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

What the statement asserts

The setting

Let $X$ be a set (of any size, possibly empty) equipped with

• a topology making $X$ a Polish space: the topology is second countable, and it is induced by some metric on $X$ that is complete;
• a σ-algebra equal to the Borel σ-algebra of that topology, that is, the σ-algebra generated by the open sets.

Throughout, "measurable" means: Borel for subsets of $X$; for subsets of $X \times X$, measurable for the product σ-algebra (generated by the rectangles $B_1 \times B_2$ with $B_1, B_2$ Borel), which for this $X$ coincides with the Borel σ-algebra of $X \times X$; for real-valued functions, measurable into $\mathbb R$ with its Borel σ-algebra. On a measurable subset $Y \subseteq X$, "measurable" refers to the trace σ-algebra $\{B \cap Y : B \text{ Borel}\}$, which consists of the Borel subsets of $X$ contained in $Y$.

Let $\mu$ be a σ-finite measure on $X$: $X$ is a union of countably many sets each of finite $\mu$-measure. Nothing else is assumed about $\mu$: it may be infinite, it may have atoms, and it may be the zero measure.

Null sets and "almost everywhere". The measure $\mu$ is applied below to arbitrary subsets of $X$, measurable or not. For a subset $S$ that is not measurable, $\mu(S)$ means the outer measure

$$\mu(S) = \inf\{\mu(B) : B \supseteq S,\ B \text{ measurable}\},$$

so $\mu(S) = 0$ exactly when $S$ is contained in some measurable set of measure $0$. "$Q(x)$ holds for $\mu$-almost every $x$" means $\mu(\{x \in X : Q(x) \text{ fails}\}) = 0$ in this sense, and "$F = G$ $\mu$-a.e." for functions $F, G : X \to \mathbb R$ means $\mu(\{x : F(x) \neq G(x)\}) = 0$.

Let $E \subseteq X \times X$ be a set of pairs, and write $x \sim y$ for $(x,y) \in E$. The following four hypotheses are assumed.

• (H1) $E$ is a measurable subset of $X \times X$.
• (H2) $\sim$ is an equivalence relation on the whole of $X$: $x \sim x$ for every $x$; $x \sim y$ implies $y \sim x$; $x \sim y$ and $y \sim z$ imply $x \sim z$.
• (H3) For every $x \in X$, the set $\{y \in X : (x,y) \in E\}$ is countable (finite or countably infinite).
• (H4) For every subset $A \subseteq X$, measurable or not, with $\mu(A) = 0$,

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

In words: the set of points related to at least one point of a $\mu$-null set is itself $\mu$-null.

None of (H1)–(H4) is vacuous: for instance the diagonal $E = \{(x,x) : x \in X\}$ satisfies all four for every $\mu$.

The claim. Under these hypotheses, if condition (A) below holds, then condition (B) below holds. This is a one-way implication; nothing is asserted in the converse direction, and nothing is asserted about whether (A) holds.

Condition (A)

There exist

• a measurable set $N \subseteq X$ with $\mu(N) = 0$, and
• a bijection $T : X \setminus N \to X \setminus N$ such that $T$ and $T^{-1}$ are both measurable (for the trace σ-algebra on $X \setminus N$),

such that for all $x, y \in X \setminus N$,

$$(x,y) \in E \iff y = T^{n}(x) \text{ for some integer } n \in \mathbb Z .$$

Here $T^0$ is the identity, $T^n$ for $n > 0$ is the $n$-fold composite $T \circ \cdots \circ T$, and $T^n$ for $n < 0$ is the $|n|$-fold composite of $T^{-1}$.

Points to note about (A):

• It constrains $E$ only on pairs with both coordinates outside $N$. Whether $(x,y) \in E$ when $x \in N$ or $y \in N$ is not addressed by (A).
• $N$ is not required to be a union of $\sim$-classes: a point of $N$ may be related to points outside $N$.
• $T$ is not required to preserve $\mu$, or to send null sets to null sets.
• $T$ may have fixed points and finite orbits; $T$ may be the identity, in which case (A) says that, off $N$, $x \sim y$ only when $x = y$.
• If $\mu$ is the zero measure (in particular, if $X$ is empty), (A) holds by taking $N = X$, so that $T$ is the empty map.

Condition (B)

Some vocabulary first, relative to $\mu$ and $E$.

