On norm almost periodic measures

Abstract

In this paper, we study norm almost periodic measures on locally compact Abelian groups. First, we show that the norm almost periodicity of \(\mu \) is equivalent to the equi-Bohr almost periodicity of \(\mu *g\) for all g in a fixed family of functions. Then, we show that, for absolutely continuous measures, norm almost periodicity is equivalent to the Stepanov almost periodicity of the Radon–Nikodym density.

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Notes

  1. 1.

    Recall that \(\mu \circledast _{\mathcal {A}}\nu \) is defined as the vague limit of \(\left( \frac{1}{|A_n|}\mu |_{A_n}*\nu _{A_n}\right) _{n\in {{\mathbb {N}}}}\), if the limit exists. Given a van Hove sequence, the limit always exists along a subsequence [16].

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Acknowledgements

The work was supported by NSERC with Grants 03762-2014 and 2020-00038 (NS) and by the German Research Foundation (DFG) via research Grant 415818660 (TS), and the authors are grateful for the support.

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Correspondence to Timo Spindeler.

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Spindeler, T., Strungaru, N. On norm almost periodic measures. Math. Z. (2021). https://doi.org/10.1007/s00209-020-02671-w

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Keywords

  • Almost periodic measures
  • Lebesgue decomposition

Mathematics Subject Classification

  • 43A05
  • 43A25
  • 52C23