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
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.
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(\bigudot^n_{i=1}A_i)=\sum^n_{i=1}\mu(A_i)[/math] | [math]\mu_0(\bigudot^n_{i=1}A_i)=\sum^n_{i=1}\mu_0(A_i)[/math] |
| Countably additive | [math]\mu(\bigudot^\infty_{n=1}A_n)=\sum^\infty_{n=1}\mu(A_n)[/math] | If [math]\bigudot^\infty_{n=1}A_n\in R[/math] then [math]\mu_0(\bigudot^\infty_{n=1}A_n)=\sum^\infty_{n=1}\mu_0(A_n)[/math] |
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]
