Integral of a simple function (measure theory)/Definition

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Quickly written. Needs to go something like "using every simple function has a standard representation" and then showing that the integral is the same for each standard representation is what needs to be done


For a simple function in its standard representation, say [ilmath]f:=\sum^n_{i=0}x_i\mathbf{1}_{A_i}[/ilmath] then the [ilmath]\mu[/ilmath]-integral, [ilmath]I_\mu:\mathcal{E}^+\rightarrow\mathbb{R} [/ilmath] is[1]:

  • [math]I_\mu(f):=\sum^n_{i=1}x_i\mu(A_i)\in[0,\infty][/math]

Note that this is independent of the particular standard representation of [ilmath]f[/ilmath].


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