An abstract and generalized formulation of a theorem by Pelc and Prikry on invariant extension of Borel measure

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