On Extension of Schwartz Distributions to the Space of Discontinuous Test Functions

The present paper is devoted to the investigation of the properties of the space R of distributions with discontinuous test functions of several variables. The consideration of the discontinuous test functions allows us to define the correct operations (i.e., the operations which are defined everywhere, continuous and coincident with the ordinary ones for the regular distributions) of integration of distributions and multiplication of distributions by discontinuous functions, which are undefined in the classical space D ′ of distributions with continuous test functions, yet necessary in many applications of the distribution theory. In particular, the consideration of the R-mixed strategies – the distributions with discontinuous test functions – allows us to provide the existence of a solution for a class of zero-sum games with discontinuous payoff functions, which may have no solution in the pure strategies or in the classical mixed strategies.