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Let $X$ and $Y$ be two random variables that are equal in distribution, $X\overset{d}{=} Y$. Suppose that $g$ is a continuous function.

How to prove that $g(X) \overset{d}{=} g(Y)$ (if this is true)? I am trying to prove this ONLY from the definition of equality in distribution, but am having some trouble.

If someone can provide a proof of this it would be very much appreciated.

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    Notice that for any Borel subset $B$ of the codomain of $g$, we have $$ \Bbb{P}(g(X) \in B) = \Bbb{P}(X \in g^{-1}(B))$$ and likewise for $Y$. What can you conclude from the assumption $X \stackrel{d}= Y$? – 2017-02-22
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    @SangchulLee Yes I had tried to prove it like this, but I am getting messed up with the fact that $\mathbb{P}(X \in (-\infty, x]) = \mathbb{P}(Y \in (-\infty,x])$ only if $x$ is a continuity point of the distribution function. How can I incorporate this fact when completing the proof with your approach? – 2017-02-22
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    @möbius I think you might be confusing equal in distribution with converges in distribution. What happens at a discontinuity point of a CDF (as opposed to a limit of CDFs) is determined by the fact that it is, well, a CDF.. it will be right-continuous. So $P(X\in (-\infty, x]) = P(Y\in (-\infty, x])$ at any point $x$, not just at continuity points. – 2017-02-23
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    @spaceisdarkgreen Ahhhhhh I see. Yes I was definitely confused on this point, but thank you very much for helping me with this. The proof is now clear to me. – 2017-02-23

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