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1 results

  • 15 - Interior-Point Theorems: Steinhaus–Weil Theory

  • N. H. Bingham, Imperial College London, Adam J. Ostaszewski, London School of Economics and Political Science
  • Book:
    Category and Measure
    Published online:
    14 January 2025
    Print publication:
    23 January 2025, pp 235-267

  • Summary

    Steinhaus’ Theorem of Chapter 9, an interior-point result, was extended from the line under Lebesgue measure to topological groups under Haar measure by Weil. The resulting Steinhaus–Weil theory, which is extensive, is presented in Chapter 15. The Simmons–Mospan converse gives the condition for the extension to hold: in a locally compact Polish group, a Borel measure has the ‘Steinhaus–Weil property’ if and only if it is absolutely continuous with respect to Haar measure. We define measure subcontinuity (adapted from Fuller’s subcontinuity for functions), and amenability at the identity. We prove Solecki’s interior-point theorem: in a Polish group, if a set E is not left Haar null, then the identity is an interior point of $E^{-1}E$. Related results on sets such as ${AB}^{-1}$ rather than ${AA}^{-1}$ are given.