  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.
