Conditional expectations and submartingale sequences of random Schwartz distributions

Let (Ω, Σ, P) be a fixed complete probability space, $\text{D}$ the real Schwartz space, and $\text{D′}$ its strong dual. $\text{D}$ and $\text{D′}$ are partially ordered by $\text{C}$ and $\text{C′}$ respectively, where $\text{C}$ is the positive cone of nonnegative functions in $\text{D}$ and $\text{C′}$ its dual in $\text{D′}$. $\text{C}$ is a strict $\text{B}$-cone and $\text{C′}$ is normal, where $\text{B}$ is the family of all bounded subsets of $\text{D}$. If X, Y are two random Schwartz distributions, then X ≤ Y if and only if Y(ω) ? X(ω) ∈ $\text{D′}$ for almost all $\text{ω ∈ Ω(P)}$. Integrability of random Schwartz distributions and properties of such integrals are discussed. The monotone convergence theorem, the dominated convergence theorem, and Fatou's lemma are proved. The existence of conditional expectations of integrable random Schwartz distributions relative to a given sub σ-field of Σ is shown. Properties of conditional expectations are discussed and the conditional form of the monotone convergence theorem is proved. Sub(super)-martingale sequences are defined via the partial order relations introduced above, and a convergence theorem is given. The notion of a potential is introduced and the Riesz decomposition theorem is proved.     

The evaluation identification in function spaces

A study is made of the natural function which maps each point x of a space X to the evaluation function e_x:Y^x→Y defined by $\text{e}{}_{\mathrm{x}}\text{(?)=?(x)}$. A consequence of the results is that βX and υX can both be considered as subspaces of spaces of continous functions from appropriate domain spaces into I or R, respectively.     

On nearly pseudocompact spaces

A completely regular space X is called nearly pseudocompact if υX?X is dense in βX?X, where βX is the Stone-?ech compactification of X and υX is its Hewitt realcompactification. After characterizing nearly pseudocompact spaces in a variety of ways, we show that X is nearly pseudocompact if it has a dense locally compact pseudocompact subspace, or if no point of X has a closed realcompact neighborhood. Moreover, every nearly pseudocompact space X is the union of two regular closed subsets X₁, X₂ such that Int X₁ is locally compact, no points of X₂ has a closed realcompact neighborhood, and $\text{Int(X}{}_1\text{?X}{}_2\text{)=?}$. It follows that a product of two nearly pseudocompact spaces, one of which is locally compact, is also nearly pseudocompact.     

Step functions from the half-open unit interval into a topological space

The set of continuous-from-the-right step functions from the half-open unit interval[0, 1[into a topological space X is denoted by X¹. Elsewhere a topology has been defined which makes X¹ a contractible, locally contractible space with the subspace of constant functions being homeomorphic to X. When X has a bounded metric ?, the topology of X¹ may be described by the metric $\text{d>(f,g)=}\text{∫}\text{0}\text{1}\text{ρ(f(t),g(t))dt}$.It is shown here that if X is separable, then X¹ is separable and if X satisfies the first (or second) axiom of countability, then X¹ satisfies it too. In contrast, it is shown that properties such as normality do not extend from X to X¹. This follows from the main result: X¹ is homeomorphic to its square, and thus contains a copy of X×X (which is closed when X is Hausdorff). The final theorem states that if X has at least two points then X¹ is not complete metrizable.     

On the square root of a positive B(X,X1)-valued function

Let (Ω, $\text{B}$, μ) be a measure space, $\text{X}$ a separable Banach space, and $\text{X}$¹ the space of all bounded conjugate linear functionals on $\text{X}$. Let f be a weak¹ summable positive B($\text{X, X}$¹)-valued function defined on Ω. The existence of a separable Hilbert space $\text{K}$, a weakly measurable B($\text{X, K}$)-valued function Q satisfying the relation $\text{Q}{}^1\text{(ω)Q(ω) = f(ω)}$ is proved. This result is used to define the Hilbert space L_2,f of square integrable operator-valued functions with respect to f. It is shown that for B⁺($\text{X, X}$¹)-valued measures, the concepts of weak¹, weak, and strong countable additivity are all the same. Connections with stochastic processes are explained.     

