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The sequence of one-parameter semigroups arising as the approximation ofinitial-boundary value problem with singularities is the object ofinvestigation of this paper. The set of limit points of the sequence ofapproximating semigroups is studied. The set of limit points of the map withvalues in a linear topological space is presented as the set of mean values ofthis map by measures on the domain of definition of the map. One to onecorrespondence betwin the semigroups generated by any approximatinginitial-boundary value problems and the pseudomeasures on the space of maps oftime semiaxe into the coordinate space is studied. The linear space ofpseudomesures endowed with the structure of Banach space and with the structureof the linear topological space such that the convergence of semigroup sequenceis equivalent to the convergence of the sequence of correspondingpseudomeasures. The desctiption of a limit point of the sequence ofapproximating semigroups is obtained by a measure on the topological vectorspace of corresponding pseudomeasures. The trajectories the limit one-parameterfamily of transformations of the space of initial data is described by the meanvalue of the random pseudomeasure.

Averaging of one-parameter semigroups and passage to the limit in the space of pseudomeasures