Welldefined Hilbert-Schmidt Operator

Question

How can I show that the general Hilbert-Schmidt-Operators as a linear map defined for $p$ and $q$ with the condition as always by

$Tf(x) := \int_Y k(x,y)f(y)d\nu (y)$

with $k(x,y)$ $\mu$x$\nu$-measurable is welldefined for $f\in L^p (\nu)$? I would be grateful for any help :)

Answer 1

Hint: Apply Holder's Inequality and Fubini-Tonelli to the product $k(x,\cdot)f$.