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 \(L^1(\Omega )+L^\infty (\Omega )\).
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Acknowledgements
The author has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant Agreement No 617747, and from the MTA Rényi Institute Lendület Limits of Structures Research Group.
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Kunszenti-Kovács, D. Counter-examples to the Dunford–Schwartz pointwise ergodic theorem on \(\varvec{L^1+L^\infty }\). Arch. Math. 112, 205–212 (2019). https://doi.org/10.1007/s00013-018-1248-z
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DOI: https://doi.org/10.1007/s00013-018-1248-z