Differentiability space of a product measure. (English. Russian original) Zbl 0744.46032

Mosc. Univ. Math. Bull. 44, No. 2, 105-108 (1989); translation from Vestn. Mosk. Univ., Ser. I 1989, No. 2, 81-84 (1989).
In his study of differentiable measures \(\mu\) on a sequentially complete locally convex space \(X\), V. I. Bogachev [Mat. Zametki 36, No. 1, 51-64 (1984; Zbl 0576.28022); Mat. Sb., Nov. Ser. 127(169), No. 3(7), 336-351 (1985; Zbl 0582.46050)] showed among others that, if \(X\) is quasi-complete, the space \(D(\mu)\) of differentiability, i.e. the subspace of \(X\) of the vectors in the directions of which \(\mu\) is differentiable is included in a Hilbert space compactly imbedded in \(X\), and, by a counterexample with a product measure, that it is not generally isomorphic to a Hilbert space. This note gives some characterizations of this space \(D(\mu)\) of a product measure \(\mu=\prod_{n=1}^ \infty \mu_ n\) on \(\mathbb{R}^ \infty=\prod_{n=1}^ \infty\mathbb{R}\) where each \(\mu_ n\) is a differentiable probability measure on \(\mathbb{R}\) with density relative to the Lebesgue measure.


46G12 Measures and integration on abstract linear spaces
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
60B11 Probability theory on linear topological spaces
58D20 Measures (Gaussian, cylindrical, etc.) on manifolds of maps