Let $Ω = (0, 1)$, $F = \mathcal{B}((0, 1))$ and $P$ be the Lebesgue measure on $(0,1)$. Let $X(ω) = − log (ω)$. Find $P_X$

Question

I'm solving questions by Athreya: Measure Theory and Probability and I'm stuck in a question:

Let $Ω = (0, 1)$, $F = \mathcal{B}((0, 1))$ and $P$ be the Lebesgue measure on $(0,1)$. Let $X(ω) = − \log (ω)$, $h(x) = x^2$ and $Y = h(X)$. Find $P_X$ and $P_Y$ and evaluate $E[Y]$ by applying the change of variables formula.

How should I find out $P_X$ in such a case? I know the definition that $P_X(A) = P(X^{-1}(A)) \forall A\in F$ but I'm not sure how to use that in this problem.