Hyperfinite Dimensional Representations of Spaces and Algebras of Measures

Abstract

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.

Gordon's conjectures 1 and 2: Pontryagin-van Kampen duality in the hyperfinite setting

“…In the present paper we will recapitulate Gordon's approach and some of his results, introducing several conceptual and notational modifications, based partly on Ziman and Zlatoš [47], and prove the first two of his conjectures for arbitrary condensing IMG triplets (G, G 0 , G f ) with an abelian hyperfinite ambient group G. What's of minor importance, working in a "sufficiently saturated" nonstandard universe, our triplets will represent all (Hausdorff) LCA groups (not just the metrizable σ -compact ones) as their observable traces. Besides the techniques of nonstandard analysis and harmonic analysis, the crucial role in our proofs will be played by the Fourier analytic methods of additive combinatorics due mainly to Green and Ruzsa [20], and Tao and Vu [44] (see Section 2.1).…”

“…Our basic nonstandard objects, by means of which we will study (sufficiently regular) topological spaces, will be ordered triples of the form (X, E, X f ) where X is an internal set, E is a monadic equivalence relation on X and X f is a galactic subset of X which is E-closed, ie, x ∈ X f and (x, y) ∈ E imply y ∈ X f , for any x, y ∈ X . We will call them alternatively bounded monadic spaces or IMG spaces like in Ziman and Zlatoš [47], or IMG triplets, indicating that we do not consider the present terminology as definitive.…”

“…The last condition (v) suggests calling the IMG triplets satisfying any (hence all) of the above conditions condensing (cf Mlček and Zlatoš [34,35] as well as Ziman and Zlatoš [47]). Obviously, the observable trace of any condensing IMG space is locally compact, and the compact subsets of X are exactly the observable traces A of internal subsets A ⊆ X f .…”

Radon-Nikodym theorem in signed Loeb space

“…In Refs. [9,10], Loeb measure had been used in hyperfinite adapted probability space and hyperfinite dimensional representations of spaces and algebras of measures, respectively.…”

Uniform liftings of continuous mappings

“…For further examples see Luxemburg [12], Henson [9], Gordon [7,8], Mlček and Zlatoš [13], and Ziman and Zlatoš [14].…”