Exercise 20, § 1

Let X, Y be compact metric spaces, T a compact operator from $\mathscr{C}(Y)$ into $\mathscr{C}(X)$.

Prove that there exists a positive measure $\mu$ on X and a measurable map N from $X\times Y$ into $\mathbf{C}$ such that

a) For every $y\in Y$, the mapping $N_y:x\mapsto N(x, y)$ from X into $\mathbf{C}$ is $\mu$-integrable ;

b) The mapping $y\mapsto N_y$ from Y into $\mathscr{L}^1(X, \mu)$ is continuous ;

c) For every $f\in \mathscr{C}$(X), and every $y\in Y$, one has