View source for Measure

Not to be confused with [[Pre-measure]]


==Definition==
A [[Sigma-ring|{{sigma|ring}}]] {{M|\mathcal{A} }} and a countably [[Additive function|additive]], [[Extended real value|extended real valued]]. non-negative [[Set function|set]] [[Function|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
{| class="wikitable" border="1"
|-
! 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>
|}

{{Definition|Measure Theory}}