Integral of a positive function (measure theory)

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• 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

Definition

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

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

Recall that:

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

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

• $\int f(x)\mu(\mathrm{d}x)$
• $\int f(x)\mathrm{d}\mu(x)$

Immediate results

[Expand]

• $$\forall f\in\mathcal{E}^+(\mathcal{A})\left[\int f\mathrm{d}\mu=I_\mu(f)\right]$$ - 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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