Monotone convergence theorem for non-negative numerical measurable functions

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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] be a sequence of measurable functions, [ilmath]f_n:X\rightarrow[0,+\infty]\eq\bar{\mathbb{R} }_{\ge 0} [/ilmath], then:

• if [ilmath]\forall n\in\mathbb{N}[f_n\le f_{n+1}][/ilmath] - 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]

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:
    1. [ilmath]f\in\mathcal{M}\big(\mathcal{A},\mathcal{B}(\bar{\mathbb{R} }_{\ge 0})\big)[/ilmath] - [ilmath]f[/ilmath] is a measurable function itself - and
    2. [math]\int f\ \mathrm{d}\mu\eq\mathop{\text{lim} }_{n\rightarrow\infty}\left(\int f_n\ \mathrm{d}\mu\right) [/math]

Proof

Grade: B

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Done on paper? Find it and post here? Do again? Can be found in before claim 9.11 which is (a form of) Fatou's lemma