# The Hopf Decomposition

Nachi Avraham-Re’em  Department of Mathematics, Chalmers University, Göteborg,
Sweden<EMAIL_ADDRESS>$\textstyle{or}$<EMAIL_ADDRESS>and
George Peterzil  Einstein Institute of Mathematics, The Hebrew University,
Israel<EMAIL_ADDRESS>In memory of George Abraham
(1934–1984)

###### Abstract.

Let $G$ be a locally compact second countable group. We present a
comprehensive treatment of the classical Hopf Decomposition (also known as the
Conservative–Dissipative Decomposition) for general nonsingular $G$-spaces,
and provide several fundamentally different characterizations of it.
Subsequently, we establish a complete structure theorem for totally
dissipative nonsingular $G$-spaces through the construction of _Krengel
$G$-spaces_, extending Krengel’s structure theorem on flows.

###### Key words and phrases:

nonsingular actions, conservative dissipative Decomposition, Hopf
decomposition, totally dissipative, completely dissipative, Neveu
decomposition

###### 2020 Mathematics Subject Classification:

Primary 37A20, 22D40; Secondary 37A15, 37A40

The research was supported by ISF grant No. 1180/22 and by the Knut and Alice
Wallenberg Foundation (KAW 2021.0258).

###### Contents

  1. 1 Introduction and Main Results
    1. 1.1 Roadmap
  2. 2 Preliminaries
  3. 3 The Hopf Decomposition: A General Form
  4. 4 Further Properties
  5. 5 Smooth Orbit Equivalence Relations
    1. 5.1 Preliminaries: Smooth Equivalence Relations
    2. 5.2 The Structure of Smooth Borel $G$-Spaces
  6. 6 Borel Description of Total Dissipativity
  7. 7 The Structure Theorem for Totally Dissipative Actions
    1. 7.1 Free Actions: Translation $G$-Spaces
    2. 7.2 Ergodic Actions: Compactly Fibered Coset $G$-Spaces
    3. 7.3 General Actions: Krengel $G$-Spaces
  8. 8 Kaimanovich-Type Characterization
    1. 8.1 Maharam Extensions
  9. 9 The Positive–Null Decomposition
  10. 10 Concluding Remarks and Further Questions
  11. A Measures on Homogeneous Spaces

##  1\. Introduction and Main Results

Let $G$ be a locally compact second countable (lcsc) group. Our main object of
study is nonsingular $G$-space: a standard measure space $\left(X,\mu\right)$,
finite or $\sigma$-finite, such that $G$ acts on $X$ in a Borel fashion and
$\mu$ is quasi-invariant, namely the action preserves the ideal of $\mu$-null
sets.

In the heart of this work lie two complementary properties of nonsingular
$G$-spaces: being conservative or being dissipative. Those dynamical
properties play a crucial role in the study of nonsingular $G$-spaces, but
their formulation in general tends to be complicated. Many authors, to say the
least, have dealt with various formulations, conditions and characterizations
of these properties; from the classical works of Poincaré, Hopf, Halmos,
Maharam and Krengel for $G=\mathbb{Z}$ (see the bibliographical notes in [24,
§1.3] and the references in [1, §1.1]) to the works of Schmidt [33, §1] and
Kaimanovich [21] for countable $G$. In fact, it was observed by Krengel that
when $G$ admits a lattice, there is a reduction, in a certain sense, to the
countable case. In recent years many studies in Ergodic Theory, Homogeneous
Dynamics, Probability, Operator Algebras and related fields have dealt with
more general acting groups, and this emphasized the necessity for developing a
thorough and detailed theory encompassing conservativity and dissipativity for
nonsingular $G$-spaces with a general lcsc acting group $G$.

The idea that one can formulate the notion of conservativity and dissipativity
for a general lcsc acting group $G$, as well as decomposing every nonsingular
$G$-space into its ’conservative part’ and its ’dissipative part’, is natural
and was used by many, and should be considered as part of the folklore.
Indeed, one may define conservativity by exploiting the following natural
approach to recurrence: starting from every point in every positive measure
set, outside every compact set in $G$ there is a group element, not belonging
to the stabilizer of the point, that returns the point into the set (see e.g.
[2, Proposition 4.29], [9, Definition 2.1], [10, §6.3]). Another natural
approach to recurrence that can be found in the literature is that, in the
previous situation, there can be found infinite-volume amount of group
elements that return the point into the set (see e.g. [30, §4.6 p.40], [31,
§1E]). While these approaches are useful and natural for a general lcsc group
$G$, and clearly coincide when it comes to a countable acting group, the
general theory which is known in countable groups for long time seems to be
unknown when it comes to lcsc groups.

Recently, two treatments of the theory were presented: one by Arano, Isono &
Marrakchi [2, §4.10], where a general form of the Hopf Decomposition was
presented as well as many of its properties (see also [6, §2], continuing
their approach), and another by Blayac, Canary, Zhu & Zimmer [7, §2.11], [8,
Appendix A], where they have treated measure preserving actions of unimodular
groups. The Hopf Decomposition is not the central focus of these studies, and
many of the fundamentals of the theory, as are fully known for countable
acting groups, are not covered in these treatments.

Our main goal is to provide a thorough treatment of the theory in full
generality. Thus, we establish the Hopf Decomposition and characterize it from
an analytic perspective, a dynamical perspective as well as a descriptive set-
theoretic perspective. In particular, we will demonstrate that the two
aforementioned approaches to recurrence yield the same decomposition. It is
noteworthy that each of these approaches will be proving itself useful in
various applications we will discuss.

For the rest of this introductory section we formulate our main results.

####  1.0.1. The Hopf Decomposition

Let $G$ be an lcsc group with a left Haar measure $\lambda$ and
$\left(X,\mu\right)$ be a nonsingular $G$-space. Our aim is to establish the
Hopf Decomposition of $\left(X,\mu\right)$ that will be denoted generally by

$X=\mathcal{C}\sqcup\mathcal{D}.$

Denote by $L_{+}^{1}\left(X,\mu\right)$ the class of (strictly) positive-
valued functions of $L^{1}\left(X,\mu\right)$, and for $f\in
L_{+}^{1}\left(X,\mu\right)$ look at the average transform

$S_{f}^{G}\left(x\right):=\int_{G}\frac{d\mu\circ
g}{d\mu}\left(x\right)f\left(g.x\right)d\lambda\left(g\right),\quad x\in X.$

Then every such $f$ yields a Borel decomposition of $X$ into

$\mathcal{C}_{f}:=\left\\{x\in
X:S_{f}^{G}\left(x\right)=+\infty\right\\}\text{ and
}\mathcal{D}_{f}:=X\backslash\mathcal{C}_{f}.$

The upcoming Theorem A is considered part of the folklore of ergodic theory
(see the formulation in [2, Theorem 4.30]).

######  Theorem A (Hopf Decomposition).

Let $G$ be an lcsc group and $\left(X,\mu\right)$ a nonsingular $G$-space.
There exists an essentially unique decomposition

$X=\mathcal{C}\sqcup\mathcal{D}$

into $G$-invariant Borel sets $\mathcal{C}$ and $\mathcal{D}$, depending only
on the measure class of $\mu$, with the property that

$\mu\left(\mathcal{C}\triangle\mathcal{C}_{f}\right)=0\text{ and
}\mu\left(\mathcal{D}\triangle\mathcal{D}_{f}\right)=0\text{ for every }f\in
L_{+}^{1}\left(X,\mu\right).$

In particular, $\mathcal{C}_{f}$ and $\mathcal{D}_{f}$ are independent of
$f\in L_{+}^{1}\left(X,\mu\right)$ modulo $\mu$.

The dependence only on the measure class is in the sense that the
$G$-invariant decomposition of equivalent measures is the same modulo $\mu$.
However, the average transform $S_{f}^{G}$ defining the decomposition does
depend on the measure itself.

Following the standard terminology (see the introductory part of [11]), a
nonsingular $G$-space $\left(X,\mu\right)$ with the Hopf Decomposition
$X=\mathcal{C}\sqcup\mathcal{D}$ is called:

$\bullet$ conservative:

if $\mu\left(\mathcal{D}\right)=0$;

$\bullet$ dissipative:

if $\mu\left(\mathcal{D}\right)>0$; and

$\bullet$ totally dissipative (or completely dissipative):

if $\mu\left(\mathcal{C}\right)=0$.

The following useful property will be deduced from Theorem A:

######  Proposition 1.1.

Let $G$ be an lcsc group and $\left(X,\mu\right)$ a nonsingular $G$-space. On
the dissipative part $\mathcal{D}$ of $\left(X,\mu\right)$ there is an
invariant measure equivalent to $\mu\mid_{\mathcal{D}}$.

Next we characterize the Hopf Decomposition from the point of view of
recurrence. We first make a useful notation that will be used extensively. For
a Borel set $A\subset X$ and $x\in X$ denote

$R_{A}\left(x\right):=\left\\{g\in G:x\in g.A\right\\}.$

A set $A$ is being recurrent when $R_{A}\left(x\right)$ is large in $G$ for
every $x\in A$. As we mentioned previously, there are two natural approaches
to formulate this:

######  Definition 1.2.

Let $G$ be an lcsc group and $\left(X,\mu\right)$ a nonsingular $G$-space. A
Borel set $A\subset X$ is called:

$\bullet$ Transient:

if $\lambda\left(R_{A}\left(x\right)\right)<+\infty$ for $\mu$-a.e. $x\in
X$.111In [30, §4.6 p.40] and [31, §1E] this is called a wandering set. Since
this does not fully align with the concept of a wandering set for countable
groups, we prefer calling it differently.

$\bullet$ Recurrent:

if $\lambda\left(R_{A}\left(x\right)\right)=+\infty$ for $\mu$-a.e. $x\in A$.

$\bullet$ Strongly transient:

if $R_{A}\left(x\right)$ is relatively compact for $\mu$-a.e. $x\in X$.

$\bullet$ Strongly recurrent:

if $R_{A}\left(x\right)$ is not relatively compact for $\mu$-a.e. $x\in A$.

One should carefully note that (strong) transience and (strong) recurrence are
not complementary; the former is defined over all of $X$ while the latter
inside the reference set $A$. Every subset of a (strongly) transient set is
(strongly) transient, thus the class of (strongly) transient set is
_hereditary_.

######  Theorem B (The Recurrence Theorem).

Let $G$ be an lcsc group and $\left(X,\mu\right)$ a nonsingular $G$-space. The
Hopf Decomposition $X=\mathcal{C}\sqcup\mathcal{D}$ is determined by:

  1. (1)

Every subset of $\mathcal{C}$ is (strongly) recurrent.

  2. (2)

Every $\mu$-positive subset of $\mathcal{D}$, if exists at all, contains a
$\mu$-positive (strongly) transient subset.

Note: Theorem B should be interpreted to mean that any of the four
combinations of recurrent/strongly recurrent and transient/strongly transient
characterizes the Hopf Decomposition.

Next we discuss a significant strengthening of the condition for
conservativity as in the Hopf Decomposition A in unimodular groups. This
characterization was discovered by Kaimanovich in [21, Theorem 29] when $G$ is
countable and $\mu$ is a probability measure, asserting that if a nonsingular
probability $G$-space $\left(X,\mu\right)$ is conservative, then for
$\mu$-a.e. $x\in X$ not only that $\sum_{g\in
G}\nabla_{g}\left(x\right)=+\infty$, but further there is $r_{x}>0$ such that

$\\#\left\\{g\in G:\nabla_{g}\left(x\right)\geq r_{x}\right\\}=+\infty.$

We establish this for all lcsc unimodular groups and all standard measure
spaces:

######  Theorem C (Kaimanovich-Type Characterization).

Let $G$ be an lcsc _unimodular_ group and $\left(X,\mu\right)$ a nonsingular
$G$-space. Then $\left(X,\mu\right)$ is conservative iff there exists $f\in
L_{+}^{1}\left(X,\mu\right)$ (equivalently, for every $f\in
L_{+}^{1}\left(X,\mu\right)$) with the following property:

> For $\mu$-a.e. $x\in X$ there exists $r=r_{\left(f,x\right)}>0$ such that
>
> $\lambda\left(g\in G:\nabla_{g}\left(x\right)f\left(g.x\right)\geq
> r\right)=+\infty.$

Kaimanovich’s theorem, when $G$ is countable and $\left(X,\mu\right)$ is a
nonsingular probability $G$-space, is recovered from Theorem C by taking the
function $f=1$. While Theorem C is presented in a straightforward manner, its
proof relies on the upcoming Theorems D and F which are of a different nature.

####  1.0.2. The Borel perspective

In the following we turn into a descriptive set-theoretic point of view. When
$G$ is countable and $\left(X,\mu\right)$ is an essentially free nonsingular
$G$-space, the dissipative part can be presented as

$\mathcal{D}=\bigcup\nolimits_{g\in G}g.W_{0},$

where $W_{0}$ is a wandering set, which is a strong form of transient set:
$\left\\{g.W_{0}:g\in G\right\\}$ are pairwise disjoint (see [24, §1.3], [1,
§1.1 & §1.6]; see also [35, 40]).222According to Aaronson’s approach in [1,
§1.6], totally dissipative nonsingular $G$-spaces are measurable unions of
wandering sets, hence they are essentially free (see [1, Proposition 1.6.1]).
Our approach uses transient sets instead, hence allows non-essentially free
nonsingular $G$-spaces to be totally dissipative. The example one should bear
in mind is the measure preserving $G$-space $\left(G/F,\\#\right)$, where $F$
is a finite subgroup of $G$ and $\\#$ is the counting measure. It admits no
wandering sets unless $F$ is trivial, but every singleton $gF\in G/F$ forms a
transient set. This subtle point was discussed extensively by Kaimanovich [21,
Remarks 19–22].

When it comes to continuous groups, of course there cannot be any
$\mu$-positive wandering set. Nevertheless, for $G=\mathbb{R}$ Krengel made
essential use of wandering sets of zero measure which are Borel transversals,
namely a Borel set that meets the orbit of every point exactly once. He then
showed that if $\left(X,\mu\right)$ is a totally dissipative nonsingular
$\mathbb{R}$-space, then it admits a Borel transversal $W_{0}$ on which one
can construct a measure $\nu_{0}$ in such a way that $\left(X,\mu\right)$
would be isomorphic (as nonsingular $G$-spaces, see Section 2 below) to the
nonsingular $G$-space

(1.0.1)
$\left(W_{0}\times\mathbb{R},\nu_{0}\otimes\mathrm{Lebesgue}\right)\text{ with
the action }g.\left(w,h\right)=\left(w,gh\right).$

Let us put this generally. The orbit equivalence relation of a nonsingular
$G$-space $\left(X,\mu\right)$ is defined to be

$E_{G}^{X}:=\left\\{\left(x,g.x\right)\in X\times X:x\in X,g\in G\right\\}.$

We say that $E_{G}^{X}$ is smooth if it admits a Borel transversal, i.e. a
Borel set $W_{0}\subset X$ that meets every orbit exactly once. This property
has a few useful characterizations that will be mentioned in Theorem 5.2
below. Nonsingular $G$-spaces whose orbit equivalence relation is smooth were
long been studied, since the celebrated works of Glimm and Effros [17, 12]
(see also [14, §5] and [41, §2.1]).

Smoothness of the orbit equivalence relation will be shown in the upcoming
theorem to be a necessary condition for dissipativity, but it is not
sufficient: compactness of the stabilizers is also required to fully capture
dissipativity. In some sources this was taken as the definition of
dissipativity (see [9, Definition 2.1(i)]), and in others it was observed that
it is sufficient for dissipativity (see [7, §2.11]).

######  Theorem D.

Let $G$ be an lcsc group and $\left(X,\mu\right)$ a nonsingular $G$-space.
TFAE:

  1. (1)

$\left(X,\mu\right)$ is totally dissipative.

  2. (2)

$E_{G}^{X}$ is smooth and all stabilizers are compact modulo $\mu$.

Since $\mathbb{Z}^{d}$ and $\mathbb{R}^{d}$ have no nontrivial compact
subgroups, we obtain:

######  Corollary 1.3 (Krengel–Rosinski).

When $A=\mathbb{Z}^{d}$ or $\mathbb{R}^{d}$, a nonsingular $A$-space
$\left(X,\mu\right)$ is totally dissipative iff $E_{A}^{X}$ is essentially
smooth, and in this case the action is essentially free.

Another corollary of Theorem D follows from Becker’s Glimm Effros–Dichotomy in
orbit equivalence relations [16, Theorem 6.5.2]. Let $E_{0}$ be the tail
equivalence relation on $\left\\{0,1\right\\}^{\mathbb{N}}$. Denote by
$\cdot\sqsubset_{\mathrm{B}}\cdot$ the relation of Borel reduction of
equivalence relations. (See [16, §6] for background and details).

######  Corollary 1.4.

Let $G$ be an lcsc group and $\left(X,\mu\right)$ a nonsingular $G$-space such
that essentially all stabilizers are compact (e.g. free actions). Then
$\left(X,\mu\right)$ is conservative iff
$E_{0}\sqsubset_{\mathrm{B}}E_{G}^{C}$ for every $\mu$-positive $G$-invariant
set $C\subset X$.

####  1.0.3. The structure of totally dissipative nonsingular $G$-spaces

The aforementioned theorem of Krengel about totally dissipative nonsingular
flows provides the general structure of such spaces: up to an isomorphism, all
totally dissipative nonsingular flows are obtained from some standard measure
space $\left(W_{0},\nu_{0}\right)$ via the later Construction (1.0.1). Our aim
is to establish a structure theorem for totally dissipative nonsingular
$G$-spaces for a general lcsc group $G$. As we shall see, when $G$ admits
nontrivial compact subgroups the general structure may be very different.

The prototype of a totally dissipative nonsingular $G$-space is
$\left(G,\lambda\right)$ itself, which becomes a measure preserving $G$-space
with the action by (left) translations. This extends to two substantially
different constructions: _translation $G$-spaces_ of the form
$\left(W_{0}\times G,\nu_{0}\otimes\lambda\right)$ (see Construction 7.1), and
_compactly fibered coset $G$-spaces_ of the form $\left(G/K,\kappa\right)$ for
a compact subgroup $K$ of $G$ (see Construction 7.3). Those constructions turn
out to be prime instances of dissipativity; the former is free and generally
non-ergodic, and the latter is typically non-free and transitive.

For free actions, Krengel’s structure theorem can be directly generalized:

######  Theorem E (Following Krengel–Rosinski).

Let $G$ be an lcsc group. An _essentially free_ nonsingular $G$-space is
totally dissipative iff it is isomorphic to a translation $G$-space
(Construction 7.1).

For ergodic actions, using Theorem D we have the following structure theorem
that is probably due to Glimm–Effros (cf. [41, Proposition 2.1.10], [14,
Definition 5.2]). A different treatment can be found in [2, Theorem 4.30].

######  Theorem F (Following Glimm–Effros).

Let $G$ be an lcsc group. An _ergodic_ nonsingular $G$-space is totally
dissipative iff it is isomorphic to a compactly fibered coset $G$-space
(Construction 7.3).

It is a well-known fact that when $G$ is countable, a dissipative nonsingular
$G$-space is never ergodic provided the measure is non-atomic. This fails in
general, since coset $G$-spaces (which can be non-atomic when $G$ is
continuous) are transitive. Recall that a nonsingular $G$-space is called
properly ergodic if it is ergodic but is not essentially transitive, i.e.
$\mu$ is not supported on a single orbit. A direct corollary of Theorem F is:

######  Corollary 1.5.

Dissipative nonsingular $G$-spaces are never properly ergodic.

