A. Ionescu Tulcea

Pointwise convergence in terms of expectations

This paper is concerned with the connection between almost sure convergence of a sequence of random variables and convergence of certain related expectations. Theorems of the kind we are interested in were proved by Meyer I-7, p. 232] and Mertens [6, p. 47-1 in the continuous-parameter case, and by Baxter 1-11 in the discrete-parameter case. For example, Baxters theorem is the following: Let (X~)~_>_I be a sequence of random variables with values in a compact metric space S, and let the set F of bounded stopping times be directed by the obvious ordering. Then (X~)n_>l converges almost surely if and only if the generalized sequence (6 ~b (X~))~r of expectations converges for every real-valued continuous function q5 on S. In the present paper we generalize this theorem in two ways: we replace S by an arbitrary complete separable metric space, and we use as few test functions ~b as possible. IfS is the real line, the single test function ~b (x) = x suffices (Theorem 2); for any complete separable metric space, a countable set of functions suffices (Theorem 3); and for a separable Banach space, there is a countable set of convex functions which suffices (Theorem 4). We have included a different proof of the key step in Baxters proof (Corollary 1), in order to make the present paper selfcontained.

On pointwise convergence, compactness and equicontinuity in the lifting topology. I

The latter defines the usual equivalence relation in Moo. We denote by f the equivalence class of each feM ~ with respect to this equivalence relation. We say that a set F carries p ifF~ and p(E-F)=O. In the case when E is a compact space and/~ a positive Radon measure on E, we denote by Supp ~ the smallest (= intersection) of all closed sets F c E, carrying p. If f: E~R and AcE, we denote by fla the restriction of f to A; finally if HcMoo we use the notation Hla for the set of all hlA, with h~H. The author is indebted to Professors Dietrich K61zow and John C. Oxtoby for several helpful remarks and relevant comments concerning the contents of this paper. We begin with the following simple but very useful result. Proposition 1. Let H c M ~ be a set which is bounded and such that the relations h 1 ~H, h 2 ~H and h I 4: h 2 imply h 1 + h 2 . Assume that H is sequentially compact for the topology of pointwise convergence on E, that is for any sequence (h,) of elements of H, there is a subsequence (h,k) and an h~H such that h,k ~ h pointwise on E. Then H is compact and metrizable for the topology of pointwise convergence on E. Proof We consider on H the topology -[-1 of mean Ll-convergence 1, and the topology T of pointwise convergence on E. We divide the proof into two steps: I) We note first that (H, T1) is compact. In fact, for any sequence (h,) in H, there is a subsequence (h,~) and an h~H such that h,~ ~ h pointwise on E. By the Lebesgue dominated convergence theorem, we also have

Liftings for abstract valued functions and separable stochastic processes

In this paper we extend the notion of lifting from real-valued functions to abstract-valued functions (functions with values in a completely regular space). We prove the existence and uniqueness of the “abstract lifting” associated with a given “real lifting”. As an application we show that given a stochastic process with values in a completely regular space, the modification of this process obtained by applying a lifting is always separable.

Ergodic properties of semi-Markovian operators on the Z1-part

SummaryIn this note we consider a semi-Markovian operator, that is a positive linear mapping T: L1 → L1 such that sup ∥Tn∥ <∞. We study the behavior of Tn on the Z1-part of the space (the “disappearing part” in Suchestons terminology). We show in particular, that if the operator T has a non-trivial conservative part in Z1, then the ratio theorem must fail.

On measurability, pointwise convergence and compactness

The starting point of this investigation is the beautiful generalization of Egorovs theorem given by P. A. Meyer in Séminaire de Probabilités. V, (Strasbourg). The material is divided as follows: §1. Setting and terminology. §2. The Generalized Egorov Theorem. §3. An application to vector-valued mappings. §4. The ((separation property)) and the notion of lifting. Proofs of most of the results contained in this paper can be found in [5], [6], [7], [8].

The existence of a lifting

Throughout this chapter we assume that N = \(\bar N\). With the exception of sections 3 and 4 we suppose that (X,N,ℛ) is strictly localizable and that l is a fixed partition of X consisting of non-negligible integrable sets satisfying sup{\(tilde K\)|Kєl} = \(tilde X\).

Domination of measures and disintegration of measures

Throughout this chapter the setting is that of compact or locally compact spaces and positive Radon measures. We make constant use of the notation and terminology of chapter 8.

On certain endomorphisms of L_R^\infty (Z,\mu )

In this chapter, unless we mention explicitly the contrary, we shall denote by ℒ1 = (Z1,µ1) and ℒ2 = (Z2,µ2) two objects, where Z1 and Z2 are locally compact spaces, µ1 ≠ 0 is a positive Radon measure on Z1 and µ1 ≠ 0 a positive Radon measure on Z2.

Integrability and measurability for abstract valued functions

In this chapter we shall discuss the integrability and measurability of functions with values in a Banach space. The definitions and results of sections 1, 2 and 3 are somewhat similar to those in chapter 1. The definitions and results of sections 4, 5, 6 are based on the notion of lifting and are essential for the applications in the next chapter.

Measure and integration

In this chapter we outline the setting for the theory of measure and integration that will be used in this book. The approach that we develop is essentially based on the notion of upper integral. It has the advantage that it provides a unified treatment for Bourbaki’s integrals (both the usual and the “essential” integral) and for the integral in abstract measure spaces. This chapter is self-contained. We give complete definitions and complete statements of the most important results in the theory. The proofs however are omitted in most cases.