Quasi-invariant and pseudo-differentiable measures on a non-Archimedean Banach space.I. Real-valued measures

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 $\bf R$. 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 also are considered. Convergence of quasi-invariant and pseudo-differentiable measures in the corresponding spaces of measures is investigated.