Axiom of finite choice

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

${\displaystyle \prod _{\alpha \in A}S_{\alpha }\neq \emptyset }$ (set-theoretic product).[1]^:14

If every set can be linearly ordered, the axiom of finite choice follows.[1]^:17

Applications

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

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

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

References

1. Herrlich, Horst (2006). The axiom of choice. Lecture Notes in Mathematics. 1876. Berlin, Heidelberg: Springer. doi:10.1007/11601562. ISBN 978-3-540-30989-5.