A class of extension properties that are not simply generated

Let $\text{P}$ be a closed-hereditary topological property preserved by products. Call a space $\text{P}$-regular if it is homeomorphic to a subspace of a product of spaces with $\text{P}$. Suppose that each $\text{P}$-regular space possesses a $\text{P}$-regular compactification. It is well-known that each $\text{P}$-regular space X is densely embedded in a unique space γ_scPX with $\text{P}$ such that if f: X → Y is continuous and Y has $\text{P}$, then f extends continuously to γ_scPX. Call $\text{P}$-pseudocompact if γ_scPX is compact.Associated with $\text{P}$ is another topological property $\text{P}$^#, possessing all the properties hypothesized for $\text{P}$ above, defined as follows: a $\text{P}$-regular space X has $\text{P}$^# if each $\text{P}$-pseudocompact closed subspace of X is compact. It is known that the $\text{P}$-pseudocompact spaces coincide with the $\text{P}$^#-pseudocompact spaces, and that $\text{P}$^# is the largest closed-hereditary, productive property for which this is the case. In this paper we prove that if $\text{P}$ is not the property of being compact and $\text{P}$-regular, then $\text{P}$^# is not simply generated; in other words, there does not exist a space E such that the spaces with $\text{P}$^# are precisely those spaces homeomorphic to closed subspaces of powers of E.     

A note on realizing homomorphisms of category algebras

The complete Boolean homomorphisms from the category algebra $\text{C}$(X) of a complete matrix space X to the category algebra $\text{C}$(Y) of a Baire topological space Y are characterized as those σ-homomorphisms which are induced by continuous maps from dense G₈-subsets of Y into X. This result is used to deduce a series of related results in topology and measure theory (some of which are well-known). Finally a similar result for the complete Boolean homomorphisms from the category algebra $\text{C}$(X) of a compact Hausdorff space X tothe category algebra $\text{C}$(Y) of a Baire topological space Y is proved.     

Spaces N ∪ R and their dimensions

About spaces N∪$\text{R}$ (see [2, Exercise 5I]), the following are proved: (1) dim $\text{N∪}\text{R}\text{= dim β(N∪}\text{R}\text{)?N∪}\text{R}\text{,(2)if|β(N∪}\text{R}\text{)?N∪}\text{R}\text{|<2}{}^{\mathrm{?}}{}^{\mathrm{o}}$, then no real-valued continuous fu ction on N∪$\text{R}$ is onto (and hence, dim $\text{N∪}\text{R}\text{=0}$), (3) any compact metric space without isolated points is homeomorphic to some $\text{β(N∪}\text{R}\text{)?N∪}\text{R}$ and (4)there are spaces X,X₁ and X₂ of the form N∪$\text{R}$ such that X=X₁∪X₂,X₂andX₂ are zero sets of X, and dim X=n, dimX₁=dimX₂=0, where n=1,2,… or ∞.     

Quasimartingales on partially ordered sets

The concept of a quasimartingale, and therefore also of a function of bounded variation, is extended to processes with a regular partially ordered index set V and with values in a Banach space. We show that quasimartingales can be described by their associated measures, defined on an inverse limit space $\text{S × Ω}$ containing $\text{V × Ω}$, furnished with the σ-algebra $\text{P}$ of the predictable sets. With the help of this measure, a Rao-Krickeberg and a Riesz decomposition is obtained, as well as a convergence theorem for quasimartingales. For a regular quasimartingale X it is proven that the spaces ($\text{S × Ω}$, $\text{P}$) and the measures associated with X are unique up to isomorphisms. In the case V = $\text{R}$₊ⁿ we prove a duality between classical (right-) quasimartingales and left-quasimartingales.     

On nagata's problem for paracompact σ-metric spaces

Let (A) be the characterization of dimension as follows: Ind X?n if and only if X has a σ-closure-preserving base $\text{W}$ such that Ind B(W)?n?1 for every W?$\text{W}$. The validity of (A) is proved for spaces X such that(i) X is a paracompact σ-metric space with a scale {X_i} such that each X_i has a uniformly approaching anti-cover, or(ii) X is a subspace of the product ΠX_i of countably many L-spaces X_i, the notion of which is due to K. Nagami.(i) and (ii) are the partial answers to Nagata's problem wheter (A) holds or not for every M₁-space X.     

Extensions and new observations of Tychonoff,Stone-Weierstrass theorems,compactifications and the realcompactification

An extension of the Tychonoff theorem is obtained in characterizing a compact space by the nets and the images induced by any family of continuous functions on it. The idea of this extension is applied to get a new process and new observations of compactifications and the realcompactification. Finally, a sufficient and necessary condition of a vector sublattice or a subalgebra of C¹(X) to be dense in (C¹(X),∥·∥) is provided in terms of the nets in X induced by C¹(X), where C¹(X) is the space of all bounded real continuous functions on a topological space X with pointwise ordering, and ∥·∥ is the supremum norm.     

