Characterization of Pre-Measures

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Theorem

Let $X$ be a set, and let $\mathcal S \subseteq \mathcal P \left({X}\right)$ be a collection of subsets of $X$.

Let $\varnothing \in \mathcal S$.

Denote $\overline \R_{\ge 0}$ for the set of positive extended real numbers.


A mapping $\mu: \mathcal S \to \overline \R_{\ge 0}$ is a pre-measure iff:

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

where $E_n \uparrow E$ denotes limit of increasing sequence of sets.


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

$(3'):\quad$ For every decreasing sequence $\left({E_n}\right)_{n \in \N}$ in $\mathcal S$ for which $\mu \left({E_1}\right)$ is finite, if $E_n \downarrow E$ for some $E \in \mathcal S$, then:
$\mu \left({E}\right) = \displaystyle \lim_{n \to \infty} \mu \left({E_n}\right)$
$(3''):\quad$ For every decreasing sequence $\left({E_n}\right)_{n \in \N}$ in $\mathcal S$ for which $\mu \left({E_1}\right)$ is finite, if $E_n \downarrow \varnothing$, then:
$\displaystyle \lim_{n \to \infty} \mu \left({E_n}\right) = 0$

where $E_n \downarrow E$ denotes limit of decreasing sequence of sets.


Proof