The condition for interchange path integral and integral over the real line

Question

The question rises from a book I'm reading: Let $D$ be a domain, we have a function $g:D\subset \mathbb{C}\to \mathbb{C}$ defined by integration over a complex-valued fucntion $f$, where $f$ is entire for each fixed value of $x$. $$g(z)=\int^{\infty}_{0}f(z,x) dx$$ And the author wants to apply Morera's theorem to prove $g$ is holomorphic on $D$. It suffices to prove that $$\int_Tg(z)=\int_T\int^{\infty}_{0}f(z,x) dx=0$$ for each triangle $T$ in $D$. The author proved that $f(z,x)\leq e^{-Ax}$ for some $A>0$ and arbitrary $z$, and then claimed that we can now interchange two integrals, which leads to the desired conclusion. I'm confused about why can we interchange two integrals with these conditions, can someone give a more detailed reasoning? Thanks a lot.

Answer 1

For any path $\gamma$ (defined on $[a,b]$, say) $\,\,\int_{\gamma} \int_0^{\infty} f(z,x)dx dz=\int_a^{b} \int_0^{\infty} f(\gamma (t),x) \gamma'(t)dx dz$. Since $\int_a^{b} \int_0^{\infty} |f(\gamma (t),x)| |\gamma'(t)|dx dz < \infty $ we can apply Fubini's Theorem to justify interchanging of the integrals.