proceedings of the american mathematical society Volume 114, Number 4, APRIL 1992 FINITELY ADDITIVE RADON-NIKODYM THEOREM AND CONCENTRATION FUNCTION OF A PROBABILITY WITH RESPECT TO A PROBABILITY PATRIZIA BERTI, EUGENIO REGAZZINI, AND PIETRO RIGO (Communicated by William D. Sudderth) Abstract. An "exact" Radon-Nikodym theorem is obtained for a pair (m, p.) of finitely-additive probabilities, using a notion of concentration function of p with respect to m . In addition, some direct consequences of that theorem are examined. 0. Introduction Throughout this paper the term probability will designate any positive, finitely- additive measure on an algebra # of subsets of il, assuming value 1 at il. We turn our attention to the case in which ß and m are probabilities on (il, J), with ß absolutely continuous with respect to (w.r.t.) m [for each e > 0 there exists S > 0 such that ß(E) < e whenever E £ $ and m(E) < ô]. Even if this condition holds, ß need not admit any "exact" Radon-Nikodym derivative w.r.t. m . Actually, under that hypothesis, Bochner [2] states that, given e > 0, there exists a simple function fi, such that | JE fe dm - ß(E)\ < e for all E in $. Later, the same theorem is proved again by various authors at various times (cf. Bhaskara Rao and Bhaskara Rao [1, Chapter 6 and Notes and Comments to Chapter 6]). A few of them deal with the problem of finding necessary and sufficient conditions for the existence of exact Radon-Nikodym derivatives. In particular, Maynard [8] provides a complete solution to the problem at issue. De Finetti [5] already had pointed out the connections between the existence of an exact Radon-Nikodym derivative and certain properties of the lower boundary of the convex hull of the range of (m, ß), under the hypothesis that 5 coin- cides with the power set of il, and m is strongly continuous [for each e > 0 there exists a partition {E\, ... , E„} of Q in # such that m(E¿) < e for every i]. Recently, de Finetti's approach has been basically taken up again by Received by the editors December 5, 1989. 1980MathematicsSubjectClassification (1985 Revision).Primary 60A10,28A25,28A60. Key words and phrases. Concentration function, extension, finitely-additive probability, Radon- Nikodym theorem. The first author's research was partially supported by MPI (40% 1985, Gruppo Nazionale "Pro- cessi Stocastici e Calcólo Stocastico"). The second author's research was partially supported by MPI (40% 1987, Gruppo Nazionale "Modelli Probabilistici") and by CNR (Progetto Strategico "Statistica dei Processi Aleatori"). © 1992 American Mathematical Society 0002-9939/92 $1.00+ $.25 per page 1069 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use