5.3 $\s$-Finite Measures

Definition 5.3.1 ($\s$-Finite Measure).label Let $(X,\cali{M},\mu)$ be a measure space, then the following are equivalent:

1. (1)

   There exists $\curl{E_n}_{n\in\N}\suf\cali{M}$ pairwise disjoint such that $\dcups{}{n\in\N}E_{n}=X$ and $\mu(E_{n})<\infty$ for all $n\in\N$

2. (2)

   There exists $\curl{E_n}_{n\in\N}\suf\cali{M}$ such that $E_{n}\nearrow X$ and $\mu(E_{n})<\infty$ for all $n\in\N$.

If the above holds, then $\mu$ is a $\s$-finite measure.