Integral of a positive function (measure theory)

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There are some problems here:
  • We don't really mean positive function, we mean non-negative. Alec (talk) 19:18, 14 April 2017 (UTC)
This is under review as a part of measure theory


Let [ilmath](X,\mathcal{A},\mu)[/ilmath] be a measure space, the [ilmath]\mu[/ilmath]-integral of a positive numerical function, [ilmath]f\in\mathcal{M}^+_{\bar{\mathbb{R} } }(\mathcal{A}) [/ilmath][Note 1][Note 2] is[1]:

  • [math]\int f\mathrm{d}\mu:=\text{Sup}\left\{I_\mu(g)\ \Big\vert\ g\le f, g\in\mathcal{E}^+(\mathcal{A})\right\}[/math][Note 3]

Recall that:

There are alternate notations, that make the variable of integration more clear, they are:

  • [ilmath]\int f(x)\mu(\mathrm{d}x)[/ilmath][1]
  • [ilmath]\int f(x)\mathrm{d}\mu(x)[/ilmath][1]

Immediate results

  • [math]\forall f\in\mathcal{E}^+(\mathcal{A})\left[\int f\mathrm{d}\mu=I_\mu(f)\right][/math] - Integrating a simple function works

Note that without this lemma we cannot be sure the integral of simple functions is well defined! Which would be really really bad if it weren't true.

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Page 70 in[1]


  1. So [ilmath]f:X\rightarrow\bar{\mathbb{R} }^+[/ilmath]
  2. Notice that [ilmath]f[/ilmath] is [ilmath]\mathcal{A}/\bar{\mathcal{B} } [/ilmath]-measurable by definition, as [ilmath]\mathcal{M}_\mathcal{Z}(\mathcal{A})[/ilmath] denotes all the measurable functions that are [ilmath]\mathcal{A}/\mathcal{Z} [/ilmath]-measurable, we just use the [ilmath]+[/ilmath] as a slight abuse of notation to denote all the positive ones (with respect to the standard order on [ilmath]\bar{\mathbb{R} } [/ilmath] - the extended reals)
  3. The [ilmath]g\le f[/ilmath] is an abuse of notation for saying that [ilmath]g[/ilmath] is everywhere less than [ilmath]f[/ilmath], we could have written:
    • [math]\int f\mathrm{d}\mu=\text{Sup}\left\{I_\mu(g)\ \Big\vert\ g\le f, g\in\mathcal{E}^+\right\}=\text{Sup}\left\{I_\mu(g)\ \Big\vert\ g\in\left\{h\in\mathcal{E}^+(\mathcal{A})\ \big\vert\ \forall x\in X\left(h(x)\le f(x)\right)\right\}\right\}[/math] instead.
    Inline with: Notation for dealing with (extended) real-valued measurable maps


  1. 1.0 1.1 1.2 1.3 Measures, Integrals and Martingales - René L. Schilling