Borel Spaces Research Articles

Borel structures on the set of Borel mappings

Let Y and Z be two Borel spaces. By B(Y,Z) we denote the set of all Borel maps of Y into Z. In Aumann (1961) [2] and Rao (1971) [10] the authors tried to generalize the results of R. Arens and J. Dugundji (see Arens and Dugundji (1951) [1]) for Borel spaces. Unfortunately as R.J. Aumann observed in Aumann (1961) [2], the results of Arens and Dugundji (1951) [1] are not true for Borel spaces, since for some of the simplest Borel spaces for example it is impossible to defined a Borel structure on the set B(Y,Z) such that the map e:B(Y,Z)×Y→Z with e(f,y)=f(y) for every f∈B(Y,Z) and y∈Y is Borel. Even if we consider the discrete structure on B(Y,Z), then e will not in general be Borel. It is for this reason that in Aumann (1961) [2] and Rao (1971) [10] the authors studied subsets F of B(Y,Z) and Borel structures on F such that the restriction of the map e on F×Y is Borel.In this paper we study the above problem and try to generalize the results presented in Arens and Dugundji (1951) [1] for Borel spaces. More precisely in Section 1 the paper preliminaries are given. In Sections 2 and 3 we give and study Borel A-splitting and A-admissible structures on B(Y,Z), where A is an arbitrary family of Borel spaces, and prove that there exists at most one Borel structure on B(Y,Z) which is both Borel splitting and admissible. When this structure exists, it coincides with the greatest Borel splitting structure, which always exists. We also present and study some special Borel structures on B(Y,Z). In Section 4 some remarks for Borel structures on B(Y,Z) are stated. In Section 5 we define and study some relations between the Borel structures of the set B(Y,Z) and the Borel structures of the set BZ(Y) consisting of all subsets f−1(B) of Y, where f∈B(Y,Z) and B is an element of the Borel structure of Z, concerning the notions of Borel A-splitting and Borel A-admissible Borel structures. Finally, some open questions for Borel structures on the set of Borel mappings are posed.

Asymptotic Normality of Discrete-Time Markov Control Processes

In this paper we study the asymptotic normality of discrete-time Markov control processes in Borel spaces, with possibly unbounded cost. Under suitable hypotheses, we show that the cost sequence is asymptotically normal. As a special case, we obtain a central limit theorem for (noncontrolled) Markov chains.

Asymptotic Normality of Discrete-Time Markov Control Processes

In this paper we study the asymptotic normality of discrete-time Markov control processes in Borel spaces, with possibly unbounded cost. Under suitable hypotheses, we show that the cost sequence is asymptotically normal. As a special case, we obtain a central limit theorem for (noncontrolled) Markov chains.

Markov control processes with pathwise constraints

This paper deals with discrete-time Markov control processes in Borel spaces, with unbounded rewards. The criterion to be optimized is a long-run sample-path (or pathwise) average reward subject to constraints on a long-run pathwise average cost. To study this pathwise problem, we give conditions for the existence of optimal policies for the problem with “expected” constraints. Moreover, we show that the expected case can be solved by means of a parametric family of optimality equations. These results are then extended to the problem with pathwise constraints.

Stochastic optimal control for small noise intensities

This paper deals with Markov Decision Processes (MDPs) on Borel spaces with an infinite horizon and a discounted total cost. It will be considered a stochastic optimal control problem which arises ...

The Discounted Method and Equivalence of Average Criteria for Risk-Sensitive Markov Decision Processes on Borel Spaces

This note concerns discrete-time controlled Markov chains with Borel state and action spaces. Given a nonnegative cost function, the performance of a control policy is measured by the superior limit risk-sensitive average criterion associated with a constant and positive risk sensitivity coefficient. Within such a framework, the discounted approach is used (a) to establish the existence of solutions for the corresponding optimality inequality, and (b) to show that, under mild conditions on the cost function, the optimal value functions corresponding to the superior and inferior limit average criteria coincide on a certain subset of the state space. The approach of the paper relies on standard dynamic programming ideas and on a simple analytical derivation of a Tauberian relation.

