A short note on the proof of the ergodic theorem. (English) Zbl 1340.37002
Let \((\Omega,\Sigma; P)\) be a probability space, \(T :\Omega\to\Omega\) a measure-preserving transformation, \(X\) an integrable random variable, and \(\operatorname{Im}\) the \(\sigma\)-field of invariant sets. Denote
\[
S_k :=\sum_{i=0}^{k-1} X\circ T^i, \;\;M := \sup_{k\in N} S_k.
\]
In the present paper, the author first proves the Maximal Ergodic Inequality
\[
\int_{[M>0]}X dP\geq 0
\]
in a short proof simpler than that of M. Keane and K. Petersen [Easy and nearly simultaneous proofs of the ergodic theorem and maximal ergodic theorem, Dynamics & Stochastics, Institute of Mathematical Statistics Lecture Notes - Monograph Series 48, 248–251 (2006; Zbl 1121.37301)]. Then this inequality is generalized as follows:
\[
\int_{A\cap[\sup_{k\in N}\frac{S_k}{k}>Y]} X dP\geq \int_{A\cap [\sup_{k\in N}\frac{S_k}{k}>Y]} Y dP
\]
for any invertible integrable random variable \(Y\) and \(A\in\operatorname{Im}\). Finally, an application of this inequality yields the Pointwise Ergodic Theorem.
Reviewer: Farruh Mukhamedov (Kuantan)
MSC:
37A30 | Ergodic theorems, spectral theory, Markov operators |
37A05 | Dynamical aspects of measure-preserving transformations |