JOINT MEASURES AND CROSS - COVARIANCE OPERATORS ( l )

Let H. (resp., H ) be a real and separable Hubert space with Borel O'field T (resp., rj, and let (H. X //-, T, X T.) be the product measurable space generated by the measurable rectangles. This paper develops relations between probability measures on (H. x H , V x T.), i.e., joint measures, and the projections of such measures on (H., T.) and (H , Y ). 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. Introduction. Let H. (resp., A7 ) be a real and separable Hubert space with inner product (•, *)j (resp.,(', -)2) and Borel o-field Tj (resp., TJ. Let Tj x T denote the o-field generated by the measurable rectangles AxB,A£rj,B£ I"^. Define H. x //, = {(u, v): u in H., v in H A. f/j x H is a real linear space, with addition and scalar multiplication defined by (u, v) + (z, y) = (u + z, v + y) and k(u, v) = (ka, k\). H. x H2 is a separable Hubert space under the inner product [•, •] defined by [(u, v), (t, z)] = (u, t)j + (v, z)2; moreover, the open sets under the norm obtained from this inner product generate Tj x T2 [10]. Let || • ||j (resp., || • ||2) denote the norm in H. (resp., H ) obtained from the inner product, and let ||| • ||| denote the norm in Hl x H2 obtained from the inner product. A probability measure on (//j x H2, Tj x T2) will be called a joint measure. A probability measure (i. on (//., T.) (i = 1 or 2) that satisfies « //fi||x||i?^.(x)<oo defines an operator R. in H. and a mean element m. of H. by <m¿, u)¿ = JH.(x, u^ft/x) Received by the editors June 3, 1971. AMS (MOS) subject classifications (1970). Primary 28A40, 60G15, 28A35, 60G3O; Secondary 94A15.