Measure
[math]\newcommand{\bigudot}{ \mathchoice{\mathop{\bigcup\mkern-15mu\cdot\mkern8mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}} }[/math][math]\newcommand{\udot}{\cup\mkern-12.5mu\cdot\mkern6.25mu\!}[/math][math]\require{AMScd}\newcommand{\d}[1][]{\mathrm{d}^{#1} }[/math]Not to be confused with Pre-measure
Contents
Definition
A [ilmath]\sigma[/ilmath]-ring [ilmath]\mathcal{A} [/ilmath] and a countably additive, extended real valued. non-negative set function [math]\mu:\mathcal{A}\rightarrow[0,\infty][/math] is a measure. That is:
- [ilmath]\mu(\emptyset)=0[/ilmath]
- [math]\mu\left(\bigudot^\infty_{n=1}A_n\right)=\sum^\infty_{n=1}\mu(A_n)[/math]
- [math]\mu(S)\ge 0\ \forall S\in\mathcal{A}[/math]
Contrast with pre-measure
Note: the family [math]A_n[/math] must be pairwise disjoint
| Property | Measure | Pre-measure |
|---|---|---|
| [math]\mu:\mathcal{A}\rightarrow[0,\infty][/math] | [math]\mu_0:R\rightarrow[0,\infty][/math] | |
| [math]\mu(\emptyset)=0[/math] | [math]\mu_0(\emptyset)=0[/math] | |
| Finitely additive | [math]\mu\left(\bigudot^n_{i=1}A_i\right)=\sum^n_{i=1}\mu(A_i)[/math] | [math]\mu_0\left(\bigudot^n_{i=1}A_i\right)=\sum^n_{i=1}\mu_0(A_i)[/math] |
| Countably additive | [math]\mu\left(\bigudot^\infty_{n=1}A_n\right)=\sum^\infty_{n=1}\mu(A_n)[/math] | If [math]\bigudot^\infty_{n=1}A_n\in R[/math] then [math]\mu_0\left(\bigudot^\infty_{n=1}A_n\right)=\sum^\infty_{n=1}\mu_0(A_n)[/math] |
Terminology
These terms apply to pre-measures to, rather [ilmath]\mathcal{A} [/ilmath] you would use the ring the pre-measure is defined on.
Complete measure
A measure is complete if for [ilmath]A\in\mathcal{A} [/ilmath] we have [math][\mu(A)=0\wedge B\subset A]\implies B\in \mathcal{A}[/math]
Finite
A set [ilmath]A\in\mathcal{A} [/ilmath] is finite if [ilmath]\mu(A)<\infty[/ilmath] - we say "[ilmath]A[/ilmath] has finite measure"
Finite measure
[ilmath]\mu[/ilmath] is a finite measure if every set [ilmath]\in\mathcal{A} [/ilmath] is finite.
Sigma-finite
A set [ilmath]A\in\mathcal{A} [/ilmath] is [ilmath]\sigma[/ilmath]-finite if [math]\exists(A_n)_{n=1}^\infty:[A\subseteq\cup^\infty_{n=1}A_n\wedge(\forall A_n,\ \mu(A_n)<\infty)][/math]
Sigma-finite measure
[ilmath]\mu[/ilmath] is [ilmath]\sigma[/ilmath]-finite if every set [ilmath]\in\mathcal{A} [/ilmath] is [ilmath]\sigma[/ilmath]-finite
Total
If [ilmath]\mathcal{A} [/ilmath] is a [ilmath]\sigma[/ilmath]-algebra rather than a ring (that is [ilmath]X\in\mathcal{A} [/ilmath] where [ilmath]X[/ilmath] is the space) then we use
Totally finite measure
If [ilmath]X[/ilmath] is finite
Totally sigma-finite measure
If [ilmath]X[/ilmath] is [ilmath]\sigma[/ilmath]-finite
Examples
Trivial measures
Given the Measurable space [ilmath](X,\mathcal{A})[/ilmath] we can define:
[math]\mu:\mathcal{A}\rightarrow\{0,+\infty\}[/math] by [math]\mu(A)=\left\{\begin{array}{lr} 0 & \text{if }A=\emptyset \\ +\infty & \text{otherwise} \end{array}\right.[/math]
Another trivial measure is:
[math]v:\mathcal{A}\rightarrow\{0\}[/math] by [math]v(A)=0[/math] for all [math]A\in\mathcal{A}[/math]
