Integral of a positive function (measure theory)
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Definition
Let [ilmath](X,\mathcal{A},\mu)[/ilmath] be a measure space, the [ilmath]\mu[/ilmath]integral of a positive numerical function, [ilmath]f\in\mathcal{M}^+_{\bar{\mathbb{R} } }(\mathcal{A}) [/ilmath]^{[Note 1]}^{[Note 2]} is^{[1]}:
 [math]\int f\mathrm{d}\mu:=\text{Sup}\left\{I_\mu(g)\ \Big\vert\ g\le f, g\in\mathcal{E}^+(\mathcal{A})\right\}[/math]^{[Note 3]}
Recall that:
 [ilmath]I_\mu(g)[/ilmath] denotes the [ilmath]\mu[/ilmath]integral of a simple function
 [ilmath]\mathcal{E}^+(\mathcal{A})[/ilmath] denotes all the positive simple functions in their standard representations from [ilmath]X[/ilmath] considered with the [ilmath]\mathcal{A} [/ilmath] [ilmath]\sigma[/ilmath]algebra.
There are alternate notations, that make the variable of integration more clear, they are:
 [ilmath]\int f(x)\mu(\mathrm{d}x)[/ilmath]^{[1]}
 [ilmath]\int f(x)\mathrm{d}\mu(x)[/ilmath]^{[1]}
Immediate results
 [math]\forall f\in\mathcal{E}^+(\mathcal{A})\left[\int f\mathrm{d}\mu=I_\mu(f)\right][/math]  Integrating a simple function works
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.
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Page 70 in^{[1]}
Notes
 ↑ So [ilmath]f:X\rightarrow\bar{\mathbb{R} }^+[/ilmath]
 ↑ Notice that [ilmath]f[/ilmath] is [ilmath]\mathcal{A}/\bar{\mathcal{B} } [/ilmath]measurable by definition, as [ilmath]\mathcal{M}_\mathcal{Z}(\mathcal{A})[/ilmath] denotes all the measurable functions that are [ilmath]\mathcal{A}/\mathcal{Z} [/ilmath]measurable, we just use the [ilmath]+[/ilmath] as a slight abuse of notation to denote all the positive ones (with respect to the standard order on [ilmath]\bar{\mathbb{R} } [/ilmath]  the extended reals)
 ↑ The [ilmath]g\le f[/ilmath] is an abuse of notation for saying that [ilmath]g[/ilmath] is everywhere less than [ilmath]f[/ilmath], we could have written:
 [math]\int f\mathrm{d}\mu=\text{Sup}\left\{I_\mu(g)\ \Big\vert\ g\le f, g\in\mathcal{E}^+\right\}=\text{Sup}\left\{I_\mu(g)\ \Big\vert\ g\in\left\{h\in\mathcal{E}^+(\mathcal{A})\ \big\vert\ \forall x\in X\left(h(x)\le f(x)\right)\right\}\right\}[/math] instead.
References
 ↑ ^{1.0} ^{1.1} ^{1.2} ^{1.3} Measures, Integrals and Martingales  René L. Schilling
