Exercise 22, § 6

Let X, Y be two compact spaces, U a continuous linear mapping of the Banach space $\mathscr{C}(X; \mathbf{R})$ into the Banach space $\mathscr{C}(Y; \mathbf{R})$, $\pi : X \to Y$ a continuous mapping. Assume that for every $y \in Y$ and every function $f \in \mathscr{C}(X; \mathbf{R})$,

Show that under these conditions there exists a vaguely continuous mapping $\sigma : y \mapsto \sigma_y$ of Y into $\mathscr{M}_+(X)$ such that, for every $y \in Y$, $\sigma_y$ is carried by $\pi^{-1}(y)$ and one has $(U \cdot f)(\pi(x)) = \langle f, \sigma_{\pi(x)} \rangle$ for all $x \in X$; converse. (Consider the transpose $^tU$.)