Generalized measures in function spaces

Various problems in the theory of stochastic processes, differential equations, quantum mechanics, etc. require the development of the measure theory in function spaces. It is quite natural from the point of view both of the theory itself and of its applications to consider not only the «ordinary» measures, but «generalized» measures as well. By generalized measures (or quasimeasures) we understand additive set functions which are neither cr-additive, nor positive. For such quasimeasures it is possible to determine a process of integration, which is meaningful for a certain class of suitably nice functionals. It permits us to interprete such quasimeasures as generalized functions on a certain space of basic functionals.
The integration with respect to quasimeasures makes it possible to obtain the integral representations of the solutions for a sufficiently wide class of equations, these representations being similar to the Katz's formula for the solution of the heat equation. In conclusion we present some problems concerning the study of quasimeasures and their use.