§2 (external, Connes–Feldman–Weiss) — an amenable relation is μ-amenable in Lodha and Moore's sense

The statement is a one-way implication. It holds for every space $X$, measure $\mu$ and relation $E$ that satisfy the standing assumptions and hypotheses (H1)–(H4) below: if $E$ has the mean property (Am) defined below, then $E$ has the single-transformation property (Z) defined below.

Standing assumptions

$X$ is a set (a type in any universe). Everything below is quantified over it, and it carries the following structure.

• A topology making $X$ a Polish space. The topology is second countable, and some metric that induces it is complete.
• A $\sigma$-algebra equal to the Borel $\sigma$-algebra. It is the $\sigma$-algebra generated by the open sets. "Measurable" for subsets of $X$ means Borel.
• A measure $\mu$ on this $\sigma$-algebra. It is countably additive, with values in $[0,\infty]$.
• $\sigma$-finiteness of $\mu$. There are subsets $S_0, S_1, S_2, \dots$ of $X$ with $\mu(S_i) < \infty$ for every $i$ and $\bigcup_i S_i = X$. The sets are not required to be measurable. Each set of finite outer measure lies inside a measurable set of the same measure, so this is the usual notion.

How $\mu$ is applied to arbitrary sets. For any subset $A \subseteq X$, measurable or not, $\mu(A)$ means the outer measure:

$$\mu(A) \;=\; \inf\{\mu(B) : B \supseteq A,\ B \text{ measurable}\}.$$

This convention applies to every occurrence of $\mu$ below. "$\mu$-almost everywhere" (a.e.) means: the set of points where the property fails has $\mu$-measure $0$ in this sense. In particular, $F = G$ a.e. means $\mu(\{x : F(x) \neq G(x)\}) = 0$.

Other $\sigma$-algebras.

• $X \times X$: the product $\sigma$-algebra, generated by the sets $B \times X$ and $X \times B$ with $B \subseteq X$ measurable. Since $X$ is second countable, this is also the Borel $\sigma$-algebra of the product topology.
• $\mathbb{R}$: the Borel $\sigma$-algebra.
• A subset $D \subseteq X$, viewed as a space in its own right: it carries the subspace $\sigma$-algebra $\{B \cap D : B \subseteq X \text{ measurable}\}$.

The relation and the hypotheses

$E \subseteq X \times X$ is a set of pairs. The following hypotheses are assumed.

• (H1) $E$ is a measurable subset of $X \times X$, for the product $\sigma$-algebra.
• (H2) The relation $x \sim y \iff (x,y) \in E$ is an equivalence relation on $X$:
  • it is reflexive: $(x,x) \in E$ for all $x$;
  • it is symmetric: $(x,y) \in E \Rightarrow (y,x) \in E$;
  • it is transitive: $(x,y),(y,z) \in E \Rightarrow (x,z) \in E$.
• (H3) For every $x \in X$, the set $\{y \in X : (x,y) \in E\}$ is countable, meaning finite or countably infinite.
• (H4) Let $A \subseteq X$ be any subset, not necessarily measurable, with $\mu(A) = 0$ (outer measure). Then

$$\mu\bigl(\{\, y \in X : \text{there is } x \in A \text{ with } (x,y) \in E \,\}\bigr) = 0 .$$

By (H2), the set inside is the union of all $\sim$-classes that meet $A$.

None of these hypotheses is impossible to satisfy. For example, the diagonal $E = \{(x,x) : x \in X\}$ satisfies (H1)–(H4) for every $\sigma$-finite $\mu$.

The mean property (Am)

Several auxiliary notions come first. Throughout, $E$ is the relation above.

Admissible functions. A function $f : X \times X \to \mathbb{R}$ is admissible if both of the following hold:

• $f$ is measurable, from the product $\sigma$-algebra to the Borel sets of $\mathbb{R}$;
• some real constant $C$ satisfies $|f(x,y)| \le C$ for all $(x,y) \in E$.

Boundedness is required only on $E$. Off $E$, $f$ is only required to be measurable.

$E$-negligible sets of pairs. A set $S \subseteq X \times X$ is $E$-negligible if

$$\mu\bigl(\pi_1(S \cap E)\bigr) = 0, \qquad \pi_1(x,y) = x .$$

In words, the set of $x$ for which some $y$ has $(x,y) \in S \cap E$ has outer measure $0$.

Partial transformations of $E$. A partial transformation $\varphi$ of $E$ consists of the following data:

• measurable sets $D, C \subseteq X$;
• a bijection $e : D \to C$ such that $e$ and $e^{-1}$ are both measurable for the subspace $\sigma$-algebras on $D$ and $C$;
• the graph condition: $(a, e(a)) \in E$ for every $a \in D$.

