Is every measure-hyperfinite countable Borel equivalence relation hyperfinite? (Kechris, Open Problem 16.4)

A countable Borel equivalence relation $E$ on a standard Borel space is hyperfinite if it is an increasing union of finite Borel equivalence relations, and measure-hyperfinite if for every Borel probability measure $\mu$ the restriction of $E$ to some $\mu$-conull Borel set is hyperfinite. By the Connes–Feldman–Weiss theorem, every amenable countable Borel equivalence relation — in particular every orbit equivalence relation of a Borel action of a countable amenable group, and more generally of a topologically amenable continuous action with amenable point stabilizers — is measure-hyperfinite; yet it is unknown whether this measure-by-measure property already forces hyperfiniteness as a global Borel relation. The question is Open Problem 16.4 in Kechris's 2025 tract on countable Borel equivalence relations, and it was invoked as motivating Question 1.1 in Oyakawa's 2025 proof of hyperfiniteness for the boundary action of virtually special groups. As of August 2026 it remains open: a positive answer would make amenability and hyperfiniteness coincide for countable Borel equivalence relations and would settle Weiss's 1984 question, while a negative answer would exhibit the first known countable Borel equivalence relation that is measure-hyperfinite (equivalently, by Connes–Feldman–Weiss, measure-amenable) yet not hyperfinite.

Background

A standard Borel space is a set carrying the Borel sigma-algebra of some complete separable metric topology (for example the real line or the Cantor space $2^{\mathbb{N}}$ of infinite binary sequences). A countable Borel equivalence relation (CBER) is a Borel equivalence relation $E$ on such a space all of whose equivalence classes are countable; by a theorem of Feldman–Moore, every CBER is the orbit equivalence relation of a Borel action of some countable group. The simplest nontrivial CBERs are the hyperfinite ones: $E$ is hyperfinite if $E=\bigcup_n F_n$ for some increasing sequence of finite Borel equivalence relations $F_0\subseteq F_1\subseteq\cdots$, each contained in $E$ (each $F_n$ Borel with finite classes). The archetype is $E_0$, eventual agreement of binary sequences: $x\,E_0\,y$ iff $x(n)=y(n)$ for all but finitely many $n$. Dougherty–Jackson–Kechris classified the aperiodic (all classes infinite) hyperfinite CBERs up to Borel isomorphism: each is Borel isomorphic to exactly one of the smooth relation $\mathbb{R}I_{\mathbb{N}}$ (a direct sum of continuum many countable classes; a relation is smooth if its classes are separated by a Borel invariant, i.e. a Borel map $f$ into a standard Borel space with $x\,E\,y\iff f(x)=f(y)$), the tail equivalence relation $E_t$ on $2^{\mathbb{N}}$ (two binary sequences are equivalent when a tail of one coincides with a tail of the other), $E_0$ itself, a finite multiple $kE_0$ with $k\geq 2$, or an infinite multiple $\mathbb{N}E_0$ or $\mathbb{R}E_0$; in particular hyperfinite relations sit at the bottom of the Borel complexity hierarchy. Hyperfinite relations are also treeable — they admit an acyclic Borel graph whose connectedness relation is $E$ — a standard fact of the theory (see Jackson–Kechris–Louveau).

Given a Borel probability measure $\mu$ on $X$, call $E$ $\mu$-hyperfinite if there is a Borel set $A$ with $\mu(X\setminus A)=0$ such that the restriction $E|A$ is hyperfinite; $E$ is measure-hyperfinite if this holds for every Borel probability measure $\mu$ on $X$. The Connes–Feldman–Weiss theorem shows that in the measure-theoretic world amenability and hyperfiniteness coincide: an amenable equivalence relation (one admitting approximately invariant systems of probability measures along orbits) is hyperfinite after discarding a null set, and is even generated by a single transformation. Consequently every amenable CBER — for instance the orbit relation of any Borel action of a countable amenable group, and more generally of any topologically amenable continuous action with amenable point stabilizers on a Polish space (a complete separable metric space; topological amenability of an action means amenability of the induced action on the space of probability measures over it) — is measure-hyperfinite. The catch is that the conull set and the approximating finite relations depend on the measure $\mu$, and no known compactness or diagonalization argument merges the measure-by-measure approximations into one Borel increasing union that works on all of $X$. Whether such a uniform upgrade is always possible is the subject of this problem.

