  Application of the functional integration methods in equilibrium statistical
mechanics of quantum Bose-systems is considered. We show that Gibbs equilibrium
averages of Bose-operators can be represented as path integrals over a special
Gauss measure defined in the corresponding space of continuous functions. This
measure arises in the Bogolyubov T-product approach and is non-Wiener. We
consider problems related to integration with respect to the Bogolyubov measure
in the space of continuous functions and calculate some functional integrals
with respect to this measure. Approximate formulas that are exact for
functional polynomials of a given degree and also some formulas that are exact
for integrable functionals belonging to a broader class are constructed. We
establish the nondifferentiability of the Bogolyubov trajectories in the
corresponding function space and prove a theorem on the quadratic variation of
trajectories and study the properties implied by this theorem for the scale
transformations. We construct some examples of semigroups related to the
Bogolyubov measure. Independent increments are found for this measure. We
consider the relation between the Bogolyubov measure and parabolic partial
differential equations. An inequality for some traces is proved.
