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Absolutely continuous transformations of Gaussian measure in Hilbert space. (English. Russian original) Zbl 0633.60058
Sov. Math. 31, No. 4, 86-90 (1987); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1987, No. 4(299), 65-68 (1987).
Let $$\mu$$ be a Gaussian measure with correlation operator R in a separable, real Hilbert space H, and (X,$${\mathcal B}_ X,\nu)$$ a probability space. On the space (H$$\times X,\sigma ({\mathcal B}_ H\times {\mathcal B}_ X),\mu \otimes \nu)$$ is considered the transformation $T(h,x)=h+K f(h,x),\quad h\in H,\quad x\in X,$ where $$K=\sqrt{R}$$, f(h,x): $$H\times X\to H$$. Let $$\lambda (A)=\mu \otimes \nu (T^{- 1}(A))$$, $$A\in {\mathcal B}_ H$$. Conditions for equivalence $$\lambda\sim \mu$$ are given and the Radon-Nikodym density is calculated.