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

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 $\mu (\{|f|>\epsilon \})=\infty$ for some $\epsilon >0$, there exists a Dunford-Schwartz operator $T$ over $(\Omega ,\mathcal{A},\mu )$ such that $\frac{1}{N}{\sum }_{n=1}^N(T^{n}f)(x)$ 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 $L^1(\Omega )+L^{\infty }(\Omega )$.

7 pages