Exercise 11, § 5

Let T be a completely regular space. Assume given, for every compact space K and every continuous mapping $f$ of T into K, a positive measure $\mu_{f, K}$ on K; assume that $g(\mu_{f, K}) = \mu_{g \circ f, L}$ for any compact spaces K and L and any continuous mappings $f : T \to K$ and $g : K \to L$; assume moreover that the measure $\mu_{f, K}$ is concentrated on $f(T)$ for any $f$ and K. Show that there exists one and only one bounded positive measure $\pi$ on T such that $\mu_{f, K} = f(\pi)$ for any $f$ and K (make use of the universal property of the Stone–Čech compactification of T).