Pre-measure
[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.
