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Setting. Let $G$ be an arbitrary group (any size, including the trivial group; no topology, finiteness or countability assumption). The statement is also parametrised by a second universe level $v$, which enters only through condition (4) below; see the remark there. For each such $G$ (and each choice of $v$), the statement asserts that the following four propositions are pairwise equivalent: for every two of them, each holds if and only if the other does.

Throughout, $[0,\infty]$ denotes the extended non-negative reals, with $a + \infty = \infty + a = \infty$. For $g \in G$ acting on a set $X$ and $S \subseteq X$, write $gS = \{\, g\cdot x : x \in S \,\}$ (the image of $S$ under $x \mapsto g\cdot x$). When $X = G$, the action is left multiplication, $g\cdot x = gx$, so $gS = \{gx : x \in S\}$.

A function $m : \mathcal P(X) \to [0,\infty]$ defined on all subsets of $X$ is called finitely additive here when

$$m(\varnothing) = 0 \quad\text{and}\quad m(S \cup T) = m(S) + m(T)\ \text{ for all disjoint } S, T \subseteq X.$$

It is called $G$-invariant when $m(gS) = m(S)$ for every $g \in G$ and every $S \subseteq X$.

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(1) Amenability (measure form). There exists a function $m : \mathcal P(G) \to [0,\infty]$, defined on all subsets of $G$, that is finitely additive, satisfies $m(G) = 1$, and is invariant under left translation:

$$m(gS) = m(S)\qquad\text{for all } g\in G,\ S \subseteq G,\ \text{where } gS = \{gx : x\in S\}.$$

(2) Invariant mean. Let $\ell^\infty(G)$ be the real vector space of bounded functions $f : G \to \mathbb R$ (bounded meaning $\sup_{h}|f(h)| < \infty$), with pointwise addition and scalar multiplication. For $g \in G$ and $f \in \ell^\infty(G)$ define the left shift

$$(g\cdot f)(h) = f(g^{-1}h)\qquad (h \in G).$$

Condition (2) says: there exists an $\mathbb R$-linear map $M : \ell^\infty(G) \to \mathbb R$ (no continuity is required) such that

• (positivity) if $f(h) \ge 0$ for all $h \in G$, then $M(f) \ge 0$;
• (normalisation) if $f(h) = 1$ for all $h\in G$, then $M(f) = 1$;
• (left invariance) $M(g\cdot f) = M(f)$ for every $g\in G$ and every $f \in \ell^\infty(G)$.

(3) $G$ is not paradoxical under its own left-multiplication action. This is the negation of the following statement:

There exist subsets $A, B \subseteq G$ such that $A \ne G$, $B \ne G$, $A \cap B = \varnothing$, $A$ is equidecomposable with $G$, and $B$ is equidecomposable with $G$.

Here, for subsets $P, Q \subseteq G$, "$P$ is equidecomposable with $Q$" (with respect to $G$ acting on itself by left multiplication $g\cdot x = gx$) means: there exist a bijection $\varphi : P \to Q$ and a finite set $F \subseteq G$ such that for every $a \in P$ there is some $g \in F$ with $\varphi(a) = ga$. (Formally $\varphi$ is recorded as a map $G\to G$ together with a map $G \to G$ that are mutually inverse between $P$ and $Q$, map $P$ into $Q$ and $Q$ into $P$; the values outside $P$, resp. $Q$, are irrelevant. The finite set $F$ is only required to exist; the pieces are implicit, namely the sets $\{a \in P : \varphi(a) = ga\}$, which need not be disjoint for different $g$.) The conditions $A \subseteq G$, $B\subseteq G$ are also stated but are automatic.

So (3) says: there are no two disjoint proper subsets $A, B$ of $G$ each of which is equidecomposable, in the above sense, with all of $G$.

(4) Invariant extension property. For every set $X$ in the universe level $\max(u, v)$ (where $u$ is the level of $G$), every action of $G$ on $X$ (a genuine group action: $1\cdot x = x$ and $(gh)\cdot x = g\cdot(h\cdot x)$), every family $\mathcal R$ of subsets of $X$, and every two functions $\mu, \nu : \mathcal P(X) \to [0,\infty]$ defined on all subsets of $X$: if

• (a) $\mathcal R$ is closed under the action: $S \in \mathcal R \Rightarrow gS \in \mathcal R$ for all $g \in G$;
• (b) $\mu$ is invariant on $\mathcal R$: $\mu(gS) = \mu(S)$ for all $g\in G$, $S \in \mathcal R$;
• (c) $\nu$ agrees with $\mu$ on $\mathcal R$: $\nu(S) = \mu(S)$ for all $S \in \mathcal R$;
• (d) $\nu$ is finitely additive on all of $\mathcal P(X)$,

then there exists $\bar\mu : \mathcal P(X) \to [0,\infty]$ that is finitely additive on all of $\mathcal P(X)$, agrees with $\mu$ on $\mathcal R$ (i.e. $\bar\mu(S) = \mu(S)$ for $S\in\mathcal R$), and is $G$-invariant on all subsets: $\bar\mu(gS) = \bar\mu(S)$ for all $g \in G$, $S\subseteq X$.

Remarks on (4):

• Nothing is assumed of $\mathcal R$ beyond (a): it need not contain $\varnothing$ or $X$, be closed under unions, complements or intersections, and may be empty. When $\mathcal R = \varnothing$, hypotheses (a)–(c) are vacuous and the conclusion only asks for some finitely additive $G$-invariant $\bar\mu$ on $\mathcal P(X)$ (the identically-zero function is one such). More generally, hypotheses (c)–(d) together require that the values of $\mu$ on $\mathcal R$ be the values of some finitely additive function on $\mathcal P(X)$; they are not a restriction on $\mu$ outside $\mathcal R$.
• $\mu$ itself is not assumed additive; only its values on $\mathcal R$ matter to the conclusion, and those values are forced by (c) to coincide with those of the finitely additive $\nu$.
• Values $\infty$ are permitted for $\mu$, $\nu$ and $\bar\mu$.
• The quantification over $X$ ranges over all types of universe level $\max(u,v)$ only, where $v$ is a free universe parameter of the whole statement. The statement is therefore a separate assertion for each $v$: for each fixed $v$, conditions (1), (2), (3) and this level-$v$ version of (4) are pairwise equivalent. It does not assert anything that quantifies over all universe levels at once. (Every level $\max(u,v)$ contains a copy of $G$ itself, so $X$ of the "size" of $G$ are always included.)

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Summary. For every group $G$ (and each universe level $v$):

$$(1) \iff (2) \iff (3) \iff (4),$$

where (1) is the existence of a left-invariant, finitely additive, $[0,\infty]$-valued set function on all subsets of $G$ with total mass $1$; (2) is the existence of a positive, normalised, left-shift-invariant real linear functional on bounded real functions on $G$; (3) is the non-existence of a paradoxical decomposition of $G$ under left multiplication, as defined above; and (4) is the invariant extension property as stated above.