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 s.t.

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

,

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

gollark: Exactly. They're weirdly biased.

gollark: "Think of a random number" does *not* produce random numbers. At all.

gollark: I really need to find a random number generation algorithm I can run in my head, it would be very convenient.

gollark: My RNG chooses `waller`.

gollark: Can I just pick at random?

References

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