What happens to a general integral of this form in two dummy variables, $x$ and $y$ \begin{equation} \langle \int\int f(x)g^*(y) dxdy \rangle \end{equation} if the expectation product $\langle f(x)g^*(y) \rangle $ = 0 for $x \neq y$?
For context, I am trying to understand how Equation (1-3) of Synthesis Imaging in Radio Astronomy II (Chapter 2), given below \begin{equation} V_\nu(r_1,r_2) = \langle \int \int \varepsilon_\nu (R_1) \varepsilon^*_\nu(R_2) \frac{e^{2\pi i \nu |R_1 - r_1|/c}}{|R_1 - r_1|} \frac{e^{2\pi i \nu |R_2 - r_2|/c}}{|R_2 - r_2|} dS_1 dS_2 \rangle \end{equation} can be simplified into \begin{equation} V_\nu(r_1,r_2) = \int \langle |\varepsilon_\nu (R)^2| \rangle |R^2| \frac{e^{2\pi i \nu |R - r_1|/c}}{|R - r_1|} \frac{e^{2\pi i \nu |R - r_2|/c}}{|R - r_2|} dS . \end{equation} when $R_1\neq R_2$.
Here $\varepsilon$ is the surface brightness distribution of sources at $R_1$ and $R_2$, $r_1$ and $r_2$ are two points where the sources are observed, and $dS$ is the unit area on an imaginary celestial sphere.