Difference between revisions of "Measure"
m |
m |
||
| Line 50: | Line 50: | ||
* [[Pre-measure]] | * [[Pre-measure]] | ||
* [[Outer-measure]] | * [[Outer-measure]] | ||
| + | * [[Lebesgue measure]] | ||
{{Definition|Measure Theory}} | {{Definition|Measure Theory}} | ||
Revision as of 18:30, 15 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
Contents
[hide]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) |
Examples
Trivial measures
Given the Measurable space (X,\mathcal{A}) we can define:
\mu:\mathcal{A}\rightarrow\{0,+\infty\} by \mu(A)=\left\{\begin{array}{lr} 0 & \text{if }A=\emptyset \\ +\infty & \text{otherwise} \end{array}\right.
Another trivial measure is:
v:\mathcal{A}\rightarrow\{0\} by v(A)=0 for all A\in\mathcal{A}
