Measure Space has Exhausting Sequence of Finite Measure iff Cover by Sets of Finite Measure

Theorem
Let $\left({X, \mathcal A, \mu}\right)$ be a measure space.

Then $\left({X, \mathcal A, \mu}\right)$ is $\sigma$-finite iff there exists a sequence $\left({A_n}\right)_{n \in \N}$ in $\mathcal A$ such that:


 * $(1):\quad \displaystyle \bigcup_{n \in \N} E_n = X$
 * $(2):\quad \forall n \in \N: \mu \left({E_n}\right) < +\infty$