We will deal with finitely additive measures on integers extending the asymptotic density. We will study their relation to the Lévy group G of permutations of N. Using a new characterization of the Lévy group G we will prove that a finitely additive measure extends density if and only if it is G-invariant.
Abstract. Let X be a locally compact topological space and (X, E, X ω ) 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 ω ⊆ X, such that all internal subsets of X ω are relatively compact in the induced topology and X is homeomorphic to the quotient X ω /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 X → * C. 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 M ω (X)/M 0 (X), for certain external subspaces M 0 (X), M ω (X) of the hyperfinite dimensional Banach space * C X , with the norm f 1 = x∈X |f (x)|. If additionally X = G is a hyperfinite group, X ω = G ω is a galactic subgroup of G, E is the equivalence corresponding to a normal monadic subgroup G 0 of G ω , and G is isomorphic to the locally compact group G ω /G 0 , then the above Banach space isomorphism preserves the convolution, as well, i.e., M(G) and M ω (G)/M 0 (G) are isometrically isomorphic as Banach algebras. IntroductionA great deal of methods of nonstandard analysis is based on embedding classical mathematical structures into somehow related hyperfinite ones. As a rule, the hyperfinite set (topological space, measure space, etc.) or hyperfinite dimensional vector space X, extending the classical object X, is subject to the inclusions X ⊆ X ⊆ * X, i.e., it is singled out from the nonstandard extension * X of X. This has the additional advantage of X naturally inheriting the structure from X, via the extension * X, and applicability of the transfer principle.The method, however, may fail to work that way in presence of some already a little bit more complex algebraic structure on X. For instance, given a group G, there need not be any hyperfinite group G subject to G ≤ G ≤ * G. On the other hand, some hyperfinite group G extending G all the same may (though still need not) exist. And similarly for associative linear algebras over some field. The situation becomes even more complicated for topological groups and Banach algebras. On the other hand, especially for the sake of applications of nonstandard methods to the study of spaces and algebras of functions or measures over G, it is desirable to have G embedded into some hyperfinite group G and relate somehow the just 1991 Mathematics Subject Classification. Primary 28E05, 43A10; Secondary 03H05, 22D15, 46S20, 54J05.
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