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Quasi-invariant and pseudo-differentiable measures with values in non-Archimedean fields on a non-Archimedean Banach space. (Russian, English) Zbl 1073.46036
Fundam. Prikl. Mat. 9, No. 1, 149-199 (2003); translation in J. Math. Sci., New York 128, No. 6, 3428-3460 (2005).
From the author’s abstract: Quasi-invariant and pseudo-differentiable measures on a Banach space \(X\) over a non-Archimedean locally compact infinite field with a nontrivial valuation are defined and constructed. Measures are considered with values in non-Archimedean fields, for example, the field \({\mathbb{Q}}_p\) of \(p\)-adic numbers. Theorems and criteria are formulated and proven about quasi-invariance and pseudo-differentiability of measures relative to linear and nonlinear operators on \(X\). Characteristic functionals of measures are studied. Moreover, the non-Archimedean analogs of the Bochner–Kolmogorov and Minlos–Sazonov theorems are investigated. Infinite products of measures are considered and the analog of the Kakutani theorem is proven. Convergence of quasi-invariant and pseudo-differentiable measures in the corresponding spaces of measures is investigated.

46G12 Measures and integration on abstract linear spaces
28C99 Set functions and measures on spaces with additional structure
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis