Let $X=[1, \infty)$, $Y=[0, 1]$, $\mu=\nu=m$-the Lebesgue measure on $X$ and $Y$ and $\mathcal A=\mathcal B=\mathcal M$-the Lebesgue $\sigma$-algebra on $X$ and $Y$. Show that $f:X\times Y\to\mathbb R$ defined by $$f(x, y)=e^{-xy}-2e^{-2xy}$$ has finite, but distinct, iterated integrals.
I'm not certain how to actually compute these integrals. I know I need to consider $|f(x,y)| = e^{-xy}+2e^{-2xy}$, which is non-negative, hence, integrable. But how do I show the integral is finite; moreover, how do I go about reversing the order of integration?