .

######  Lemma 5.9.

Let $(U_{i},\varphi_{i})$, $i=1,2$, be fragment-wise charts of dimensions
$n_{i}$. Denote by $|\cdot|_{i,x}$ and $\mathrm{d}_{x}^{i}f$ the pointwise
norms and the differential of $f\in\operatorname{LIP}(X)$ with respect to
$(U_{i},\varphi_{i})$, respectively, given by Theorem 1.2, for $\mu$-a.e.
$x\in U_{i}$. If $\mu(U_{1}\cap U_{2})>0$, then $n_{1}=n_{2}=:n$ and there
exists a linear isometric bijection
$\Phi_{1,2,x}:((\mathbb{R}^{n})^{*},|\cdot|_{1,x})\to((\mathbb{R}^{n})^{*},|\cdot|_{2,x})$
such that $\displaystyle\Phi_{1,2,x}(\mathrm{d}_{x}^{1}f)=\mathrm{d}_{x}^{2}f$
$\mu$-a.e. $x\in U_{1}\cap U_{2}$ for any $f\in\operatorname{LIP}(X)$.
Moreover the collection $(\Phi_{i,j,x})$ satisfies the cocycle condition.

It also follows from Lemma 5.9 that, given $f\in\operatorname{LIP}(X)$, the
collection $\\{d_{U_{i},\varphi_{i}}f\\}$ is a section of $T^{*}X$, which we
denote $\mathrm{d}f\in\Gamma_{\infty}(T^{*}X)$ and call the _fragment-wise
differential_ of $f$.

######  Remark 5.10.

It is possible to equip $\Gamma_{p}(T^{*}X)$ and $\Gamma_{p}(TX)$ with the
structure of an $L^{p}$-normed module, cf. [22, Section 5.2] and [12]. It
follows from Lemma 5.6 that $\Gamma_{\infty}(T^{*}X)=\Gamma_{1}(TX)^{*}$ and
from Theorem 5.8 that the fragment-wise differential is w∗-continuous in the
following sense. Further, from the proof of Theorem 5.8 we get that if
$(f_{j})\subset\operatorname{LIP}(X)$ converges pointwise to
$f\in\operatorname{LIP}(X)$ and $\sup_{j}\operatorname{LIP}(f_{j})<\infty$,
then $\mathrm{d}f_{j}\stackrel{{\scriptstyle
w^{*}}}{{\rightharpoonup}}\mathrm{d}f$ in $\Gamma_{\infty}(T^{*}X)$. We will
not need this fact in this paper and leave the details to the interested
reader.

##  6\. Density of Directions

Recall the definition of the cone $C(w,\vartheta)$ from Section 2.5, and
notice that $v\notin C(w,\vartheta)$ if and only if $|w\cdot
v|\leq\cot\vartheta|\pi^{\perp}_{w}(v)|$. We will use this observation
repeatedly.

######  Lemma 6.1.

Let $w\in S^{n-1}$, $\vartheta\in(0,\pi/2)$,
$\varphi\in\operatorname{LIP}(X;\mathbb{R}^{n})$ and let $U\subset X$ be a
Borel set of positive $\mu$-measure. Suppose there is no 1-plan $\mathbb{P}$
such that $(\varphi\circ\gamma)_{t}^{\prime}\in C(w,\vartheta)$ for
$\bar{\mathbb{P}}$-a.e. $(\gamma,t)\in e^{-1}(U)$. Then there exists a compact
set $K\subset U$ of positive $\mu$-measure and, for any $\delta>0$, a sequence
$(f_{j})\subset\operatorname{LIP}(X)$ with $\operatorname{LIP}(f_{j})\leq
1+\cot\vartheta\operatorname{LIP}(\varphi)$ satisfying the following
properties.

  * (i)

$f_{j}\to w\cdot\varphi$ uniformly in $K$;

  * (ii)

for every $j$ there exists $r_{j}>0$ so that

(6.1) $\displaystyle|f_{j}(x)-f_{j}(y)|\leq\delta
d(x,y)+\cot\vartheta|\pi_{w}^{\perp}(\varphi(x)-\varphi(y))|$

for all $x,y\in K$ with $d(x,y)<r_{j}$.

###### Proof.

Let $\mathbb{P}$ be a 1-plan and $D\subset X$ a Borel set as in 4.5, i.e.

