Measure-Class-Preserving Equivalence Relations

• Measure-class-preserving equivalence relations are measured relations that maintain null sets rather than fixed measures, providing a flexible non-singular framework.
• They utilize left/right counting measures to capture the nuances between unimodular and non-unimodular dynamics in group actions and orbit equivalence.
• These relations are pivotal in analyzing rigidity, coamenability, and geometric refinements in measured groupoids, with implications for operator algebras and descriptive set theory.

Measure-class-preserving equivalence relations are measured equivalence relations in which the basic symmetry preserves null sets rather than necessarily preserving a fixed probability measure. In the countable Borel setting, this is the non-singular framework: a countable Borel equivalence relation $\mathcal R$ on $(X,\mu)$ is non-singular if for every Borel isomorphism $\varphi:A\to B$ with graph in $\mathcal R$, one has $\varphi_*(\mu|_A)\sim \mu|_B$; it is probability-measure-preserving only when the normalized restricted measures agree. This distinction is decisive in orbit equivalence theory, in measured groupoids attached to actions of locally compact groups, in the passage from unimodular to non-unimodular settings, and in operator-algebraic and descriptive-set-theoretic classifications (Koivisto et al., 2018).

1. Basic notions and the non-singular framework

A countable Borel equivalence relation $E$ on a standard Borel space $X$ is one for which each class $[x]_E$ is countable. In the invariant-measure formulation, a Borel measure $\mu$ is $E$-invariant iff whenever $(X,\mu)$0 are Borel and there is a Borel bijection $(X,\mu)$1 with graph contained in $(X,\mu)$2, one has $(X,\mu)$3. A measure is $(X,\mu)$4-ergodic if every $(X,\mu)$5-invariant Borel set has measure $(X,\mu)$6 or full measure (Chen, 2018).

For discrete measured equivalence relations, a complementary formulation uses the left and right counting measures

$(X,\mu)$7

The measure $(X,\mu)$8 is $(X,\mu)$9-quasi-invariant if $\varphi:A\to B$0 and $\varphi:A\to B$1 are in the same measure class, and $\varphi:A\to B$2 is measure-preserving if $\varphi:A\to B$3. This is the measured-equivalence-relation analogue of replacing exact invariance by preservation of measure class (Bowen, 2015).

For actions, the same distinction appears as non-singular versus pmp. A non-singular action satisfies

$\varphi:A\to B$4

while a pmp action satisfies

$\varphi:A\to B$5

This is not a cosmetic weakening. In the locally compact setting, several naturally associated equivalence relations are only measure-class-preserving, and the pmp formalism becomes intrinsic only under additional hypotheses such as unimodularity (Koivisto et al., 2017).

2. Orbit relations, action groupoids, and cross sections

For countable groups, orbit equivalence relations are the standard source of measured equivalence relations. If $\varphi:A\to B$6, then

$\varphi:A\to B$7

is a countable Borel pmp equivalence relation. More generally, Feldman–Moore show that every countable Borel equivalence relation arises from a Borel action of a countable group, and when the action is pmp one gets a countable pmp equivalence relation (Ioana, 2009).

For non-discrete lcsc groups, the orbit relation on the whole space is not countable, and the correct object is the action groupoid

$\varphi:A\to B$8

with

$\varphi:A\to B$9

Two essentially free ergodic non-singular actions are orbit equivalent iff their action groupoids are isomorphic; stable orbit equivalence corresponds to similarity of measured groupoids (Koivisto et al., 2018).

The reduction from continuous-group orbit theory to countable measured equivalence relations is achieved by cross sections. For an essentially free non-singular action $\mathcal R$0, a cross section is a Borel subset $\mathcal R$1 such that there is an open neighborhood $\mathcal R$2 with $\mathcal R$3 injective and $\mathcal R$4 conull. The associated relation

$\mathcal R$5

is countable Borel. In the unimodular pmp case it carries a canonical invariant probability measure $\mathcal R$6, and the action $\mathcal R$7 is ergodic iff $\mathcal R$8 is ergodic (Koivisto et al., 2017).