$D$ and $C$ may be empty. For such a $\varphi$ and functions $f : X \times X \to \mathbb{R}$ and $F : X \to \mathbb{R}$, define

$$f^{\varphi}(x,y) = \begin{cases} f\bigl(e^{-1}(x),\, y\bigr) & x \in C,\\ 0 & x \notin C,\end{cases} \qquad F^{\varphi}(x) = \begin{cases} F\bigl(e^{-1}(x)\bigr) & x \in C,\\ 0 & x \notin C.\end{cases}$$

Only the first coordinate is moved. In $f^\varphi$ the second coordinate $y$ is left unchanged.

Definition of (Am). $E$ has property (Am) with respect to $\mu$ if there is an operator $P$ that sends every function $f : X \times X \to \mathbb{R}$, admissible or not, to a function $P f : X \to \mathbb{R}$, and that satisfies all seven of the following conditions.

• (M1) Almost-everywhere measurability. For every admissible $f$ there is a Borel-measurable $g : X \to \mathbb{R}$ with $P f = g$ $\mu$-a.e.
• (M2) Dependence only on $E$, up to negligible sets. Let $f$ and $g$ be admissible, and suppose the set $\{p \in X \times X : f(p) \neq g(p)\}$ is $E$-negligible. Then $P f = P g$ $\mu$-a.e.
• (M3) Additivity. For all admissible $f, g$: $P(f + g) = P f + P g$ $\mu$-a.e. The sums are pointwise.
• (M4) Homogeneity. For every $c \in \mathbb{R}$ and admissible $f$: $P(c f) = c \cdot P f$ $\mu$-a.e. Here $(cf)(p) = c\,f(p)$.
• (M5) Positivity. Let $f$ be admissible with $f(p) \ge 0$ for every $p \in E$; no sign condition is imposed off $E$. Then $P f(x) \ge 0$ for $\mu$-a.e. $x$.
• (M6) Normalisation. Let $\mathbf 1$ denote the constant function $1$ on $X \times X$. Then $P \mathbf 1 = 1$ $\mu$-a.e., where the right side is the constant function $1$ on $X$.
• (M7) Invariance. For every partial transformation $\varphi$ of $E$ and every admissible $f$:

$$P\bigl(f^{\varphi}\bigr) \;=\; (P f)^{\varphi} \quad \mu\text{-a.e.}$$

No admissibility of $f^\varphi$ is assumed; $P$ is simply applied to it.

Fine print about (Am).

• In each of (M1)–(M7), the exceptional null set may depend on the functions involved, on the scalar $c$ and on $\varphi$. No single null set is required to work for all of them simultaneously.
• $P f$ itself need not be measurable. Only (M1) is required.
• Apart from being the input in (M7), non-admissible functions are unconstrained: $P$ may send them anywhere.

The single-transformation property (Z)

$E$ has property (Z) with respect to $\mu$ if there exist:

• a measurable set $N \subseteq X$ with $\mu(N) = 0$;
• a bijection $T : X \setminus N \to X \setminus N$ such that $T$ and $T^{-1}$ are both measurable for the subspace $\sigma$-algebra on $X \setminus N$. Because $N$ is measurable, this $\sigma$-algebra consists of the measurable subsets of $X$ contained in $X \setminus N$.

These must satisfy, for all $x, y \in X \setminus N$,

$$(x,y) \in E \iff \text{there is } n \in \mathbb{Z} \text{ with } T^{n}(x) = y .$$

Here the powers are taken in the group of bijections of $X \setminus N$ under composition:

• $T^0$ is the identity;
• $T^{n} = T \circ \dots \circ T$ ($n$ factors) for $n > 0$;
• $T^{-n} = (T^{-1})^{n}$.

In other words, restricted to pairs of points both lying outside the null set $N$, the relation $E$ coincides exactly with the orbit relation of the single map $T$. Both directions are required:

• every $E$-related pair outside $N$ is joined by some power of $T$;
• every $T$-orbit pair is $E$-related.

Fine print about (Z).

• Nothing is required of $N$ beyond being measurable and null. In particular it is not required to be a union of $\sim$-classes. Points outside $N$ may be $E$-related to points inside $N$, and (Z) says nothing about such pairs.
• Nothing is required of $T$ regarding $\mu$. $T$ is not asked to preserve $\mu$, nor to map null sets to null sets.
• $n = 0$ is allowed, which covers the pairs $(x,x)$.

Degenerate cases included by the quantifiers

• $\mu = 0$, or more generally $\mu(X) = 0$, is permitted. Such a $\mu$ is $\sigma$-finite. Then every "$\mu$-a.e." condition in (M1)–(M7) holds trivially, so any $P$ witnesses (Am). In (Z), the choice $N = X$ is then available, and it leaves $X \setminus N$ empty and $T$ the empty map.
• $X = \varnothing$ is permitted.
• $\sim$-classes may be finite, including singletons, because (H3) allows finite sets.