Pre-measure

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(\bigcup^\infty_{n=1}A_n)=\sum^\infty_{n=1}\mu_0(A_n)[/math] where the [math]A_n[/math] are pairwise disjoint

Immediately one should think "but [math]\bigcup^\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.