Some open mapping theorems for measures

Abstract:

Given a compact Hausdorff space X, let $C(X)$ be the Banach space of continuous real valued functions on X with sup norm and let $M(X)$ be its dual considered as finite regular Borel measures on X. Let $U(X)$ denote the closed unit ball of $M(X)$ and let $P(X)$ denote the nonnegative measures in $M(X)$ of norm 1. A continuous map $\varphi$ of X onto another compact Hausdorff space Y induces a natural linear transformation $\pi$ of $M(X)$ onto $M(Y)$ defined by setting $\pi (\mu )(g) = \mu (g \circ \varphi )$ for $\mu \in M(X)$ and $g \in C(Y)$. It is shown that $\pi$ is norm open on $U(X)$ and on $A \cdot P(X)$ for any subset A of the real numbers. If $\varphi$ is open, then $\pi$ is $\mathrm {weak}^*$ open on $A \cdot P(X)$. Several examples are given which show that generalization in certain directions is not possible. The paper concludes with some remarks about continuous selections.