Krengel $G$-spaces
$\scriptstyle{+\text{free}}$$\scriptstyle{+\text{transitive}}$ Translation
$G$-spaces
$\scriptstyle{+\text{transitive}}$$\textstyle{\boxed{\parbox{116.65646pt}{\centering{Totally
dissipative}\\\ {nonsingular $G$-spaces}\@add@centering}}}$ Compactly fibered
coset $G$-spaces $\scriptstyle{+\text{free}}$
$G\curvearrowright\left(G,\lambda\right)$ Figure 1. An arrow
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{+\text{P}}$$\textstyle{B}$
presents the passage from class $A$ of totally dissipative $G$-spaces to its
subclass $B$ that satisfies the restrictive property P.

In the next we unify Theorems E and F into a general structure theorem. To
this end we make use of a construction that we call Krengel $G$-spaces, which
simultaneously generalizes translation $G$-spaces and compactly fibered coset
$G$-spaces, as illustrated in Figure 1.

Recall that by Theorem D, one may associate with a totally dissipative
nonsingular $G$-space $\left(X,\mu\right)$ a standard measure space
$\left(W_{0},\nu_{0}\right)$ and a map
$\psi_{0}:W_{0}\to\mathbf{K}_{\mathrm{gr}}\left(G\right)$, where
$\mathbf{K}_{\mathrm{gr}}\left(G\right)$ is the space of compact subgroups of
$G$. Indeed, $W_{0}$ would be a Borel transversal, $\nu_{0}$ the image measure
of $\mu$ via a Borel selector $X\to W_{0}$, and $\psi_{0}$ the usual
stabilizer map. The construction of Krengel $G$-space is designed to recover
$\left(X,\mu\right)$ out of $\left(W_{0},\nu_{0}\right)$ and $\psi_{0}$ in a
canonical way up to an isomorphism.

The general construction will be introduced in detail as Construction 7.6 in
Section 7.3, where a measure preserving isomorphism will be constructed, and
here we describe a softer version, where a nonsingular isomorphism is
constructed. The input of the construction, called Krengel $G$-datum, consists
of two ingredients:

  * •

A standard measure space $\left(W_{0},\nu_{0}\right)$ (possibly atomic);

  * •

An arbitrary Borel function
$\psi_{0}:W_{0}\to\mathbf{K}_{\mathrm{gr}}\left(G\right)$.333In a sense,
$\psi_{0}$ is a random compact subgroup, but the measure $\nu_{0}$ may be
infinite.

With it we associate a measure preserving $G$-space
$\left(X_{0},\mu_{0}\right)$, which is called Krengel $G$-space, by the factor
map

$\left(W_{0}\times
G,\nu_{0}\otimes\lambda\right)\to\left(X_{0},\mu_{0}\right)\text{ via
}\left(w,g\right)\mapsto\left(w,g\psi_{0}\left(w\right)\right),$

where the action on $W_{0}\times G$ is the aforementioned (1.0.1). Thus, the
points of $X_{0}$ are pairs consist of a point of $W_{0}$ and a coset of the
compact subgroup associated with it by $\psi_{0}$. Then
$\left(X_{0},\mu_{0}\right)$ has the following basic properties:

  1. (1)

$W_{0}\cong\left\\{\left(w,\psi_{0}\left(w\right)\right):w\in W_{0}\right\\}$
is a Borel transversal for $E_{G}^{X_{0}}$.

  2. (2)

The stabilizers of $\left(w,g\psi_{0}\left(w\right)\right)\in X_{0}$ is the
conjugation of $\psi_{0}\left(w\right)$ by $g$.

  3. (3)

$\left(W_{0},\nu_{0}\right)$ is the space of the ergodic components of
$\left(X_{0},\mu_{0}\right)$: a point $w\in W_{0}$ corresponds to a compactly
fibered coset $G$-space $G/\psi_{0}\left(w\right)$.

######  Theorem G.

Let $G$ be an lcsc group. A nonsingular $G$-space is totally dissipative iff
it is isomorphic to the Krengel $G$-space arising from its natural Krengel
$G$-datum.

###  1.1. Roadmap

After reviewing general preliminaries in Section 2, in Section 3 we prove the
Hopf Decomposition A, and lay the foundations for the proof of the Recurrence
Theorem B. Then Section 4 focuses on further properties, including the
recurrence theorems of Poincaré and Maharam, maximal transient sets, Hopf
Decomposition under ergodic decomposition and with respect to subactions of
closed subgroups. Section 5 is dedicated to Borel dynamics and includes a
structure theorem for Borel $G$-spaces whose orbit equivalence relation is
smooth (without a measure). In Section 6 we carry out the proof of Theorem D,
and complete the proof of the Recurrence Theorem B. The general structure
theory of totally dissipative nonsingular $G$-spaces, including Construction
7.6 of Krengel $G$-spaces and the proofs of Theorems E, F and G, will be
carried out in Section 7. Then Section 8 goes back to prove Theorem C, and
using it to provide a new proof to the conservativity of Maharam Extensions in
unimodular groups. The final Section 9 is complementary and presents a
treatment of the related Positive–Null Decomposition.

### Acknowledgments

We extend our heartfelt thanks to Michael Björklund for a few illuminating
discussions, and particularly for his Remark 7.7. We also thank Sasha
Danilenko for answering several questions and for pointing to us the reference
[14]. We are deeply grateful to Zemer Kosloff who initiated this project, for
his advice, dedicated mentorship and invaluable support. We finally thank
Johanna Steinmeyer for her assistance in creating the figures.

##  2\. Preliminaries

####  2.0.1. Locally compact second countable groups.

Throughout this work, $G$ stands for a locally compact second countable (lcsc)
group and $\lambda$ for a fixed version of its left Haar measure. While
typically our results will be nontrivial when $G$ is non-compact, we do not
exclude the case when $G$ is compact. We also do not exclude countable
discrete groups. The identity element of $G$ will be denoted by $e$. We denote
the modular function of $G$ and $\lambda$ by $\varDelta:G\to\mathbb{R}_{+}$.
Thus, for every test function $\varphi:G\to\mathbb{R}$,

$\int_{G}\varphi\left(g\right)\varDelta\left(g\right)d\lambda\left(g\right)=\int_{G}\varphi\left(g^{-1}\right)d\lambda\left(g\right).$

Recall that by Struble’s theorem [38], every lcsc group $G$ admits a proper
compatible metric, namely a metric that induces its topology for which balls
are relatively compact. Thus, if $B_{r}$ is the ball of radius $r>0$ around
$e$ with respect to such a metric on $G$, then not only that $B_{r}\nearrow G$
as $r\nearrow+\infty$, but also every compact set $K\subset G$ is eventually
contained in $B_{r}$.

We denote by

$\mathbf{K}_{\mathrm{gr}}\left(G\right)\subset\mathbf{K}\left(G\right)\text{
and }\mathbf{F}_{\mathrm{gr}}\left(G\right)\subset\mathbf{F}\left(G\right)$

the spaces of compact subgroups of $G$ and of compact subsets of $G$, and the
spaces of closed subgroups of $G$ and of closed subsets of $G$, respectively.
These are known to be standard Borel spaces in the Chabauty–Fell topology.
(see e.g. [22, Exercise (12.7) & §12.E(2)]).

####  2.0.2. Borel $G$-spaces.

By a standard Borel space we refer to a measurable space $X$ whose
$\sigma$-algebra is the Borel $\sigma$-algebra of some Polish topology
(separable, completely metrizable) on $X$. This $\sigma$-algebra on $X$ is
fixed and will remain implicit. If $G$ is an lcsc group, a Borel $G$-space is
a standard Borel space $X$ with a jointly Borel map $G\times X\to X$,
$\left(g,x\right)\mapsto g.x$, such that $e.x=x$ and $gh.x=g.\left(h.x\right)$
for every $g,h\in G$ and $x\in X$. The orbit equivalence relation associated
with a Borel $G$-space $X$ is the equivalence relation

$E_{G}^{X}:=\left\\{\left(x,g.x\right):x\in X,g\in G\right\\}.$

Since $G$ is lcsc $E_{G}^{X}$ is Borel, namely a Borel subset of $X\times X$,
and for every point $x\in X$ the orbit $G.x:=\left\\{g.x:g\in G\right\\}$ is
Borel (see [16, Exercise 3.4.6, Theorem 3.3.2]). We denote the stabilizer of a
point $x\in X$ by

$G_{x}:=\left\\{g\in G:g.x=x\right\\}.$

The following fact will be essential for us (see [16, Theorems 3.3.2 &
8.2.1]).

######  Theorem 2.1 (Miller; Becker–Kechris).

Let $G$ be an lcsc group and $X$ a Borel $G$-space. Then the stabilizers of
all points in $X$ are closed subgroups of $G$, and the map
$X\to\mathbf{F}_{\mathrm{gr}}\left(G\right)$, $x\mapsto G_{x}$, is a Borel
map.

####  2.0.3. Nonsingular $G$-spaces.

A standard measure space is a measure space $\left(X,\mu\right)$, where $X$ is
a standard Borel space and $\mu$ is a Borel measure which is either a
probability measure or an infinite $\sigma$-finite measure. Generally
speaking, we do not exclude the case when $\mu$ has atoms. We will say that a
Borel set $A\subset X$ is $\mu$-null if $\mu\left(A\right)=0$ and that it is
$\mu$-conull if $\mu\left(X\backslash A\right)=0$. Similarly, we will say that
a Borel set $A\subset X$ is $\mu$-positive if $\mu\left(A\right)>0$. We may
also use the convenient writing of $A=B$ modulo $\mu$ for Borel sets
$A,B\subset X$ when $A\triangle B$ is a $\mu$-null set. Here $\triangle$
denotes symmetric difference of sets. We use similar terminology for
containment of Borel sets, equality and inequality of Borel functions, etc.
When the measure is clear from the context, we will say that a certain
property essentially occurs when it occurs on a conull set (e.g. essentially
free for actions).

The following is a classical lemma about standard measure spaces. A proof can
be found in [24, p. 17] and [1, Lemma 1.0.7].

######  Lemma 2.2 (The Exhaustion Lemma).

Let $\left(X,\mu\right)$ be a standard measure space. Every _hereditary_
(closed to taking subsets) collection $\mathfrak{H}$ of Borel sets in $X$
admits a _measurable union_ , namely a Borel set $\mathcal{H}\subset X$ such
that:

  1. (1)

Every element of $\mathfrak{H}$ is a subset of $\mathcal{H}$ modulo $\mu$.

  2. (2)

Every $\mu$-positive subset of $\mathcal{H}$ contains a $\mu$-positive element
of $\mathfrak{H}$.

Moreover, $\mathcal{H}$ is the union of countably many disjoint elements of
$\mathfrak{H}$.

Let $\nu,\mu$ be Borel measures on the same standard Borel space. We say that
$\nu$ is absolutely continuous with respect to $\mu$, denoting it $\nu\ll\mu$,
if every $\mu$-null set is also $\nu$-null. We say that $\nu$ and $\mu$ are
equivalent, denoting it $\nu\sim\mu$, if they are mutually absolutely
continuous: $\nu\ll\mu$ and $\mu\ll\nu$.

A nonsingular $G$-space is a standard measure space $\left(X,\mu\right)$ such
that $X$ is a Borel $G$-space and $\mu$ is quasi-invariant to the action; that
is,

$\mu\circ g\sim\mu\text{ for every }g\in G,$

where $\mu\circ g$ denotes the measure $A\mapsto\mu\left(g.A\right)$ (each $g$
defines an invertible mapping of $X$). When further $\mu\circ g=\mu$ for every
$g\in G$ we will call $\left(X,\mu\right)$ by a measure preserving $G$-space.
Occasionally we deal with the case where $\mu$ is a probability measure, in
which we refer to $\left(X,\mu\right)$ as a nonsingular probability $G$-space
or a probability preserving $G$-space.

A Borel set $A\subset X$ is said to be $G$-invariant if $g.A=A$ for each $g\in
G$, and it is said to be $\mu$-almost $G$-invariant if $\mu\left(g.A\triangle
A\right)=0$ for each $g\in G$. The following is a well-known fact that will be
essential to our discussion. For a proof see [5, Theorems 1.2.8–1.2.9] (cf.
[13, Proposition 8.3]).

######  Theorem 2.3.

Let $G$ be an lcsc group and $\left(X,\mu\right)$ a nonsingular $G$-space. For
every $\mu$-almost $G$-invariant Borel set $A\subset X$ there is a
$G$-invariant Borel set $A^{\prime}\subset X$ with $\mu\left(A\triangle
A^{\prime}\right)=0$.

Finally, two nonsingular $G$-spaces $\left(X,\mu\right)$ and
$\left(Y,\nu\right)$ are said to be isomorphic if there are $G$-invariant
Borel sets $X_{o}\subset X$ and $Y_{o}\subset Y$, the first is $\mu$-conull
and the second is $\nu$-conull, and a Borel bijection $\varphi:X_{o}\to
Y_{o}$, such that $\varphi_{\ast}\mu\sim\nu$ and such that for every $g\in G$
we have that $\varphi\left(g.x\right)=g.\varphi\left(x\right)$ for $\mu$-a.e.
$x\in X_{o}$. Similarly, two measure preserving $G$-spaces
$\left(X,\mu\right)$ and $\left(Y,\nu\right)$ are said to be isomorphic if
they are isomorphic as nonsingular $G$-spaces via $\varphi$ as above with the
further property that $\varphi_{\ast}\mu=\nu$.

####  2.0.4. The Radon–Nikodym cocycle.

For a nonsingular $G$-space $\left(X,\mu\right)$, we have the collection of
Radon–Nikodym derivatives

$\frac{d\mu\circ g}{d\mu}\left(\cdot\right)\in L^{1}\left(X,\mu\right),\quad
g\in G.$

By the properties of Radon–Nikodym derivatives, for every $g,h\in G$ we have

$\frac{d\mu\circ gh}{d\mu}\left(x\right)=\frac{d\mu\circ
g}{d\mu}\left(h.x\right)\cdot\frac{d\mu\circ h}{d\mu}\left(x\right)\text{ for
}\mu\text{-a.e. }x\in X.$

By the Mackey Cocycle Theorem (see e.g. [39, Lemma 5.26, p. 179]), there
exists a (pointwise defined) cocycle

$\nabla:G\times
X\to\mathbb{R}_{+},\quad\nabla:\left(g,x\right)\mapsto\nabla_{g}\left(x\right),$

namely

$\nabla_{e}\left(x\right)=1\text{ and
}\nabla_{gh}\left(x\right)=\nabla_{g}\left(h.x\right)\cdot\nabla_{h}\left(x\right)\text{
for every }g,h\in G\text{ and \emph{every} }x\in X,$

such that for every $g\in G$,

$\nabla_{g}\left(x\right)=\frac{d\mu\circ g}{d\mu}\left(x\right)\text{ for
}\mu\text{-a.e. }x\in X.$

A fundamental property of $\nabla$, which is a direct computation using the
Fubini Theorem and the basic property of the Radon–Nikodym derivatives, is the
following:

($\dagger$) $\displaystyle\iint_{G\times
X}\nabla_{g}\left(x\right)f_{0}\left(g.x\right)f_{1}\left(x\right)\varphi\left(g\right)d\lambda\otimes\mu\left(g,x\right)$
$\displaystyle\qquad\quad=\iint_{G\times
X}f_{0}\left(x\right)f_{1}\left(g^{-1}.x\right)\varphi\left(g\right)d\lambda\otimes\mu\left(g,x\right),$
$\text{ for all Borel functions
}f_{0},f_{1}:X\to\left[0,\infty\right),\,\varphi:G\to\left[0,\infty\right).$

##  3\. The Hopf Decomposition: A General Form

Let $G$ be an lcsc group and $\left(X,\mu\right)$ a nonsingular $G$-space.
Using the associated Radon–Nikodym cocycle $\nabla$ as in 2.0.4, define an
averaging transform of Borel functions $f:X\to\left[0,+\infty\right)$ by

$S^{G}:f\mapsto S_{f}^{G},\quad
S_{f}^{G}\left(x\right):=\int_{G}\nabla_{g}\left(x\right)f\left(g.x\right)d\lambda\left(g\right).$

Denote by $L_{+}^{1}\left(X,\mu\right)$ the class of (strictly) positive-
valued elements of $L^{1}\left(X,\mu\right)$. For $f\in
L_{+}^{1}\left(X,\mu\right)$ consider the sets

$\mathcal{C}_{f}:=\left\\{x\in
X:S_{f}^{G}\left(x\right)=+\infty\right\\}\text{ and
}\mathcal{D}_{f}:=X\backslash\mathcal{C}_{f}.$

These are Borel sets (see [22, Theorem (17.25)]).

######  Definition 3.1.

A nonsingular $G$-space $\left(X,\mu\right)$ is said to have the Hopf
Decomposition if there is a decomposition

$X=\mathcal{C}\sqcup\mathcal{D}$

of $X$ into $G$-invariant disjoint Borel sets $\mathcal{C}$ and $\mathcal{D}$,
such that

$\mu\left(\mathcal{C}\triangle\mathcal{C}_{f}\right)=0\text{ and
}\mu\left(\mathcal{D}\triangle\mathcal{D}_{f}\right)=0\text{ for every }f\in
L_{+}^{1}\left(X,\mu\right).$

######  Theorem 3.2.

Let $G$ be an lcsc group and $\left(X,\mu\right)$ be a nonsingular $G$-space.
Fix an arbitrary $f\in L_{+}^{1}\left(X,\mu\right)$ and an arbitrary Borel set
$A\subset X$. TFAE:

  1. (1)

$A\subset\mathcal{C}_{f}$ modulo $\mu$.

  2. (2)

For every $\mu$-positive set $B\subset A$ it holds that

$\int_{G}1_{B}\left(g^{-1}.x\right)d\lambda\left(g\right)=+\infty\text{ for
}\mu\text{-positively many }x\in X.$

  3. (3)

For every $\mu$-positive set $B\subset A$ it holds that

$\int_{G}1_{B}\left(g^{-1}.x\right)d\lambda\left(g\right)=+\infty\text{ for
}\mu\text{-a.e. }x\in B.$

Using the notation we introduced before, for a Borel set $A\subset X$ and
$x\in X$ write

$R_{A}\left(x\right):=\left\\{g\in G:x\in g.A\right\\},\text{ so that
}\lambda\left(R_{A}\left(x\right)\right)=\int_{G}1_{A}\left(g^{-1}.x\right)d\lambda\left(g\right).$

Observe the basic identity

(3.0.1) $R_{A}\left(g.x\right)=gR_{A}\left(x\right),\quad x\in X,g\in G.$

Recall the notations of transience and recurrence for sets as in Definition
1.2. Thus, a Borel set $T\subset X$ is called transient when

$\lambda\left(R_{T}\left(x\right)\right)<+\infty\text{ for every }x\in X.$

Let $\mathfrak{T}$ be the collection of transient sets in
$\left(X,\mu\right)$. Since in general if $A\subset B$ then
$R_{A}\left(x\right)\subset R_{B}\left(x\right)$ for every $x\in X$, we see
that $\mathfrak{T}$ is hereditary, so by the Exhaustion Lemma 2.2 it admits a
measurable union that we denote by

$\mathcal{T}.$

######  Proposition 3.3.

For every $f\in L_{+}^{1}\left(X,\mu\right)$ it holds that
$\mu\left(\mathcal{T}\triangle\mathcal{D}_{f}\right)=0$.

