Some open mapping theorems for marginals

Abstract:

Let S and T be compact Hausdorff spaces and let $P(S),P(T)$ and $P(S \times T)$ denote the collection of probability measures on S, T and $S \times T$, respectively. Given a probability measure $\mu$ on $S \times T$, set $\pi \mu = (\alpha ,\beta )$ where $\alpha$ and $\beta$ are the marginals of $\mu$ on S and T. We prove that the mapping $\pi :P(S \times T) \to P(S) \times P(T)$ is norm open and ${\text {weak}^\ast }$ open. An analogous result for ${L_1}(X \times Y,\mu \times \nu )$ where $(X,\mu )$ and $(Y,\nu )$ are probability spaces is established.