Some Properties of Functional Integrals with Respect to the Bogoliubov Measure

We consider problems related to integration with respect to the Bogoliubov 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. An inequality for traces is proved, and an upper estimate is derived for the Gibbs equilibrium mean square of the coordinate operator in the case of a one-dimensional nonlinear oscillator with a positive symmetric interaction.