James K. Brooks

Strong additivity, absolute continuity and compactness in spaces of measures

In this paper we shall examine strong additivity, absolute continuity, and compactness (weak and strong) in the space of vector measures and discuss various relationships among these concepts. Many of the theorems cast new light on the structure of measures, even in the scalar case. Some of the results presented here have been announced in Brooks and Dinculeanu [S, 91. Variations on the above themes are also treated in detail in Brooks [7]. In Section 1, the main concept of uniform strong additivity of a family of vector measures X is introduced, namely, m(E,)-+O uniformly for m E -X, when (EJ is a disjoint sequence of sets. This notion is inextricably tied up with weak and strong compactness of %” in different topological settings. The existence of a positive control measure p such that X < p, when X is uniformly strongly additive is presented in Section 2. A local control measure is constructed in Section 3 by means of establishing a “synthesis theorem” which allows us to piece together locally equivalent families of positive measures. This theorem is also used to prove the existence of a local control measure for relatively weakly compact sets in the space of vector measures with local finite variation (Section 4). Criteria for weak compactness in this locally convex space (Theorem 4.2) extends the work of Dieudonne [14], who considered a special case, viz, the space of locally integrable functions on a locally compact space. Conditions concerning compactness (weak and strong) with respect to the quasivariation norm are presented in Sections 5

Conditional expectations and weak and strong compactness in spaces of Bochner integrable functions

In [6, theorem IV.8.18], relatively norm compact sets K in Lp([mu]) are characterized by means of strong convergence of conditional expectations, E[pi]f --> f in Lp([mu]), uniformly for f [set membership, variant] K, where (E[pi]) is the family of conditional expectations corresponding to the net of all finite measurable partitions. In this paper we extend the above result in several ways: we consider nets of not necessarily finite partitions; we consider spaces of vector valued pth power Bochner integrable functions (and spaces M([Sigma], E) of vector valued measures with finite variation); we characterize relatively strong compact sets K in by means of uniform strong convergence E[pi]f --> f, as well as relatively weak compact sets Kby means of uniform weak convergence E[pi]f --> f. Previously, in [4], uniform strong convergence (together with some other conditions) was proved to be sufficient (but not necessary) for relative weak compactness.

Representing Yosida-Hewitt decompositions for classical and non-commutative vector measures

Abstract Let m be a bounded, real valued measure on a field of sets. Then, by the Yosida-Hewitt theorem, m has a unique decomposition into the sum of a countably additive and a singular measure. We show here that, in contrast to the classical arguments, this decomposition can be achieved by constructing the countably additive component. From this we obtain a simple formula for the countably additive part of a (strongly bounded) vector measure. We develop these ideas further by considering a weakly compact operator T on a von Neumann algebra M. It turns out that T has a unique decomposition into TN +TS, where TS is singular, TN is completely additive on projections and, for each x in M, there exists an increasing sequence of projections (pn)(n = 1,2…), such that T N ( x ) = lim ⁡ T ( p n xp n ) . When M has a faithful representation on a separable Hilbert space, then we can fix a sequence of projections (pn)(n = 1,2…) such that the above equation holds for every choice of x in M. For general M, there exists an increasing net of projections such that, for every y in M, lim ⁡ F ∥ T N ( y ) − T ( q F yq F ) ∥ = 0.

Operator algebras and a theorem of Dieudonne

LetA be aC*-algebra with second dualA″. Let (φn)(n=1,...) be a sequence in the dual ofA such that limφn(a) exists for eacha εA. In general, this does not imply that limφn(x) exists for eachx εA″. But if limφn(p) exists whenever p is the range projection of a positive self-adjoint element of the unit ball ofA, then it is shown that limφn(x) does exist for eachx inA″. This is a non-commutative generalisation of a celebrated theorem of Dieudonné. A new proof of Dieudonné’s theorem, for positive measures, is given here. The proof of the main result makes use of Dieudonné’s original theorem.

