Equivalence of Definitions of Sigma-Finite Measure
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 $n \in \N$.
This needs considerable tedious hard slog to complete it. In particular: include Definition 4 To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Finish}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |