Interchange of limits in computation with Gamma

Question

On page 164 of Stein and Shakarchi's Complex Analysis there is the following computation: (here $0 < s < 1$, and line 1 -> 2 is a preceding lemma) \begin{align*} \Gamma(1 - s)\Gamma(s) &= \int_0^{\infty} e^{-t} t^{s-1} \Gamma(1 - s)dt \\ &= \int_0^{\infty} e^{-t} t^{s-1} \left(t \int_0^{\infty} e^{-vt} (vt)^{-s} dv\right) dt \\ &= \int_0^{\infty} \int_0^{\infty} e^{-t[1+v]} v^{-s} dv dt \\ &= \int_0^{\infty} \frac{v^{-s}}{1 + v} dv \\ &= \frac{\pi}{\sin \pi(1 - s)} \\ &= \frac{\pi}{\sin \pi s}. \end{align*} My question is about going from line 3 to line 4: of course this follows by swapping the order of integration, but I'm curious about how we can justify the interchange. Actually, if I'm not mistaken it follows from the Fubini-Tonelli theorem, but since this textbook does not assume Lebesgue theory, I was wondering if there is a reasonable way to justify the interchange with only Riemann integration theory? (taking the infinite integrals to mean something like $\lim_{\epsilon \to 0} \int_{\epsilon}^{1/\epsilon} \ldots$)