###### Proof of Proposition 3.3.

We start by showing that
$\mu\left(\mathcal{D}_{f}\backslash\mathcal{T}\right)=0$. Let
$A\subset\mathcal{T}^{\complement}$ be an arbitrary Borel set and put

$A_{0}:=\left\\{x\in A:S_{f}^{G}\left(x\right)<+\infty\right\\}.$

Suppose toward a contradiction that $A_{0}$ is $\mu$-positive. Passing to some
$\mu$-positive subset $B\subset A_{0}$ with $\mu(B)<+\infty$, using the
formula ($\dagger$ ‣ 2.0.4) we obtain

$\displaystyle+\infty>\int_{B}S_{f}^{G}\left(x\right)d\mu\left(x\right)$
$\displaystyle\qquad\quad=\iint_{G\times
X}\nabla_{g}\left(x\right)f\left(g.x\right)1_{B}\left(x\right)d\lambda\otimes\mu\left(g,x\right)$
$\displaystyle\qquad\quad=\iint_{G\times
X}f\left(x\right)1_{B}\left(g^{-1}.x\right)d\lambda\otimes\mu\left(g,x\right)$
$\displaystyle\qquad=\int_{X}f\left(x\right)\lambda\left(R_{B}\left(x\right)\right)d\mu\left(x\right).$

Since $f$ is positive on a $\mu$-conull set, it follows that
$\lambda\left(R_{B}\left(x\right)\right)<+\infty$ for $\mu$-a.e. $x\in X$.
Thus, $A\supset B\in\mathfrak{T}$ which is a contradiction to that
$A\subset\mathcal{T}^{\complement}$. We deduce that
$S_{f}^{G}\left(x\right)=+\infty$ for $\mu$-a.e. $x\in A$, namely
$A\subset\mathcal{C}_{f}$. Since $A\subset\mathcal{T}^{\complement}$ is an
arbitrary $\mu$-positive set it readily follows that
$\mu\left(\mathcal{T}^{\complement}\backslash\mathcal{C}_{f}\right)=0$, hence
$\mu\left(\mathcal{D}_{f}\backslash\mathcal{T}\right)=0$.

We now show that $\mu\left(\mathcal{T}\backslash\mathcal{D}_{f}\right)=0$. Fix
an arbitrary transient set $T\in\mathfrak{T}$. For $r>0$ denote

$X_{r}:=\left\\{x\in X:\lambda\left(R_{T}\left(x\right)\right)\leq
r\right\\}\text{ and }T_{r}:=T\cap X_{r}.$

These are Borel sets (see [22, Theorem (17.25)]). Since $T\in\mathfrak{T}$ it
follows that $X_{r}\nearrow X$ as $r\nearrow+\infty$ modulo $\mu$, hence
$T_{r}\nearrow T$ as $r\nearrow+\infty$ modulo $\mu$.

For a fixed $r>0$ and every $x\in X$, using (3.0.1) and the invariance of
$\lambda$ we have

$\displaystyle g\in R_{T_{r}}\left(x\right)$
$\displaystyle\iff\left(g^{-1}.x\in
T\right)\wedge\left(\lambda\left(R_{T}\left(g.x\right)\right)\leq r\right)$
$\displaystyle\iff\left(g^{-1}.x\in
T\right)\wedge\left(\lambda\left(R_{T}\left(x\right)\right)\leq r\right),$

hence

$R_{T_{r}}\left(x\right)=\begin{cases}R_{T}\left(x\right)&\lambda\left(R_{T}\left(x\right)\right)\leq
r\\\ \emptyset&\text{otherwise}\end{cases}.$

This readily implies that $\lambda\left(R_{T_{r}}\left(x\right)\right)\leq r$
for every $x\in X$. Then using the formula ($\dagger$ ‣ 2.0.4) we deduce that
for every $r>0$,

(3.0.2) $\displaystyle\int_{T_{r}}S_{f}^{G}\left(x\right)d\mu\left(x\right)$
$\displaystyle\qquad=\iint_{G\times
X}\nabla_{g}\left(x\right)f\left(g.x\right)1_{T_{r}}\left(x\right)d\lambda\otimes\mu\left(g,x\right)$
$\displaystyle\qquad=\iint_{G\times
X}f\left(x\right)1_{T_{r}}\left(g^{-1}.x\right)d\lambda\otimes\mu\left(g,x\right)$
$\displaystyle\qquad=\int_{X}f\left(x\right)\lambda\left(R_{T_{r}}\left(x\right)\right)d\mu\left(x\right)$
$\displaystyle\leq r\left\|f\right\|_{L^{1}\left(X,\mu\right)}<+\infty.$

It follows that $S_{f}^{G}\left(x\right)<+\infty$ for $\mu$-a.e. $x\in T_{r}$.
Since $r$ is arbitrary it follows that $S_{f}^{G}\left(x\right)<+\infty$ for
$\mu$-a.e. $x\in T$. By the definition of $\mathcal{D}_{f}$ this implies that
$\mu\left(T\backslash\mathcal{D}_{f}\right)=0$. Since $T\in\mathfrak{T}$ is
arbitrary we deduce that
$\mu\left(\mathcal{T}\backslash\mathcal{D}_{f}\right)=0$. ∎

###### Proof of Theorem 3.2.

Let $A\subset X$ be a $\mu$-positive set. By Proposition 3.3 we have
$\mu\left(A\cap\mathcal{D}_{f}\right)=0\iff\mu\left(A\cap\mathcal{T}\right)=0$.
Thus, $\mu\left(A\cap\mathcal{D}_{f}\right)=0$ iff $A$ contains no
$\mu$-positive transient set. This is precisely condition (2) for $A$ as in
the theorem. Thus, we have established the equivalence of conditions (1) and
(2) for $A$ as in the theorem.

It is obvious that the failure of condition (2) for $A$ implies the failure of
condition (3) for $A$. Let us show the converse. Thus, suppose that $A$ is a
$\mu$-positive set containing no transient set, and let $B\subset A$ be an
arbitrary $\mu$-positive set. Consider the set

$T_{B}:=\left\\{x\in
B:\lambda\left(R_{B}\left(x\right)\right)<+\infty\right\\}.$

As the notation suggests, we claim that $T_{B}$ is a transient set. Indeed,
let $x\in X$ be arbitrary and, as we argued in the proof of Proposition 3.3
with the identity (3.0.1), one directly verifies that

$R_{T_{B}}\left(x\right)=\begin{cases}R_{B}\left(x\right)&\lambda\left(R_{B}\left(x\right)\right)<+\infty\\\
\emptyset&\text{otherwise}\end{cases},$

so that $\lambda\left(R_{T_{B}}\left(x\right)\right)<+\infty$ for every $x\in
T_{B}$, concluding that $T_{B}$ is a transient set. As $T_{B}\subset B\subset
A$ and $A$ contains no $\mu$-positive transient set,
$\lambda\left(T_{B}\right)=0$. This is precisely condition (3) for $A$. ∎

###### Proof of Theorem A.

Recall the identity (3.0.1) from which it follows that if $T$ is a transient
set then, for every $g\in G$, also $g.T$ is a transient set. This readily
implies that $\mathcal{T}$ is almost $\mu$-invariant. By Theorem 2.3 there is
a $G$-invariant set $\mathcal{T}^{\prime}\subset X$ with
$\mu\left(\mathcal{T}\triangle\mathcal{T}^{\prime}\right)=0$, thus
$\mathcal{T}^{\prime}$ is also a measurable union of the class of transients
sets. We then put

$\mathcal{D}=\mathcal{T}^{\prime}\text{ and
}\mathcal{C}=X\backslash\mathcal{D}.$

It follows from Proposition 3.3 directly that this $G$-invariant Borel
decomposition constitutes the Hopf Decomposition of $\left(X,\mu\right)$ as in
Definition 3.1. The fact that this decomposition depends only on the measure
class of $\mu$ follows directly from the Recurrence Theorem 3.2. ∎

As for Theorem B, its version with transient sets (rather than strongly
transient) follows from Theorem 3.2. The complete proof will be given in the
end of Section 6. We end this discussion by proving Proposition 1.1.

###### Proof of Proposition 1.1.

Fix any $f_{0}\in L_{+}^{1}\left(X,\mu\right)$ and define the function

$f_{0}^{\ast}:X\to\mathbb{R}_{+},\quad
f_{0}^{\ast}\left(x\right):=S_{f_{0}}^{G}\left(x\right)=\int_{G}\nabla_{g}\left(x\right)f_{0}\left(g.x\right)d\lambda\left(g\right).$

Assuming for simplicity that $X=\mathcal{D}$, we have $f_{0}^{\ast}<+\infty$
on a $\mu$-conull set, so define a measure $\eta_{0}$ on $X$ by

$d\eta_{0}\left(x\right)=f_{0}^{\ast}\left(x\right)d\mu\left(x\right).$

Since $f_{0}^{\ast}$ is positive on a $\mu$-conull set, $\eta_{0}$ is
equivalent to $\mu$. In order to show that $\eta_{0}$ is $G$-invariant note
that by the formula ($\dagger$ ‣ 2.0.4), for every nonnegative Borel function
$f_{1}:X\to\mathbb{R}$ we have

$\displaystyle\int_{X}f_{1}\left(x\right)d\eta_{0}\left(x\right)$
$\displaystyle=\iint_{G\times
X}\nabla_{g}\left(x\right)f_{0}\left(g.x\right)f_{1}\left(x\right)d\lambda\otimes\mu\left(g,x\right)$
$\displaystyle=\iint_{G\times
X}f_{0}\left(x\right)f_{1}\left(g^{-1}.x\right)d\lambda\otimes\mu\left(g,x\right).$

In follows by the invariance of the Haar measure together with the Fubini
Theorem that, for an arbitrary $h\in G$, if we replace $f_{1}$ by $f_{1}\circ
h$ this integral remains unchanged, namely $\eta_{0}$ is $G$-invariant. ∎

##  4\. Further Properties

Here we gather a few general facts about conservativity and dissipativity,
some are known and others would be new in the generality of lcsc groups.

####  4.0.1. Recurrence Theorems in measure preserving $G$-spaces

######  Theorem 4.1 (Poincaré Recurrence Theorem).

For every lcsc non-compact group $G$, totally dissipative nonsingular
$G$-spaces admit no absolutely continuous $G$-invariant probability measure.

One can easily deduce this from Theorems A and B. The following more general
theorem was proved by Maharam for $G=\mathbb{Z}$ (see [1, Theorem 1.1.7]).

######  Theorem 4.2 (Maharam’s Recurrence Theorem).

Let $\left(X,\mu\right)$ be a _measure preserving_ $G$-space. If there exists
a Borel set $A_{o}\subset X$ with

$0<\mu\left(A_{o}\right)<+\infty\text{ and
}\lambda\left(R_{A_{o}}\left(x\right)\right)=+\infty\text{ for }\mu\text{-a.e.
}x\in X,$

(in particular $G$ is non-compact) then $\left(X,\mu\right)$ is conservative.

###### Proof.

Suppose that $T\subset X$ is a transient set. With the sets

$T_{r}:=\left\\{x\in T:\lambda\left(R_{T}\left(x\right)\right)\leq
r\right\\},\quad r>0,$

as in the previous proof, the computation (3.0.2) for $f=1_{A_{o}}$ (which was
applied there to positive functions but holds true for nonnegative functions)
shows that for every $r>0$,

$\int_{T_{r}}S_{1_{A_{o}}}^{G}\left(x\right)d\mu\left(x\right)\leq
r\mu\left(A_{o}\right)<+\infty\text{, so
}S_{1_{A_{o}}}^{G}\left(x\right)<+\infty\text{ for }\mu\text{-a.e. }x\in
T_{r}.$

However, as we are in the measure preserving case and by the assumption,

$S_{1_{A_{o}}}^{G}\left(x\right)=\lambda\left(R_{A_{o}}\left(x\right)\right)=+\infty,$

hence $\mu\left(T_{r}\right)=0$. As $T_{r}\nearrow T$ as $r\nearrow+\infty$ we
deduce that $\mu\left(T\right)=0$. ∎

####  4.0.2. Maximal transient set

In the case of a countable group $G$, it is known that there exists a
wandering set $W_{\max}\subset X$ with $\mathcal{D}=G.W_{\max}$ modulo $\mu$
(see [1, Proposition 1.6.1]). The following is the analog for transient sets:

######  Proposition 4.3.

Let $G$ be an lcsc group and $\left(X,\mu\right)$ a nonsingular $G$-space.
There exists a transient set $T_{\max}\subset X$ with $\mathcal{D}=G.T_{\max}$
modulo $\mu$. Moreover, when $\mu\left(\mathcal{D}\right)>0$ one may assume
further that $\mu\left(T_{\max}\right)>0$.

The following proof is essentially due to Aaronson [1, Proposition 1.6.1].

###### Proof.

W may assume that $\mu$ is a probability measure. Recall the hereditary
collection $\mathfrak{T}$ of transient sets of $\left(X,\mu\right)$. Let us
proceed in an inductive way as follows. First, if
$\mu\left(\mathcal{D}\right)=0$ set $T_{\max}:=\emptyset$. Otherwise

$\alpha_{1}:=\sup\left\\{\mu\left(T\right):T\in\mathfrak{T}\right\\}>0,$

so pick $T_{1}\in\mathfrak{T}$ with $\mu\left(T_{1}\right)>\alpha_{1}/2$.
Suppose $T_{1},\dots,T_{n}\in\mathfrak{T}$ were defined. If
$\mu\left(\mathcal{D}\setminus G.\left(T_{1}\sqcup\dotsm\sqcup
T_{n}\right)\right)=0$ set $T_{\max}=:T_{1}\sqcup\dotsm\sqcup T_{n}$.
Otherwise set

$\alpha_{n}:=\sup\left\\{\mu(T):T\in\mathfrak{T},T\subset\mathcal{D}\setminus
G.\left(T_{1}\sqcup...\sqcup T_{n}\right)\right\\}.$

If this process is indefinite, so that $T_{\max}$ is undefined, set
$T_{\max}:=T_{1}\sqcup T_{2}\sqcup\dotsm$.

We now claim that $T_{\max}$ is a maximal transient set. For every $x\in X$,
either $R_{T_{\max}}\left(x\right)=\emptyset$ or
$R_{T_{\max}}\left(x\right)=R_{T_{n_{x}}}\left(x\right)$ for some
$n_{x}\in\mathbb{N}$, hence
$\lambda\left(R_{T_{\max}}\left(x\right)\right)<+\infty$ and
$T_{\max}\in\mathfrak{T}$. We note that for every $g\in G$, since
$\mu\left(T_{\max}\right)>0$ also $\mu\left(g.T_{\max}\right)>0$, and since
$g.T_{\max}\in\mathfrak{T}$ we have $g.T_{\max}\subset\mathcal{D}$ modulo
$\mu$. This implies that $S.T_{\max}\subset\mathcal{D}$ modulo $\mu$ for
whatever countable set $S\subset G$. By the continuity of the mapping
$G\to\left[0,1\right]$, $g\mapsto\mu\left(g.T_{0}\right)$, the separability of
$G$ and using that $\mu\left(T_{\max}\right)>0$, we deduce that
$G.T_{\max}\subset\mathcal{D}$ modulo $\mu$. Finally, since
$T_{1},T_{2},\dotsc$ are disjoint we have

$\sum\nolimits_{n\in\mathbb{N}}\alpha_{n}=2\cdot\sum\nolimits_{n\in\mathbb{N}}\mu\left(T_{n}\right)\leq
2\cdot\mu\left(X\right)<+\infty,$

so we have $\mu\left(\mathcal{D}\backslash
G.T_{\max}\right)\leq\alpha_{n}\xrightarrow[n\to\infty]{}0$, thus
$\mathcal{D}\subset G.T_{\max}$ modulo $\mu$. ∎

####  4.0.3. Dissipativity and ergodic decomposition

Another consequence of the Recurrence Theorem B is that dissipativity is
preserved when passing to ergodic components. Recall that by the ergodic
decomposition theorem (see [19, Theorem 1.1]),444While this is formulated for
nonsingular probability $G$-spaces, one can derive the general case by passing
to an equivalent probability measure. for a nonsingular $G$-space
$\left(X,\mu\right)$ there is a standard probability space
$\left(E,\upsilon\right)$ and a family $\left\\{\mu_{e}:e\in E\right\\}$ of
Borel measures on $X$ such that:

  1. (1)

For each $e\in E$, the space $\left(X,\mu_{e}\right)$ is an _ergodic_
nonsingular $G$-space.

  2. (2)

For every Borel set $A\subset X$, the map $E\to\left[0,+\infty\right]$,
$e\mapsto\mu_{e}\left(A\right)$, is Borel and
$\mu\left(A\right)=\int_{E}\mu_{e}\left(A\right)d\upsilon\left(e\right)$.

######  Proposition 4.4.

A nonsingular $G$-space is totally dissipative iff essentially each of its
ergodic components is totally dissipative.

###### Proof.

For an arbitrary Borel set $A\subset X$ let

$A_{\infty}:=\left\\{x\in
X:\lambda\left(R_{A}\left(x\right)\right)=+\infty\right\\}.$

With the ergodic decomposition, $\mu\left(A_{\infty}\right)=0$ iff
$\mu_{e}\left(A_{\infty}\right)=0$ for $\upsilon$-a.e. $e\in E$, so the
proposition follows directly from the Recurrence Theorem 3.2. ∎

Combining Theorem F and Proposition 4.4, we obtain the characterization of
totally dissipative nonsingular $G$-spaces as those whose each ergodic
component is isomorphic to a compactly fibered $G$-space.

####  4.0.4. Subactions of closed subgroups

The coming proposition relates the Hopf Decomposition when is taken with
respect to a closed subgroup of the acting group. Halmos showed that for every
nonsingular transformation $\tau$, the Hopf decomposition of $\tau$ coincides
with that of its self-iteration $\tau^{p}$, for any positive integer $p$ (see
[1, Corollary 1.1.4]). Krengel observed that the analog statement holds for
nonsingular group actions and their lattices subactions (see [1, Theorem
1.6.4]).

######  Proposition 4.5.

Let $G$ be an lcsc group and $\left(X,\mu\right)$ a nonsingular $G$-space. If
$H$ is a closed subgroup of $G$ the following hold.

  1. (1)

If $\left(X,\mu\right)$ is totally dissipative, then it is also totally
dissipative with respect to $H$ (in the subaction of $H$).

  2. (2)

If $H$ is further a lattice then the converse of (1) is also true, and the
Hopf Decomposition with respect to $G$ coincides with the one with respect to
$H$.

###### Proof.

We start to prove part (1). By Proposition 4.4 it is sufficient to show that
every ergodic component of $\left(X,\mu\right)$ is dissipative as a
nonsingular $H$-space, and by Theorem F every such ergodic component is
isomorphic to a compactly fibered coset $G$-spaces $\left(G/K,\kappa\right)$.
Note that the stabilizers of the action of $H$ on $G/K$ are of the form $H\cap
K^{g}$ for some $g\in G$, hence are all compact. We also apply Theorem A.2, by
which the orbit equivalence relation $E_{G/K}^{H}$ is smooth. Then from
Theorem D it follows that $\left(G/K,\kappa\right)$ is totally dissipative
also as a nonsingular $H$-space.

As for part (2), when $H$ is a lattice we may fix a fundamental domain
$\Omega\subset G$ for $H$, so that $0<\lambda\left(\Omega\right)<+\infty$ and
the translations $\left\\{\omega\Omega:\omega\in H\right\\}$ of $\Omega$ are
pairwise disjoint and $G=\bigcup_{\omega\in H}\omega\Omega$. Fix any $f\in
L_{+}^{1}\left(X,\mu\right)$ and let

$f_{\Omega}:X\to\mathbb{R}_{+},\quad
f_{\Omega}\left(x\right):=\int_{\Omega}\nabla_{g}\left(x\right)f\left(g.x\right)d\lambda\left(g\right).$

Note that $f_{\Omega}\in L_{+}^{1}\left(X,\mu\right)$ since
$\lambda\left(\Omega\right)<+\infty$. We now have the relation

$\displaystyle S_{f}^{G}\left(x\right)$
$\displaystyle=\sum\nolimits_{\omega\in
H}\int_{\omega\Omega}\nabla_{g}\left(x\right)f\left(g.x\right)d\lambda\left(g\right)$
$\displaystyle=\sum\nolimits_{\omega\in
H}\int_{\Omega}\nabla_{g\omega}\left(x\right)f\left(g\omega.x\right)d\lambda\left(g\right)$
$\displaystyle=\sum\nolimits_{\omega\in
H}\nabla_{\omega}\left(x\right)\int_{\Omega}\nabla_{g}\left(\omega.x\right)f\left(g\omega.x\right)d\lambda\left(g\right)$
$\displaystyle=\sum\nolimits_{\omega\in
H}\nabla_{\omega}\left(x\right)f_{\Omega}\left(\omega.x\right)=S_{f_{\Omega}}^{H}\left(x\right),$

where the last equality is nothing but the very definition of the averaging
transform $S^{H}$ for the discrete countable acting group $H$ with its Haar
(counting) measure. This completes the proof. ∎

##  5\. Smooth Orbit Equivalence Relations

Our aim here is to present a structure theorem for Borel $G$-spaces whose
orbit equivalence relation is smooth. This result will serve us well in
proving the general structure theorem of totally dissipative nonsingular
$G$-spaces, but it will be formulated without measures and it is of its own
interest.