A note on standard borel and related spaces

The following is a fundamental construction in the theory of point processes: For a measurable space (X, ɛ) let X◃ denote the set of all measures on (X, ɛ) taking only values in the set ℕ (and so each p ∈ X◃ is a finite measure, since p(X) ∈ ℕ); put ɛ◃ = σ(ɛ◊), where ɛ◊ is the set of all subsets of X◃ having the form {p ∈ X◃: p(E) = k} with E ∈ ɛ and k ∈ ℕ.

A duality involving Borel spaces

The purpose of the paper is to show that there is a dual equivalence between sober Borel spaces and spatial Boolean σ-frames. We show that a discrete space is sober, if and only if, its cardinality is non-measurable and also show that many well known Borel spaces are sober.

When do cylinder [formula omitted]-algebras equal Borel [formula omitted]-algebras in Polish spaces?

We give a sufficient and necessary condition to test whether a cylinder σ -algebra equals the Borel σ -algebra in a Polish space. We apply the result to several examples to illustrate the unified approach. A generalization to Borel spaces is also noted.

Extreme points of the convex set of joint probability distributions with fixed marginals

By using a quantum probabilistic approach we obtain a description of the extreme points of the convex set of all joint probability distributions on the product of two standard Borel spaces with fixed marginal distributions.

An Infinite-Dimensional Linear Programming Algorithm for Deterministic Semi-Markov Decision Processes on Borel Spaces

We devise an algorithm for solving the infinite-dimensional linear programs that arise from general deterministic semi-Markov decision processes on Borel spaces. The algorithm constructs a sequence of approximate primal-dual solutions that converge to an optimal one. The innovative idea is to approximate the dual solution with continuous piecewise linear ridge functions that naturally represent functions defined on a high-dimensional domain as linear combinations of functions defined on only a single dimension. This approximation gives rise to a primal/dual pair of semi-infinite programs, for which we show strong duality. In addition, we prove various properties of the underlying ridge functions.

Markov Decision Processes with Variance Minimization: A New Condition and Approach

This article deals with the limiting average variance criterion for discrete-time Markov decision processes in Borel spaces. The costs may have neither upper nor lower bounds. We propose another set of conditions under which we prove the existence of a variance minimal policy in the class of average expected cost optimal stationary policies. Our conditions are weaker than those in the previous literature. Moreover, some sufficient conditions for the existence of a variance minimal policy are imposed on the primitive data of the model. In particular, the stochastic monotonicity condition in this paper has been first used to study the limiting average variance criterion. Also, the optimality inequality approach provided here is different from the “optimality equation approach” widely used in the previous literature. Finally, we use a controlled queueing system to illustrate our results.

Markov control processes with randomized discounted cost

In this paper we consider Markov Decision Processes with discounted cost and a random rate in Borel spaces. We establish the dynamic programming algorithm in finite and infinity horizon cases. We provide conditions for the existence of measurable selectors. And we show an example of consumption-investment problem.

Average optimality for Markov decision processes in borel spaces: a new condition and approach

In this paper we study discrete-time Markov decision processes with Borel state and action spaces. The criterion is to minimize average expected costs, and the costs may have neither upper nor lower bounds. We first provide two average optimality inequalities of opposing directions and give conditions for the existence of solutions to them. Then, using the two inequalities, we ensure the existence of an average optimal (deterministic) stationary policy under additional continuity-compactness assumptions. Our conditions are slightly weaker than those in the previous literature. Also, some new sufficient conditions for the existence of an average optimal stationary policy are imposed on the primitive data of the model. Moreover, our approach is slightly different from the well-known ‘optimality inequality approach’ widely used in Markov decision processes. Finally, we illustrate our results in two examples.

Average optimality for Markov decision processes in borel spaces: a new condition and approach

In this paper we study discrete-time Markov decision processes with Borel state and action spaces. The criterion is to minimize average expected costs, and the costs may have neither upper nor lower bounds. We first provide two average optimality inequalities of opposing directions and give conditions for the existence of solutions to them. Then, using the two inequalities, we ensure the existence of an average optimal (deterministic) stationary policy under additional continuity-compactness assumptions. Our conditions are slightly weaker than those in the previous literature. Also, some new sufficient conditions for the existence of an average optimal stationary policy are imposed on the primitive data of the model. Moreover, our approach is slightly different from the well-known ‘optimality inequality approach’ widely used in Markov decision processes. Finally, we illustrate our results in two examples.

