# zbMATH — the first resource for mathematics

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