Shift-invariant measures on infinite-dimensional spaces: integrable functions and random walks

Averaging of random shift operators on a space of the square integrable by shift-invariant measure complex-valued functions on linear topological spaces has been studied. The case of the $l_\infty$ space has been considered as an example.
A shift-invariant measure on the $l_\infty$ space, which was constructed by Caratheodory's scheme, is $\sigma$-additive, but not $\sigma$-finite. Furthermore, various approximations of measurable sets have been investigated. One-parameter groups of shifts along constant vector fields in the $l_\infty$ space and semigroups of shifts to a random vector, the distribution of which is given by a collection of the Gaussian measures, have been discussed. A criterion of strong continuity for a semigroup of shifts along a constant vector field has been established.
Conditions for collection of the Gaussian measures, which guarantee the semigroup property and strong continuity of averaged one-parameter collection of linear operators, have been defined.