Pre-measure

$$\newcommand{\bigudot}{ \mathchoice{\mathop{\bigcup\mkern-15mu\cdot\mkern8mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}} }$$ $$\newcommand{\udot}{\cup\mkern-12.5mu\cdot\mkern6.25mu\!}$$ $$\require{AMScd}\newcommand{\d}[1][]{\mathrm{d}^{#1} }$$

Definition

A pre-measure (often denoted $\mu_0$) is the precursor to a measure which are often denoted $\mu$

A pre-measure is a ring $R$ together with an extended real valued, non-negative set function $$\mu_0:R\rightarrow[0,\infty]$$ that is countably additive with $$\mu_0(\emptyset)=0$$

To sum up:

• $$\mu_0:R\rightarrow [0,\infty]$$
• $$\mu_0(\emptyset)=0$$
• $$\mu_0(\bigudot_{n\in\mathbb{N}}A_n)=\sum_{n\in\mathbb{N}}\mu_0(A_n)$$ where the $$A_n$$ are pairwise disjoint (as implied by the $$\bigudot$$ )

Notice that given a finite family of pairwise disjoint sets $$\{F_i\}_{i=1}^n$$ we can define the countably infinite family of sets $$\{A_n\}_{n\in\mathbb{N}}$$ by: $$A_i=\left\{\begin{array}{lr}F_i&\text{if }i\le n\\ \emptyset & \text{otherwise} \end{array}\right.$$ as clearly $$\bigudot_{i\in\mathbb{N}}A_i=\bigudot_{i=1}^nF_n$$

Immediately one should think "but $$\bigudot^\infty_{n=1}A_n$$ is not always in $R$" this is true - but it is sometimes. When it is $$\in R$$ that is when it is defined.

Note that for any finite sequence we can make it infinite by just bolting $$\emptyset$$ on indefinitely.