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The statement

Let $A$ be a subring of $\mathbb{R}$ (a subset containing $0$ and $1$ and closed under addition, negation and multiplication). Assume

• (countability) $A$ is a countable set, and
• (density) $A$ is dense in $\mathbb{R}$ with its usual topology.

These are the only hypotheses; there are no other variables. (They can be met: for instance $A=\mathbb{Q}$.)

Conclusion. The orbit relation $R_A$ of $\mathrm{SL}_2(A)$ acting on the projective line $\mathbb{R}\cup\{\infty\}$ by Möbius transformations is not amenable with respect to the measure $\lambda_\infty$, in the precise sense of "amenable relation" spelled out below. That is: there is no map $P$ satisfying the seven conditions (a)–(g) of the definition below for $X=\mathbb{R}\cup\{\infty\}$, $\mu=\lambda_\infty$, $R=R_A$.

The rest of this account defines every object in that sentence.

The space, its σ-algebra and its measure

The space. $X=\mathbb{R}\cup\{\infty\}$ is the one-point compactification of $\mathbb{R}$ (a single point $\infty$ is adjoined; neighbourhoods of $\infty$ are complements of compact subsets of $\mathbb{R}$, together with $\infty$). As a topological space it is a circle.

The σ-algebra. $X$ carries its Borel σ-algebra (generated by the open sets of the one-point compactification topology). A subset $E\subseteq X$ is Borel exactly when $E\cap\mathbb{R}$ is a Borel subset of $\mathbb{R}$; the point $\infty$ may or may not belong to $E$. The product $X\times X$ carries the product σ-algebra (generated by rectangles $E_1\times E_2$ of Borel sets), and $\mathbb{R}$ carries its Borel σ-algebra.

The measure. $\lambda_\infty$ is the push-forward of Lebesgue measure $\lambda$ on $\mathbb{R}$ along the inclusion $\mathbb{R}\hookrightarrow X$. For a Borel set $E\subseteq X$,

$$\lambda_\infty(E)=\lambda(E\cap\mathbb{R}),$$

so $\lambda_\infty(\{\infty\})=0$ and $\lambda_\infty(X)=\infty$ (the measure is σ-finite, not finite). For an arbitrary, possibly non-Borel, set $S\subseteq X$, "$\lambda_\infty(S)$" means the outer measure

$$\lambda_\infty(S)=\inf\{\lambda_\infty(E) : E\supseteq S,\ E \text{ Borel}\}.$$

"$\lambda_\infty$-almost every $x$" means: for all $x$ outside some set of $\lambda_\infty$-measure $0$.

The group and its action

$\mathrm{SL}_2(A)$ is the set of $2\times 2$ matrices

$$g=\begin{pmatrix} a & b\\ c & d\end{pmatrix},\qquad a,b,c,d\in A,\qquad ad-bc=1 .$$

Each such $g$ is viewed as a real invertible matrix and acts on $X$ by the Möbius transformation $x\mapsto g\cdot x$, given explicitly by

$$g\cdot x=\begin{cases}\dfrac{ax+b}{cx+d} & \text{if } x\in\mathbb{R} \text{ and } cx+d\neq 0,\\[2mm] \infty & \text{if } x\in\mathbb{R} \text{ and } cx+d=0,\end{cases} \qquad g\cdot\infty=\begin{cases}a/c & \text{if } c\neq 0,\\ \infty & \text{if } c=0.\end{cases}$$

(This is the action on lines in $\mathbb{R}^2$, with $x\in\mathbb{R}$ identified with the line through the column vector $(x,1)^{T}$, $\infty$ with the line through $(1,0)^{T}$, and $g$ acting by matrix–column-vector multiplication.) It is a group action: $g\cdot(h\cdot x)=(gh)\cdot x$ and $I\cdot x=x$.

The relation.

$$R_A=\{(x,y)\in X\times X \;:\; \text{there is } g\in \mathrm{SL}_2(A) \text{ with } g\cdot x=y\}.$$

This is the orbit equivalence relation of the action. No measurability of $R_A$ is asserted or assumed by the statement.

