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I am looking for a counterexample to the following theorem when $p=\infty$:

$L^p$ Ergodic Theorem of Von Neumann. Let $1\leq p<\infty$ and let $T$ be a measure-preserving transformation of the probability space $(X,\mathfrak{B},m)$. If $f\in L^p(m)$ there exists $f^*\in L^p(m)$ with $f^*\circ T=f^*$ a.e. and $\left\lVert (1/n)\sum_{i=0}^{n-1}f(T^ix)-f^*(x)\right\rVert_p\to 0$.

Specifically, I want to find a probability space and $f\in L^{\infty}(m)$, such that there doesn't exist a $f^*$ preserved by $T$ such that the convergence holds in $L^\infty$ norm.

Davide Giraudo
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No One
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1 Answers1

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Consider an i.i.d. sequence $\left(\varepsilon_i\right)_{i\geq 0}$ of random variables taking the values $1$ and $-1$ with probability $1/2$. This can be written in the language of dynamical systems. Indeed, let $\mathbb N$ be the set of non-negative integers, $\Omega:=\left\{-1,1\right\}^{\mathbb N}$ (the set of sequences with values in $\left\{-1,1\right\}$). Let $\mu$ be the probability measure such that for all integer $n$ and all $\left(a_i\right)_{i=0}^n\in\left\{-1,1\right\}^{n+1}$, $\mu\left\{x=\left(x_i\right)_{i\geqslant 0} \in\Omega\mid x_i=a_i\mbox{ for each }0\leqslant i\leqslant n \right\}=2^{-n-1}$. Let $T\colon \left(x_i\right)_{i\geqslant 0} \in\Omega\mapsto \left(x_{i+1}\right)_{i\geqslant 0} \in\Omega$; then $T$ is measure preserving. Then define $f\colon \left(x_i\right)_{i\geqslant 0} \in\Omega\mapsto x_0$.

In this context, $n^{-1}\left\lVert \sum_{i=0}^{n-1}\varepsilon_i\right\rVert_\infty=1$, as the partial sum $\sum_{i=0}^{n-1}\varepsilon_i$ takes the value one with a positive probability, namely, $2^{-n}$.

Davide Giraudo
  • 172,925