Let X and Y be independent variables with densities f and g concentrated on $(0, \infty)$. If E(X) < $\infty$ , show that the ratio X/Y has a finite expectation iff
$$ \int_0^1 \frac{1}{y} g(y)dy < \infty $$
I know that I have show both sides. Can I just use the expectation formula for continuous variables
$$ \int x f(x) dx $$ for a density f(x) of the variable X?
Thanks!