###  5.1. Preliminaries: Smooth Equivalence Relations

An equivalence relation on a set $X$ is a set $E\subset X\times X$ such that
$x\sim x^{\prime}\iff\left(x,x^{\prime}\right)\in E$ defines an equivalence
relation on $X$. When $X$ is a standard Borel space, such an equivalence
relation is called Borel if it belongs to the product Borel $\sigma$-algebra
of $X\times X$.

######  Definition 5.1.

An equivalence relation $E$ on a standard Borel space $X$ is smooth, if there
is a standard Borel space $Y$ and a Borel function $s:X\rightarrow Y$ such
that

$\left(x,x^{\prime}\right)\in E\iff
s\left(x\right)=s\left(x^{\prime}\right)\text{ for all }x,x^{\prime}\in X.$

The function $s$ is referred to as a reduction map (in that it is a reduction
of $E$ to the equivalence relation of equality). Note that a smooth
equivalence relation is necessarily Borel, as it is the inverse image of the
diagonal of $Y$ under the map $X\times X\to Y\times Y$,
$\left(x,x^{\prime}\right)\mapsto\left(s\left(x\right),s\left(x^{\prime}\right)\right)$.
In the following we collect a few well-known characterizations for smoothness.

######  Theorem 5.2.

Let $G$ be an lcsc group and $X$ a Borel $G$-space. TFAE:

  1. (1)

$E_{G}^{X}$ is smooth.

  2. (2)

$E_{G}^{X}$ is _countably separated_ : there exists a countable collection of
Borel functions $u_{1},u_{2},\dotsc$ from $X$ to some standard Borel space,
such that

$\left(x,x^{\prime}\right)\in
E\iff\left(\forall_{n},\,u_{n}\left(x\right)=u_{n}\left(x^{\prime}\right)\right)\text{
for all }x,x^{\prime}\in X.$

  3. (3)

$E_{G}^{X}$ admits a _Borel selector_ : a Borel function $s:X\to X$ such that

$\left(x,s\left(x\right)\right)\in E\text{ and }\left(x,x^{\prime}\right)\in
E\iff s\left(x\right)=s\left(x^{\prime}\right)\text{ for all }x,x^{\prime}\in
X.$

  4. (4)

$E_{G}^{X}$ admits a _Borel transversal_ : a Borel subset of $X$ that
intersects every orbit in exactly one point.

  5. (5)

The space of orbits $X/E_{G}^{X}$ is standard Borel.555By definition, the
measurable sets in $X/E_{G}^{X}$ are those collection of orbits whose union
set is a Borel set in $X$.

We refer to the proofs of the various parts of Theorem 5.2. The equivalence
$\left(1\right)\iff\left(2\right)$ is [16, Proposition 5.4.4]. The equivalence
$\left(1\right)\iff\left(3\right)$ is the most difficult part of this theorem
and a version of this in the presence of a measure on $X$ is attributed to von
Neumann (see [41, Appendix A] and in particular Theorem A.9). Without a
measure this is a result of Burgess, which holds even for Polish groups (see
[16, Corollary 5.4.12]); for a proof due to Kechris see [16, Proposition
5.4.10 and Theorem 5.4.11]. The equivalence $\left(3\right)\iff\left(4\right)$
appears in the proof of [16, Proposition 3.4.5]. For the implication
$\left(4\right)\implies\left(5\right)$, note that whenever $W_{0}$ is a
transversal for $E_{G}^{X}$, the natural map $W_{0}\to X/E_{G}^{X}$ that takes
a point to its orbit is a bijection, and thus when $W_{0}$ is a Borel set this
becomes a Borel bijection inducing a standard Borel structure on
$X/E_{G}^{X}$. Finally, the implication $\left(5\right)\implies\left(1\right)$
follows since there is a natural reduction $s:X\to X/E_{G}^{X}$.

###  5.2. The Structure of Smooth Borel $G$-Spaces

######  Definition 5.3.

Let $G$ be an lcsc group. A Borel $G$-datum is a pair

$\left(W_{0},\psi_{0}\right)$

consists of a standard Borel space $W_{0}$ and a Borel map
$\psi_{0}:W_{0}\to\mathbf{F}_{\mathrm{gr}}\left(G\right)$.

######  Construction 5.4.

For a Borel $G$-datum $\left(W_{0},\psi_{0}\right)$, define the space

$X_{0}=\left\\{\left(w,g\psi_{0}\left(w\right)\right):w\in W_{0},g\in
G\right\\},$

with the action

$g.\left(w,h\psi_{0}\left(w\right)\right)=\left(w,gh\psi_{0}\left(w\right)\right),\quad
g\in G,\left(w,h\psi_{0}\left(w\right)\right)\in X.$

######  Lemma 5.5.

The space $X_{0}$ becomes a (standard) Borel $G$-space.

###### Proof.

Consider the Borel map

$\Psi_{0}:W_{0}\times G\to
W_{0}\times\mathbf{F}\left(G\right),\quad\Psi_{0}\left(w,g\right)=\left(w,g\psi_{0}\left(w\right)\right).$

Thus, $X_{0}=\Psi_{0}\left(W_{0}\times G\right)$. In order to deduce that
$X_{0}$ is a Borel subset of $W_{0}\times\mathbf{F}\left(G\right)$, note that
the fiber of every point
$\left(w_{0},g_{0}\psi_{0}\left(w_{0}\right)\right)\in
W_{0}\times\mathbf{F}\left(G\right)$ under $\Psi_{0}$ is

$\displaystyle\Psi_{0}^{-1}\left(w_{0},g_{0}\psi_{0}\left(w_{0}\right)\right)$
$\displaystyle=\left\\{\left(w,g\right):\left(w,g\psi_{0}\left(w\right)\right)=\left(w_{0},g_{0}\psi_{0}\left(w_{0}\right)\right)\right\\}$
$\displaystyle=\left\\{w_{0}\right\\}\times\left\\{g\in
G:g\psi_{0}\left(w_{0}\right)=g_{0}\psi_{0}\left(w_{0}\right)\right\\}$
$\displaystyle=\left\\{w_{0}\right\\}\times g_{0}\psi_{0}\left(w_{0}\right).$

Pick any $\sigma$-compact Polish topology on $W_{0}$ that induces its Borel
structure (which always exists by the isomorphism theorem of standard Borel
spaces), and consider the product topology of this with the given
($\sigma$-compact) topology of $G$. Then with this product topology the space
$W_{0}\times G$ becomes $\sigma$-compact and Polish. It is also clear that
$\left\\{w_{0}\right\\}\times g_{0}\psi_{0}\left(w_{0}\right)$ is closed
$W_{0}\times G$, hence $\sigma$-compact. We thus found that all fibers of
$\Psi$ are $\sigma$-compact in some Polish topology that is compatible with
the Borel structure of $W_{0}\times G$, and by Kallman’s theorem (a strong
generalization of the fact the injective image of a Borel set is Borel; see
[41, Corollary A.6]) it follows that the image of $W_{0}\times G$ under
$\Psi$, namely $X_{0}$, is a Borel set. ∎

A crucial feature of the Borel $G$-space $X_{0}$ in Construction 5.4 is that
the stabilizers are explicit in terms of $\psi_{0}$:

######  Observation 5.6.

The stabilizer of a point $\left(w,h\psi_{0}\left(w\right)\right)\in X_{0}$ is

(5.2.1) $G_{\left(w,h\psi_{0}\left(w\right)\right)}=\left\\{g\in
G:gh\psi_{0}\left(w\right)=h\psi_{0}\left(w\right)\right\\}=\psi_{0}\left(w\right)^{h},$

which is also the stabilizer of $h\psi_{0}\left(w\right)$ in the coset
$G$-space $G/\psi_{0}\left(w\right)$.

We now come to the structure theorem of Borel $G$-spaces with smooth orbit
equivalence relation:

######  Theorem 5.7.

Let $G$ be an lcsc group and $X$ a Borel $G$-space. TFAE:

  1. (1)

$E_{G}^{X}$ is smooth.

  2. (2)

There exists a Borel $G$-datum $\left(W_{0},\psi_{0}\right)$ whose associated
Borel $G$-space is isomorphic to $X$.

We will start with a technical lemma. While it follows from [41, Lemma B.8],
we give a more direct proof and highlight its basic properties.

######  Lemma 5.8.

Let $X$ be a Borel $G$-space such that $E_{G}^{X}$ is smooth and fix a Borel
transversal $W_{0}$ for $E_{G}^{X}$. There exists a Borel map $r:X\to G$ such
that $r\left(x\right).x\in W_{0}$ for $\mu$-a.e. $x\in X$, and the map

$s:X\to W_{0},\quad s\left(x\right):=r\left(x\right).x,$

is a Borel selector. Every such map $r$ satisfies the following properties:

  1. (1)

$r\left(g.x\right)g.x=r\left(x\right).x$ for every $g\in G$ and $x\in X$.

  2. (2)

$r\left(g.x\right)gr\left(x\right)^{-1}\in G_{r\left(x\right).x}$ for every
$g\in G$ and $x\in X$.

###### Proof.

Extending the notation $G_{x}$ for stabilizers, for $x,x^{\prime}\in X$ let us
denote

$G_{x,x^{\prime}}=\left\\{g\in G:g.x=x^{\prime}\right\\}.$

Note that either $G_{x,x^{\prime}}=\emptyset$ or that for an arbitrary $g\in
G_{x,x^{\prime}}$ we have $G_{x,x^{\prime}}=gG_{x}$, then in view of Theorem
2.1 we have that $G_{x}$ and $G_{x,x^{\prime}}$ are closed for all
$x,x^{\prime}\in X$. Consequently, we may see each $G_{x,x^{\prime}}$ as an
element of the standard Borel space $\mathbf{F}\left(G\right)$ of closed
subsets of $G$. Consider the standard Borel subspace
$\mathbf{F}^{\ast}\left(X\right)=\mathbf{F}\left(X\right)\backslash\left\\{\emptyset\right\\}$.
By the Kuratowski–Ryll-Nardzewski Selection Theorem (see [16, Theorem 1.4.6])
there is a Borel map

$\xi:\mathbf{F}^{\ast}\left(G\right)\to G\text{ with }\xi\left(C\right)\in
C\text{ for every }C\in\mathbf{F}^{\ast}\left(G\right).$

Let us fix a Borel selector $s:X\to X$ for $E_{G}^{X}$. Observing that
$G_{x,s\left(x\right)}\neq\emptyset$ for every $x\in X$, we put

$r:X\to G,\quad r\left(x\right)=\xi\left(G_{x,s\left(x\right)}\right).$

We claim that this is a Borel map. Indeed, $r$ is the composition of mappings

$X\xrightarrow{x\mapsto\left(x,s\left(x\right)\right)}X\times
X\xrightarrow{\left(x,x^{\prime}\right)\mapsto
G_{x,x^{\prime}}}\mathbf{F}\left(G\right)\xrightarrow{C\mapsto\xi\left(C\right)}G;$

the first and the third are clearly Borel maps, and the second is a Borel map
by Becker–Kechris Theorem (see e.g. [16, Theorem 8.2.1]). It is then evident
that $r$ is the desired mapping.

Finally, property (1) follows from the fact that $r\left(g.x\right)g.x$ and
$r\left(x\right).x$ both lie in the same orbit and at the same time both
belong to the Borel transversal $W_{0}$. Property (2) follows from property
(1) since

$r\left(x\right).x=r\left(g.x\right)g.x=r\left(g.x\right)gr\left(x\right)^{-1}r\left(x\right).x.\qed$

###### Proof of Theorem 5.7.

First, if $X$ is (isomorphic to) the Borel $G$-space $X_{0}$ that is
constructed from a Borel $G$-datum $\left(W_{0},\psi_{0}\right)$, then the
graph

$\left\\{\left(w,\psi_{0}\left(w\right)\right):w\in W_{0}\right\\}\subset X$

of $\psi_{0}$ is clearly a transversal for $\left(X,\mu\right)$, so
$E_{G}^{X}$ is smooth in light of Theorem 5.2. Let us prove the converse.

Thus, suppose $X$ is a Borel $G$-space for which $E_{G}^{X}$ is smooth, so it
admits a Borel transversal $W_{0}\subset X$. Define the Borel $G$-datum
$\left(W_{0},\psi_{0}\right)$ by

$\psi_{0}:W_{0}\to\mathbf{F}_{\mathrm{gr}}\left(G\right),\quad\psi_{0}\left(w\right)=G_{w},\quad
w\in W_{0}.$

By Theorem 2.1 this is a Borel map. Let $X_{0}$ be the Borel $G$-space
associated with the Borel $G$-datum $\left(W_{0},\psi_{0}\right)$, and we show
that $X$ and $X_{0}$ are isomorphic as Borel $G$-spaces. Let us pick a Borel
function $r:X\to G$ corresponding to $W_{0}$ as in Lemma 5.8, and define the
Borel map

$\iota:X\to X_{0}\subset
W_{0}\times\mathbf{F}\left(G\right),\quad\iota\left(x\right)=\left(r\left(x\right).x,r\left(x\right)^{-1}G_{r\left(x\right).x}\right).$

In proving that $\iota$ is the desired isomorphism, we will constantly use the
properties of $r$ as mentioned in Lemma 5.8.

  * •

$\iota$ is equivariant. For every $g\in G$ and $x\in X$ we have

$\displaystyle\iota\left(g.x\right)$
$\displaystyle=\left(r\left(g.x\right)g.x,r\left(g.x\right)^{-1}G_{r\left(g.x\right)g.x}\right)$
$\displaystyle=\left(r\left(x\right).x,gr\left(x\right)^{-1}G_{r\left(x\right).x}\right)=g.\iota\left(x\right).$

  * •

$\iota$ is surjective. To this end note that for every $w\in W_{0}$ and $g\in
G$, the properties of $r$ as in Lemma 5.8 become

$w=r\left(w\right).w=r\left(g.w\right).w=r\left(g.w\right)g.w,$

and in particular $r\left(g.w\right)\in G_{w}$. We obtain that

$\displaystyle\iota\left(g.w\right)$
$\displaystyle=\left(r\left(g.w\right)g.w,r\left(g.w\right)^{-1}G_{r\left(g.w\right)g.w}\right)$
$\displaystyle=\left(w,gr\left(w\right)^{-1}G_{r\left(w\right).w}\right)=\left(w,gG_{w}\right)=\left(w,g\psi_{0}\left(w\right)\right).$

  * •

$\iota$ is injective. Define $\jmath:X_{0}\to X$ by

$\quad\jmath\left(w,g\psi_{0}\left(w\right)\right)=\alpha\left(g,w\right)=g.w,\quad\left(w,g\psi_{0}\left(w\right)\right)\in
X_{0}.$

We verify that $\jmath$ is well-defined. If
$\left(w,g\psi_{0}\left(w\right)\right)=\left(w^{\prime},g^{\prime}\psi_{0}\left(w^{\prime}\right)\right)$
then
$g\psi_{0}\left(w\right)=g^{\prime}\psi_{0}\left(w^{\prime}\right)=g^{\prime}\psi_{0}\left(w\right)$
hence $g.w=g^{\prime}.w$ so that

$\jmath\left(w,g\psi_{0}\left(w\right)\right)=g.w=g^{\prime}.w=g^{\prime}.w^{\prime}=\jmath\left(w^{\prime},g^{\prime}\psi_{0}\left(w^{\prime}\right)\right).$

We then obtain that $\jmath\circ\iota=\mathrm{Id}_{X}$: for every $x\in X$,

$\jmath\left(\iota\left(x\right)\right)=\jmath\left(r\left(x\right).x,r\left(x\right)^{-1}G_{r\left(x\right).x}\right)=r\left(x\right)^{-1}r\left(x\right).x=x.\qed$

Once we have established a canonical correspondence between Borel $G$-spaces
with smooth orbit equivalence relation and Borel $G$-datum, it is of interest
to identify their isomorphism type in terms of Borel $G$-datum:

######  Theorem 5.9.

For $i\in\left\\{0,1\right\\}$, let $\left(W_{i},\psi_{i}\right)$ be a Borel
$G$-datum and denote by $X_{i}$ the associated Borel $G$-space as in
Construction 5.4. TFAE:

  1. (1)

$X_{0}$ and $X_{1}$ are isomorphic as Borel $G$-spaces.

  2. (2)

There is a Borel isomorphism $\sigma:W_{0}\to W_{1}$ as well as a Borel map
$\tau:W_{0}\to G$, such that
$\psi_{0}\left(w\right)^{\tau\left(w\right)}=\psi_{1}\left(\sigma\left(w\right)\right)$
for all $w\in W_{0}$.

  3. (3)

There is a Borel isomorphism $\sigma:W_{0}\to W_{1}$ such that
$\psi_{0}\left(w\right)$ and $\psi_{1}\left(\sigma\left(w\right)\right)$ are
conjugated for all $w\in W_{0}$.

###### Proof.

(1)$\implies$(2): If $\pi:X_{0}\to X_{1}$ is an isomorphism of Borel
$G$-spaces, define

$V_{0}:=\left\\{\left(w,\psi_{0}\left(w\right)\right):w\in
W_{0}\right\\}\text{ and }V_{1}:=\pi\left(V_{0}\right).$

Since $V_{0}$ is a Borel transversal for $X_{0}$, $V_{1}$ is a Borel
transversal for $X_{1}$. Define

$\sigma:W_{0}\to
W_{1},\quad\sigma:W_{0}\xrightarrow{w\mapsto\left(w,\psi_{0}\left(w\right)\right)}V_{0}\xrightarrow{\pi}V_{1}\xrightarrow{x\mapsto
s_{1}\left(x\right)}W_{1},$

where $s_{1}:X_{1}\to W_{1}$ is a Borel selector for the transversal $W_{1}$.
Let us fix a Borel map $r_{1}:X_{1}\to V_{1}$ corresponding to $s_{1}$ as in
Lemma 5.8, and put

$\tau:W_{0}\to
G,\quad\tau\left(w\right)=r_{1}\left(\pi\left(w,\psi_{0}\left(w\right)\right)\right),\quad
w\in W_{0}.$

We obtain that

$\displaystyle\psi_{1}\left(\sigma\left(w\right)\right)$
$\displaystyle\quad=\psi_{1}\left(s_{1}\left(\pi\left(w,\psi_{0}\left(w\right)\right)\right)\right)$
$\displaystyle\quad=\mathrm{Stab}_{G}\left(s_{1}\left(\pi\left(w,\psi_{0}\left(w\right)\right)\right)\right)$
$\displaystyle\quad=\mathrm{Stab}_{G}\left(\pi\left(w,\psi_{0}\left(w\right)\right)\right)^{r_{1}\left(\pi\left(w,\psi_{0}\left(w\right)\right)\right)}$
$\displaystyle=\psi_{0}\left(w\right)^{\tau\left(w\right)},$

where in the last equality we used that $\pi$ is a Borel $G$-isomorphism so
that
$\psi_{0}\left(w\right)=\mathrm{Stab}_{G}\left(w,\psi_{0}\left(w\right)\right)=\mathrm{Stab}_{G}\left(\pi\left(w,\psi_{0}\left(w\right)\right)\right)$.

