Category:Sigma-Finite Measures

This category contains results about $\sigma$-finite measures.
Definitions specific to this category can be found in Definitions/Sigma-Finite Measures.

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.