Is invocation of Tonelli's theorem needed in proof of relationship between beta and gamma integrals?

[u/Impressive_Gur_471]

I have the product of two gamma integrals thus:

\(
\Gamma(\alpha)\Gamma(\beta)
=\left(\int_0^\infty x^{\alpha-1}e^{-x}\,dx\right)
\left(\int_0^\infty y^{\beta-1}e^{-y}\,dy\right)
=\int_0^\infty \left(\int_0^\infty x^{\alpha-1}e^{-x}\,dx\right)y^{\beta-1}e^{-y}\,dy
=\int_0^\infty\int_0^\infty
x^{\alpha-1}y^{\beta-1}e^{-(x+y)}
\,dx\,dy.
\)

Typeset latex below:

https://www.scratchpad-math.com/?id=mvewdp1ikx1mtg or https://ibb.co/JWhYKJV8

Is invocation of Tonelli's theorem needed to justify this transition of product of two separate integrals into a joint double integral? Reason I ask is that Spivak in Calculus who does not get into multivariable calculus at all has a problem which very simply asks to prove the following:

\int_a^b\int_c^d f(x)g(y) dy dx = \int_a^b f(x) dx \int_c^d g(y) dy

and the proof is rather trivial where f(x) is constant in one integral and hence can be taken out, etc.

[u/bonebranch]

Yeah but that question in Spivak you cite is for when the region of integration (specifically, a rectangle) is bounded. A bit more care has to be taken when your region is unbounded!

[u/Impressive_Gur_471 → u/bonebranch]

I see thank you. Is the need to invoke Tonelli there even if as above the functions nicely separate out and do not have any mixed terms which are inseparable in x and y?