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]