Pre-measure

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:

• $$\mu_0:R\rightarrow [0,\infty]$$
• $$\mu_0(\emptyset)=0$$
• $$\mu_0(\bigcup^\infty_{n=1}A_n)=\sum^\infty_{n=1}\mu_0(A_n)$$

Immediately one should think "but $$\bigcup^\infty_{n=1}A_n$$ 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.