Difference between revisions of "Measure"
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| − | Not to be confused with [[Pre-measure]] | + | {{Extra Maths}}Not to be confused with [[Pre-measure]] |
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| Finitely additive | | Finitely additive | ||
| − | | <math>\mu(\ | + | | <math>\mu(\bigudot^n_{i=1}A_i)=\sum^n_{i=1}\mu(A_i)</math> |
| − | | <math>\mu_0(\ | + | | <math>\mu_0(\bigudot^n_{i=1}A_i)=\sum^n_{i=1}\mu_0(A_i)</math> |
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| Countably additive | | Countably additive | ||
| − | | <math>\mu(\ | + | | <math>\mu(\bigudot^\infty_{n=1}A_n)=\sum^\infty_{n=1}\mu(A_n)</math> |
| − | | If <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> |
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{{Definition|Measure Theory}} | {{Definition|Measure Theory}} | ||
Revision as of 22:32, 13 March 2015
\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}} }\newcommand{\udot}{\cup\mkern-12.5mu\cdot\mkern6.25mu\!}\require{AMScd}\newcommand{\d}[1][]{\mathrm{d}^{#1} }Not to be confused with Pre-measure
Definition
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.
Contrast with pre-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(\bigudot^n_{i=1}A_i)=\sum^n_{i=1}\mu(A_i) | \mu_0(\bigudot^n_{i=1}A_i)=\sum^n_{i=1}\mu_0(A_i) |
| Countably additive | \mu(\bigudot^\infty_{n=1}A_n)=\sum^\infty_{n=1}\mu(A_n) | If \bigudot^\infty_{n=1}A_n\in R then \mu_0(\bigudot^\infty_{n=1}A_n)=\sum^\infty_{n=1}\mu_0(A_n) |
