# 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 $\displaystyle{ (S_\alpha)_{\alpha \in A} }$ 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 such that

$\displaystyle{ \int_\Omega |f| d \nu \lt \infty }$,

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.