The question is intertwined with two classical open problems. Jackson–Kechris–Louveau list as a notoriously difficult open problem whether every amenable CBER is hyperfinite; and Weiss asked in 1984 whether every Borel action of a countable amenable group generates a hyperfinite orbit equivalence relation. Since amenability implies measure-hyperfiniteness, a positive answer to the present question would settle both affirmatively. Four decades of partial results cover special classes: abelian group actions (Gao–Jackson); Borel graphs of finite Borel asymptotic dimension — a Borel analogue of Gromov's asymptotic dimension (a large-scale dimension invariant of metric spaces and finitely generated groups) introduced by Conley–Jackson–Marks–Seward–Tucker-Drob, whose finiteness forces hyperfiniteness of the generated equivalence relation; Borel graphs of pointwise volume growth (bounded ball sizes around each point) at most $\exp(O(r^{\gamma}))$ for the small explicit constant $\gamma\approx 0.1523$ (Grebík–Marks–Rozhoň–Shinko); and equivalence relations generated by countably many pairwise commuting bounded-to-one Borel functions (Shinko–Weilacher–Yu). Directly on the measure-hyperfinite question, Naryshkin–Vaccaro proved that a measure-hyperfinite orbit relation of a free continuous action of a countable virtually free group (one with a free subgroup of finite index) on a $\sigma$-compact Polish space is hyperfinite, that a treeable orbit relation of a Borel action of a countable amenable group is hyperfinite, and that a treeable, amenable, Borel bounded relation is hyperfinite (Borel bounded, in the sense of Boykin–Jackson: every Borel function $\varphi:X\to\mathbb{N}^{\mathbb{N}}$ is pointwise dominated, up to eventual agreement, by a function that is itself eventually constant on each $E$-class). A separate stream of evidence comes from geometric group theory: non-amenable groups can act topologically amenably on natural boundaries of nonpositively curved spaces — Oyakawa's own motivation, since by Connes–Feldman–Weiss the orbit relation of a topologically amenable continuous action with amenable point stabilizers is measure-hyperfinite — and these boundary orbit relations have been confirmed hyperfinite case by case: the free group $F_2$ on its Gromov boundary (the boundary at infinity of a hyperbolic group, i.e. a group whose Cayley graph satisfies Gromov's thin-triangle condition), where the orbit relation is tail equivalence on infinite reduced words (Dougherty–Jackson–Kechris); cubulated hyperbolic groups (hyperbolic groups acting properly and cocompactly on a CAT(0) cube complex, a simply connected cell complex glued from Euclidean cubes) on their Gromov boundary (Huang–Sabok–Shinko); all hyperbolic groups (Marquis–Sabok); and virtually special groups (groups with a finite-index subgroup that embeds into a right-angled Artin group via such a cube-complex action) on the Roller boundary (a canonical compact boundary of a CAT(0) cube complex, built from its hyperplanes) (Oyakawa). None of these techniques removes the structural hypotheses, so the general implication remains unproved.

Problem Statement

Let $X$ be a standard Borel space (a set carrying the Borel sigma-algebra of some complete separable metric topology) and $E$ a countable Borel equivalence relation on $X$ (a Borel equivalence relation all of whose classes are countable). Assume $E$ is measure-hyperfinite: for every Borel probability measure $\mu$ on $X$ there exists a Borel set $A\subseteq X$ with $\mu(X\setminus A)=0$ such that the restriction $E|A$ is hyperfinite, i.e. $E|A$ is an increasing union of finite Borel equivalence relations on $A$. Must $E$ be hyperfinite on all of $X$ — that is, must there exist an increasing sequence $F_0\subseteq F_1\subseteq\cdots$ of finite Borel equivalence relations on $X$, each contained in $E$, with $\bigcup_n F_n=E$? (By the Dougherty–Jackson–Kechris classification this amounts to asking whether the aperiodic part of $E$ — the union of its infinite classes — must be Borel isomorphic to one of the canonical hyperfinite relations $\mathbb{R}I_{\mathbb{N}}$, $E_t$ (tail equivalence: two binary sequences are equivalent when a tail of one coincides with a tail of the other), $E_0$ (eventual agreement of binary sequences), a finite multiple $kE_0$, $\mathbb{N}E_0$, or $\mathbb{R}E_0$, with the remaining finite-class part smooth — its classes separated by a Borel invariant map.) This is Open Problem 16.4 in Kechris's The Theory of Countable Borel Equivalence Relations, posed as Question 1.1 in Oyakawa's Hyperfiniteness of the boundary action of virtually special groups (arXiv:2509.04613). The question is whether the implication holds for every countable Borel equivalence relation — equivalently, whether some measure-hyperfinite countable Borel equivalence relation fails to be hyperfinite.

References

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