Equivalence of Definitions of Sigma-Finite Measure

Definition 1 equivalent to Definition 2
This is Measure Space has Exhausting Sequence of Finite Measure iff Cover by Sets of Finite Measure.

Definition 3 implies Definition 2
Let $\sequence {E_n}_{n \mathop \in \N}$ be a partition of $X$ in $\Sigma$ with:


 * $\map \mu {E_n} < \infty$ for each $n \in \N$.

Then:


 * $E_n \cap E_m = \O$ for $n \ne m$

and:


 * $\ds X = \bigcup_{n \mathop = 1}^\infty E_n$

with:


 * $\map \mu {E_n} < \infty$ for each $n \in \N$.

In particular, $\sequence {E_n}_{n \mathop \in \N}$ is a cover of $X$ in $\Sigma$ with:


 * $\map \mu {E_n} < \infty$ for each $n \in \N$.

Definition 2 implies Definition 3
Let $\sequence {E_n}_{n \mathop \in \N}$ be a cover of $X$ in $\Sigma$ with:


 * $\map \mu {E_n} < \infty$ for each $n \in \N$.

By Countable Union of Measurable Sets as Disjoint Union of Measurable Sets there exists a sequence of measurable sets $\sequence {F_n}_{n \mathop \in \N}$ such that:


 * $F_n \subseteq E_n$ for each $n \in \N$

and:


 * $\ds X = \bigcup_{n \mathop = 1}^\infty E_n = \bigcup_{n \mathop = 1}^\infty F_n$

From Measure is Monotone we have:


 * $\map \mu {F_n} < \infty$ for each $n \in \N$.

So $\sequence {F_n}_{n \mathop \in \N}$ is a partition of $X$ in $\Sigma$ with:


 * $\map \mu {F_n} < \infty$ for each $n \in \N$.