Admissible functions. Call $f : X \times X \to \mathbb R$ admissible if

• $f$ is measurable, and
• there is a real number $C$ with $|f(p)| \le C$ for every $p \in E$.

The bound is required only on $E$; off $E$, $f$ may be unbounded.

$E$-negligible sets. Call a set $S \subseteq X \times X$ $E$-negligible if

$$\mu\bigl(\pi_1(S \cap E)\bigr) = 0, \qquad \text{where } \pi_1(x,z) = x .$$

The projection $\pi_1(S \cap E)$ need not be measurable; $\mu$ of it is the outer measure described above.

Partial transformations of $E$. A partial transformation $\varphi$ of $E$ consists of

• two measurable sets $D, C \subseteq X$ (its domain and codomain), and
• a bijection $e : D \to C$ such that $e$ and $e^{-1}$ are both measurable (for the trace σ-algebras on $D$ and $C$),

such that $(a, e(a)) \in E$ for every $a \in D$. The empty case $D = C = \varnothing$ is included.

A partial transformation $\varphi$ acts on functions $f : X \times X \to \mathbb R$ by moving the first coordinate only:

$$(\varphi \cdot f)(x, z) = \begin{cases} f\bigl(e^{-1}(x),\, z\bigr) & \text{if } x \in C,\\ 0 & \text{if } x \notin C,\end{cases}$$

and on functions $F : X \to \mathbb R$ by

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

Condition (B) asserts: there exists a map $P$ that assigns to every function $f : X \times X \to \mathbb R$ (admissible or not) a function $Pf : X \to \mathbb R$, such that all seven of the following hold.

1. Measurability. For every admissible $f$, the function $Pf$ agrees $\mu$-a.e. with some measurable function $X \to \mathbb R$.
2. Insensitivity to negligible changes. For all admissible $f, g$: if the set $\{p \in X \times X : f(p) \neq g(p)\}$ is $E$-negligible, that is, if

$$\mu\Bigl(\pi_1\bigl(\{p \in E : f(p) \neq g(p)\}\bigr)\Bigr) = 0,$$

then $Pf = Pg$ $\mu$-a.e. 3. Additivity. For all admissible $f, g$: $P(f + g) = Pf + Pg$ $\mu$-a.e. (sums taken pointwise). 4. Homogeneity. For every real number $c$ and every admissible $f$: $P(c f) = c\, Pf$ $\mu$-a.e. (scalar multiples taken pointwise). 5. Positivity. For every admissible $f$ with $f(p) \ge 0$ for all $p \in E$: $Pf(x) \ge 0$ for $\mu$-almost every $x$. 6. Normalization. $P\mathbf 1 = \mathbf 1$ $\mu$-a.e., where on the left $\mathbf 1$ is the constant function $1$ on $X \times X$ and on the right it is the constant function $1$ on $X$. 7. Invariance. For every partial transformation $\varphi$ of $E$ (with domain $D$, codomain $C$, bijection $e$) and every admissible $f$: $P(\varphi \cdot f) = \varphi \cdot (Pf)$ $\mu$-a.e. Spelled out: for $\mu$-almost every $y \in X$,

$$P(\varphi \cdot f)(y) = \begin{cases} (Pf)\bigl(e^{-1}(y)\bigr) & \text{if } y \in C,\\ 0 & \text{if } y \notin C.\end{cases}$$

Points to note about (B):

• Each "$\mu$-a.e." in items 1–7 is asserted separately for each choice of $f$, $g$, $c$ and $\varphi$. The exceptional null set may depend on those choices; no single null set is required to serve for all of them at once.
• The conditions constrain $P$ only through its values at admissible functions (the constant function $\mathbf 1$ of item 6 is one) and, in item 7, at the transformed functions $\varphi \cdot f$. Item 7 applies to $P(\varphi \cdot f)$ whatever $\varphi \cdot f$ is; it does not itself require $\varphi \cdot f$ to be admissible. On other functions $P$ is unconstrained.
• Linearity, positivity and invariance are required only up to $\mu$-null sets, never pointwise everywhere.
• Positivity in item 5 needs $f \ge 0$ only on $E$, not on all of $X \times X$.
• If $\mu$ is the zero measure, every requirement in items 1–7 is an "almost everywhere" requirement for a measure under which every set is null, and any map $P$ meets it.