# Integral of a positive function (measure theory)

This page is provisional and the information it contains may change before this notice is removed (in a backwards incompatible way). This usually means the content is from one source and that source isn't the most formal, or there are many other forms floating around. It is on a to-do list for being expanded.The message provided is:
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

## 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][Note 1][Note 2] is:

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

## Immediate results

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