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 [math]\displaystyle{ (S_\alpha)_{\alpha \in A} }[/math] is a family of non-empty finite sets, then

[math]\displaystyle{ \prod_{\alpha \in A} S_\alpha \neq \emptyset }[/math] (set-theoretic product).

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

Applications

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

[math]\displaystyle{ \int_\Omega |f| d \nu \lt \infty }[/math],

then [math]\displaystyle{ f(\omega) \neq 0 }[/math] for at most countably many [math]\displaystyle{ \omega \in \Omega }[/math].