Some open mapping theorems for measures

Abstract

Given a compact Hausdorff space X, let C ( X ) C(X) be the Banach space of continuous real valued functions on X with sup norm and let M ( X ) M(X) be its dual considered as finite regular Borel measures on X. Let U ( X ) U(X) denote the closed unit ball of M ( X ) M(X) and let P ( X ) P(X) denote the nonnegative measures in M ( X ) M(X) of norm 1. A continuous map φ \varphi of X onto another compact Hausdorff space Y induces a natural linear transformation π \pi of M ( X ) M(X) onto M ( Y ) M(Y) defined by setting π ( μ ) ( g ) = μ ( g ∘ φ ) \pi (\mu )(g) = \mu (g \circ \varphi ) for μ ∈ M ( X ) \mu \in M(X) and g ∈ C ( Y ) g \in C(Y) . It is shown that π \pi is norm open on U ( X ) U(X) and on A ⋅ P ( X ) A \cdot P(X) for any subset A of the real numbers. If φ \varphi is open, then π \pi is w e a k ∗ \mathrm {weak}^* open on A ⋅ P ( X ) 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.