(2)$\implies$(1): Suppose Borel maps $\sigma$ and $\tau$ as in (2) are given.
Define a map $\pi:X_{0}\to X_{1}$ by

$\pi\left(w,g\psi_{0}\left(w\right)\right):=\left(\sigma\left(w\right),g\psi_{0}\left(w\right)\tau\left(w\right)^{-1}\right)=\left(\sigma\left(w\right),g\tau\left(w\right)^{-1}\psi_{1}\left(\sigma\left(w\right)\right)\right).$

One can directly verify that $\pi$ is well-defined and is equivariant. Using
that $\sigma$ is a Borel isomorphism, one can routinely verify that $\pi$ is
also a Borel isomorphism.

(2)$\iff$(3): One implication is trivial so we prove the converse. Let
$\sigma:W_{0}\to W_{1}$ be as in (3). Using Lemma 5.8, pick a Borel map
$r_{1}:X_{1}\to G$ for the Borel transversal
$\left\\{\left(w,\psi_{1}\left(w\right)\right):w\in W_{1}\right\\}$ of
$X_{1}$, and define

$\tau:W_{0}\to
G,\quad\tau\left(w\right)=r_{1}\left(\sigma\left(w\right),\psi_{1}\left(\sigma\left(w\right)\right)\right),\quad
w\in W_{0}.$

We claim that the bare assumption that $\psi_{0}$ and $\psi_{1}\circ\sigma$
are conjugated, implies that $\tau$ conjugates them. Since $\tau$ is a Borel
map this will complete the proof. Fix an arbitrary $w^{\prime}\in W_{0}$. As
$\psi_{0}\left(w^{\prime}\right)$ and
$\psi_{1}\left(\sigma\left(w^{\prime}\right)\right)$ are conjugated, say by
$g^{\prime-1}\in G$, we have that

$\displaystyle\left(\sigma\left(w^{\prime}\right),\psi_{0}\left(w^{\prime}\right)g^{\prime}\right)$
$\displaystyle=\left(\sigma\left(w^{\prime}\right),g^{\prime}\psi_{1}\left(\sigma\left(w^{\prime}\right)\right)\right)$
$\displaystyle=g^{\prime}.\left(\sigma\left(w^{\prime}\right),\psi_{1}\left(\sigma\left(w^{\prime}\right)\right)\right)$
$\displaystyle\quad\in
G.\left(\sigma\left(w^{\prime}\right),\psi_{1}\left(\sigma\left(w^{\prime}\right)\right)\right)\subset
X_{1},$

and it follows from the transversal property that

$\tau\left(w^{\prime}\right).\left(\sigma\left(w^{\prime}\right),\psi_{0}\left(w^{\prime}\right)g^{\prime}\right)=\left(\sigma\left(w^{\prime}\right),\psi_{1}\left(\sigma\left(w^{\prime}\right)\right)\right).$

Since the action is leftwise,
$\mathrm{Stab}_{G}\left(\sigma\left(w^{\prime}\right),\psi_{0}\left(w^{\prime}\right)g^{\prime}\right)=\psi_{0}\left(w^{\prime}\right)$.
We deduce

$\displaystyle\psi_{1}\left(\sigma\left(w^{\prime}\right)\right)$
$\displaystyle=\mathrm{Stab}_{G}\left(\sigma\left(w^{\prime}\right),\psi_{1}\left(\sigma\left(w^{\prime}\right)\right)\right)$
$\displaystyle=\mathrm{Stab}_{G}\left(\tau\left(w^{\prime}\right).\left(\sigma\left(w^{\prime}\right),\psi_{0}\left(w^{\prime}\right)g^{\prime}\right)\right)=\psi_{0}\left(w^{\prime}\right)^{\tau\left(w^{\prime}\right)}.\qed$

The following theorem was proved by Kosloff & Soo in the context of
nonsingular $G$-spaces [23, Theorems 11 & 14]. In lights of Theorem 5.7 and
Lemma 5.8 we formulate a Borel version of this theorem.

######  Theorem 5.10 (Following Kosloff–Soo).

Let $G$ be an lcsc group and $X$ a _free_ Borel $G$-space. TFAE:

  1. (1)

$X/E_{G}^{X}$ is an uncountable standard Borel space.

  2. (2)

$X$ is _$G$ -factor universal_: it factors onto any other Borel $G$-space.

###### Proof.

(1)$\implies$(2): By Theorem 5.2, $X/E_{G}^{X}$ being a standard Borel space
is equivalent to that $E_{G}^{X}$ is smooth. Then by Theorem 5.7 we deduce
that $X$ is isomorphic to the Borel $G$-space that is constructed out of the
Borel $G$-datum $\left(W_{0},\psi_{0}\right)$, where $W_{0}$ is an uncountable
standard Borel space and $\psi_{0}\equiv\left\\{e\right\\}$ (as the action is
free). That is, $X\cong W_{0}\times G$.

Let $Y$ be any Borel $G$-space and consider the diagonal Borel $G$-space
$X\times Y$. Since the action on $X$ is free then so is the action on $X\times
Y$. Furthermore, we claim that since the action of $G$ on $X$ is free and
since $E_{G}^{X}$ is smooth, then also $E_{G}^{X\times Y}$ is smooth. Indeed,
if $s:X\to X$ is a Borel selector for $E_{G}^{X}$, using Lemma 5.8 we may
write $s\left(x\right)=r\left(x\right).x$ for all $x\in X$, and define

$S:X\times Y\to X\times Y,\quad
S\left(x,y\right)=\left(r\left(x\right).x,r\left(x\right).y\right),\quad\left(x,y\right)\in
X\times Y.$

One can easily verify, using that the action on $X$ is free, that $S$ is a
Borel selector for $E_{G}^{X\times Y}$. Then by Theorem 5.7, $X\times Y$ is
isomorphic to the space constructed from the Borel $G$-datum
$\left(W_{1},\psi_{1}\right)$, where $W_{1}$ is some standard Borel space and
$\psi_{1}\equiv\left\\{e\right\\}$. That is, $X\times Y\cong W_{1}\times G$.

Since $X/E_{G}^{X}$ is uncountable, $W_{0}$ is an uncountable standard Borel
space, hence there is a surjective Borel map $f:W_{0}\to W_{1}$. It is then
evident that the desired factor map is obtain by the composition

$X\cong W_{0}\times
G\xrightarrow{\left(w,g\right)\mapsto\left(f\left(w\right),g\right)}W_{1}\times
G\cong X\times Y\xrightarrow{\left(x,y\right)\mapsto y}Y.$

(2)$\implies$(1): Pick any free Borel $G$-space $Y$ such that $E_{G}^{Y}$ is
smooth and has uncountably many orbits, e.g. $Y=\left[0,1\right]\times G$. As
$X$ is factor universal, there is a factor map $\pi:X\to Y$. Since $E_{G}^{Y}$
admits uncountably many orbits and since $\pi$ maps orbit to orbit, it follows
that $E_{G}^{X}$ admits uncountably many orbits. Let us show that $E_{G}^{X}$
is smooth. Pick some Borel selector $s:Y\to Y$ for $E_{G}^{Y}$, and using
Lemma 5.8 we may write $s\left(y\right)=r\left(y\right).y$ for all $y\in Y$.
Define

$S:X\to X,\quad S\left(x\right)=r\left(\pi\left(x\right)\right).x,\quad x\in
X.$

Of course $S\left(x\right)\in G.x$ for every $x\in X$. The selector property
can be verified using the freeness of the action as well as the basic
properties of $r$ as in Lemma 5.8. ∎

##  6\. Borel Description of Total Dissipativity

The following theorem in the case $G=\mathbb{R}$ was proved by Krengel, and in
the case $G=\mathbb{R}^{d}$ by Rosinski. In the following we formulate this
generally, and the proof we provide is greatly inspired by Rosinski’s proof
[32, Theorem 2.2].

######  Theorem 6.1 (Following Krengel–Rosinski).

If $\left(X,\mu\right)$ is a totally dissipative nonsingular $G$-space, then
$E_{G}^{X}$ is essentially smooth.

###### Proof.

We prove that on the $\mu$-conull set $\mathcal{T}$, the measurable union of
the transient sets as in Proposition 3.3, $E_{G}^{X}$ is countably separated
by Borel functions that we denote $\left\\{u_{n,m}:n,m\in\mathbb{N}\right\\}$.
We carry this in a few steps.

#### Step 1:

(constructing $\left\\{u_{n,m}:n,m\in\mathbb{N}\right\\}$).

We recall that $\mathcal{T}$ is the measurable union of the class of transient
sets of $\left(X,\mu\right)$. As such, the Exhaustion Lemma 2.2 guarantees
that $\mathcal{T}$ is the union of countably many transient sets, say
$\left\\{T_{1},T_{2},\dotsc\right\\}$. Let us also apply Varadarajan’s compact
model theorem [39, Theorem 5.7, p. 160]. Thus, there is a compact $G$-space
$\widebar{X}$ (i.e. $\widebar{X}$ is a compact metric space with a jointly
continuous action of $G$), and a Borel $G$-embedding of $X$ in $\widebar{X}$.
We thus can fix a countable base
$\mathcal{U}=\left\\{U_{1},U_{2},\dotsc\right\\}$ for the topology of
$\widebar{X}$, and we may further assume that it is closed to finite unions
and finite intersections. We now put, for every $n,m\in\mathbb{N}$, the
function

$u_{n,m}:X\to\mathbb{R},\quad u_{n,m}\left(x\right):=\lambda\left(R_{U_{n}\cap
T_{m}}\left(x\right)\right)=\int_{G}1_{U_{n}\cap
T_{m}}\left(g^{-1}.x\right)d\lambda\left(g\right).$

We recall that each $U_{n}\cap T_{m}$, being a subset of a transient set is
again a transient set, so that $u_{n,m}$ takes only finite values on the
$\mu$-conull set $\mathcal{T}$.

#### Step 2:

(a positivity property of $\left\\{u_{n,m}:n,m\in\mathbb{N}\right\\}$).

We will show that for every $x_{0}\in\mathcal{T}$ there are
$n_{0},m_{0}\in\mathbb{N}$ such that $u_{n_{0},m_{0}}\left(x_{0}\right)>0$.
Since $\left\\{U_{n}\cap T_{m}:n,m\in\mathbb{N}\right\\}$ covers
$\mathcal{T}$, for every $g\in G$ also $\left\\{g.U_{n}\cap
g.T_{m}:n,m\in\mathbb{N}\right\\}$ covers $\mathcal{T}$. Thus, for every
$x_{0}\in\mathcal{T}$ we have

$\sum\nolimits_{n,m\in\mathbb{N}}1_{U_{n}\cap
T_{m}}\left(g^{-1}.x_{0}\right)\geq 1.$

It then follows that

$\sum\nolimits_{n,m\in\mathbb{N}}u_{n,m}\left(x_{0}\right)=\int_{G}\left(\sum\nolimits_{n,m\in\mathbb{N}}1_{U_{n}\cap
T_{m}}\left(g^{-1}.x_{0}\right)\right)d\lambda\left(g\right)=+\infty,$

which readily implies the desired property.

#### Step 3:

(each $u_{n,m}$ is constant on along orbits).

Let $\left(x_{0},h.x_{0}\right)\in E_{G}^{X}$ for some $x_{0}\in X$ and $h\in
G$. By the identity (3.0.1) it follows that for every $n,m\in\mathbb{N}$,

$\displaystyle u_{n,m}\left(h.x_{0}\right)$
$\displaystyle=\lambda\left(R_{U_{n}\cap T_{m}}\left(h.x_{0}\right)\right)$
$\displaystyle=\lambda\left(h.R_{U_{n}\cap
T_{m}}\left(x_{0}\right)\right)=\lambda\left(R_{U_{n}\cap
T_{m}}\left(x_{0}\right)\right)=u_{n,m}\left(x_{0}\right).$

#### Step 4:

($E_{G}^{X}$ is separated by $\left\\{u_{n,m}:n,m\in\mathbb{N}\right\\}$).

Let $\left(x_{0},x_{1}\right)\notin E_{G}^{X}$. Using Step 2 we may pick some
$n_{0},m_{0}\in\mathbb{N}$ for $x_{0}$ with
$u_{n_{0},m_{0}}\left(x_{0}\right)>0$. If
$u_{n_{0},m_{0}}\left(x_{0}\right)\neq u_{n_{0},m_{0}}\left(x_{1}\right)$ we
are done, so assume otherwise that

$c:=u_{n_{0},m_{0}}\left(x_{0}\right)=u_{n_{0},m_{0}}\left(x_{1}\right)>0.$

By the inner regularity of $\lambda$, there is a compact symmetric set
$K\subset G$ such that

$\int_{K}1_{U_{n_{0}}\cap T_{m_{0}}}\left(g^{-1}.x_{i}\right)>c/2,\quad
i\in\left\\{0,1\right\\}.$

By the continuity of the action on $\overline{X}$, the sets $K.x_{0}$ and
$K.x_{1}$ are compact in $\overline{X}$ and, since
$\left(x_{0},x_{1}\right)\notin E_{G}^{X}$, necessarily $K.x_{0}\cap
K.x_{1}=\emptyset$.

As $\mathcal{U}$ is closed to finite unions, we may pick $n_{1}\in\mathbb{N}$
such that

$K.x_{0}\subset U_{n_{1}}\text{ and }K.x_{1}\cap U_{n_{1}}=\emptyset.$

As $\mathcal{U}$ is closed to finite intersections, we may pick
$n_{2}\in\mathbb{N}$ such that

$U_{n_{0}}\cap U_{n_{1}}=U_{n_{2}}.$

We then claim that $u_{n_{2},m_{0}}$ assigns different values to $x_{0}$ and
$x_{1}$.

On one hand, from $K.x_{0}\cap U_{n_{0}}\subset U_{n_{0}}\cap
U_{n_{1}}=U_{n_{2}}$ we have

$\displaystyle u_{n_{2},m_{0}}\left(x_{0}\right)$
$\displaystyle=\int_{G}1_{U_{n_{2}}\cap
T_{m_{0}}}\left(g^{-1}.x_{0}\right)d\lambda\left(g\right)$
$\displaystyle\geq\int_{G}1_{K.x_{0}\cap U_{n_{0}}\cap
T_{m_{0}}}\left(g^{-1}.x_{0}\right)d\lambda\left(g\right)$
$\displaystyle\geq\int_{K}1_{U_{n_{0}}\cap
T_{m_{0}}}\left(g^{-1}.x_{0}\right)d\lambda\left(g\right)>c/2.$

On the other hand, from $K.x_{1}\cap U_{n_{2}}\subset K.x_{1}\cap
U_{n_{1}}=\emptyset$ we have

$\int_{K}1_{U_{n_{2}}\cap
T_{m_{0}}}\left(g^{-1}.x_{1}\right)d\lambda\left(g\right)\leq\int_{G}1_{K.x_{1}\cap
U_{n_{2}}\cap T_{m_{0}}}\left(g^{-1}.x_{1}\right)d\lambda\left(g\right)=0,$

hence

$\displaystyle u_{n_{2},m_{0}}\left(x_{1}\right)$
$\displaystyle=\int_{G\backslash K}1_{U_{n_{2}}\cap
T_{m_{0}}}\left(g^{-1}.x_{1}\right)d\lambda\left(g\right)$
$\displaystyle\leq\int_{G\backslash K}1_{U_{n_{0}}\cap
T_{m_{0}}}\left(g^{-1}.x_{1}\right)d\lambda\left(g\right)$
$\displaystyle=u_{n_{0},m_{0}}\left(x_{1}\right)-\int_{K}1_{U_{n_{0}}\cap
T_{m_{0}}}\left(g^{-1}.x_{1}\right)d\lambda\left(g\right)$
$\displaystyle<c-c/2=c/2.$

Thus, $u_{n_{2},m_{0}}\left(x_{0}\right)\neq
u_{n_{2},m_{0}}\left(x_{1}\right)$, concluding the proof of Step 4.

Finally, by the virtue of Steps 3 and 4, $E_{G}^{X}$ is countably separated by
Borel functions, and from Theorem 5.2 we deduce that it is smooth. ∎

Dissipative nonsingular $G$-spaces may not be free, as the coming Construction
7.3 suggests. However, a fundamental phenomenon in dissipative nonsingular
$G$-spaces is that all stabilizers are compact:

######  Theorem 6.2.

If $\left(X,\mu\right)$ is a totally dissipative nonsingular $G$-space, then
the ($G$-invariant) set $X_{\mathrm{c}}\subset X$ of points whose stabilizers
are compact is $\mu$-conull.

For the proof of Theorem 6.2 we will need the following elementary fact,
generalizing that an lcsc group with finite Haar measure is compact. The proof
is an adaptation of [28, Chapter II, §5, Proposition 4].

######  Lemma 6.3.

Let $H$ be an lcsc group with a left Haar measure $\lambda_{H}$. If
$\psi:H\to\mathbb{R}_{+}$ is a continuous multiplicative group homomorphism
with $\int_{H}\psi d\lambda_{H}<+\infty$, then $\psi$ is the trivial
homomorphism and $H$ is compact.

###### Proof.

If $\psi$ is bounded it is necessarily the trivial homomorphism hence
$\lambda_{H}\left(H\right)=\int_{H}\psi d\lambda_{H}<+\infty$, in which case
$H$ is compact. Suppose toward a contradiction that $\psi$ is unbounded and,
since it is continuous, $H$ is non-compact and $\psi$ is unbounded outside
every compact set in $H$. Fix some compact set $K\subset H$ such that

$I_{K}:=\int_{K}\psi d\lambda_{H}\in\left(0,+\infty\right).$

Construct a sequence $h_{1},h_{2},\dotsc\in H$ inductively as follows. After
defining $h_{1},\dotsc,h_{n}$ such that $h_{1}K,\dotsc,h_{n}K$ are pairwise
disjoint and $\psi\left(h_{i}\right)\geq 1$ for $i=1,\dotsc,n$, pick $h_{n+1}$
so that $h_{n+1}K$ is disjoint from $h_{1}K\cup\dotsc\cup h_{n}K$ ($H$ is non-
compact) and $\psi\left(h_{n+1}\right)\geq 1$ ($\psi$ is unbounded outside
$h_{1}K\cup\dotsc\cup h_{n}K$).

Let $n\in\mathbb{N}$ be arbitrary. By the construction of $h_{1},h_{2},\dotsc$
we have

$\sum\nolimits_{i=1}^{n}1_{K}\left(h_{i}^{-1}h\right)\leq 1,\quad h\in H,$

and, using the left invariance of $\lambda_{H}$ and that $\psi$ is a
homomorphism, we get

$\displaystyle+\infty$
$\displaystyle>\int_{H}\psi\left(h\right)d\lambda_{H}\left(h\right)$
$\displaystyle\geq\sum\nolimits_{i=1}^{n}\int_{H}1_{K}\left(h_{i}^{-1}h\right)\psi\left(h\right)d\lambda_{H}\left(h\right)$
$\displaystyle=\int_{K}\psi\left(h\right)d\lambda_{H}\left(h\right)\cdot\sum\nolimits_{i=1}^{n}\psi\left(h_{i}\right)\geq
I_{K}\cdot n.$

Since $I_{K}\in\left(0,+\infty\right)$ and $n\in\mathbb{N}$ is arbitrary we
got to a contradiction. ∎

In the following proof we make use of the construction of _measures on
homogeneous spaces_ and _Weil’s formula_. See Theorem A.3 in Appendix A.

