Axiom of finite choice

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

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

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