Counter-examples to the Dunford–Schwartz pointwise ergodic theorem on L1 + L∞

Kunszenti-Kovács, Dávid (2019) Counter-examples to the Dunford–Schwartz pointwise ergodic theorem on L1 + L∞. ARCHIV DER MATHEMATIK, 112. pp. 205-212. ISSN 0003-889X

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Abstract

Extending a result by Chilin and Litvinov, we show by construction that given any $\sigma$-finite infinite measure space $(\Omega,\mc{A}, \mu)$ 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 $(\Omega,\mc{A}, \mu)$ 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 $L^1(\Omega)+L^\infty(\Omega)$.

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