ABSTRACT. The present paper is devoted to generalizations of the Dieudonnd theorem claiming that the convergence of sequences of regular Borelian measures is preserved under the passage from a system of open subsets of a compact metric space to the class of all Borelian subsets of this space. The Dieudonnd theorem is proved in the case for which the set functions are weakly regular, nonadditive, defined on an algebra of sets that contains the class of open subsets of an arbitrary a-topological space, and take values in a uniform space.

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Conditions for the uniform exhaustivity of a family of regular set functions defined on an algebra £ of subsets of a cr-topological space and taking values in arbitrary topological space are found.

The uniform exhaustivity criteria are proved for a sequence of exhaustive outer measures defined on the non-sigma-complete class of sets and taking values in an Abelian topological group.

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