Equivalence of Definitions of Sigma-Finite Measure

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Theorem

The following definitions of the concept of Sigma-Finite Measure are equivalent:

Definition 1

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


We say that $\mu$ is a $\sigma$-finite (or sigma-finite) measure if and only if there exists an exhausting sequence $\sequence {E_n}_{n \mathop \in \N}$ in $\Sigma$ such that:

$\forall n \in \N: \map \mu {E_n} < \infty$

Definition 2

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


We say that $\mu$ is a $\sigma$-finite (or sigma-finite) measure if and only if there exists a cover $\sequence {E_n}_{n \mathop \in \N}$ of $X$ in $\Sigma$ such that:

$\forall n \in \N: \map \mu {E_n} < \infty$

Definition 3

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


We say that $\mu$ is a $\sigma$-finite (or sigma-finite) measure if and only if there exists a partition $\sequence {E_n}_{n \mathop \in \N}$ of $X$ in $\Sigma$ such that:

$\forall n \in \N: \map \mu {E_n} < \infty$

Definition 4

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


We say that $\mu$ is a $\sigma$-finite (or sigma-finite) measure if and only if it is the countable union of sets of finite measure.


Proof

Definition 1 equivalent to Definition 2

This is Measure Space has Exhausting Sequence of Finite Measure iff Cover by Sets of Finite Measure.

$\Box$


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 $n \in \N$.