I'm trying to solve an additive RHP and in order to do that I need to prove a Cauchy transform is analytic. In particular, given an analytic function $f : U \rightarrow \mathbb{C}$, where $U \subset \mathbb{C}$ is a domain, I'm trying to prove the Cauchy's transform of $f(z)$ in the contour $\gamma = \mathbb{R}$ (given below) is analytic
$$ Cf(z) = \frac{1}{2\pi i}\int_\mathbb{R} \frac{f(s)}{s-z}ds. $$
So far, I tried to apply Morera's theorem here: given a simple closed contour $\Gamma$, prove that
$$ \frac{1}{2\pi i}\int_\Gamma \int_\mathbb{R} \frac{f(s)}{s-z}dsdz = 0. $$
If somehow we could apply Fubini in the integral above then
$$ \frac{1}{2\pi i}\int_\Gamma \int_\mathbb{R} \frac{f(s)}{s-z}dsdz = \frac{1}{2\pi i} \int_\mathbb{R} f(s) \int_\Gamma \frac{1}{s-z}dz ds = 0 $$
because $\frac{1}{s-z}$ is analytic in a region containing $\Gamma$, so by Morera's theorem the inside integral is 0 and we are done.
I already know this Cauchy transform is analytic, so in fact the double integral is 0 and we can apply Fubini, but what is the argument to solve it properly?