Difference between revisions of "Measure"

Revision as of 18:08, 13 March 2015

Not to be confused with Pre-measure

A $\sigma$ -ring $\mathcal{A}$ and a countably additive, extended real valued. non-negative set function $\mu:\mathcal{A}\rightarrow[0,\infty]$ is a measure.

Note: the family $A_n$ must be pairwise disjoint

Property	Measure	Pre-measure
	$\mu:\mathcal{A}\rightarrow[0,\infty]$	$\mu_0:R\rightarrow[0,\infty]$
	$\mu(\emptyset)=0$	$\mu_0(\emptyset)=0$
Finitely additive	$\mu(\bigcup^n_{i=1}A_i)=\sum^n_{i=1}\mu(A_i)$	$\mu_0(\bigcup^n_{i=1}A_i)=\sum^n_{i=1}\mu_0(A_i)$
Countably additive	$\mu(\bigcup^\infty_{n=1}A_n)=\sum^\infty_{n=1}\mu(A_n)$	If $\bigcup^\infty_{n=1}A_n\in R$ then $\mu_0(\bigcup^\infty_{n=1}A_n)=\sum^\infty_{n=1}\mu_0(A_n)$

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 ===Contrast with pre-measure===
+'''Note:''' the family <math>A_n</math> must be pairwise disjoint
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