Invariant mean on almost periodic functions

Remarks

Von Neumann's proof shows that the closed convex hull of the left translates of $f$ contains exactly one constant function, and $M(f)$ is that constant. A shorter proof uses the Bohr compactification $\beta \colon G \to bG$. Every $f \in AP(G)$ is $F \circ \beta$ for one continuous $F$ on $bG$, and $M(f)$ is the integral of $F$ against the normalized Haar measure of $bG$. For $G = \R$ the mean is $$M(f) = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} f(x) \, dx.$$ For instance, $M(e^{i\lambda x}) = 0$ for $\lambda \ne 0$, and $M(\abs{\cos x + \cos(\sqrt{2} x)}^2) = 1$. For a compact group $G$, $M$ is the integral against normalized Haar measure.

Going deeper

For a student

The mean $M$ is an invariant mean on the subspace $AP(G)$ of bounded functions, and it exists for every topological group, amenable or not. It turns $AP(G)$ into a pre-Hilbert space with inner product $M(f \bar g)$. In that space the coefficients of inequivalent irreducible finite-dimensional unitary representations satisfy Schur orthogonality relations, which leads to von Neumann's Fourier theory of almost periodic functions. The same uniqueness holds for weakly almost periodic functions, by the Ryll-Nardzewski fixed-point theorem.