This construction underlies the main orbit-equivalence dictionary for lcsc groups. For non-discrete lcsc groups $\mathcal R$9, the following are equivalent: $\varphi_*(\mu|_A)\sim \mu|_B$0 and $\varphi_*(\mu|_A)\sim \mu|_B$1 are measure equivalent; $\varphi_*(\mu|_A)\sim \mu|_B$2 and $\varphi_*(\mu|_A)\sim \mu|_B$3 admit orbit equivalent essentially free ergodic pmp actions; and they admit essentially free ergodic pmp actions whose cross section equivalence relations are stably orbit equivalent. In the unimodular case, this identifies measure equivalence of groups with stable orbit equivalence of cross section relations; in the non-unimodular case, the same bridge exists, but the cross section relation is often only non-singular rather than pmp (Koivisto et al., 2017, Koivisto et al., 2018).

3. Unimodularity, modular data, amenability, and type

The passage from pmp to merely measure-class-preserving relations is governed, in the lcsc setting, by the modular function. For a pmp action $\varphi_*(\mu|_A)\sim \mu|_B$4, the modular function of the action groupoid is

$\varphi_*(\mu|_A)\sim \mu|_B$5

Hence the group-level modular data passes directly to the measured groupoid. The action groupoid is unimodular iff $\varphi_*(\mu|_A)\sim \mu|_B$6 is unimodular, and this is the fundamental obstruction to producing pmp cross section relations in the non-unimodular case (Koivisto et al., 2018).

This obstruction is visible even when the original action is pmp. For a free mixing pmp action of a non-unimodular group $\varphi_*(\mu|_A)\sim \mu|_B$7, a cross section $\varphi_*(\mu|_A)\sim \mu|_B$8 yields a countable equivalence relation $\varphi_*(\mu|_A)\sim \mu|_B$9 that is not pmp, because

$E$0

Accordingly, $E$1 is not of type $E$2. The measured relation extracted from a cross section is therefore genuinely only measure-class-preserving (Koivisto et al., 2018).

Amenability remains the boundary between hyperfinite and non-hyperfinite behavior. For amenable lcsc groups, the associated cross section relations are hyperfinite; among amenable lcsc groups there are exactly three measure-equivalence classes: compact groups, non-compact unimodular amenable groups, and non-unimodular amenable groups. In equivalence-relation terms, this yields three regimes: trivial finite-measure behavior, hyperfinite pmp relations, and hyperfinite non-singular relations of type III determined by modular data (Koivisto et al., 2018).

Modular data can also appear as a stable orbit-equivalence invariant for pmp actions. For ergodic essentially free pmp actions of non-amenable Baumslag–Solitar groups $E$3, a canonical associated flow is constructed from the modular homomorphism

$E$4

and the isomorphism class of this flow is invariant under weak orbit equivalence. Although the original actions are pmp, the invariant is explicitly of type-III flavor and is extracted through cocycles, Mackey ranges, and modular groupoid data (Kida, 2011).

4. Rigidity, coamenability, and indecomposability

A basic rigidity phenomenon for countable pmp relations is Ioana’s dichotomy for the relation induced by $E$5: every ergodic subequivalence relation $E$6 is either hyperfinite or rigid. Rigidity is characterized by the nonexistence of off-diagonal probability measures $E$7 on $E$8 with fixed marginals, asymptotic concentration on the diagonal, and asymptotic invariance under the full group. This gives a purely ergodic-theoretic formulation of Popa rigidity for free ergodic pmp actions (Ioana, 2009).

A different rank-one rigidity appears for relations acting on bundles of hyperbolic spaces. If an ergodic pmp equivalence relation $E$9 satisfies Bowen’s hyperbolic bundle hypotheses, then every aperiodic hyperfinite subequivalence relation is contained in a unique maximal hyperfinite subequivalence relation. The same framework yields a boundary-measure classification of subrelations and full-group elements into parabolic, loxodromic, and mixed types, and a measured analogue of Tits’ alternative: every non-hyperfinite such relation contains an ergodic non-hyperfinite treeable subequivalence relation (Bowen, 2015).

Relative amenability is captured by coamenability of inclusions $X$0. For discrete pmp relations, coamenability is equivalent to the existence of a global invariant mean on $X$1, to almost invariant vectors in $X$2, to Følner sets in the quotient bundle, and to a Kesten-type spectral criterion

$X$3

for every countably supported symmetric generating $X$4 and every positive-measure $X$5-invariant $X$6. The stronger notion of everywhere coamenability is equivalent to pointwise almost sure equality $X$7 for the cospectral radius (Hayes, 2024).

Indecomposability also occurs at the level of product decompositions. Orbit equivalence relations arising from essentially free ergodic pmp actions of Zariski dense discrete subgroups of $X$8-isotropic almost $X$9-simple algebraic $[x]_E$0-groups are strongly prime. As a consequence, direct products of higher-rank lattices admit existence and uniqueness of prime factorization at the level of orbit equivalence relations, and, under the property $[x]_E$1 hypotheses in the product case, product decompositions of the relation come from genuine product decompositions of the action itself (Drimbe et al., 2024).

5. Full groups, kernels, quotients, and geometric refinements

Normality for subequivalence relations admits a kernel characterization parallel to group theory. For an ergodic discrete pmp equivalence relation $[x]_E$2, a subequivalence relation $[x]_E$3 is normal iff there exists a discrete Borel groupoid $[x]_E$4 and a Borel homomorphism

$[x]_E$5

with

$[x]_E$6

When $[x]_E$7, the quotient is described by

$[x]_E$8

This allows one to formulate measured analogues of simple and large groups. Bowen constructs ergodic pmp equivalence relations with no proper ergodic normal subequivalence relations and no proper ergodic finite-index subequivalence relations, and also proves that under treeability and cost $[x]_E$9, together with a primitive proper ergodic subrelation, a relation surjects onto every countable group (Bowen, 2015).

The full group itself encodes substantial measured structure. For an aperiodic pmp equivalence relation $\mu$0, Le Maître proves

$\mu$1

where $\mu$2 is the conditional cost, i.e. the measurable field of costs of the ergodic components. He also introduces the metric

$\mu$3

for which $\mu$4 is connected and has automatic continuity. This yields an algebraic characterization of aperiodicity: $\mu$5 is aperiodic iff $\mu$6 has no nontrivial morphisms into totally disconnected separable groups. In the hyperfinite case,

$\mu$7

(Maître, 2014).

The orbit relation can also be refined by adding leafwise geometry. For pmp actions of finitely generated groups with chosen generating systems, the associated graphings carry graph metrics on almost every orbit. Isometric orbit equivalence asks for a measurable isometry of these orbitwise graph metrics. For essentially free actions, this is equivalent to orbit equivalence with length-preserving cocycles, and it is strictly stronger than ordinary orbit equivalence. The free-group examples show that mixing is not invariant under isometric orbit equivalence, so the measured equivalence relation does not determine the chosen graph metric on classes (Joseph, 2022).

6. Decomposition spaces, definability, and regularity of classes

The invariant-measure theory of countable Borel equivalence relations can itself be topologized. For a countable Borel equivalence relation $\mu$8 induced by a countable Borel group action, Chen proves that for cofinally many compatible Polish topologies, the topological ergodic decomposition

$\mu$9

realizes the uniform measure-theoretic ergodic decomposition. On each component $E$0, there is at most one non-totally-singular $E$1-invariant $E$2-finite measure up to scaling, and the components supporting such measures form a clopen set $E$3. This supplies a topological parameter space for ergodic invariant $E$4-finite measure classes (Chen, 2018).

At the projective level, smallness of classes is tied to regularity properties of sets. For a $E$5-ideal $E$6, $E$7 asserts that every $E$8 on a Borel $E$9-positive set, with all $(X,\mu)$00-classes $(X,\mu)$01-small, has perfectly many classes. For $(X,\mu)$02 provably ccc ideals,

$(X,\mu)$03

By contrast, the analogous statement for $(X,\mu)$04 relations fails: for the meager ideal,

$(X,\mu)$05

Thus the perfect-set phenomenon for definable equivalence relations with small classes is controlled by ideal-regularity and forcing absoluteness rather than by countability or group-generation hypotheses (Drucker, 2016).

A concrete descriptive model of change-of-density equivalence appears on the Polish space $(X,\mu)$06 of strictly positive probability measures on $(X,\mu)$07. For $(X,\mu)$08,

$(X,\mu)$09

and

$(X,\mu)$10

The relation

$(X,\mu)$11

is definable and admits a strong approximation by the turbulent Polish group action of

$(X,\mu)$12

on

$(X,\mu)$13

This gives a descriptive-set-theoretic model of a measure-class-type relation defined by positive Radon–Nikodym densities (Sofronidis, 2014).

Across these settings, measure-class-preserving equivalence relations form the natural enlargement of pmp measured equivalence relations. In the unimodular pmp regime they often reduce to invariant countable relations with canonical probability measures; in the non-unimodular and modular regimes they carry essential type-III data; in rigidity theory they support prime factorization, coamenability, and hyperbolic rank-one phenomena; and in descriptive set theory their small-class behavior is controlled by ideal measurability and turbulence rather than by measure preservation alone.