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

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 $L^1(\Omega )+L^\infty (\Omega )$.