Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 26K / a / Solution

Let ${\mu }_X$ and ${\mu }_Y$ be the probability laws of $X$ and $Y$. The hypothesis saysfor every compactly supported continuous function $f$. The theorem that compactly supported continuous functions determine a finite Borel measure gives ${\mu }_X={\mu }_Y$. Hence $X$ and $Y$ have the same law.