Birman's definition of the integral trace of a nuclear operator as an integral over the diagonal is linked to the recent concept of virtually continuous measurable functions of several variables [2, 3]. Namely, it is shown that the construction of Birman is a special case of the general integration of virtually continuous functions over polymorphisms (or bistochastic measures), which in particular makes it possible to integrate such functions over some submanifolds of zero measure. Virtually continuous functions have similar application to embedding theorems (see [2]).