Difference between revisions of "Integral of a positive function (measure theory)/Definition"
From Maths
m |
m |
||
| Line 8: | Line 8: | ||
* {{M|I_\mu(g)}} denotes the [[integral of a simple function (measure theory)|{{M|\mu}}-integral of a simple function]] | * {{M|I_\mu(g)}} denotes the [[integral of a simple function (measure theory)|{{M|\mu}}-integral of a simple function]] | ||
* {{M|\mathcal{E}^+(\mathcal{A})}} denotes all the positive [[simple function (measure theory)|simple functions]] in their [[standard representation (measure theory)|standard representations]] from {{M|X}} considered with the {{M|\mathcal{A} }} [[sigma-algebra|{{sigma|algebra}}]]. | * {{M|\mathcal{E}^+(\mathcal{A})}} denotes all the positive [[simple function (measure theory)|simple functions]] in their [[standard representation (measure theory)|standard representations]] from {{M|X}} considered with the {{M|\mathcal{A} }} [[sigma-algebra|{{sigma|algebra}}]]. | ||
| − | |||
<noinclude> | <noinclude> | ||
| + | {{Todo|Link to {{M|\mathcal{E} }} somewhere, are they numeric or real valued?}}{{Todo|Can every simple function be made into a standard representation, thus what is {{M|\mathcal{E} }} exactly and what is the domain of {{M|I_\mu}} exactly?}} | ||
==Notes== | ==Notes== | ||
<references group="Note"/> | <references group="Note"/> | ||
Latest revision as of 16:59, 17 March 2016
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.
TODO: Link to [ilmath]\mathcal{E} [/ilmath] somewhere, are they numeric or real valued?
TODO: Can every simple function be made into a standard representation, thus what is [ilmath]\mathcal{E} [/ilmath] exactly and what is the domain of [ilmath]I_\mu[/ilmath] exactly?
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.
