Observation on some similarity between Riesz-Markov Theorem and Birkhoff theorem

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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 $X$.


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.