Characterization of Measures

Theorem
Let $\left({X, \mathcal A}\right)$ be a measurable space.

Denote $\overline{\R}_{\ge0}$ for the set of positive extended real numbers.

A mapping $\mu: \mathcal A \to \overline{\R}_{\ge0}$ is a measure iff:


 * $(1):\quad \mu \left({\varnothing}\right) = 0$
 * $(2):\quad \mu$ is finitely additive
 * $(3):\quad$ For every increasing sequence $\left({A_n}\right)_{n \in \N}$ in $\mathcal A$, if $A_n \uparrow A$, then:
 * $\mu \left({A}\right) = \displaystyle \lim_{n \to \infty} \mu \left({A_n}\right)$

where $A_n \uparrow A$ denotes limit of increasing sequence of sets.

Alternatively, and equivalently, $(3)$ may be replaced by either of:


 * $(3'):\quad$ For every decreasing sequence $\left({A_n}\right)_{n \in \N}$ in $\mathcal A$ for which $\mu \left({A_1}\right)$ is finite, if $A_n \downarrow A$, then:
 * $\mu \left({A}\right) = \displaystyle \lim_{n \to \infty} \mu \left({A_n}\right)$
 * $(3''):\quad$ For every decreasing sequence $\left({A_n}\right)_{n \in \N}$ in $\mathcal A$ for which $\mu \left({A_1}\right)$ is finite, if $A_n \downarrow \varnothing$, then:
 * $\displaystyle \lim_{n \to \infty} \mu \left({A_n}\right) = 0$

where $A_n \downarrow A$ denotes limit of decreasing sequence of sets.