Axiom of finite choice

Short description: Axiom in set theory

In mathematics, the axiom of finite choice is a weak version of the axiom of choice which asserts that if $(S_{\alpha }{)}_{\alpha \in A}$ is a family of non-empty finite sets, then

$\prod _{\alpha \in A}{S}_{\alpha }\ne \mathrm{\varnothing }$ (set-theoretic product).

If every set can be linearly ordered, the axiom of finite choice follows.

Applications

An important application is that when $(\mathrm{\Omega },2^{\mathrm{\Omega }},\nu )$ is a measure space where $\nu$ is the counting measure and $f:\mathrm{\Omega }\to \mathbb{ℝ}$ is a function such that

$\underset{\mathrm{\Omega }}{\int }|f|d\nu <\mathrm{\infty }$,

then $f(\omega )\ne 0$ for at most countably many $\omega \in \mathrm{\Omega }$.