Observation on some similarity between Riesz-Markov Theorem and Birkhoff theorem
Published:
Two theorems at a glance.
Riesz - Markov Theorem
Theorem: Let $X$ be a locally compact Hausdorff space. For any positive linear functional $\psi$ on $C(X)$, there is a unique regular Borel measure $\mu$ on $X$ such that
\[\begin{equation} \forall f \in C(X), \psi(f) = \int_X f(x) d\mu(x) \end{equation}\]Birkhoff’s ergodic theorem
Theorem: Let $(X, \mathcal{B}, \mu, T)$ be a measure-preserving system ($T: X \mapsto X$ is measure-preserving transformation). For any $f \in \mathcal{L}_{\mu}^1$,
\[\begin{equation} \lim_{n\rightarrow \infty} \frac{1}{n} \sum_{i=0}^{n-1} f\circ T^i(x) = \int_X f d\mu, \end{equation}\]is true almost everywhere in $X$.
Discussions
First, both right hand sides are the same.
If we take the finite approximation of the left hand side on the second equation, denote $\mathcal{K}$ as the Koopman operator on the measure-preserving system associated with $T$, then we have
\[\begin{equation} \frac{1}{n} \sum_{i=0}^{n-1} f \circ T^i(x) = \left(\frac{1}{n} \sum_{i=0}^{n-1} \mathcal{K}^i\right) f \triangleq \bar{\mathcal{K}}_n f. \end{equation}\]Since $\bar{\mathcal{K}_n}$ is a linear operator (so as the corresponding limit) rather than a positive linear functional, one cannot directly apply RMT to obtain Birkhoff theorem. However, I guess the ergodic nature makes the linear operator evaluated pointwise resembles a linear functional. But there is still some difference that makes them quite different.
Just to take the note here to not confuse one with another.