(6.2)
$|D(\lambda\cdot\varphi)|(x)=\chi_{D}(x)\left\|\frac{\lambda\cdot(\varphi\circ\gamma)_{t}^{\prime}}{|\gamma_{t}^{\prime}|}\right\|_{L^{\infty}(\bar{\mathbb{P}}_{x})},\quad\lambda\in\mathbb{R}^{n},$

for $\mu$-a.e. $x\in U$. By assumption and the inner regularity of $\mu$ there
exists a compact set $K\subset U$ of positive measure such that

(6.3)
$\displaystyle|w\cdot(\varphi\circ\gamma)_{t}^{\prime}|\leq\cot\vartheta|\pi^{\perp}_{w}((\varphi\circ\gamma)_{t}^{\prime})|\quad\bar{\mathbb{P}}_{x}-\textrm{a.e.
}(\gamma,t)$

for every $x\in K$. Consider the metric $\tilde{d}(x,y)=\delta
d(x,y)+\cot\vartheta|\pi^{\perp}_{w}(\varphi(x)-\varphi(y))|$ on $X$ for
$\varepsilon>0$. This metric is bi-Lipschitz to $d$ and thus the classes of
1-plans and curve fragments remain unchanged and in particular the
representation above remains valid for possibly a different plan. Denote the
minimal $\ast$-upper gradients and metric differentials with respect to the
metric $\bar{d}$ by adding a subscript. By averaging the two plans, and
adjusting the notation, we can assume that we have in addition to (6.2) the
following facts. First, we have

$\displaystyle|\gamma_{t}^{\prime}|_{\tilde{d}}=\delta|\gamma_{t}^{\prime}|+\cot\vartheta|\pi^{\perp}_{w}((\varphi\circ\gamma)_{t}^{\prime})|\quad
a.e.\ t\in\operatorname{dom}(\gamma)$

for any $\gamma$. For $\mu$-a.e. $x\in K$, (6.3) and the choice of plan yields

$\displaystyle|D(w\cdot\varphi)|_{\tilde{d}}(x)$
$\displaystyle=\left\|\frac{w\cdot(\varphi\circ\gamma)_{t}^{\prime}}{|\gamma_{t}^{\prime}|_{\tilde{d}}}\right\|_{L^{\infty}(\bar{\mathbb{P}}_{x})}=\left\|\frac{w\cdot(\varphi\circ\gamma)_{t}^{\prime}}{\delta|\gamma_{t}^{\prime}|+\cot\vartheta|\pi^{\perp}_{w}((\varphi\circ\gamma)_{t}^{\prime})|}\right\|_{L^{\infty}(\bar{\mathbb{P}}_{x})}$
$\displaystyle\leq\left\|\frac{|w\cdot(\varphi\circ\gamma)_{t}^{\prime}|}{\delta|\gamma_{t}^{\prime}|+|w\cdot(\varphi\circ\gamma)_{t}^{\prime}|}\right\|_{L^{\infty}(\bar{\mathbb{P}}_{x})}<1.$

By Theorem 1.6 there is a uniformly Lipschitz sequence $(f_{j})$ converging
pointwise to $w\cdot\varphi$ such that
$\operatorname{Lip}_{a,\tilde{d}}f_{j}\to|D(w\cdot\varphi)|_{\tilde{d}}$. Up
to passing to a smaller set, and a subsequence, (with positive measure) we
have that there exist $r_{j}\downarrow 0$ so that

$|f_{j}(y)-f_{j}(z)|\leq\delta
d(y,z)+\cot\vartheta|\pi^{\perp}_{w}(\varphi(y)-\varphi(z))|,\quad y,z\in K,\
d(y,z)<r_{j},$

as claimed. ∎

######  Remark 6.2.

Functions satisfying an estimate of the type (6.1) appear in [8] and [35]
where they were used to construct non-differentiable functions.

To derive a contradiction from (6.1) we use the following lemma analogous to
[8, Lemma 9.1].

######  Lemma 6.3.

Let $\varphi:X\to\mathbb{R}^{n}$ be Lipschitz and $U\subset X$ Borel so that
there exists a function $\bm{\delta}:X\to(0,\infty)$ with $\bm{\delta}>0$
$\mu$-a.e. in $U$ and such that $\mu|_{U}$ admits $n$ $\varphi$-independent
$\xi$-separated Alberti representations
$\mathcal{A}_{j}=(\mu_{j},\mathbb{P}_{j})$ with
$|(\varphi\circ\gamma)_{t}^{\prime}|\geq\bm{\delta}_{\gamma_{t}}\gamma_{t}^{\prime}|$
for $\mathbb{P}_{j}$-a.e. $\gamma$ and a.e. $t\in\operatorname{dom}(\gamma)$,
with $j=1,\dots,n$. Suppose moreover that $f\in\operatorname{LIP}(X)$ admits a
fragment-wise differential $\mathrm{d}f$ with respect to $(U,\varphi)$. Then,
given $w\in S^{n-1}$ and any set $S\subset U$, for $\mu$-a.e. $x\in S$ there
exists a sequence $(x_{k})\subset S$ converging to $x$ such that