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).[1]^: 14

If every set can be linearly ordered, the axiom of finite choice follows.[1]^: 17

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 $\omega \in \Omega$ .

References

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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[herrlich-1] 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.