###### Proof of Theorem 6.2.

Pick some $f\in L_{+}^{1}\left(X,\mu\right)$ and let $x\in\mathcal{D}_{f}$ be
fixed from now until (nearly) the end of the proof. In light of Proposition
1.1 we may assume that $\mu$ is $G$-invariant, and thus that $\nabla_{g}\equiv
1$ for every $g\in G$. We apply the construction of measures on homogeneous
spaces as in Theorem A.3 to the group pair $G_{x}\lessdot G$ (recall Theorem
2.1). Thus, $G$ has the Haar measure $\lambda$ and the corresponding modular
function $\varDelta$, and for $G_{x}$ we fix a Haar measure $\lambda_{x}$ and
the corresponding modular function $\varDelta_{x}$. Pick a rho-function
$\rho_{x}:G\to\mathbb{R}_{+}$ for $G_{x}\lessdot G$ and a quasi-invariant
measure $\kappa_{x}$ on $G/G_{x}$ with respect to $\rho_{x}$. Applying Weil’s
formula (A.0.1) to the function $G\to\mathbb{R}_{+}$, $g\mapsto
f\left(g.x\right)$, we obtain

$\displaystyle S_{f}^{G}\left(x\right)$
$\displaystyle=\int_{G}f\left(g.x\right)d\lambda\left(g\right)$
$\displaystyle=\int_{G/G_{x}}\left[\int_{G_{x}}f\left(gc.x\right)\varDelta\left(c\right)\varDelta_{x}\left(c\right)^{-1}d\lambda_{x}\left(c\right)\right]\rho\left(g\right)^{-1}d\kappa_{x}\left(gG_{x}\right)$
$\displaystyle=\left[\int_{G/G_{x}}f\left(g.x\right)\rho\left(g\right)^{-1}d\kappa_{x}\left(gG_{x}\right)\right]\cdot\left[\int_{G_{x}}\varDelta\left(c\right)\varDelta_{x}\left(c\right)^{-1}d\lambda_{x}\left(c\right)\right].$

Since $x\in\mathcal{D}_{f}$ we have $S_{f}^{G}\left(x\right)<+\infty$, and
since all terms in the above expression are strictly positive we obtain that

$\int_{G_{x}}\varDelta\left(c\right)\varDelta_{x}\left(c\right)^{-1}d\lambda_{x}\left(c\right)<+\infty.$

The map $G_{x}\to\mathbb{R}_{+}$,
$c\mapsto\varDelta\left(c\right)\varDelta_{x}\left(c\right)^{-1}$, is a
continuous group homomorphism, so by Lemma 6.3, applied to $H=G_{x}$ and
$\psi:c\mapsto\varDelta\left(c\right)/\varDelta_{x}\left(c\right)$, we deduce
that $G_{x}$ is compact.

Letting $X_{\mathrm{c}}$ be the set of points in $X$ whose stabilizer is
compact (which is Borel by Theorem 2.1), we have shown that
$\mathcal{D}_{f}\subset X_{\mathrm{c}}$ modulo $\mu$. Since
$\left(X,\mu\right)$ is totally dissipative we deduce that $X_{\mathrm{c}}$ is
$\mu$-conull. ∎

###### Proof of Theorem D.

The implication (1)$\implies$(2) is the virtue of Theorems 6.1 and 6.2. Let us
show the implication (2)$\implies$(1). By the assumption, the $G$-invariant
set $X_{\mathrm{c}}$ of points whose stabilizers are compact is $\mu$-conull,
and by the other assumption we may find a Borel transversal $W_{0}$ on some
$\mu$-conull subset of $X_{\mathrm{c}}$, thus the stabilizer of _every_ point
in $W_{0}$ is compact. Recall that by identity (3.0.1), a translation of a
transient set is again transient. Then by the property of $W_{0}$ as a
transversal, in order to prove that $\mu$-a.e. $x\in X$ is contained in a
$\mu$-positive transient set it is sufficient to show that this is true for
every point in $W_{0}$. Since $G.W_{0}=X$ and $G$ is lcsc, it is sufficient to
show that $C.W_{0}$ is a transient set for every compact set $C\subset G$.
Indeed, note the identity

$R_{C.W_{0}}\left(w\right)=G_{w}C^{-1},\quad w\in W_{0},$

by which for $\mu$-a.e. $x\in X$ there is $g_{0}\in G$ with $g_{0}.x\in
W_{0}$, hence

$R_{C.W_{0}}\left(x\right)=g_{0}^{-1}R_{C.W_{0}}\left(g_{0}.x\right)=g_{0}^{-1}G_{g_{0}.x}C^{-1}.$

Since $G_{g_{0}.x}$ and $C^{-1}$ are both compact so is
$g_{0}^{-1}G_{g_{0}.x}C^{-1}$, and we conclude that $C.W_{0}$ is strongly
transient and a fortiori transient. This shows that $\left(X,\mu\right)$ is
totally dissipative, completing the proof of Theorem D. ∎

The fact that in the previous proof we obtained strongly transient sets
enables us to complete the proof of Theorem B:

###### Proof of Theorem B.

As we mentioned before, the proof of the version with recurrent and transient
sets follows directly from Theorem 3.2, so we prove now the version with
strongly recurrent and strongly transient sets. Let $\mathfrak{T}$ and
$\mathfrak{S}$ be the collections of transient sets and strongly transient
sets, respectively. Since both are hereditary they admit measurable unions
that we denote $\mathcal{T}$ and $\mathcal{S}$, respectively. As we proved in
Theorem 3.2, the Hopf Decomposition $X=\mathcal{C}\sqcup\mathcal{D}$ is
determined by the property that
$\mu\left(\mathcal{D}\triangle\mathcal{T}\right)=0$, so it is sufficient to
show that $\mathcal{T}=\mathcal{S}$ modulo $\mu$. One one hand, as every
strongly transient set is transient it is clear that
$\mathcal{S}\subset\mathcal{D}$ modulo $\mu$. Conversely, in the proof of
Theorem 3.2 we have shown that $\mu$-a.e. point in $\mathcal{D}$ is contained
in a $\mu$-positive strongly transient set, so $\mathcal{D}$ is a measurable
union of the strongly transient sets, thus $\mathcal{D}=\mathcal{S}$ modulo
$\mu$.

From this we immediately see that every $\mu$-positive subset of $\mathcal{D}$
has a $\mu$-positive strongly transient subset. We need to show that every
subset of $\mathcal{C}$ is strongly recurrent. Given a Borel set
$A\subset\mathcal{C}$, we consider the set

$T_{A}:=\left\\{x\in A:R_{T}\left(x\right)\text{ is relatively
compact}\right\\},$

and for $r>0$ consider the set

$T_{A,r}:=\left\\{x\in T_{A}:R_{T_{A}}\left(x\right)\subset B_{r}\right\\},$

where $B_{r}$ is the ball of radius $r>0$ with respect to some proper
compatible metric on $G$. We then see that

$g\in R_{T_{A,r}}\left(x\right)\iff\left(g^{-1}.x\in
T_{A}\right)\wedge\left(R_{T_{A}}\left(x\right)\subset B_{r}\right),$

hence

$R_{T_{A,r}}\left(x\right)=\begin{cases}R_{T_{A}}\left(x\right)&R_{T_{A}}\left(x\right)\subset
B_{r}\\\ \emptyset&\text{otherwise},\end{cases}$

and in any case $R_{T_{A,r}}\left(x\right)\subset B_{r}$, thus $T_{A,r}$ is
strongly transient and is contained in $A$, and by the assumption
$\mu\left(T_{A,r}\right)=0$. Since $T_{A,r}\nearrow T_{A}$ as
$r\nearrow+\infty$ we deduce that $\mu\left(T_{A}\right)=0$. Thus, $A$ is
strongly recurrent. ∎

##  7\. The Structure Theorem for Totally Dissipative Actions

###  7.1. Free Actions: Translation $G$-Spaces

A classical theorem of Krengel asserts that every totally dissipative
nonsingular flow (i.e. $\mathbb{R}$-space) is isomorphic to a translation
flow. The following construction is the obvious generalization of translation
flows from $\mathbb{R}$ to general lcsc groups.

######  Construction 7.1.

A translation $G$-space is a measure preserving $G$-space of the form
$\left(W_{0}\times G,\nu_{0}\otimes\lambda\right)$, for some standard measure
space $\left(W_{0},\nu_{0}\right)$, with the action that is given by

$g.\left(w,h\right)=\left(w,gh\right),\quad\left(w,h\right)\in W_{0}\times
G,\quad g\in G.$

We will refer to it as the translation $G$-space attached to
$\left(W_{0},\nu_{0}\right)$.

It is clear that every translation $G$-spaces is free and generally non-
ergodic. Indeed, a translation $G$-space that is attached to a nontrivial
measure space (i.e. not all Borel sets are null or conull) is never ergodic.
Krengel’s theorem identify translation flows as the only source for totally
dissipative nonsingular $\mathbb{R}$-spaces, and this was generalized by
Rosinski to $\mathbb{R}^{d}$. In particular, this shows that nonsingular
$\mathbb{R}^{d}$-spaces are essentially free, and in light of Theorem 6.2 this
should not come as a surprise. As for general lcsc groups, we will generalize
this in showing that non-freeness is the only obstruction to being (isomorphic
to) a translation $G$-space.

######  Theorem 7.2.

Let $G$ be an lcsc group and $\left(X,\mu\right)$ an _essentially free_
nonsingular $G$-space with Hopf Decomposition
$X=\mathcal{C}\sqcup\mathcal{D}$. TFAE:

  1. (1)

$\left(X,\mu\right)$ is totally dissipative.

  2. (2)

$E_{G}^{X}$ is essentially smooth and $\mu$ admits a $G$-invariant equivalent
measure.

  3. (3)

$\left(X,\mu\right)$ is isomorphic to a translation $G$-space.

###### Proof of Theorem 7.2.

We will follow the lines of the proof in [32, Theorem 2.2] with the assistence
of Theorem 5.2.

#### (1)$\implies$(2)

This holds in general by Proposition 1.1 and Theorem 6.1.

#### (2)$\implies$(3)

Since $E_{G}^{X}$ is essentially smooth, by Theorem 5.2 it admits a Borel
transversal $W_{0}\subset X$. Consider the restriction of the action map to
$W_{0}$, namely

$\Phi:W_{0}\times G\to X,\quad\Phi\left(w,g\right)=g.w.$

Using the transversal property of $W_{0}$ and that the action of $G$ on $S$ is
free, $\Phi$ is one-to-one Borel map, so it admits a Borel inverse that we
denote

$\Phi^{-1}\left(x\right)=\left(\omega\left(x\right),\gamma\left(x\right)\right)\text{
where }\omega:X\to W_{0}\text{ and }\gamma:X\to G.$

We claim that

$\omega\left(g.x\right)=\omega\left(x\right)\text{ and
}\gamma\left(g.x\right)=g\gamma\left(x\right)\text{ for all }x\in X,\,\,g\in
G.$

Indeed, the first identity is by the transversal property, and this implies

$g\gamma\left(x\right).\omega\left(x\right)=g.x=\gamma\left(g.x\right).\omega\left(g.x\right)=\gamma\left(g.x\right).\omega\left(x\right),$

which implies the second identity since the action is free.

Fix some invariant measure $\eta_{0}$ equivalent to $\mu$. Consider the
measure $\mu_{1}:=\eta_{0}\circ\Phi$. Then for every Borel sets $A\subset
W_{0}$ and $B\subset G$, and every $g\in G$,

$\displaystyle\mu_{1}\left(A\times g^{-1}B\right)$
$\displaystyle\quad=\eta_{0}\left(\Phi\left(A\times g^{-1}B\right)\right)$
$\displaystyle\quad=\eta_{0}\left(x\in X:\omega\left(x\right)\in
A,g\omega\left(x\right)\in B\right)$ $\displaystyle\quad=\eta_{0}\left(x\in
X:\omega\left(g.x\right)\in A,\omega\left(g.x\right)\in B\right)$
$\displaystyle\quad=\eta_{0}\left(x\in X:\omega\left(x\right)\in
A,\omega\left(x\right)\in B\right)$ $\displaystyle=\mu_{1}\left(A\times
B\right).$

It follows that for every Borel set $A\subset W_{0}$, the map
$B\mapsto\mu_{0}\left(A\times B\right)$ is a left invariant Borel
$\sigma$-finite measure on $G$, so it is $\lambda$ up to a positive constant
(see Theorem A.4(2)). Thus, there is a positive constant
$\mu_{0}\left(A\right)$ such that

$\mu_{1}\left(A\times B\right)=\mu_{0}\left(A\right)\lambda\left(B\right).$

As $\mu_{1}$ is a measure, $A\mapsto\mu_{0}\left(A\right)$ defines a measure
on $W_{0}$ so we deduce (3).

#### (3)$\implies$(1)

For a translation $G$-space attached to $\left(W_{0},\nu_{0}\right)$, since it
is measure preserving we may assume that $\nabla\left(w,h\right)=1$ for every
$\left(w,h\right)\in W_{0}\times G$, so for every $f\in
L_{+}^{1}\left(W_{0}\times G,\nu_{0}\otimes\lambda\right)$ we have

$S_{f}^{G}\left(w,h\right)=\int_{G}f\left(w,gh\right)d\lambda\left(g\right).$

By the Fubini Theorem and the integrability of $f$ we have

$\displaystyle\int_{W_{0}}S_{f}^{G}\left(w,h\right)d\nu_{0}\left(w\right)$
$\displaystyle=\iint_{W_{0}\times
G}f\left(w,gh\right)d\nu_{0}\otimes\lambda\left(w,h\right)$
$\displaystyle=\iint_{W_{0}\times
G}f\left(w,h\right)d\nu_{0}\otimes\lambda\left(w,h\right)<+\infty.$

It follows that $S_{f}^{G}\left(w,h\right)<+\infty$ for
$\nu_{0}\otimes\lambda$-a.e. $\left(w,h\right)\in W_{0}\times G$, concluding
that the translation $G$-space attached to $\left(W_{0},\nu_{0}\right)$ is
totally dissipative. ∎

###  7.2. Ergodic Actions: Compactly Fibered Coset $G$-Spaces

We now come to the construction of compactly fibered coset $G$-spaces,
following the construction of measures on homogeneous spaces (Appendix
LABEL:Appendix:_Measures_on_homogeneous_spaces).

######  Construction 7.3.

A coset $G$-space is a nonsingular $G$-space of the form
$\left(G/K,\kappa\right)$, where $K$ is a closed subgroup of $G$, with the
action

$g.hK=ghK,\quad g\in G,\,hK\in G/K,$

and $\kappa$ is a quasi-invariant measure (which is in fact unique up to
measure class). When $K$ is compact we will call $\left(G/K,\kappa\right)$ a
compactly fibered coset $G$-space.

The stabilizers of a point $hK\in G/K$ is the conjugation $K^{h}=h^{-1}Kh$,
thus the action is not essentially free as soon as $K$ is nontrivial. As for
dissipativity we have:

######  Lemma 7.4.

A coset $G$-space is totally dissipative iff it is compactly fibered.

###### Proof.

In the nonsingular $G$-space $\left(G/K,\kappa\right)$, all the stabilizers
are compact exactly when $K$ is compact. Thus, if $\left(G/K,\kappa\right)$ is
totally dissipative then from Theorem 6.2 it follows that $K$ is compact.
Conversely, clearly $E_{G}^{G/K}$ is smooth (it admits a one point Borel
transversal), so if $K$ is compact then every stabilizer is compact, so from
Theorem D it follows that $\left(G/K,\kappa\right)$ is totally dissipative. ∎

We can now prove Theorem F:

###### Proof of Theorem F.

We have shown in Lemmas 7.4 that a compactly fibered coset $G$-space is
totally dissipative. Let us show the converse, thus $\left(X,\mu\right)$ is an
ergodic nonsingular $G$-space which is totally dissipative. We start by
showing that $\left(X,\mu\right)$ is essentially transitive. By Theorem 5.2,
since $E_{G}^{X}$ is smooth the space $X/E_{G}^{X}$ is standard Borel and the
quotient map $\pi:X\rightarrow X/E_{G}^{X}$ is a Borel map. It is then
sufficient to show that the pushforward measure $\pi_{\ast}\mu$ on
$X/E_{G}^{X}$ is a Dirac measure. Indeed, if $A,B\subset X/E_{G}^{X}$ are
$\pi_{\ast}\mu$-positive disjoint Borel sets, then
$\pi^{-1}\left(A\right),\pi^{-1}\left(B\right)$ are $\mu$-positive disjoint
Borel sets, and they are clearly $G$-invariant, a contradiction to ergodicity.

Now that we came to that $\left(X,\mu\right)$ is essentially transitive, it is
isomorphic as a Borel $G$-space to $\left(G.x_{0},\mu\right)$ for some
$x_{0}\in X$ (orbits are always is $G$-invariant). Then the map $q:G.x_{0}\to
G/G_{x_{0}}$, $q:g.x_{0}\mapsto gG_{x_{0}}$, is an isomorphism of the measure
preserving $G$-spaces $\left(G.x_{0},\mu\right)$ and
$\left(G/G_{x_{0}},q_{\ast}\mu\right)$. However, by Theorem 6.2 the stabilizer
$G_{x_{0}}$ is a compact subgroup of $G$, so by Theorem A.4(1) there is a
unique measure class of quasi-invariant measures on $G/G_{x_{0}}$, namely
$q_{\ast}\mu$ is equivalent to the measure $\kappa$ that forms the compactly
fibered coset $G$-space $\left(G/G_{x_{0}},\kappa\right)$. All together, we
deduce that the nonsingular $G$-spaces $\left(X,\mu\right)$ and
$\left(G/G_{x_{0}},\kappa\right)$ are isomorphic. ∎

###  7.3. General Actions: Krengel $G$-Spaces

We now come to the construction that generalizes translation $G$-spaces and
compactly fibered coset $G$-spaces simultaneously, aiming to obtain a full
structure theorem of totally dissipative nonsingular $G$-spaces. For an lcsc
group $G$, recall the standard Borel spaces

$\mathbf{K}_{\mathrm{gr}}\left(G\right)\subset\mathbf{K}\left(G\right)$

of compact subgroups of $G$ and of compact subsets of $G$, respectively. These
are known to have a structure of standard Borel space. Write
$\mathcal{M}\left(G\right)$ for the space of Radon measures on $G$, considered
as a standard Borel space with the vague topology. We generalize Construction
5.4 from Borel $G$-spaces to nonsingular $G$-spaces as follows.

######  Definition 7.5.

Let $G$ be an lcsc group. A Krengel $G$-datum is a triplet

$\left(\left(W_{0},\nu_{0}\right),\psi_{0},\lambda_{0}\right)$

that consists of the following ingredients:

  * •

A standard measure space $\left(W_{0},\nu_{0}\right)$ (possibly atomic).

  * •

An arbitrary Borel map
$\psi_{0}:W_{0}\to\mathbf{K}_{\mathrm{gr}}\left(G\right)$.

  * •

An arbitrary Borel map $\lambda_{0}:W_{0}\to\mathcal{M}\left(G\right)$ such
that each $\lambda_{0}\left(w\right)$ is a positive constant multiple of the
fixed Haar measure $\lambda$.

Thus, a Krengel $G$-datum is a Borel $G$-datum as in Definition 5.3 together
with the extra measure $\nu_{0}$ and the map $\lambda_{0}$.

