Difference between revisions of "Integral (measure theory)"

Revision as of 16:59, 17 March 2016

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.

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.

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].

Notes

1. ↑ So [ilmath]f:X\rightarrow\bar{\mathbb{R} }^+[/ilmath]
2. ↑ 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)
3. ↑ 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.
   Inline with: Notation for dealing with (extended) real-valued measurable maps

References

1. ↑ ^{1.0 1.1 1.2} Measures, Integrals and Martingales - René L. Schilling