Counter-examples to the Dunford-Schwartz pointwise ergodic theorem on $L^1+L^\infty$

TL;DR

This paper constructs counterexamples showing that the Dunford-Schwartz pointwise ergodic theorem does not hold for certain functions in $L^1+L^ fty$ on infinite measure spaces, highlighting limitations of the theorem.

Contribution

It extends previous results by explicitly constructing Dunford-Schwartz operators where pointwise convergence fails for a broad class of functions.

Findings

Counterexamples for $L^1+L^ ablafty$ functions on infinite measure spaces.

Failure of pointwise convergence is prevalent for a residual set of functions.

Constructed operators demonstrate divergence for almost every point.

Abstract

Extending a result by Chilin and Litvinov, we show by construction that given any $\sigma$-finite infinite measure space $(\Omega,\mathcal{A}, \mu)$ and a function $f\in L^1(\Omega)+L^\infty(\Omega)$ with $\mu(\{|f|>\varepsilon\})=\infty$ for some $\varepsilon>0$, there exists a Dunford-Schwartz operator $T$ over $(\Omega,\mathcal{A}, \mu)$ such that $\frac{1}{N}\sum_{n=1}^N (T^nf)(x)$ fails to converge for almost every $x\in\Omega$. In addition, for each operator we construct, the set of functions for which pointwise convergence fails almost everywhere is residual in $L^1(\Omega)+L^\infty(\Omega)$.