"Amenable relation": the definition being negated

Let $X$ be a set with a σ-algebra, $\mu$ a measure on it, and $R\subseteq X\times X$ any subset. (Here $X=\mathbb{R}\cup\{\infty\}$, $\mu=\lambda_\infty$, $R=R_A$.)

Bounded measurable on $R$. A function $f:X\times X\to\mathbb{R}$ is called admissible if it is measurable on all of $X\times X$ (product σ-algebra to Borel sets of $\mathbb{R}$) and there is a real $C$ with $|f(p)|\le C$ for every $p\in R$. No bound is required off $R$.

$R$-null. A set $S\subseteq X\times X$ is $R$-null if

$$\mu\bigl(\pi_1(S\cap R)\bigr)=0,$$

where $\pi_1(x,y)=x$ and $\mu$ of a possibly non-measurable set is its outer measure.

Partial transformations of $R$. A partial transformation $\varphi$ consists of two measurable sets $D,C\subseteq X$ (either may be empty) and a bijection $\varphi:D\to C$ such that $\varphi$ and $\varphi^{-1}$ are both measurable (for the σ-algebras on $D$, $C$ consisting of traces $E\cap D$, $E\cap C$ of measurable $E$), and such that $(a,\varphi(a))\in R$ for every $a\in D$. For such $\varphi$ define, for $f:X\times X\to\mathbb{R}$ and $F:X\to\mathbb{R}$,

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

Only the first coordinate of $f$ is moved.

Left-invariant mean. A map $P$ assigning to every function $f:X\times X\to\mathbb{R}$ a function $P f:X\to\mathbb{R}$ (no linearity, measurability or other structure is presupposed beyond what follows) is a left-invariant mean for $(\mu,R)$ if all seven conditions hold:

• (a) measurability. For every admissible $f$, $Pf$ is $\mu$-almost-everywhere measurable (agrees $\mu$-a.e. with a measurable function $X\to\mathbb{R}$).
• (b) insensitivity to $R$-null changes. For admissible $f,g$ such that the set $\{p\in X\times X: f(p)\neq g(p)\}$ is $R$-null, $Pf=Pg$ $\mu$-a.e.
• (c) additivity. For admissible $f,g$: $P(f+g)=Pf+Pg$ $\mu$-a.e. (sums pointwise).
• (d) homogeneity. For every real $c$ and admissible $f$: $P(cf)=c\,Pf$ $\mu$-a.e.
• (e) positivity. For admissible $f$ with $f(p)\ge 0$ for all $p\in R$: $Pf(x)\ge 0$ for $\mu$-a.e. $x$.
• (f) normalisation. $P\mathbf 1=\mathbf 1$ $\mu$-a.e., where the input $\mathbf 1$ is the constant function $1$ on all of $X\times X$ and the output $\mathbf 1$ the constant function $1$ on $X$.
• (g) invariance. For every partial transformation $\varphi$ of $R$ and every admissible $f$:

$$P(\varphi_* f)=\varphi_*(Pf)\quad \mu\text{-a.e.}$$

(Here $\varphi_* f$ itself is not required to be admissible.)

In each of (a)–(g) the exceptional $\mu$-null set may depend on all the data of that condition ($f$, $g$, $c$, $\varphi$).

$R$ is amenable with respect to $\mu$ if some left-invariant mean $P$ for $(\mu,R)$ exists.

The statement, restated

For every countable dense subring $A\subseteq\mathbb{R}$: there does not exist any map $P$ from real functions on $X\times X$ to real functions on $X$ satisfying (a)–(g) for $\mu=\lambda_\infty$ (push-forward of Lebesgue measure to $\mathbb{R}\cup\{\infty\}$) and $R=R_A=\{(x,g\cdot x): x\in X,\ g\in\mathrm{SL}_2(A)\}$ (Möbius action as above).