Kolmogorov integral, infinite dimensional pseudo-differential operators and the Weyl secondary quantization

Using the notion of the indefinite Kolmogorov integral we define pseudo-differential operators with the Weyl symbols (PDOW) acting in a space of smooth enough countably-additive complex-valued Borel measures on the infinite-dimensional Hilbert space Q, and also PDOW acting in a space of smooth enough functions on the same Q. We introduce and prove the symmetric property of these PDOs with real symbols.