Difference between revisions of "Measure"
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==Definition== | ==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. | + | 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. That is: |
| + | * {{M|1=\mu(\emptyset)=0}} | ||
| + | * <math>\mu\left(\bigudot^\infty_{n=1}A_n\right)=\sum^\infty_{n=1}\mu(A_n)</math> | ||
| + | * <math>\mu(S)\ge 0\ \forall S\in\mathcal{A}</math> | ||
===Contrast with pre-measure=== | ===Contrast with pre-measure=== | ||
Revision as of 22:59, 2 May 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. That is:
- \mu(\emptyset)=0
- \mu\left(\bigudot^\infty_{n=1}A_n\right)=\sum^\infty_{n=1}\mu(A_n)
- \mu(S)\ge 0\ \forall S\in\mathcal{A}
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\left(\bigudot^n_{i=1}A_i\right)=\sum^n_{i=1}\mu(A_i) | \mu_0\left(\bigudot^n_{i=1}A_i\right)=\sum^n_{i=1}\mu_0(A_i) |
| Countably additive | \mu\left(\bigudot^\infty_{n=1}A_n\right)=\sum^\infty_{n=1}\mu(A_n) | If \bigudot^\infty_{n=1}A_n\in R then \mu_0\left(\bigudot^\infty_{n=1}A_n\right)=\sum^\infty_{n=1}\mu_0(A_n) |
Terminology
These terms apply to pre-measures to, rather \mathcal{A} you would use the ring the pre-measure is defined on.
Complete measure
A measure is complete if for A\in\mathcal{A} we have [\mu(A)=0\wedge B\subset A]\implies B\in \mathcal{A}
Finite
A set A\in\mathcal{A} is finite if \mu(A)<\infty - we say "A has finite measure"
Finite measure
\mu is a finite measure if every set \in\mathcal{A} is finite.
Sigma-finite
A set A\in\mathcal{A} is \sigma-finite if \exists(A_n)_{n=1}^\infty:[A\subseteq\cup^\infty_{n=1}A_n\wedge(\forall A_n,\ \mu(A_n)<\infty)]
Sigma-finite measure
\mu is \sigma-finite if every set \in\mathcal{A} is \sigma-finite
Total
If \mathcal{A} is a \sigma-algebra rather than a ring (that is X\in\mathcal{A} where X is the space) then we use
Totally finite measure
If X is finite
Totally sigma-finite measure
If X is \sigma-finite
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}
