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 $$\omega \in \Omega$$.