Suppose $(X,M,\mu)$ and $(Y,N,v)$ are $\sigma-$finite measure spaces. Is the following calculation correct?
$\int \chi_E(x,y) d\mu=\int (\chi_E)^y(x) d\mu=\int \chi_{E^y}(x) d\mu=\mu(E^y)$ for a given $E\subset X\times Y$. It is given that $E^y$ is $\mu-$ measurable.
This is just for confirmation, because I am not sure whether my calculation is correct.