On means with countably additive continuities

Abstract

Some notions of countable additivity, meaningful for an expectation whose domain is an arbitrary linear space of bounded, real-valued functions, are studied. For linear spaces or ordered linear spaces which do not possess the structure of a vector lattice important in the development of the Lebesgue-Daniell integralthere is to our knowledge no study of continuous or countably additive linear functionals other than the seminal contributions of de Finetti [1 and 2]. The present paper was inspired mainly by de Finetti's writings. For simplicity, linear subspaces L of lo (Q), the space of bounded R-valued (realvalued) functions defined on a nonempty set Q, will be considered and, on L, only linear functions Q, necessarily nonnegative, which satisfy f C L and a < f < b everywhere imply a < Qf < b. Call such a functional a prevision, a term borrowed from de Finetti [1 and 2]. With the help of the Hahn-Banach extension theorem, it is simple to verify that each prevision on L is the restriction to L of a prevision P on lo. Let Ap be the collection of all L such that P restricted to L is continuous or countably additive. In [1, Chapter 5.34 and 2, Vol. 2, Appendix 18.3], de Finetti introduces Ap and initiates the study of its structure. He observes L' c L C Ap implies L' C Ap, and he goes on to state L1 and L2 belong (to Ap), then so does L1 + L2 (the linear space of sums X1 + X2, X1 E L1 and X2 c L2). The present paper developed in large part from the observation that the quoted assertion is erroneous. Several counterexamples, which also illustrate other phenomena, are offered. For the first, [0,1] designates the closed unit interval, C = C[0,11] the space of all continuous R-valued functions with [0,1] as their domain, and A is the usual Lebesgue integral. The indicator of a set is, as usual, the function that assumes the value 1 on the set 0 off the set. The useful convention introduced by de Finetti of designating a set and its indicator by the same letter will usually be adopted here. EXAMPLE 1. Let Wp be an open dense subset of [0,1] with Ap < 1, L1 = C[0, 1], L2 = {t: t C R}, L = L1 + L2 and P(O + t) = AO + t, 0 e C[0, 1]. To formulate the strong sense in which Example 1 is a counterexample, it is necessary to articulate two definitions. If Pfn -? 0 for every decreasing sequence fC e L that converges pointwise to 0, call P m-continuous on L (m for 'monotone'). Let v(L) designate the smallest sigma field of subsets of Q such that f-1 (B) C a(L) Received by the editors February 21, 1983 and, in revised form, August 29, 1983. 1980 Mathematics Subject Cossification. Primary 28C05, 46G12, 60A05.