View source for Integral of a positive function (measure theory)

{{Provisional page|grade=A*|msg=There are some problems here:
* We don't really mean positive function, we mean non-negative. [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 19:18, 14 April 2017 (UTC)
This is under review as a part of measure theory}}
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==Definition==
{{:Integral of a positive function (measure theory)/Definition}}
There are alternate notations, that make the ''variable of integration'' more clear, they are:
* {{M|\int f(x)\mu(\mathrm{d}x)}}<ref name="MIAMRLS"/>
* {{M|\int f(x)\mathrm{d}\mu(x)}}<ref name="MIAMRLS"/>
==Immediate results==
{{Begin Inline Theorem}}
* {{MM|1=\forall f\in\mathcal{E}^+(\mathcal{A})\left[\int f\mathrm{d}\mu=I_\mu(f)\right]}} - Integrating a [[simple function (measure theory)|simple function]] works
{{Begin Inline Proof}}
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.
{{Requires proof|Page 70 in<ref name="MIAMRLS"/>}}
{{End Proof}}{{End Theorem}}
==Notes==
<references group="Note"/>
==References==
<references/>
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{{Definition|Measure Theory}}
{{Theorem Of|Measure Theory}}