  The sequence of one-parameter semigroups arising as the approximation of
initial-boundary value problem with singularities is the object of
investigation of this paper. The set of limit points of the sequence of
approximating semigroups is studied. The set of limit points of the map with
values in a linear topological space is presented as the set of mean values of
this map by measures on the domain of definition of the map. One to one
correspondence betwin the semigroups generated by any approximating
initial-boundary value problems and the pseudomeasures on the space of maps of
time semiaxe into the coordinate space is studied. The linear space of
pseudomesures endowed with the structure of Banach space and with the structure
of the linear topological space such that the convergence of semigroup sequence
is equivalent to the convergence of the sequence of corresponding
pseudomeasures. The desctiption of a limit point of the sequence of
approximating semigroups is obtained by a measure on the topological vector
space of corresponding pseudomeasures. The trajectories the limit one-parameter
family of transformations of the space of initial data is described by the mean
value of the random pseudomeasure.
