Hint Required for :
$\int_{[0,\infty[}(\int_{[0,\infty[} e^{-xy}\sin{x} \sin{y} d\lambda(x)) d\lambda(y)$
My ideas:
Looking at the first inetgral $\int_{[0,\infty[} e^{-xy}\sin{x} \sin{y} d\lambda(x)$
I would say $\int_{[0,\infty[} e^{-xy}\sin{x} \sin{y} d\lambda(x)=\sin{y}\int_{[0,\infty[}e^{-xy}\sin{x}d\lambda(x)$
And we know $\sin(x)=\frac{e^{ix}-e^{-ix}}{2i}$
So $\sin{y}\int_{[0,\infty[}e^{-xy}\sin{x}d\lambda(x)=\sin{y}\int_{[0,\infty[}e^{-xy}\frac{e^{ix}-e^{-ix}}{2i}d\lambda(x)$
and then I do not know what to do. Should I use this route or is partial integration better suited?
I've just started with multivariable integration, so any pointers would be of great assistance.