The representation of Popper measures

Abstract

The paper shows that all and only a-additive Popper measures can be, indeed uniquely, represented by so called dimensionally well-ordered families of a-additive probability measures. Probabilities conditional on events having probability 0 cannot be defined within standard probability theory. Many considered this to be a serious defect and tried to do better. Essentially four ways to deal with this problem have emerged. One method is due to Carnap; it consists in imposing his regularity condition on probability measures (cf. Carnap (1971, 1980), Section 7 and 21). However, this does not solve the problem; it is only a way of avoiding it as far as possible within standard probability theory. But the problem is unavoidable for measures on uncountable o-fields. The second method is the standard one within mathematics. It consists in generalizing the concept of conditional probability, i.e. in defining probabilities (and expectations) conditional on e-fields (cf., e.g., Lo~ve (1960), w w 24, 25). This method meets most of the needs of applied mathematics and should make us aware of the fact that the problem is a rather theoretical one. However , the probabilities conditional on oYfields are determined only almost uniquely (in the technical sense). Thus, I think, this method does not count as a fundamental solution of our problem. Another method is to do probability theory within the framework of nonstandard analysis (cf. Loeb, 1979). Within this framework division by infinitesimals is defined. Thus, if non-empty events get at least infinitesimal instead of zero probability, probabilities conditional on them can also be defined. Without doubt, this method should be taken seriously. But nonstandard analysis is a very intricate matter, and I won' t dwell upon this method here. The fourth method, developed independently by Popper and R~nyi, is the one I want to deal with. It consists in construing conditional probability not as a derived, but as a fundamental concept and in looking for a suitable set of axioms for it; absolute probabilities are then defined as probabilities conditional on the sure event. The simplest axiomatics that has emerged is comprised in the following definition: DEFINITION 1. Let ~2 be a non-empty set and S / , the set of events, a o-field of subsets of~2. Then (~2, > U , . ~ , P) is called a conditional probability space (e.p.s.} iff ~r the set of conditions, is a non-empty subset o f s U { 0 } a n d P is a function from . W ' x ~ into the closed interval [0, 1]' such that the following holds: (a) for each B E ~ the function P( . I B) is a o-additive probability measure on ~ / w i t h P ( B I B) -1, (b) for each A, B, C E ~ / w i t h C, B n C E~-~:J we have P ( A n B I C) = P ( A I B n C) . P (B I C). (fY, ~ / , . ~ , P) is called an additive conditional probability spaee iff we have moreover: (c) i fA , B E ~ , then alsoA u B E ~ 5 r. (~2, ~ / , 2 ,P) is a Popper space iff in addition to (a) and (b) we have (d) for each A C S ' , i f P ( A [B) > 0 for some B E ~ , then A E 3 ~ . Finally, let's call ( ~ 2 , ~ , ~ , P ) a full conditional probability space iff in addition to (a) and (b) (e) . ~ T = . ~ ~ 0}. The function P of a c.p.s, is called, respectively, a (additive, full} conditional probability measure (c.p.m.) or a Popper measure. Obviously, every full c.p.s, is a Popper space, and every Popper space is an additive c.p.s. Some history: The definition of c.p.s.s is due to R~nyi (1955). It scarcely needs any explanation; the point of clause (b) is, of course, to ensure the compatibility of probabilities with respect to varying conditions. The concept of an additive c.p.s, has also been invented by R6nyi; its role will become clear in Theorem 1 below. (This use of 'additive' has nothing to do with the usual addititivity of measures.) Popper spaces have, of course, been introduced by Popper; the axioms given by him in Popper (1959), new appendices *ii *v, have been simplified by Stalnaker (1970), Harper (1976), and van Fraassen (1976). I have taken the axioms stated by van Fraassen (1976), p. 420. Topoi 5 (1986), 69-74. 9 1986 by D. Reidel Publishing Company.