# Monotone convergence theorem for non-negative numerical measurable functions

From Maths

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## Contents

## Statement

Let [ilmath](X,\mathcal{A},\mu)[/ilmath] be a measure space and let [ilmath](f_n)_{n\in\mathbb{N} }\subseteq\mathcal{M}\big(\mathcal{A},\mathcal{B}(\bar{\mathbb{R} }_{\ge 0})\big)[/ilmath]^{[Note 1]} be a sequence of measurable functions, [ilmath]f_n:X\rightarrow[0,+\infty]\eq\bar{\mathbb{R} }_{\ge 0} [/ilmath], then^{[1]}^{[2]}:

- if [ilmath]\forall n\in\mathbb{N}[f_n\le f_{n+1}][/ilmath]
^{[Note 2]}-*i.e.*[ilmath](f_n)_{n\in\mathbb{N} } [/ilmath] is a non-decreasing sequence - then:- [math]\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) [/math]
^{[Note 3]}

- [math]\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) [/math]

This could be phrased differently; as an alternative statement:

- Define [ilmath]f:X\rightarrow[0,+\infty][/ilmath] by [math]f:x\mapsto\mathop{\text{lim} }_{n\rightarrow\infty}\Big(f_n(x)\Big)[/math], this limit exists forall [ilmath]x\in X[/ilmath] as we allow the value [ilmath]+\infty[/ilmath].
- Then we have:
- [ilmath]f\in\mathcal{M}\big(\mathcal{A},\mathcal{B}(\bar{\mathbb{R} }_{\ge 0})\big)[/ilmath]
^{[Todo 1]}- [ilmath]f[/ilmath] is a measurable function itself - and - [math]\int f\ \mathrm{d}\mu\eq\mathop{\text{lim} }_{n\rightarrow\infty}\left(\int f_n\ \mathrm{d}\mu\right) [/math]

- [ilmath]f\in\mathcal{M}\big(\mathcal{A},\mathcal{B}(\bar{\mathbb{R} }_{\ge 0})\big)[/ilmath]

- Then we have:

## Proof

Grade: B

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^{[1]}before*claim 9.11*which is (a form of) Fatou's lemma## Notes

- ↑ Suppose we had:
- [ilmath]f\in\mathcal{M}\big(\mathcal{A},\mathcal{B}(\bar{\mathbb{R} }_{\ge 0})\big)[/ilmath] and
- [ilmath]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\} [/ilmath]

- I've seen (but not read) a proof and trust the source - Alec -17/April/2017 - 0908

- ↑ [ilmath]f\le g[/ilmath] is short for [ilmath]\forall x\in X[f(x)\le g(x)][/ilmath]
- ↑ 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".

## References

- ↑
^{1.0}^{1.1}Measures, Integrals and Martingales - René L. Schilling - ↑ A Guide To Advanced Real Analysis - Gerald B. Folland

## Tasks

- ↑ TODO: Link to specific proof!

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