Integral (measure theory)
Contents
Definition
Given a measure space, [ilmath](X,\mathcal{A},\mu)[/ilmath] and a function [ilmath]f:X\rightarrow\bar{\mathbb{R} } [/ilmath], we say that [ilmath]f[/ilmath] is [ilmath]\mu[/ilmath]-integrable if[1]:
- [ilmath]f[/ilmath] is a measurable map, an [ilmath]\mathcal{A}/\bar{\mathcal{B} } [/ilmath]-measurable map; and if
- The integrals [ilmath]\int f^+\mathrm{d}\mu,\ \int f^-\mathrm{d}\mu<\infty[/ilmath], then:
We define the [ilmath]\mu[/ilmath]-integral of [ilmath]f[/ilmath] to be:
- [math]\int f\mathrm{d}u:=\int f^+\mathrm{d}\mu-\int f^-\mathrm{d}\mu[/math]
Where:
- [math]\int f^\pm\mathrm{d}\mu[/math] are the integrals of positive functions.
Reminder: Integration of positive functions
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.
Reminder: Integration of simple functions
For a simple function in its standard representation, say [ilmath]f:=\sum^n_{i=0}x_i\mathbf{1}_{A_i}[/ilmath] then the [ilmath]\mu[/ilmath]-integral, [ilmath]I_\mu:\mathcal{E}^+\rightarrow\mathbb{R} [/ilmath] is[1]:
- [math]I_\mu(f):=\sum^n_{i=1}x_i\mu(A_i)\in[0,\infty][/math]
Note that this is independent of the particular standard representation of [ilmath]f[/ilmath].
Reminder: Definition of simple function
A simple function [ilmath]f:X\rightarrow\mathbb{R} [/ilmath] on a measurable space [ilmath](X,\mathcal{A})[/ilmath] is a[1]:
- function of the form [ilmath]\sum^N_{i=1}x_i\mathbf{1}_{A_i}(x)[/ilmath] for
- finitely many sets, [ilmath]A_1,\ldots,A_N\in\mathcal{A} [/ilmath] and
- finitely many [ilmath]x_1,\ldots,x_n\in\mathbb{R} [/ilmath]
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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