I am stuck on a question from Chapter 2 Section 5 Question 21 of Erhan Cinlar's "Probability and Stochastics". This is the question:
Let X and Y be positive random variables. Then, X and Y are independent if and only if their joint Laplace transform is the product of their Laplace transforms, that is, if and only if $\mathbb{E}(e^{-pX-qY})=\mathbb{E}(e^{-pX})\mathbb{E}(e^{-qY})$, $p,q\in\mathbb{R}_+$. Show this recalling that the joint Laplace transforms determine the joint distributions.
I am working on independence$\implies\mathbb{E}(e^{-pX-qY})=\mathbb{E}(e^{-pX})\mathbb{E}(e^{-qY})$
I did not understand the hint. So I tried to use Stone-Weierstrass Theorem with sum exponential functions to approximate continuous function. And I managed to show that for any continuous functions $f_X$ and $f_Y$, $\mathbb{E}((f_X\circ X)(f_Y\circ Y))=\mathbb{E}(f_X\circ X)\mathbb{E}(f_Y\circ Y)$.
But to show independence we need $\mathbb{E}((f_X\circ X)(f_Y\circ Y))=\mathbb{E}(f_X\circ X)\mathbb{E}(f_Y\circ Y)$ for positive Borel functions, not just continuous. How may I continue from here?
Is there an easier way using the hint?
If $\mu_X, \mu_Y$ and $\mu_{X,Y}$ are the probability measures induced by $X,Y$ and $(X,Y)$ respectively then the hypothesis says that the Laplace transform of the product measure $\mu_{X,Y}$ is same as the Laplace transform of $\mu_X \times \mu_Y$. This implies that $\mu_{X,Y}=\mu_X \times \mu_Y$ which is precisely what it means to say that $X$ and $Y$ are independent.