Difference between revisions of "Integral of a positive function (measure theory)"
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| + | * We don't really mean positive function, we mean non-negative. [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 19:18, 14 April 2017 (UTC) | ||
| + | This is under review as a part of measure theory}} | ||
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==Definition== | ==Definition== | ||
{{:Integral of a positive function (measure theory)/Definition}} | {{:Integral of a positive function (measure theory)/Definition}} | ||
Latest revision as of 19:18, 14 April 2017
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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:
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
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