Exercise 3, § 2

Let X be a locally compact topological space, $\mu$ a positive measure on X and $Y\subset X$ a locally $\mu$-negligible subset. Let N be the characteristic function of $X\times Y$.

a) We have

for $f,g\in L^2(X, \mu)$.

b) If $f\in L^2(X, \mu)$- $L^1(X, \mu)$, the set of $y\in X$ such that $x\mapsto N(x, y)f(x)$ is $\mu$-integrable coincides with Y.