Axiom of finite choice

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

$\prod _{\alpha \in A}S_{\alpha }\neq \emptyset$ (set-theoretic product).

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

Applications

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

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

then $f(\omega )\neq 0$ for at most countably many $\omega \in \Omega$.