Suppose that $X$ and $Y$ are sets with measures $\mu$ and $\nu$ defined on them. Suppose $K(x, y)$ is $\mu × \nu$ measurable and that there is a constant $C > 0$ such that $\int_{X}|K(x, y)| dµ(x) ≤ C$ for $\nu$ a.e. $y$.
If $f$ is a measurable function on $Y$ , let $Tf$ be the function on $X$ defined by $Tf(x) = \int_{Y} K(x, y)f(y) dν(y)$.
Is $Tf$ $\mu$ measurable?
Thank you in advance