Let $$\psi(x) = \sum_{n = 1}^{\infty} e^{-n^{2} \pi x}$$ be a theta function. Can it be shown that that
$$\int_{0}^{\infty} \psi(x) \cdot dx < \infty$$ without invoking Fubini-Tonelli’s Theorem about interchanging the sum and integral? If so, this would be a way to prove Riemann’s functional equation, $$\pi^{-s/2}\Gamma(s/2)\zeta(s) = \pi^{(1-s)/2}\Gamma((1-s)/2)\zeta(1-s)$$ using only the dominated convergence theorem and properties of the theta function described above.