Difference between revisions of "Pre-measure"
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==Definition== | ==Definition== | ||
A pre-measure (often denoted {{M|\mu_0}}) is the precursor to a [[Measure|measure]] which are often denoted {{M|\mu}} | A pre-measure (often denoted {{M|\mu_0}}) is the precursor to a [[Measure|measure]] which are often denoted {{M|\mu}} | ||
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* <math>\mu_0:R\rightarrow [0,\infty]</math> | * <math>\mu_0:R\rightarrow [0,\infty]</math> | ||
* <math>\mu_0(\emptyset)=0</math> | * <math>\mu_0(\emptyset)=0</math> | ||
| − | * <math>\mu_0(\ | + | * <math>\mu_0(\bigudot_{n\in\mathbb{N}}A_n)=\sum_{n\in\mathbb{N}}\mu_0(A_n)</math> where the <math>A_n</math> are pairwise disjoint (as implied by the <math>\bigudot</math>) |
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| + | Notice that given a finite family of pairwise disjoint sets <math>\{F_i\}_{i=1}^n</math> we can define the countably infinite family of sets <math>\{A_n\}_{n\in\mathbb{N}}</math> by: <math>A_i=\left\{\begin{array}{lr}F_i&\text{if }i\le n\\ \emptyset & \text{otherwise} \end{array}\right.</math> as clearly <math>\bigudot_{i\in\mathbb{N}}A_i=\bigudot_{i=1}^nF_n</math> | ||
| − | Immediately one should think "but <math>\ | + | Immediately one should think "but <math>\bigudot^\infty_{n=1}A_n</math> is not always in {{M|R}}" this is true - but it is sometimes. When it is <math>\in R</math> that is when it is defined. |
Note that for any finite sequence we can make it infinite by just bolting <math>\emptyset</math> on indefinitely. | Note that for any finite sequence we can make it infinite by just bolting <math>\emptyset</math> on indefinitely. | ||
Revision as of 21:03, 14 March 2015
[math]\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}} }[/math][math]\newcommand{\udot}{\cup\mkern-12.5mu\cdot\mkern6.25mu\!}[/math][math]\require{AMScd}\newcommand{\d}[1][]{\mathrm{d}^{#1} }[/math]
Definition
A pre-measure (often denoted [ilmath]\mu_0[/ilmath]) is the precursor to a measure which are often denoted [ilmath]\mu[/ilmath]
A pre-measure is a ring [ilmath]R[/ilmath] together with an extended real valued, non-negative set function [math]\mu_0:R\rightarrow[0,\infty][/math] that is countably additive with [math]\mu_0(\emptyset)=0[/math]
To sum up:
- [math]\mu_0:R\rightarrow [0,\infty][/math]
- [math]\mu_0(\emptyset)=0[/math]
- [math]\mu_0(\bigudot_{n\in\mathbb{N}}A_n)=\sum_{n\in\mathbb{N}}\mu_0(A_n)[/math] where the [math]A_n[/math] are pairwise disjoint (as implied by the [math]\bigudot[/math])
Notice that given a finite family of pairwise disjoint sets [math]\{F_i\}_{i=1}^n[/math] we can define the countably infinite family of sets [math]\{A_n\}_{n\in\mathbb{N}}[/math] by: [math]A_i=\left\{\begin{array}{lr}F_i&\text{if }i\le n\\ \emptyset & \text{otherwise} \end{array}\right.[/math] as clearly [math]\bigudot_{i\in\mathbb{N}}A_i=\bigudot_{i=1}^nF_n[/math]
Immediately one should think "but [math]\bigudot^\infty_{n=1}A_n[/math] is not always in [ilmath]R[/ilmath]" this is true - but it is sometimes. When it is [math]\in R[/math] that is when it is defined.
Note that for any finite sequence we can make it infinite by just bolting [math]\emptyset[/math] on indefinitely.
