View source for Monotone convergence theorem for non-negative numerical measurable functions/Statement

<noinclude>
{{Provisional page|grade=A*|msg=There are many forms, this form matches what I'll be examined on}}
__TOC__
==Statement==
</noinclude>
Let {{M|(X,\mathcal{A},\mu)}} be a [[measure space]] and let {{M|(f_n)_{n\in\mathbb{N} }\subseteq\mathcal{M}\big(\mathcal{A},\mathcal{B}(\bar{\mathbb{R} }_{\ge 0})\big)}}<ref group="Note">Suppose we had:
# {{M|f\in\mathcal{M}\big(\mathcal{A},\mathcal{B}(\bar{\mathbb{R} }_{\ge 0})\big)}} and
# {{M|f\in\mathcal{L}^+:\eq\left\{f:X\rightarrow\bar{\mathbb{R} }\ \Big\vert\ \forall x\in X[0\le f(x)]\wedge f\in\mathcal{M}\big(\mathcal{A},\mathcal{B}(\bar{\mathbb{R} })\big)\right\} }}
Then these are the same requirements for {{M|f}}, that is {{M|\mathcal{M}\big(\mathcal{A},\mathcal{B}(\bar{\mathbb{R} }_{\ge 0})\big)\eq\mathcal{L}^+ }}
* I've seen (but not read) a proof and trust the source - Alec -17/April/2017 - 0908</ref> be a [[sequence]] of [[measurable functions]], {{M|f_n:X\rightarrow[0,+\infty]\eq\bar{\mathbb{R} }_{\ge 0} }}, then{{rMIAMRLS}}{{rAGTARAF}}:
* if {{M|\forall n\in\mathbb{N}[f_n\le f_{n+1}]}}<ref group="Note">{{M|f\le g}} is short for {{M|\forall x\in X[f(x)\le g(x)]}}</ref> - {{ie}} {{M|(f_n)_{n\in\mathbb{N} } }} is a [[non-decreasing sequence]] - then:
** {{MM|\int \mathop{\text{lim} }_{n\rightarrow\infty} \Big(f_n\Big)\ \mathrm{d}\mu\eq\mathop{\text{lim} }_{n\rightarrow\infty}\left(\int f_n\ \mathrm{d}\mu\right) }}<ref group="Note">Note that for the [[integral of a non-negative numerical measurable function]] to be even defined that (as the name suggests) the function must be a [[measurable function]]. This is covered in the "alternative statement".</ref>
This could be phrased differently; as an alternative statement:
* Define {{M|f:X\rightarrow[0,+\infty]}} by {{MM|f:x\mapsto\mathop{\text{lim} }_{n\rightarrow\infty}\Big(f_n(x)\Big)}}, this [[limit]] exists forall {{M|x\in X}} as we allow the value {{M|+\infty}}.
** Then we have:
**# {{M|f\in\mathcal{M}\big(\mathcal{A},\mathcal{B}(\bar{\mathbb{R} }_{\ge 0})\big)}}<ref group="Todo">{{XXX|Link to specific proof!}}</ref> - {{M|f}} is a [[measurable function]] itself - and
**# {{MM|\int f\ \mathrm{d}\mu\eq\mathop{\text{lim} }_{n\rightarrow\infty}\left(\int f_n\ \mathrm{d}\mu\right) }}<noinclude>
==Notes==
<references group="Note"/>
==References==
<references/>
==Tasks==
<references group="Todo"/>
{{Theorem Of|Measure Theory|Analysis|Functional Analysis}}
</noinclude>