Axiom of finite choice

Axiom in set theory

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

∏ α ∈ A S α ≠ ∅ {\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

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An important application is that when  ( Ω , 2 Ω , ν ) {\displaystyle (\Omega ,2^{\Omega },\nu )} is a measure space where  ν {\displaystyle \nu } is the counting measure and  f : Ω -> R {\displaystyle f:\Omega \to \mathbb {R} } is a function such that

∫ Ω | f | d ν < ∞ {\displaystyle \int _{\Omega }|f|d\nu <\infty } ,

then  f ( ω ) ≠ 0 {\displaystyle f(\omega )\neq 0} for at most countably many ω ∈ Ω {\displaystyle \omega \in \Omega } .

## References

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

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