Measure

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