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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.

MSC:
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
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