Simple function (measure theory)

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TODO: Cross reference with Halmos' book


A simple function [ilmath]f:X\rightarrow\mathbb{R} [/ilmath] on a measurable space [ilmath](X,\mathcal{A})[/ilmath] is a[1]:

  • function of the form [ilmath]\sum^N_{i=1}x_i\mathbf{1}_{A_i}(x)[/ilmath] for
  • finitely many sets, [ilmath]A_1,\ldots,A_N\in\mathcal{A} [/ilmath] and
  • finitely many [ilmath]x_1,\ldots,x_n\in\mathbb{R} [/ilmath]

Standard representation

Standard representation (measure theory)/Definition


  1. Measures, Integrals and Martingales - René L. Schilling