Difference between revisions of "Measure"
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Revision as of 18:08, 13 March 2015
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] |
