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1 results

  • 1 - Preliminaries

  • 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 24-33

  • Summary

    The text proper of the book begins with Littlewood’s three principles. The first – ‘any measurable set is nearly a finite union of intervals’ – is essentially regularity of Lebesgue measure. The second – ‘any measurable function is nearly continuous’ – is Lusin’s Theorem. The third – ‘any convergent sequence of measurable functions is nearly uniformly continuous’ – is Egorov’s Theorem. Then what will be needed from general topology is summarised, with references, going as far as para-compactness. Modes of convergence – in measure (in probability), almost everywhere (almost sure), etc. – are discussed. The Borel hierarchy – the result of applying, to (say) the open sets, the sigma and delta operations (union and intersection) alternately – is developed, as far as the Souslin operation. Analytic sets – much used in the book – are briefly treated here.