transactions of the american mathematical society Volume 269, Number 2, February 1982 A LOEB-MEASURE APPROACH TO THEOREMS BY PROHOROV, SAZONOVAND GROSS BY TOM L. LINDSTR0M Abstract. We use the Loeb-measure of nonstandard analysis to prove three classical results on limit measures: Let {n¡)¡ei be a projective system of Radon measures, we use the Loeb-measure L(jiE) for an infinite E £ */ and a standard part map to construct a Radon limit measure on the projective limit (Prohorov's Theorem). Using the Loeb-measures on hyperfinite dimensional linear spaces, we characterize the Fourier-transforms of measures on Hubert spaces (Sazonov*s Theorem), and extend cylindrical measures on Hubert spaces to o-additive mea- sures on Banach spaces (Gross' Theorem). I. Introduction. It has been clear for some time now that the Loeb-measure of nonstandard analysis [11] may be useful in the construction of different kinds of limit measures. Indeed, Anderson's nonstandard construction of a Brownian mo- tion [1] may be regarded as a direct construction of a weak limit measure (compare Billingsley [4]). Work on weak convergence from a nonstandard point of view has been carried on by Anderson and Rashid [3], and Loeb [12]. In another direction, Helms and Loeb [7], Hurd [9], and Helms [6] have used the Loeb-measure to obtain limit equilibrium measures in statistical mechanics. In this paper we shall work with other-but related-concepts of limit measures, and we hope to show the efficiency of the Loeb-measure approach by giving simple proofs of three classical theorems. The first of these theorems is due to Prohorov [14]: Given a projective system <A„ t„ *«>,je/ of Hausdorff spaces endowed with a cylindrical measure { p,},e/ of Radon measures on the A„ it gives a characterization of when there is a limit Radon measure on the projective limit. The idea of the proof is to construct the limit measure from the Loeb-measure of p¡ for an infinite i G *I, using a standard part map 9¡: X¡ —* X. The second theorem is due to Sazonov [16]; it characterizes the functions that are Fourier-transforms of probability measures on Hubert spaces. The idea of the proof is that even when the measure does not exist on the Hubert space 77, measures exist on the hyperfinite dimensional subspaces of *77 and we can perform the necessary calculations on these spaces. The last theorem is by Gross [5] and is concerned with the extension of cylindrical measures on Hubert spaces to measures on Banach spaces where the Received by the editors April 21, 1980 and, in revised form, December 31, 1980. 1980 Mathematics Subject Classification Primary 60B10, 60E10; Secondary 03H05. Key words and phrases. Nonstandard analysis, Loeb-measure, cylindrical measures, projective limits, Fourier-transforms, measurable norms. © 1982 American Mathematical Society 0002-9947/82/0000-0770/S04.S0 521