Hyperfinite Dimensional Representations of Spaces and Algebras of Measures

Abstract

Let X be a locally compact topological space and (X, E, X?) be any triple consisting of a hyperfinite set X in a sufficiently saturated nonstandard universe, a monadic equivalence relation E on X, and an E-closed galactic set X? ? X, such that all internal subsets of X? are relatively compact in the induced topology and X is homeomorphic to the quotient X?/E. We will show that each regular complex Borel measure on X can be obtained by pushing down the Loeb measure induced by some internal function . The construction gives rise to an isometric isomorphism of the Banach space M(X) of all regular complex Borel measures on X, normed by total variation, and the quotient , for certain external subspaces of the hyperfinite dimensional Banach space , with the norm ?f?1 = ?x ? X |f(x)|. If additionally X = G is a hyperfinite group, X? = G? is a galactic subgroup of G, E is the equivalence corresponding to a normal monadic subgroup G0 of G?, and G is isomorphic to the locally compact group G?/G0, then the above Banach space isomorphism preserves the convolution, as well, i.e., M(G) and are isometrically isomorphic as Banach algebras