Projections and regularity of abstract processes

In this note we shall prove the existence of optional and predictable projections of stochastic processes X taking values in a Banach space E. Furthermore, if the range of X is contained in a compact set and if X is cadlag (respectively caglad), then the optional (respectively predictable) projection possesses the same property. Finally, we shall prove that every E-valued martingale has a cadlag modification

CHAPTER 10 – Stochastic Processes and Stochastic Integration in Banach Spaces

This chapter examines the stochastic processes and stochastic integration in Banach spaces. The stochastic integral in Banach spaces is developed with the aid of a vector bilinear integral. The chapter provides convergence theorems and has applied them to establish Itos formula, the essential tool used in stochastic calculus. The summable processes in the theory play the role of the square integrable martingales in the classical theory. It turns out that every Hilbert valued square integrable martingale is summable. The advantage and purpose of establishing a Lebesgue space for the bilinear vector integral is the possibility to examine weak completeness and weak compactness. The regularity and the Doob-Meyer decomposition of abstract quasimartingales are elaborated in the chapter. One of the main theorems concerning the existence of cadlag modifications of a quasimartingale is presented in the chapter. It is found that for each stopping time, the stopped process is a quasimartingale on satisfying the regularity condition. The existence of right continuous or cadlag modifications is also examined in the chapter.

The Optional Stochastic Integral

In this paper we shall study the optional (or compensated) stochastic integral CH·X. The two main problems connected with this integral will be considered. First, we wish to express HC·X in terms of an ordinary predictable stochastic integral H’·X, where H’ is a suitable predictable process associated with the optional process H. An attempt in this direction was first undertaken by Yor [8]; however, even for bounded, scalar H, the problem remained open. We shall show in this case that HC·X — H’·X exists as a certain limit in M2, the space of cadlag (Hilbert-valued) square integrable martingales, (cf. §3). Secondly, we shall develop HC·X for processes H and X which take their values in a separable Hilbert space. These integrals, in turn, will allow us in a later paper to develop HC·X for certain nuclear-valued processes. Full details of the proofs of the theorems presented here will appear elsewhere.

CHAPTER 7 – One-Dimensional Diffusions and Their Convergence in Distribution

This chapter examines the one-dimensional diffusions and their convergence in distribution. The normalized Brownian motion is analyzed in the chapter. To obtain one-dimensional Brownian motion, fix an axis and project the motion of the particle on this axis. The Brownian traveler starts afresh with respect to path shifts induced by random times. One of the best ways to appreciate the wilderness that is filled with exotic forms is to delve into the study of the sample path properties of Brownian motion. The universal nature of Brownian motion is revealed in the sense that this process can be used to capture random walks. The scaled random walk derived from fair coin tossing converges weakly to normalized Brownian motion. This not only gives insight into Brownian motion, but it displays the power of weak convergence. Diffusion in natural scale has the same state space features as Brownian motion. The problem of describing the speed of diffusion in natural scale in terms of its speed measure is elaborated in the chapter. The chapter also analyzes the local time for Brownian motion.

H1 and BMO Spaces of Abstract Martingales

Abstract stochastic processes have been considered in various contexts by a number of authors. See, for example, Burkholder [2], Da Pratto [3], Kallianpur and Wolpert [14] and Metivier [15]. In this paper we shall examine the structure of H1 and its dual BMO, for martingales taking their values in a Banach space E, and we use this to characterize weakly compact subsets in the former space. These results extend the theory for these Banach spaces developed by Dellacherie, Meyer, Yor and Mokobodzki [5]. The condition imposed on E is that it have the Radon-Nikodym property (RNP), which is not unexpected since this is a necessary and sufficient condition that the martingale convergence theorem holds in E. The connection between RNP and the geometry of E has been under intense study for over fifteen years in functional analysis. However, even without this assumption, by using the theory of lifting [13], a cepresentation theorem for elements in (H E 1 )’ is (Theorem 3) (notation is given below). More precisely, every element of (H E 1 )’ is of the form \( X \to E\left( {\int {\left\langle {{X_t},d{A_t}} \right\rangle } } \right) \), where the optional process A with integrable variation has E’ as its range.