Quasi-invariant and pseudo-differentiable measures with values in non-Archimedean fields on a~non-Archimedean Banach space

S. V. Lyudkovskii

Fundamentalʹnaâ i prikladnaâ matematika, Tome 9 (2003) no. 1, pp. 149-199.

Résumé

Quasi-invariant and pseudo-differentiable measures on a Banach space $X$ over a non-Archimedean locally compact infinite field with a non-trivial valuation are defined and constructed. Measures are considered with values in non-Archimedean fields, for example, the field $\mathbf Q_p$ of $p$-adic numbers. Theorems and criteria are formulated and proved about quasi-invariance and pseudo-differentiability of measures relative to linear and non-linear 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 proved. Convergence of quasi-invariant and pseudo-differentiable measures in the corresponding spaces of measures is investigated.

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