On the value of a stopped set function process

For certain types of stochastic processes {X_n | n ∈ $\text{N}$}, which are integrable and adapted to a nondecreasing sequence of σ-algebras $\text{F}$_n on a probability space (Ω, $\text{F}$, P), several authors have studied the following problems: IfSdenotes the class of all stopping times for the stochastic basis {$\text{F}$_n | n ∈ $\text{N}$}, when is$\text{sup}{}_{\mathrm{s}}\text{∫}{}_{\mathrm{\Omega }}\text{| X}{}_{\mathrm{\sigma }}\text{| dP}$finite, and when is there a stopping time $\text{σ}$ for which this supremum is attained? In the present paper we set the problem in a measure theoretic framework. This approach turns out to be fruitful since it reveals the root of the problem: It avoids the use of such notions as probability, null set, integral, and even σ-additivity. It thus allows a considerable generalization of known results, simplifies proofs, and opens the door to further research.     

Properties of Hida processes on R2. 2. Prediction and interpolation problems for processes on R2

The properties of N-Hida processes Part 1 (B. Prum, 1984, J. Multivar. Anal.15, 336–360) are studied when the indices set is $\text{R}$². First, the past of a point (s, t) of $\text{R}$² is extended to $\text{G}$_st = σ{γ_uv, u ≤ s or v ≤ t}. The dimension of the linear space generated by the conditional expectations of an N-Hida process γ_z when z goes over a p × q lattice is bounded by N(p + q ? 1). The same problem is then considered when the expectations are taken conditionally to the field generated by the process outside of a rectangle, and the bound of the dimension of the linear space generated on a lattice is also given. Special attention is devoted to the case when γ_z is a combination of strong martingales.     

An extension theorem for D-normal spaces

A second countable developable T₁-space $\text{D}$₁ is defined which has the following properties: (1) $\text{D}$₁ is an absolute extensor for the class of perfect spaces. (2) $\text{D}$₁^?₀ is a universal space for second countable developable T₁-spaces.     

The orderability of nonarchimedean spaces

Let X be a nonarchimedean space and C be the union of all compact open subsets of X. The following conditions are listed in increasing order of generality. (Conditions 2 and 3 are equivalent.) 1. X is perfect; 2. C is an F_σ in X; 3. C? is metrizable; 4. X is orderable. It is also shown that X is orderable if $\text{C}\text{?}\text{?C}$ is scattered or X is a GO space with countably many pseudogaps. An example is given of a non-orderable, totally disconnected, GO space with just one pseudogap.     

On D-Paracompact spaces

Following Pareek a topological space X is called D-paracompact if for every open cover $\text{A}$ of X there exists a continuous mapping f from X onto a developable T₁-space Y and an open cover $\text{B}$ of Y such that { f^-1[B]|B ∈ $\text{B}$ } refines $\text{A}$. It is shown that a space is D-paracompact if and only if it is subparacompact and D-expandable. Moreover, it is proved that D-paracompactness coincides with a covering property, called dissectability, which was introduced by the author in order to obtain a base characterization of developable spaces.     

Convergence of Riesz space martingales

We prove a martingale convergence for sub and super martingales on Riesz spaces. As a consequence we can form Krickeberg and Riesz like decompositions. The minimality of the Krickeberg decomposition yields a natural ordered lattice structure on the space of convergent martingales making this space into a Dedekind complete Riesz space. Finally we show that the Riesz space of convergent martingales is Riesz isomorphic to the order closure of the union of the ranges of the conditional expectations in the filtration. Consequently we can characterize the space of order convergent martingales both in Riesz spaces and in the setting of probability spaces.     

On projections and limit mappings of inverse systems of compact spaces

A compactificaton αX of a completely regular space X is “determined” by a subset F of C¹(X) if αX is the smallest compactificaton of X to which each element of F extends, and is “generated” by F if the evaluation map $\text{e}{}_{\mathrm{F}}\text{:X →}\text{R}{}^{\mathrm{n}}\text{,n = |F|}$, is an embedding and $\text{αX =}\text{e}{}_{\mathrm{F}}\text{(X)}$. Evidently, if F either determines or generates αX, then every elements of F has an extension to αX; whenever F satisfies this latter condition, the set of all such extensions is denoted F^α.A major results of our previous paper is that F determines αX if and only if F^α separates points of αX ? X. A major result of the present paper is that F generates αX if and only if F^α separates points of αX.     

The construction of all locally compact extensions

The paper presents one of the ways to construct all the locally compact extensions of a given Tychonoff space T. First, there proved the “local” variant of the Stone-C?ech theorem on “completely regular” Riesz spaces X(T) of continuous bounded functions on T with no unit function, in general, but with a collection of local units. In Theorem 1 it is proved that all the functions from X(T) can be “completely regularly” extended on the largest locally compact extension β_xT. Theorem 3 states, that β_xT are presenting, in fact, all the locally compact extensions of T.