Let X and Y be random variables with distributions F and G; and write the laplace transforms of them as $\displaystyle \hat F , \hat G $

Show that $\displaystyle E[e^{- \lambda xy }] = \int ^ \infty _0 \hat F (\lambda y)dG(y) = \int ^ \infty _0 \hat G( \lambda y)dF(y) $

I'm just confused about the expected value of two different variables, does that means I should use dF or dG as the probability density function for it?

Thanks!