######  Construction 7.6.

A Krengel $G$-space is a measure preserving $G$-space
$\left(X_{0},\mu_{0}\right)$ constructed out of a Krengel $G$-datum
$\left(\left(W_{0},\nu_{0}\right),\psi_{0},\lambda_{0}\right)$ as follows.
Define a Borel measure $\Lambda_{0}$ on $W_{0}\times G$ by

$\Lambda_{0}\left(A\right)=\int_{W_{0}}\lambda_{0}\left(w\right)\left(A_{w}\right)d\nu_{0}\left(w\right)\text{
for a Borel set }A\subset W_{0}\times G,$

where for $w\in W_{0}$ we denote $A_{w}=\left\\{g\in G:\left(w,g\right)\in
A\right\\}$. Put the Borel map

$\Psi_{0}:W_{0}\times G\to
W_{0}\times\mathbf{K}\left(G\right),\quad\Psi_{0}\left(w,g\right)=\left(w,g\psi_{0}\left(w\right)\right).$

Define a measure space $\left(X_{0},\mu_{0}\right)$ to be the image of
$\left(W_{0}\times G,\Lambda_{0}\right)$ under $\Psi_{0}$:

$X_{0}:=\Psi_{0}\left(W_{0}\times
G\right)=\left\\{\left(w,h\psi_{0}\left(w\right)\right):w\in W_{0},h\in
G\right\\}\text{ and }\mu_{0}:=\Psi_{0\ast}\left(\Lambda_{0}\right),$

with the structure of a measure preserving $G$-space via

$g.\left(w,h\psi_{0}\left(w\right)\right)=\left(w,gh\psi_{0}\left(w\right)\right),\quad
g\in G,\,\left(w,h\psi_{0}\left(w\right)\right)\in X_{0}.$

######  Remark 7.7.

An alternative approach to define Krengel $G$-spaces was suggested to us by
Michael Björklund. Let $W_{0}$ be a standard Borel space and
$\psi_{0}:W_{0}\to\mathbf{K}_{\mathrm{gr}}\left(G\right)$ a Borel map.
Consider $W_{0}\times G$ as a translation $G$-space, and let $\nu_{0}$ be any
invariant measure on $W_{0}\times G$. Define on $W_{0}\times G$ the
$G$-invariant Borel equivalence relation

$E_{\psi_{0}}=\left\\{\left(\left(w,g\right),\left(w,g^{\prime}\right)\right)\in\left(W_{0}\times
G\right)^{2}:g\psi_{0}\left(w\right)=g^{\prime}\psi_{0}\left(w\right)\right\\},$

and look at the $\sigma$-algebra $\mathcal{I}_{\psi_{0}}$ of
$E_{\psi_{0}}$-invariant Borel sets. By Mackey’s point-realization theorem,
the Boolean $G$-space corresponding to $\mathcal{I}_{\psi_{0}}$ admits a
point-realization, i.e. a measure preserving $G$-space
$\left(X_{0},\mu_{0}\right)$ such that $\mathcal{I}_{\psi_{0}}$ can be
realized as a measure preserving factor map

$\left(W_{0}\times G,\nu_{0}\right)\to\left(X_{0},\mu_{0}\right).$

It can be shown that $\left(X_{0},\mu_{0}\right)$ is naturally a Krengel
$G$-space.

######  Proposition 7.8.

Let $\left(X_{0},\mu_{0}\right)$ be the Krengel $G$-space associated with a
Krengel $G$-datum
$\left(\left(W_{0},\nu_{0}\right),\psi_{0},\lambda_{0}\right)$. Then:

  1. (1)

$\left(X_{0},\mu_{0}\right)$ is a (standard) measure preserving $G$-space.

  2. (2)

The ergodic decomposition of $\left(X_{0},\mu_{0}\right)$ is given on the
space $\left(W_{0},\nu_{0}\right)$ by the measures $\left\\{\delta_{w}\otimes
q_{w\ast}\lambda_{0}\left(w\right):w\in W_{0}\right\\}$, where $q_{w}:G\to
G/\psi_{0}\left(w\right)$ is the canonical projection for each $w\in W_{0}$.

  3. (3)

$\left(X_{0},\mu_{0}\right)$ is totally dissipative.

###### Proof.

We have shown in Lemma 5.5 that $X_{0}$ is a standard Borel space, and we need
to show that $\mu_{0}$ is $\sigma$-finite. To this end fix an arbitrary Borel
set $A\subset W_{0}$ with $\nu_{0}\left(A\right)<+\infty$ and some compact set
$K\subset G$. For $r>0$ let

$K_{r}:=\left\\{w\in W_{0}:\lambda_{0}\left(w\right)\left(K\right)\leq
r\right\\},$

so that

$\Lambda_{0}\left(\left(A\cap K_{r}\right)\times
K\right)\leq\nu_{0}\left(A\right)r<+\infty.$

Since each $\lambda_{0}\left(w\right)$ is a Radon measure, $\left(A\cap
K_{r}\right)\times K\nearrow A\times K$ as $r\nearrow+\infty$, so that
$\Lambda_{0}$ is $\sigma$-finite on $A\times K$. As $A$ and $K$ are arbitrary
this readily implies that $\Lambda_{0}$ is $\sigma$-finite. The $G$-invariance
of $\Lambda_{0}$ follows from the $G$-invariant of each
$\lambda_{0}\left(w\right)$.

Regarding the ergodic decomposition, we note that for every $w\in W_{0}$ the
measure $\lambda_{0}\left(w\right)$ on $G$ is a $G$-invariant, and since $G$
is a transitive $G$-space it is necessarily ergodic. It then follows that the
measure $\delta_{w}\otimes\lambda_{0}\left(w\right)$ on the space $W_{0}\times
G$ is ergodic with the given action, hence the pushforward
$\Psi_{0\ast}\left(\delta_{w}\otimes\lambda_{0}\left(w\right)\right)$ is
ergodic on $X_{0}$. This readily implies that the ergodic decomposition of
$\left(W_{0}\times G,\Lambda_{0}\right)$ has the form

$\Lambda_{0}=\int_{W_{0}}\delta_{w}\otimes\lambda_{0}\left(w\right)d\nu_{0}\left(w\right).$

As
$\Psi_{0\ast}\left(\delta_{w}\otimes\lambda_{0}\left(w\right)\right)=\delta_{w}\otimes
q_{w\ast}\lambda_{0}\left(w\right)$ for every $w\in W_{0}$, we conclude that

$\displaystyle\mu_{0}\left(A\right)$
$\displaystyle=\Lambda_{0}\left(\Psi_{0}^{-1}\left(A\right)\right)$
$\displaystyle=\int_{W_{0}}\left(\delta_{w}\otimes\lambda_{0}\left(w\right)\right)\left(\Psi_{0}^{-1}\left(A\right)\right)d\nu_{0}\left(w\right)$
$\displaystyle=\int_{W_{0}}\left(\delta_{w}\otimes
q_{w\ast}\lambda_{0}\left(w\right)\right)\left(A\right)d\nu_{0}\left(w\right),$

establishing the desired form of the ergodic decomposition of
$\left(X_{0},\mu_{0}\right)$.

Finally, the stabilizers in $\left(X_{0},\mu_{0}\right)$ are nothing but
conjugacy classes of $\psi_{0}$ hence all compact, and also
$\left\\{\left(w,\psi_{0}\left(w\right)\right):w\in W_{0}\right\\}$ is a Borel
transversal so $E_{X_{0}}^{G}$ is smooth. From Theorem D it follows that
$\left(X_{0},\mu_{0}\right)$ is totally dissipative. ∎

We now move to the proof of Theorem G, which relies on the fact that in
totally dissipative nonsingular $G$-spaces, the ergodic decomposition is given
by the disintegration theorem along the Borel transversal:

######  Proposition 7.9.

Let $\left(X,\mu\right)$ be a totally dissipative nonsingular $G$-space. There
exists a model for the ergodic decomposition of $\left(X,\mu\right)$ on a
standard measure space $\left(W_{0},\nu_{0}\right)$ with Borel measures
$\left\\{\mu_{w}:w\in W_{0}\right\\}$, such that:

  1. (1)

$W_{0}\subset X$ and is a Borel transversal for $E_{G}^{X}$.

  2. (2)

For $\nu_{0}$-a.e. $w\in W_{0}$, $\mu_{w}$ is $G$-invariant and is supported
on the orbit $G.w$.

  3. (3)

For $\nu_{0}$-a.e. $w\in W_{0}$ the measure preserving $G$-space
$\left(G.w,\mu_{w}\right)$ is isomorphic (up to positive constant multiple of
the measures) to the compactly fibered coset $G$-space
$\left(G/G_{w},\kappa_{w}\right)$ via

$q_{w}:G.w\to G/G_{w},\quad q_{w}\left(g.w\right)=gG_{w}.$

###### Proof.

Since $\left(X,\mu\right)$ is totally dissipative, by Proposition 1.1 and
Theorem 6.1 we may assume that $\mu$ is $G$-invariant and $E_{G}^{X}$ is
essentially smooth. Then by Theorem 5.2 there exists a Borel transversal
$W_{0}$ for $E_{G}^{X}$. Pick a Borel selector $s:X\to W_{0}$ and consider the
measure $\nu_{0}:=s_{\ast}\mu$, so we have a measure preserving factor map
$s:\left(X,\mu\right)\to\left(W_{0},\nu_{0}\right)$. Using the disintegration
theorem (see e.g. [1, Theorem 1.0.8] and [36, §6]), there is a collection
$\left\\{\mu_{w}:w\in W_{0}\right\\}$ of Borel measures on $X$ such that

  1. (1)

For $\nu_{0}$-a.e. $w\in W_{0}$ the measure $\mu_{w}$ is supported on
$s^{-1}\left(\left\\{w\right\\}\right)$.

  2. (2)

The map $W_{0}\to\mathbb{R}_{\geq 0}$, $w\mapsto\mu_{w}\left(A\right)$, is
Borel and

$\mu\left(A\right)=\int_{W_{0}}\mu_{w}\left(A\right)d\nu_{0}\left(w\right)\text{
for every Borel set }A\subset X.$

For $\nu_{0}$-a.e. $w\in W_{0}$, the transversal property guarantees that
$s^{-1}\left(\left\\{w\right\\}\right)=G.w$, hence the measure $\mu_{w}$ is
supported on a transitive $G$-space. Since $s$ is a Borel selector,
$s\left(g.x\right)=s\left(x\right)$ for every $g\in G$ and $x\in X$. This
means that if we consider $\left(W_{0},\nu_{0}\right)$ as a trivial measure
preserving $G$-space, then $s:\left(X,\mu\right)\to\left(W_{0},\nu_{0}\right)$
is a factor map of measure preserving $G$-spaces, and it follows from [1,
Proposition 1.0.11] that for $\nu_{0}$-a.e. $w\in W$ the measure $\mu_{w}$ is
$G$-invariant. Finally, for $\nu_{0}$-a.e. $w\in W_{0}$, since
$\psi_{0}\left(w\right)$ is compact in $G$, by Theorem A.4(3) the canonical
isomorphism $q_{w}:G.w\to G/G_{w}$ is equivariant and $q_{w\ast}\mu_{w}$ is
equivalent to $\kappa_{w}$. ∎

###### Proof of Theorem G.

We have shown in Proposition 7.8 that every Krengel $G$-space is totally
dissipative. Suppose $\left(X,\mu\right)$ is a totally dissipative nonsingular
$G$-space and we prove that it is isomorphic to a Krengel $G$-space. We start
by associating with $\left(X,\mu\right)$ a Krengel $G$-datum

$\left(\left(W_{0},\nu_{0}\right),\psi_{0},\lambda_{0}\right)$

as follows. First, let $\left(W_{0},\nu_{0}\right)$ be a standard measure
space that supports the ergodic decomposition $\left\\{\mu_{w}:w\in
W_{0}\right\\}$ of $\left(X,\mu\right)$ as in Proposition 7.9. Let also
$\psi_{0}:W_{0}\to\mathbf{K}_{\mathrm{gr}}\left(G\right)$ be the stabilizer
map, $\psi_{0}\left(w\right)=G_{w}$. Note that by Theorem 6.2 we may assume
that $G_{x}$ is compact for every $x\in X$, so indeed the image of $\psi_{0}$
lies in $\mathbf{K}_{\mathrm{gr}}\left(G\right)$. For $\nu_{0}$-a.e. $w\in
W_{0}$ we have the compactly fibered coset $G$-space
$\left(G.w,\mu_{w}\right)$, so let

$\lambda_{0}\left(w\right)=\jmath_{w\ast}\mu_{w}\text{, for }\jmath_{w}:G\to
G.w,\,\,\jmath_{w}\left(g\right)=g.w.$

Since $\mu_{w}$ is $G$-invariant and $\jmath_{w}$ is equivariant, indeed
$\lambda_{0}\left(w\right)$ is $G$-invariant.

From the Krengel $G$-datum
$\left(\left(W_{0},\nu_{0}\right),\psi_{0},\lambda_{0}\right)$ that we have
just associated with $\left(X,\mu\right)$ we obtain the Krengel $G$-space
$\left(X_{0},\mu_{0}\right)$ as in Construction 7.6. We will finish the proof
by showing that $\left(X,\mu\right)$ is isomorphic to
$\left(X_{0},\mu_{0}\right)$ as a nonsingular $G$-space. As we already showed
in Theorem 5.7, $X$ and $X_{0}$ are isomorphic as Borel $G$-spaces via the map

$\iota:X\to
X_{0},\quad\iota\left(x\right)=\left(r\left(x\right).x,r\left(x\right)^{-1}G_{x}\right),\quad
x\in X.$

As for the measures, from Propositions 7.8 and 7.9, both ergodic
decompositions of $\left(X_{0},\mu_{0}\right)$ and $\left(X,\mu\right)$ are
given on the space $\left(W_{0},\nu_{0}\right)$; the first one has the ergodic
components $\left\\{\delta_{w}\otimes q_{w\ast}\lambda_{0}\left(w\right):w\in
W_{0}\right\\}$, each of which is an invariant measure on
$\left\\{w\right\\}\times\left\\{g\psi_{0}\left(w\right):g\in G\right\\}$, and
the second has the ergodic components $\left\\{\kappa_{w}:w\in
W_{0}\right\\}$, each of which is an invariant measure on
$G/\psi_{0}\left(w\right)$. Both types of ergodic components, for
$\nu_{0}$-a.e. $w\in W_{0}$, are transitive $G$-spaces with the structure of a
compactly fibered coset $G$-space with respect to the compact subgroup
$\psi_{0}\left(w\right)$. Under the Borel isomorphism $\iota$, for
$\nu_{0}$-a.e. $w\in W_{0}$ the component
$\left\\{w\right\\}\times\left\\{g\psi_{0}\left(w\right):g\in G\right\\}$
corresponds to the component $G.w\cong G/\psi_{0}\left(w\right)$. This
completes the proof. ∎

##  8\. Kaimanovich-Type Characterization

Here we prove Theorem C. Our approach will be first to prove it in essentially
free actions and from this case, utilizing an idea due to Arano, Isono &
Marrakchi in their proof of [2, Theorem 4.30], we deduce the general case from
it.

From now and until the end of this section, we fix an lcsc _unimodular_ group
$G$, a nonsingular $G$-space $\left(X,\mu\right)$ and a function $f\in
L_{+}^{1}\left(X,\mu\right)$. We will set some convenient notations. In this
setup define a new cocycle

$\Psi:G\times
X\to\mathbb{R}_{+},\quad\Psi_{g}\left(x\right)=\nabla_{g}\left(x\right)f\left(g.x\right)/f\left(x\right),\quad\left(g,x\right)\in
G\times X.$

For $x\in X$ and $r>0$ denote the set

$D_{r}\left(x\right):=\left\\{g\in G:\Psi_{g}\left(x\right)\geq r\right\\}.$

Note that for every $x\in X$, a number $r>0$ satisfies the property presented
in Theorem C precisely when
$\lambda\left(D_{rf\left(x\right)}\left(x\right)\right)=+\infty$.

In the essentially free case we have:

######  Lemma 8.1.

Let $G$ be an lcsc unimodular group and $\left(X,\mu\right)$ an _essentially
free_ nonsingular $G$-space such that for $\mu$-a.e. $x\in X$, for every $r>0$
we have $\lambda\left(D_{r}\left(x\right)\right)<+\infty$. Then
$\left(X,\mu\right)$ is totally dissipative.

###### Proof.

In the first part of the proof we fix $r>0$, aiming to construct a transient
set $T_{r}\subset X$. Define

$\tau_{r}\left(x\right):X\to\mathbb{R}_{+},\quad\tau_{r}\left(x\right)=\inf\left\\{s>0:\lambda\left(D_{s}\left(x\right)\right)\leq
r\right\\}.$

This is a Borel function by [22, Theorem (17.25)]. For $\mu$-a.e. $x\in X$ we
have $\lambda\left(D_{s}\left(x\right)\right)\nearrow+\infty$ as $s\searrow
0$, hence $\tau_{r}\left(x\right)$ is strictly positive on a $\mu$-conull set.
For $x\in X$ consider the set

$\displaystyle G_{r}\left(x\right)$
$\displaystyle:=D_{\tau_{r}\left(x\right)}\left(x\right)$
$\displaystyle\,\,=\left\\{g\in
G:\Psi_{g}\left(x\right)\geq\tau_{r}\left(x\right)\right\\}$
$\displaystyle\,\,=\left\\{g\in G:\lambda\left(h\in
G:\Psi_{h}\left(x\right)\geq\Psi_{g}\left(x\right)\right)\leq r\right\\}.$

We claim that for $\mu$-a.e. $x\in X$ we have the properties

(8.0.1) $\lambda\left(G_{r}\left(x\right)\right)<+\infty\text{ and
}G_{r}\left(g_{o}.x\right)=G_{r}\left(x\right)g_{o}^{-1}\text{ for every
}g_{o}\in G.$

The first property follows from the assumption, since
$\tau_{r}\left(x\right)>0$. As for the second property, note that for every
$g_{o}\in G$, if $g\in G_{r}\left(x\right)$ then

$\displaystyle\lambda\big{(}h\in
G:\Psi_{h}\left(g_{o}.x\right)\geq\Psi_{gg_{o}^{-1}}\left(g_{o}.x\right)\big{)}$
$\displaystyle\qquad=\lambda\big{(}h\in
G:\Psi_{h}\left(g_{o}.x\right)\geq\Psi_{g}\left(x\right)\Psi_{g_{o}^{-1}}\left(g_{o}.x\big{)}\right)$
$\displaystyle\qquad=\lambda\left(h\in
G:\Psi_{hg_{o}}\left(x\right)\geq\Psi_{g}\left(x\right)\right)$
$\displaystyle\qquad=\lambda\left(h\in
G:\Psi_{h}\left(x\right)\geq\Psi_{g}\left(x\right)\right)$
$\displaystyle\qquad\leq r.$

In the last equality we used the unimodularity in that $\lambda$ is right
invariant. This shows that $gg_{o}^{-1}\in G_{r}\left(g_{o}.x\right)$, thus
$G_{r}\left(g_{o}.x\right)\supset G_{r}\left(x\right)g_{o}^{-1}$ for every
$g_{o}\in G$. By symmetry (apply this to $x\mapsto g_{o}.x$ and $g_{o}\mapsto
g_{o}^{-1}$) also the converse containment follows, establishing the second
part of (8.0.1). We now define the set

$T_{r}:=\bigcup\nolimits_{x\in X}G_{r}\left(x\right).x.$

Setting $r$ free, we proceed by showing that $T_{r}\nearrow X$ as
$r\nearrow+\infty$ modulo $\mu$ while each $T_{r}$ is a $\mu$-measurable
transient set:

  * •

For every $r>0$, $T_{r}$ is $\mu$-measurable. Indeed, $T_{r}$ is the image of
the set

$\left\\{\left(g,x\right)\in G\times X:g\in
G_{r}\left(x\right)\right\\}=\left\\{\left(g,x\right)\in G\times
X:\Psi_{g}\left(x\right)\geq\tau_{r}\left(x\right)\right\\}$

under the action map $G\times X\to X$ and, since this is a Borel set in
$G\times X$, we deduce that $T_{r}$ is analytic hence $\mu$-measurable (see
[22, Theorem (21.10)]).

  * •

For every $r>0$, $T_{r}$ is a transient set. Since the action is free and
using (8.0.1), one can routinely verify that for every $x\in X$,

$x\in T_{r}\iff x\in G_{r}\left(x\right).x.$

Using again that the action is free it follows that

$R_{T_{r}}\left(x\right)^{-1}=\left\\{g\in G:g.x\in
T_{r}\right\\}=\left\\{g\in G:g.x\in
G_{r}\left(x\right).x\right\\}=G_{r}\left(x\right).$

Then from (8.0.1) and the unimodularity we deduce that $T_{r}$ is transient.

