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Q - the additive group of rationals in the discrete topology. a. for (Q. : '''A nonstandard analysis approach to Fourier analysis, contributions to nonstandards analysis'" , Amsterdam, North Holland 1972, p. 16 - 39. : "'On Fourier transform in nonstandard analysis"', Izv. Vyssh. Uchebn. Zaved. Math. 1989, N 2 p. 17 - 25 (in Russian). : "'Hyperfinite approximations of locally compact abelian groups"', Soviet Math. Dokl. 1991, Vol. 42, N 2 p. 567 - 571. : '''Nonstandard analysis and compact abelian groups"', Siberian Math.

Cutland 36 References [1] A. Bensoussan, A model of stochastic differential equation in Hilbert space applicable to Navier-Stokes equation in dimension 2, in: Stochastic Analysis, Liber Amicorum for Moshe Zakai, eds. 51-73. Temam, Equations stochastiques du type Navier-Stokes, J. Functional Analysis 13 (1973), 195-222. Cutland, Statistical solutions of Navier-Stokes equations by nonstandard densities, Mathematical Models and Methods in Applied Sciences 1:4 (1991), 447-460. Cutland, Stochastic Navier-Stokes equations, Acta Applicanda Mathematicae 25 (1991), 59-85.

It is easy to see that there is a hyperfinite subgroup F ~* Sk such that (9) Consider now the hyperfinite group M M = {< K-1(f) ~ L such that + X(d), 1 > If E F, dE D1, l E L2, 02(1) = d}. (10) The homomorphism, is surjective. So we can find an internal set G1 = {gmlm E M} such that Vm E M ,(gm) = m. Define the binary operation +1 on G1 by the formula gml +1 gm2 = gml +m2' so that ,(gml + gm2) = ,(gml) +1 ,(gm2)' It is easy to see that < G1, +1 > is a hyperfinite abelian group. Lemma 1. i) Vm1, m2 E M gml +1 gm2 ii) Vm E M (-lgm) ~ ~ gml + gm2 -gm The proof follows immediately from the diagram (6).