Integral of a positive function (measure theory)

Definition

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] is:

• [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]

Recall that:

• [ilmath]I_\mu(g)[/ilmath] denotes the [ilmath]\mu[/ilmath]-integral of a simple function
• [ilmath]\mathcal{E}^+(\mathcal{A})[/ilmath] denotes all the positive simple functions in their standard representations from [ilmath]X[/ilmath] considered with the [ilmath]\mathcal{A} [/ilmath] [ilmath]\sigma[/ilmath]-algebra.

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

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

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

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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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