# Applying Fubini to $\int_0^\infty \int_0^\infty |f(t)|^2 e^{-st} \sqrt{\frac{t}{s}}\;\mathrm dt \, \mathrm ds,$

Consider the double integral $$\int_0^\infty \int_0^\infty |f(t)|^2 e^{-st} \sqrt{\frac{t}{s}}\;\mathrm dt \, \mathrm ds,$$ where $$f \in L^2(0,\infty)$$. This double integral appears when computing the norm of the Laplace transform as a map of $$L^2(0,\infty) \to L^2(0,\infty)$$.

To continue with the computation, I must interchange the order of integration, i.e. I would like to apply Fubini's theorem. However, I'm not seeing why Fubini's theorem can be applied here.

I was thinking of splitting the domain of integration, and then considering each sub-integral separately, but I would guess that there is a better way of seeing it. Could someone help me with this?

• Do you know Tonelli's theorem? – zhw. Nov 4 '18 at 20:33
• @zhw. I didn't, until now. Thanks for the comment! – MisterRiemann Nov 4 '18 at 20:37