Integral (measure theory)

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Definition

Given a measure space, [ilmath](X,\mathcal{A},\mu)[/ilmath] and a function [ilmath]f:X\rightarrow\bar{\mathbb{R} } [/ilmath], we say that [ilmath]f[/ilmath] is [ilmath]\mu[/ilmath]-integrable if:

• [ilmath]f[/ilmath] is a measurable map, an [ilmath]\mathcal{A}/\bar{\mathcal{B} } [/ilmath]-measurable map; and if
• The integrals [ilmath]\int f^+\mathrm{d}\mu,\ \int f^-\mathrm{d}\mu<\infty[/ilmath], then:

We define the [ilmath]\mu[/ilmath]-integral of [ilmath]f[/ilmath] to be:

• [math]\int f\mathrm{d}u:=\int f^+\mathrm{d}\mu-\int f^-\mathrm{d}\mu[/math]

Where:

• [math]\int f^\pm\mathrm{d}\mu[/math] are the integrals of positive functions.

Integration of positive functions

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.