View source for Integral (measure theory)

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==Definition==
Given a [[measure space]], {{M|(X,\mathcal{A},\mu)}} and a function {{M|f:X\rightarrow\bar{\mathbb{R} } }}, we say that {{M|f}} is ''{{M|\mu}}-integrable'' if{{rMIAMRLS}}:
* {{M|f}} is a [[measurable map]], an {{M|\mathcal{A}/\bar{\mathcal{B} } }}-measurable map; and if
* The integrals {{M|\int f^+\mathrm{d}\mu,\ \int f^-\mathrm{d}\mu<\infty}}, then:
We define the ''{{M|\mu}}-integral'' of {{M|f}} to be:
::* {{MM|1=\int f\mathrm{d}u:=\int f^+\mathrm{d}\mu-\int f^-\mathrm{d}\mu}}
Where:
* {{MM|\int f^\pm\mathrm{d}\mu}} are the [[integral of a positive function (measure theory)|integrals of positive functions]].
{{Begin Inline Theorem}}
'''Reminder: ''' [[Integral of a positive function (measure theory)|Integration of positive functions]]
{{Begin Inline Proof}}
{{:Integral of a positive function (measure theory)/Definition}}
{{Begin Inline Theorem}}
'''Reminder: ''' [[Integral of a simple function (measure theory)|Integration of simple functions]]
{{Begin Inline Proof}}
{{:Integral of a simple function (measure theory)/Definition}}
{{Begin Inline Theorem}}
'''Reminder: ''' [[Simple function (measure theory)|Definition of simple function]]
{{Begin Inline Proof}}
{{:Simple function (measure theory)/Definition}}
{{End Proof}}{{End Theorem}}<div style="height:0.4em"></div>
{{End Proof}}{{End Theorem}}<div style="height:0.4em"></div>
{{End Proof}}{{End Theorem}}
==Notes==
<references group="Note"/>
==References==
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