Measure

Not to be confused with Pre-measure

Definition

A $\sigma$-ring $\mathcal{A}$ and a countably additive, extended real valued. non-negative set function $$\mu:\mathcal{A}\rightarrow[0,\infty]$$ is a measure.

Contrast with pre-measure

Note: the family $$A_n$$ must be pairwise disjoint

Property	Measure	Pre-measure
	$$\mu:\mathcal{A}\rightarrow[0,\infty]$$	$$\mu_0:R\rightarrow[0,\infty]$$
	$$\mu(\emptyset)=0$$	$$\mu_0(\emptyset)=0$$
Finitely additive	$$\mu(\bigcup^n_{i=1}A_i)=\sum^n_{i=1}\mu(A_i)$$	$$\mu_0(\bigcup^n_{i=1}A_i)=\sum^n_{i=1}\mu_0(A_i)$$
Countably additive	$$\mu(\bigcup^\infty_{n=1}A_n)=\sum^\infty_{n=1}\mu(A_n)$$	If $$\bigcup^\infty_{n=1}A_n\in R$$ then $$\mu_0(\bigcup^\infty_{n=1}A_n)=\sum^\infty_{n=1}\mu_0(A_n)$$