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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.

Abstract. Let d be a metric on a nonempty set A . The ternary betweenness relation T¡¡ induced by d on A is defined by Td{x,y,z) o d{x,y) + d{y, z) = d(x, z) for x, y, z e A. Allowing the range of d to vary over some "reasonable" ordered additive algebraic structures (not just the real numbers), we will prove that the class JK of all metrizable ternary structures, i.e., the class of all structures (A, Tj), where d is some metric on A, is an elementary class which can be axiomatized by a set of universal Horn sentences. Further, using an algorithm of linear programming, we will show that the first-order theory of Jt is recursively axiomatizable and its universal part is decidable. On the other hand, the theory of J( is not finitely axiomatizable and the theory of finite members of JK is hereditarily undecidable.In his fundamental work Grundlagen der Geometrie [H 1899] David Hubert introduced the ternary relation of betweenness which he used in formulating the second group of his axioms, namely the axioms of order. (In fact " y is between x and z " in Hilbert's formulation corresponds to " y lies between x and z and x ^ y ^ z" in ours.) Later on Alfred Tarski [T 1959] used the betweenness relation and the quaternary equidistance or congruence relation (i.e., "the distance of x, y and z, u is the same") to obtain a complete and decidable axiomatization of elementary geometry. Since the 1920s the notion of betweenness became of interest in its own right, as it naturally occurred in diverse mathematical structures, like linearly and partially ordered sets and lattices (see [B 1967[B ], [Bl 1953, metric spaces (see [M 1928[M ], [Bl 1953, linear spaces over ordered fields and normed linear spaces (see [Sm 1943], [Bl 1953]), etc. However, because of its intuitively clear geometrical meaning acquired in some "nice", though by far not all, metric spaces, the metric betweenness, defined as above, still has kept its central position. So far the paper [Mo 1977] by M. Moszyñska seems to be the only contribution to the study of the first-order theory of betweenness and equidistance relations in metric spaces.

Using the ideas of E. I. Gordon we present and farther advancean approach, based on nonstandard analysis, to simultaneousapproximations of locally compact abelian groups and their dualsby (hyper)finite abelian groups, as well as to approximations ofvarious types of Fourier transforms on them by the discrete Fouriertransform. Combining some methods of nonstandard analysis andadditive combinatorics we prove the three Gordon's Conjectureswhich were open since 1991 and are crucial both in the formulationsand proofs of the LCA groups and Fourier transform approximationtheorems

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