Let be a probability space, where is the space of random outcomes, the -algebra of measurable events, and the probability measure.
Given a random variable , define as:
Theorem. is measurable and for all . Equality holds if and only if there exists a sequence such that , and as .
Proof. Let , and denote: so that .
Assume first that there exists , that is to say . We can show that:
is a measurable, zero probability set, which proves the thesis for this particular case. As a matter of fact, for any , if and , then we must have:
If and is a sequence in such that as , then:
the right hand side being a probability zero set.
If is empty, let , and let be a sequence in such that as . Then:
so that, from the particular case proved earlier, we have:
as was to be proved.
Citation
BibTeX citation:
@online{gherardi2023,
author = {Gherardi, Valerio},
title = {P-Values and Measure Theory},
date = {2023-06-07},
url = {https://vgherard.github.io/posts/2023-06-07-p-values-and-measure-theory/},
langid = {en}
}
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