Lemma 1

The set K of means on X is the subset of the unit ball of the dual of the Banach space $E = \mathscr{B}(X; \mathbf{R})$ whose elements are the linear forms $\mu$ such that $\mu(1) = 1$. It is a subset of $E'$ which is convex and compact for $\sigma(E', E)$.

Let $\mu$ be a linear form on E, such that $\mu(1) = 1$. For every function $f \in E$, we define the function $f' \in E$ by $f'(x) = \|f\| - f(x)$ ($x \in X$). First assume that $\mu$ is a mean; for every $f \in E$, we have $f' \geq 0$, hence $\mu(f') \geq 0$, i.e. $\mu(f) \leq \|f\|$; therefore $\|\mu\| \leq 1$.

Conversely, suppose $\mu$ belongs to $E'$, and that $\| \mu \| \leq 1$; for every positive function $f \in E$, we have $\mu(f') \leq \| f' \|$, hence

and finally $\mu(f) \geq 0$; consequently, $\mu$ is a mean.

It is clear that $K$ is convex; that it is compact for $\sigma(E', E)$ follows from cor. 3 of III, p. 17.

Q.E.D.

Let $\Gamma$ be a set of continuous mappings from $X$ into $X$ which commute pairwise. Let $\gamma \in \Gamma$. For every function $f \in E$, we have $f \circ \gamma \in E$; hence we can define an affine transformation $u_{\gamma}$ on the set $K$ of means on $X$, by

If $K$ is assigned the topology induced by $\sigma(E', E)$, the mapping $u_{\gamma}$ is continuous. If $\gamma$ is a homeomorphism, $u_{\gamma} \mu$ can be deduced from $\mu$ by transport of structure. Finally, we have $u_{\gamma} u_{\gamma'} = u_{\gamma' \gamma} u_{\gamma}$ for all $\gamma, \gamma'$ in $\Gamma$. By the Markoff-Kakutani th. (IV, p. 39, th. 1), there exists a mean $\mu$ on $X$, such that $u_{\gamma} \mu = \mu$ for all $\gamma \in \Gamma$; in other words, $\mu$ satisfies the relation $\mu(f) = \mu(f \circ \gamma)$ for $f \in E$ and $\gamma \in \Gamma$.

In an analogous way, the corollary of th. 1 (IV, p. 40) implies the following result: