# $X$ and $Y$ Random Variables Same Distribution

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

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.