Axiom of finite choice explained

In mathematics, the axiom of finite choice is a weak version of the axiom of choice which asserts that if

(S _{\alpha)} _{\alpha} is a family of non-empty finite sets, then\prod _{\alpha} S _{\alpha} ≠ \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 R is a function such that\int _{\Omega} |f| d \nu < inf ty ,then f(\omega) ≠ 0 for at most countably many \omega \in \Omega .

References

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Notes and References

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