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Archiv der Mathematik

, Volume 112, Issue 2, pp 205–212 | Cite as

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

  • Dávid Kunszenti-KovácsEmail author
Article
  • 11 Downloads

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

Keywords

Pointwise ergodic theorem Dunford–Schwartz operator Infinite measure 

Mathematics Subject Classification

Primary 47A35 Secondary 37A30 47B38 

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Notes

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.

References

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.MTA Alfréd Rényi Institute of MathematicsBudapestHungary

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