Characterization of Pre-Measures

Theorem

Let $X$ be a set, and let $\SS \subseteq \powerset X$ be a collection of subsets of $X$.

Let $\O \in \SS$.

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

A mapping $\mu: \SS \to \overline \R_{\ge 0}$ is a pre-measure if and only if:

$(1):\quad \map \mu \O = 0$

$(2):\quad \mu$ is finitely additive

$(3):\quad$ For every increasing sequence $\sequence {E_n}_{n \mathop \in \N}$ in $\SS$, if $E_n \uparrow E$ for some $E \in \SS$, then:

$\ds \map \mu E = \lim_{n \mathop \to \infty} \map \mu {E_n}$

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 $\sequence {E_n}_{n \mathop \in \N}$ in $\SS$ for which $\map \mu {E_1}$ is finite, if $E_n \downarrow E$ for some $E \in \SS$, then:

$\ds \map \mu E = \lim_{n \mathop \to \infty} \map \mu {E_n}$

$(3):\quad$ For every decreasing sequence $\sequence {E_n}_{n \mathop \in \N}$ in $\SS$ for which $\map \mu {E_1}$ is finite, if $E_n \downarrow \O$, then:

$\ds \lim_{n \mathop \to \infty} \map \mu {E_n} = 0$

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

Proof

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