Joint measures and cross-covariance operators

Abstract

Let ${H_1}$ (resp., ${H_2}$) be a real and separable Hilbert space with Borel $\sigma$-field ${\Gamma _1}$ (resp., ${\Gamma _2}$), and let $({H_1} \times {H_2},{\Gamma _1} \times {\Gamma _2})$ be the product measurable space generated by the measurable rectangles. This paper develops relations between probability measures on $({H_1} \times {H_2},{\Gamma _1} \times {\Gamma _2})$, i.e., joint measures, and the projections of such measures on $({H_1},{\Gamma _1})$ and $({H_2},{\Gamma _2})$. In particular, the class of all joint Gaussian measures having two specified Gaussian measures as projections is characterized, and conditions are obtained for two joint Gaussian measures to be mutually absolutely continuous. The cross-covariance operator of a joint measure plays a major role in these results and these operators are characterized.