Definition:Sigma-Finite Measure
Definition
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.
Also see
- Results about $\sigma$-finite measures can be found here.