0
$\begingroup$

If $X$ and $Y$ are random variables with the same distribution (their CDF's are the same), is it true that $P(X \in A) = P'(Y \in A)$ for any Borel set $A$? I was reading a theorem in Durrett and it seemed to indicate this (at least for $A$ with induced measure $0$). Clearly its true for $A$ of the form ($-\infty, a$] but I am not sure how to prove it for a general Borel set.

Thanks

$\endgroup$
1
$\begingroup$

This is immediate from uniqueness of extensions. Finite disjoint unions of intervals of the type $(a,b]$ form a field which generates the Borel sigma field, so if two measures agree on this field they agree on all Borel sets.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.