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

Theorem
Let $\struct {X, \Sigma, \mu}$ be a measure space.

Then $\struct {X, \Sigma, \mu}$ is $\sigma$-finite there exists a sequence $\sequence {E_n}_{n \mathop \in \N}$ in $\Sigma$ such that:


 * $(1): \quad \ds \bigcup_{n \mathop \in \N} E_n = X$
 * $(2): \quad \forall n \in \N: \map \mu {E_n} < +\infty$

Necessary Condition
Let $\mu$ be a $\sigma$-finite measure.

Let $\sequence {F_n}_{n \mathop \in \N}$ be an exhausting sequence in $\Sigma$ such that:


 * $\forall n \in \N: \map \mu {F_n} < +\infty$

Then as $\sequence {F_n}_{n \mathop \in \N}$ is exhausting, have:


 * $\ds \bigcup_{n \mathop \in \N} F_n = X$

It follows that the sequence $\sequence {F_n}_{n \mathop \in \N}$ satisfies $(1)$ and $(2)$.

Sufficient Condition
Let $\mu$ be any measure.

Let $\sequence {E_n}_{n \mathop \in \N}$ be a sequence satisfying $(1)$ and $(2)$.

Define $F_n := \ds \bigcup_{k \mathop = 1}^n E_k$.

Then by Sigma-Algebra Closed under Union:
 * $F_n \in \Sigma$ for all $n \in \N$

Also, by Set is Subset of Union:
 * $F_{n+1} = F_n \cup E_{n+1}$, hence $F_n \subseteq F_{n + 1}$

The definition of the $F_n$ assures that:
 * $X = \ds \bigcup_{n \mathop \in \N} E_n = \bigcup_{n \mathop \in \N} F_n$

Hence $\sequence {F_n}_{n \mathop \in \N}$ is an exhausting sequence in $\Sigma$.

Furthermore, compute, for any $n \in \N$:

Hence, by definition, $\mu$ is $\sigma$-finite.

Thus, $\struct {X, \Sigma, \mu}$ is also $\sigma$-finite.