14.4 $\sigma$-Finite Measures

Definition 14.4.1 ($\sigma$-Finite Measure). Let $(X, \cm, \mu)$ be a measure space, then the following are equivalent:

1. There exists $\seq{E_n}\subset \cm$ pairwise disjoint such that $\bigsqcup_{n \in \nat}E_{n} = X$ and $\mu(E_{n}) < \infty$ for all $n \in \nat$.

2. There exists $\seq{E_n}\subset \cm$ such that $E_{n} \upto X$ and $\mu(E_{n}) < \infty$ for all $n \in \nat$.

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