  * •

$T_{r}\nearrow X$ as $r\nearrow+\infty$ modulo $\mu$. To see this note that
for $\mu$-a.e. $x\in X$ we have $\tau_{r}\left(x\right)\searrow 0$ as
$r\nearrow+\infty$, so pick $r_{x}>0$ sufficiently large with
$\tau_{r_{x}}\left(x\right)<1$ and, since
$\Psi_{e}\left(x\right)=\nabla_{e}\left(x\right)=1$, we have $e\in
G_{r_{x}}\left(x\right)$, thus $x\in T_{r_{x}}\left(x\right)$.

To sum up, fixing any sequence of positive numbers $r_{n}\nearrow+\infty$, by
the $\mu$-measurability of $T_{r_{n}}$ we may pick a Borel set
$\widebar{T}_{n}\subset T_{r_{n}}$ with
$\mu\left(T_{r_{n}}\backslash\widebar{T}_{n}\right)=0$ (see [22, Theorem
(17.10)]) and, as a subset of a transient set, $\widebar{T}_{n}$ is transient.
From $X=\bigcup_{n=1}^{+\infty}\widebar{T}_{n}$ modulo $\mu$ we get that $X$
is totally dissipative. ∎

The following fact is the main ingredient in reducing the general case of
Theorem C to the essentially free case:

######  Fact 8.2.

Every lcsc non-compact group $G$ admits a probability preserving $G$-space
$\left(\Pi_{G},\varrho_{G}\right)$ which is essentially free and mixing.

By a theorem of Schmidt & Walters [34, Theorem 2.3], as
$\left(\Pi_{G},\varrho_{G}\right)$ is mixing it is in particular _mild mixing_
, i.e. for all ergodic nonsingular probability $G$-space $\left(X,\mu\right)$,
the diagonal nonsingular $G$-space
$\left(X\times\Pi_{G},\mu\otimes\varrho_{G}\right)$ (which is automatically
essentially free) is ergodic.

######  Remark 8.3.

When $G$ is countably infinite Fact 8.2 can be justified by the classical
Bernoulli shift $\big{(}\left\\{0,1\right\\}^{G},\left(1/2,1/2\right)^{\otimes
G}\big{)}$. When $G$ is continuous and non-compact, the natural way to justify
Fact 8.2 is by the Poisson suspension construction, which is the continuous
analog of the classical Bernoulli shift (following Ornstein & Weiss [29,
III§4]; a detailed construction can be found in [3, §3 & §4]). Thus, the
Poisson suspension built over the measure preserving $G$-space
$\left(G,\lambda\right)$ suffices: it can be proved to be mixing in an
analogous way to that the classical Bernoulli shift is mixing. Another
approach is using a construction of Gaussian actions following Adams, Elliot &
Giordano (see [2, Remark 5.10] and the references therein).

We can now complete the proof of Theorem C.

###### Proof of Theorem C.

By Theorem A, if the condition in the theorem is satisfied then
$\left(X,\mu\right)$ is conservative, so we prove the converse. We then assume
that for $\mu$-a.e. $x\in X$ and for every $r>0$ we have
$\lambda\left(D_{r}\left(x\right)\right)<+\infty$, aiming to prove that
$\left(X,\mu\right)$ is totally dissipative. By Proposition 4.4 it is
sufficient to assume that $\left(X,\mu\right)$ is ergodic. Pick an essentially
free mixing probability preserving $G$-space
$\left(\Pi_{G},\varrho_{G}\right)$ as in Fact 8.2, and by the aforementioned
[34, Theorem 2.3] the diagonal nonsingular $G$-space
$\left(X\times\Pi_{G},\mu\otimes\varrho_{G}\right)$ becomes essentially free
and ergodic. Since the second coordinate is probability preserving, the
Radon–Nikodym cocycle
$\left(x,p\right)\mapsto\nabla^{\varrho_{G}}\left(x,p\right)$ of the diagonal
action is essentially independent of its $p$-variable and is given by

$\nabla_{g}^{\varrho_{G}}\left(x,p\right)=\nabla_{g}\left(x\right)\text{ for
all }g\in G\text{ and }\left(x,p\right)\in X\times\Pi_{G}.$

Since $\left(\Pi_{G},\varrho_{G}\right)$ is a probability space, we can
naturally view $f$ as a function of
$L_{+}^{1}\left(X\times\Pi_{G},\mu\otimes\varrho_{g}\right)$ via
$\left(x,p\right)\mapsto f\left(x\right)$. Letting $\Psi^{\varrho_{G}}$ and
$D_{r}^{\varrho_{G}}$ be defined analogously for
$\left(X\times\Pi_{G},\mu\otimes\varrho_{g}\right)$, we see that

$\Psi_{g}^{\varrho_{G}}\left(x,p\right)=\Psi_{g}\left(x\right),\text{ hence
}D_{r}^{\varrho_{G}}\left(x,p\right)=D_{r}\left(x\right),\text{ hence
}\lambda\left(D_{r}^{\varrho_{G}}\left(x,p\right)\right)<+\infty,$

for all $g\in G$ and $\mu\otimes\varrho_{G}$-a.e. $\left(x,p\right)\in
X\times\Pi_{G}$ and every $r>0$. As
$\left(X\times\Pi_{G},\mu\otimes\varrho_{G}\right)$ is essentially free, from
Lemma 8.1 we deduce that it is totally dissipative. Since it is ergodic, by
Theorem F we deuce that $\left(X\times\Pi_{G},\mu\otimes\varrho_{G}\right)$ is
(isomorphic to a compactly fibered $G$-spaces hence) essentially transitive.
This immediately implies that its factor $\left(X,\mu\right)$ is also
essentially transitive and in particular $E_{G}^{X}$ is essentially smooth
(admitting a one point Borel transversal).

Next we show that essentially all the stabilizers in $\left(X,\mu\right)$ are
compact. Let $x\in X$ be fixed in some appropriate $\mu$-conull set. Note that
since $\Psi$ is a cocycle, the map $G_{x}\to\mathbb{R}_{+}$,
$g\mapsto\Psi_{g}\left(x\right)$, forms a multiplicative group homomorphism,
so that

$\Psi_{g}\left(x\right)\geq 1\,\,\iff\,\,\Psi_{g^{-1}}\left(x\right)\leq
1\text{ for every }g\in G_{x}.$

It follows that if $G_{x}$ is a non-compact closed subgroup, it is symmetric
and unbounded, so we can extract a sequence $g_{1},g_{2},\dotsc\in G_{x}$ with
$g_{n}\nearrow+\infty$ as $n\nearrow+\infty$ such that
$\Psi_{g_{n}}\left(x\right)\geq 1$ for every $n\in\mathbb{N}$. Now by our
assumption there can be found $r_{x}>0$ such that
$0<\lambda\left(D_{r_{x}}\left(x\right)\right)<+\infty$, and using the inner
regularity of $\lambda$ there can be found some compact set $K_{x}\subset
D_{r_{x}}\left(x\right)$ with $0<\lambda\left(K_{x}\right)<+\infty$. Thus,
passing to a subsequence of $g_{1},g_{2},\dotsc$ if necessary, we may assume
that $K_{x}g_{1},K_{x}g_{2},\dotsc$ are pairwise disjoint. For every
$n\in\mathbb{N}$ we have

$\forall_{g\in
D_{r_{x}}\left(x\right)},\,\Psi_{gg_{n}}\left(x\right)=\Psi_{g}\left(x\right)\Psi_{g_{n}}\left(x\right)\geq
r,$

implying that

$K_{x}g_{n}\subset D_{r_{x}}\left(x\right)g_{n}\subset
D_{r_{x}}\left(x\right).$

Since $K_{x}g_{1},K_{x}g_{2},\dotsc$ are pairwise disjoint and $G$ is
unimodular, this implies that
$\lambda\left(D_{r_{x}}\left(x\right)\right)=+\infty$, contradicting our
assumption. Thus, $G_{x}$ is necessarily compact.

We came to that $\left(X,\mu\right)$ is essentially transitive and the
stabilizer of $\mu$-a.e. point in it is compact, so by Theorem D it is totally
dissipative. ∎

###  8.1. Maharam Extensions

Here we use Theorem C to give a direct proof to the conservativity of Maharam
Extensions for lcsc unimodular groups. Recall that for an lcsc group $G$ and a
nonsingular $G$-space $\left(X,\mu\right)$, the associated Maharam Extension
is the standard measure space

$\big{(}\widetilde{X},\widetilde{\mu}\big{)}:=\left(X\times\mathbb{R},\mu\otimes\eta\right),\text{
where }d\eta\left(t\right):=e^{t}dt,$

considered as a Borel $G$-space via

$g.\left(x,t\right)=\left(g.x,t-\log\nabla_{g}\left(x\right)\right),\quad g\in
G,\,\left(x,t\right)\in\widetilde{X}.$

One can verify that $\big{(}\widetilde{X},\widetilde{\mu}\big{)}$ becomes an
infinite measure preserving $G$-space.

For $G=\mathbb{Z}$, it was famously proved by Maharam [27, Theorem 2] that if
$\left(X,\mu\right)$ is conservative then so is
$\big{(}\widetilde{X},\widetilde{\mu}\big{)}$. Maharam’s proof utilized the
notion of _incompressibility_ , which does not seem to directly apply for
general groups. A proof in the case of countable groups was presented by
Schmidt [33, Theorem 4.2 (p. 47), Theorem 5.5 (p. 56)]. For general lcsc
groups it was presented by Arano, Isono & Marrakchi [2, Proposition 4.34].

######  Proposition 8.4.

Let $G$ be an lcsc _unimodular_ group. A nonsingular $G$-space
$\left(X,\mu\right)$ is conservative iff its Maharam Extension
$\big{(}\widetilde{X},\widetilde{\mu}\big{)}$ is conservative.

###### Proof.

Note that if $A\subset X$ is a transient set in $\left(X,\mu\right)$, then
$A\times B\subset\widetilde{X}$ is a transient set in
$\big{(}\widetilde{X},\widetilde{\mu}\big{)}$ for whatever Borel set
$B\subset\mathbb{R}$. Hence, by Theorem B, if
$\big{(}\widetilde{X},\widetilde{\mu}\big{)}$ is conservative then so is
$\left(X,\mu\right)$. Let us show the converse.

Since conservativity is unchanged when passing to an equivalent measure, we
may replace $\widetilde{\mu}$ by the probability measure $\widehat{\mu}$ that
is given by

$d\widehat{\mu}\left(x,t\right)=d\mu\left(x\right){\textstyle\frac{1}{2}}e^{-\left|t\right|}dt.$

Now $\big{(}\widetilde{X},\widehat{\mu}\big{)}$ is no longer measure
preserving but a nonsingular probability $G$-space. The Radon–Nikodym cocycle
$\widehat{\nabla}$ with respect to $\widehat{\mu}$ takes the form

$\widehat{\nabla}_{g}\left(x,t\right)=\nabla_{g}\left(x\right)e^{\left|t\right|-\left|t+\log\nabla_{g}\left(x\right)\right|},\quad\left(x,t\right)\in\widetilde{X},$

and by the triangle inequality

$\widehat{\nabla}_{g}\left(x,t\right)\geq\nabla_{g}\left(x\right)e^{-\left|\log\nabla_{g}\left(x\right)\right|}\geq\min\big{\\{}1,\nabla_{g}\left(x\right)^{2}\big{\\}}.$

By Theorem C, given $x\in X$ which is a member of an appropriate $\mu$-conull
set, there is $0<r_{x}\leq 1$ such that
$\lambda\left(D_{r_{x}}\left(x\right)\right)=+\infty$. Then for every
$t\in\mathbb{R}$,

$\int_{G}\widehat{\nabla}_{g}\left(x,t\right)d\lambda\left(g\right)\geq\int_{D_{r_{x}}\left(x\right)}\min\big{\\{}1,\nabla_{g}\left(x\right)^{2}\big{\\}}d\lambda\left(g\right)\geq
r_{x}^{2}\lambda\left(D_{r_{x}}\left(x\right)\right)=+\infty.$

It follows from Theorem A that $\big{(}\widetilde{X},\widehat{\mu}\big{)}$ is
conservative, hence so is $\big{(}\widetilde{X},\widetilde{\mu}\big{)}$. ∎

##  9\. The Positive–Null Decomposition

In this complementary discussion we deal with another general decomposition of
nonsingular $G$-spaces, namely the Positive–Null Decomposition, that regards
the problem of whether a nonsingular $G$-space $\left(X,\mu\right)$ admits an
invariant measure absolutely continuous with respect to $\mu$. This problem is
fundamental in ergodic theory and it is of great interest already for
$G=\mathbb{Z}$ (see the bibliographical notes in [24, §3.4, pp. 144–146 and
§6.3, pp. 220–221] and the many references in [11]). Let us formulate the most
important object of the current discussion.

######  Definition 9.1.

Let $G$ be an lcsc group and $\left(X,\mu\right)$ a nonsingular $G$-space. A
Borel $\sigma$-finite measure is called a.c.i.m of $\left(X,\mu\right)$ if it
is both $G$-invariant and absolutely continuous with respect to $\mu$. An
a.c.i.m of $\left(X,\mu\right)$ which is a _probability_ measure will be
called a.c.i.p of $\left(X,\mu\right)$.

We emphasize that an a.c.i.m of $\left(X,\mu\right)$ need not be equivalent to
$\mu$ but merely absolutely continuous with respect to $\mu$. We also stress
that an a.c.i.m is finite or infinite, thus every a.c.i.p is in particular an
a.c.i.m. We will follow the convention that all finite measures are normalized
to be probability measures.

The Poincaré Recurrence Theorem 4.1 asserts that only the conservative part
may support an a.c.i.p. It then can be one’s goal to decompose the
conservative part further according to the existence of an a.c.i.p. The
Positive–Null Decomposition for $G=\mathbb{Z}^{d}$ is classical [24, §3.4,
Theorem 4.6, §6.3, Theorem 3.9], and for general lcsc groups it can be derived
using the machinery of weakly wandering functions (see [24, §3.4; §3.5 Theorem
4.9; §6.3 Theorem 3.10], [18], [4, §4]). The following is a refined version of
this classical decomposition, in that we also take into account absolutely
continuous invariant infinite measures.

######  Theorem 9.2 (Positive–Null Decomposition).

Let $G$ be an lcsc group and $\left(X,\mu\right)$ a nonsingular $G$-space.
There is a $G$-invariant Borel decomposition

$X=\mathcal{P}_{1}\sqcup\mathcal{P}_{\infty}\sqcup\mathcal{N},$

unique modulo $\mu$, such that:

  1. (1)

$\left(\mathcal{P}_{1},\mu\mid_{\mathcal{P}_{1}}\right)$, unless $\mu$-null,
admits an equivalent a.c.i.p.

  2. (2)

$\left(\mathcal{P}_{\infty},\mu\mid_{\mathcal{P}_{\infty}}\right)$, unless
$\mu$-null, admits an equivalent a.c.i.m but no a.c.i.p.

  3. (3)

$\left(\mathcal{N},\mu\mid_{\mathcal{N}}\right)$ admits no a.c.i.m.

The relations of the Positive–Null Decomposition with the Hopf Decomposition
are illustrated in Figure 2.

$\mathcal{C}\cap\mathcal{P}_{1}$$\mathcal{C}\cap\mathcal{P}_{\infty}$$\mathcal{D}\cap\mathcal{P}_{\infty}$$\mathcal{C}\cap\mathcal{N}$
Figure 2. The relations between the Hopf Decomposition
$X=\mathcal{C}\sqcup\mathcal{D}$ and the Positive–Null Decomposition
$X=\mathcal{P}_{1}\sqcup\mathcal{P}_{\infty}\sqcup\mathcal{N}$.
The component $\mathcal{D}\cap\mathcal{P}_{1}$ is always empty (Poincaré
Recurrence Theorem 4.1), and also the component $\mathcal{D}\cap\mathcal{N}$
is always empty (Proposition 1.1).

It is important to note that $\mathcal{P}_{1}$ may support an infinite a.c.i.m
in addition to an a.c.i.p. Thus, the part

$\mathcal{P}:=\mathcal{P}_{1}\sqcup\mathcal{P}_{\infty},$

ought to be the maximal part of $\left(X,\mu\right)$ that supports an a.c.i.m
(finite or infinite). Our approach in proving Theorem 9.2 is the following: in
the first stage we decompose $X=\mathcal{P}\sqcup\mathcal{N}$, with the
property that $\mathcal{P}$ fully supports an a.c.i.m and $\mathcal{N}$ does
not support an a.c.i.m, and in the second stage we decompose
$\mathcal{P}=\mathcal{P}_{1}\sqcup\mathcal{P}_{\infty}$, with the property
that $\mathcal{P}_{1}$ fully supports an a.c.i.p and $\mathcal{P}_{\infty}$
does not support an a.c.i.p.

The classical Neveu Decomposition deals with a.c.i.p’s only and thus, in our
terminology, it refers to the coarser decomposition

$X=\mathcal{P}_{1}\sqcup X\backslash\mathcal{P}_{1}.$

We mention here a characterization of the Neveu Decomposition due to Hajian &
Ito. A Borel set $W\subset X$ in a nonsingular $G$-space $\left(X,\mu\right)$
is called weakly wandering if there exists a countably infinite set $S\subset
G$ such that $\left\\{s.W:s\in S\right\\}$ are pairwise disjoint. The
collection of weakly wandering sets is hereditary, so the following theorem
follows from [20, Theorem 1] with the Exhaustion Lemma 2.2.

######  Theorem 9.3 (Hajian–Ito).

Let $G$ be an lcsc group and $\left(X,\mu\right)$ a nonsingular $G$-space. The
decomposition $X=\mathcal{P}_{1}\sqcup\left(X\backslash\mathcal{P}_{1}\right)$
is determined by:

  1. (1)

$\mathcal{P}_{1}$ contains no weakly wandering set modulo $\mu$.

  2. (2)

Every $\mu$-positive set in $X\backslash\mathcal{P}_{1}$ contains a
$\mu$-positive weakly wandering subset.

Toward proving Theorem 9.2 we establish some convenient terminology.

######  Definition 9.4.

For an a.c.i.m $\nu$ of a nonsingular $G$-space $\left(X,\mu\right)$ define:

$\bullet$ Support:

A Borel set $P\subset X$ with $\nu\mid_{P}=\nu$ and $\nu\mid_{X\backslash
P}=0$.

$\bullet$ Compatible support:

A $G$-invariant support $P$ with $\mu\mid_{P}\sim\nu\mid_{P}$.

$\bullet$ maximal compatible support:

a compatible support that contains, modulo $\mu$, the compatible support of
any other a.c.i.m.

$\bullet$ $1$-maximal compatible support:

a compatible support that contains, modulo $\mu$, the compatible support of
any other a.c.i.p.

######  Lemma 9.5.

Every a.c.i.m admits a compatible support.

###### Proof.

For an a.c.i.m $\nu$ of $\left(X,\mu\right)$ look at the Radon–Nikodym
derivative $d\nu/d\mu\in L^{1}\left(X,\mu\right)$. Pick a pointwise version
$\phi:X\to\mathbb{R}$ of $d\nu/d\mu$ and set

$P:=\left\\{x\in X:\phi\left(x\right)>0\right\\}.$

Clearly $P$ supports $\nu$. We also have that $\mu\mid_{P}\sim\nu\mid_{P}$:
clearly $\nu\mid_{P}\ll\mu\mid_{P}$, and also $\mu\mid_{P}\ll\nu\mid_{P}$
since if $\nu\left(E\right)=0$ then

$\mu\left(E\cap P\right)=\int_{E\cap
P}1/\phi\left(x\right)d\nu\left(x\right)=0.$