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Choose $M<\infty$ such that $|X|,|Y|\le M$ almost surely. Equality of the positive moments, together with equality of the zeroth moments $\mathbb{E}[1]=1$, givesfor every polynomial $p$. By the Weierstrass approximation theorem, every continuous function on $[-M,M]$ is a uniform limit of polynomials. Expectations preserve this limit, so $X$ and $Y$ have equal expectations for every continuous function on their common compact support. The moment determinacy on a compact interval therefore gives equality of their laws.

The Laplace transform of a nonnegative random variable determines its probability law. Indeed, $U=e^{-X}$ takes values in $(0,1]$, and $\mathbb{E}[U^n]={\psi }_X(n)$. Thus the transform determines every moment of $U$; moment determinacy on a compact interval determines the law of $U$, and the injective inverse $X=-\log U$ determines the law of $X$.