Английская Википедия:Axiom of finite choice
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Шаблон:Short description In mathematics, the axiom of finite choice is a weak version of the axiom of choice which asserts that if <math>(S_\alpha)_{\alpha \in A}</math> is a family of non-empty finite sets, then
- <math>\prod_{\alpha \in A} S_\alpha \neq \emptyset </math> (set-theoretic product).[1]Шаблон:Rp
If every set can be linearly ordered, the axiom of finite choice follows.[1]Шаблон:Rp
Applications
An important application is that when <math>(\Omega, 2^\Omega, \nu)</math> is a measure space where <math>\nu</math> is the counting measure and <math>f: \Omega \to \mathbb R</math> is a function such that
- <math>\int_\Omega |f| d \nu < \infty</math>,
then <math>f(\omega) \neq 0</math> for at most countably many <math>\omega \in \Omega</math>.
References
