Denote by \(\lambda\) Lebesgue measure on \((0, \infty)\). Prove that the following iterated integrals exist and that

\[\int_{(0, \infty)} \int_{(0, \infty)} e^{-x y} \sin x \sin y \lambda(d x) \lambda(d y)=\int_{(0, \infty)} \int_{(0, \infty)} e^{-x y} \sin x \sin y \lambda(d y) \lambda(d x) .\]

Does this imply that the double integral exists?