AN ABSTRACT AND GENERALIZED FORMULATION OF A THEOREM BY PELC AND PRIKRY ON INVARIANT EXTENSION OF BOREL

Abstract

There are certain countably generated $\sigma$-algebras of sets in the real line
which do not admit any non-zero,$\sigma$ -finite, diused (or, continuous) measure. Such countably generated $\sigma$-algebras can be obtained by the use of some special types of infnite matrix known as the Banach-Kuratowski matrix and the same may be used in deriving a generalized version of Pelc and Prikry's theorem as shown by Kharazishvili. Here, in this paper, we develop an abstract and generalized formulation of Pelc and Prikry's theorem in spaces with transformation groups, where instead of using measure type functionals as done by Kharazishvili, we utilize a newly introduced concept which is that of an admissible, diffused k-additive algebra where k is an arbitrary infinite cardinal.