Another set of conditions for Markov decision processes with average sample-path costs

This paper deals with discrete-time Markov decision processes with average sample-path costs (ASPC) in Borel spaces. The costs may have neither upper nor lower bounds. We propose new conditions for the existence of ε-ASPC-optimal (deterministic) stationary policies in the class of all randomized history-dependent policies. Our conditions are weaker than those in the previous literature. Moreover, some sufficient conditions for the existence of ASPC optimal stationary policies are imposed on the primitive data of the model. In particular, the stochastic monotonicity condition in this paper has first been used to study the ASPC criterion. Also, the approach provided here is slightly different from the “optimality equation approach” widely used in the previous literature. On the other hand, under mild assumptions we show that average expected cost optimality and ASPC-optimality are equivalent. Finally, we use a controlled queueing system to illustrate our results.

Existence of Optimal Policies for Semi-Markov Decision Processes Using Duality for Infinite Linear Programming

Semi-Markov decision processes on Borel spaces with deterministic kernels have many practical applications, particularly in inventory theory. Most of the results from general semi-Markov decision processes do not carry over to a deterministic kernel since such a kernel does not provide "smoothness." We develop infinite dimensional linear programming theory for a general stochastic semi-Markov decision process. We give conditions, general enough to allow deterministic kernels, for solvability and strong duality of the resulting linear programs. By using the developed linear programming theory we give conditions for the existence of a stationary deterministic policy for deterministic kernels, which is optimal among all possible policies.

Extreme Points of Sets of Randomized Strategies in Constrained Optimization and Control Problems

This paper concerns the existence and characterization of optimal randomized strategies for some constrained optimization and control problems. We first present a characterization of the extreme points of a set of randomized strategies that satisfy n moment-like constraints. Conditions are given under which those extreme points are randomizations of at most n+1 deterministic strategies. This result is then applied to obtain the existence and characterization of optimal strategies for a class of deterministic, allocation-like, optimization problems and their Young relaxations. Similar results are obtained for constrained Markov control processes in Borel spaces.

The Scalarization Approach to Multiobjective Markov Control Problems: Why Does It Work?

This paper concerns discrete-time multiobjective Markov control processes on Borel spaces and unbounded costs. Under mild assumptions, it is shown that the usual “scalarization approach” to obtain Pareto policies for the multiobjective control problem is in fact equivalent to solving the dual of a certain multiobjective infinite-dimensional linear program. The latter program is obtained from a multiobjective measure problem which is also used to prove the existence of strong Pareto policies, that is, Pareto policies whose cost vector is the closest to the control problem’s virtual minimum.

Gabriel Debs and Jean Saint Raymond. Compact covering mappings and cofinal families of compact subsets of a Borel set. Fundamenta Mathematicae, vol. 167, no. 3 (2001), pp. 213–249. - Gabriel Debs and Jean Saint Raymond. Compact covering mappings between Borel spaces. Acta Universitatis Carolinae. Mathematica et Physica, vol. 40, no. 2 (1999), pp. 53–64. - Gabriel Debs and Jean Saint

Gabriel Debs and Jean Saint Raymond. Compact covering mappings and cofinal families of compact subsets of a Borel set. Fundamenta Mathematicae, vol. 167, no. 3 (2001), pp. 213–249. - Gabriel Debs and Jean Saint Raymond. Compact covering mappings between Borel spaces. Acta Universitatis Carolinae. Mathematica et Physica, vol. 40, no. 2 (1999), pp. 53–64. - Gabriel Debs and Jean Saint Raymond. Cofinal and subsets of ωω . Fundamenta Mathematicae, vol. 159, no. 2 (1999), pp. 161–193. - Gabriel Debs and Jean Saint Raymond. Compact-covering-properties of finite-to-one mappings. Topology and its Applications, vol. 81, no. 1 (1997), pp. 55–84. - Gabriel Debs and Jean Saint Raymond. Some applications of game determinacy. Acta Universitatis Carolinae. Mathematica et Physica, vol. 37, no. 2 (1996), pp. 7–23. - Gabriel Debs and Jean Saint Raymond. Compact covering and game determinacy. Topology and its Applications, vol. 68, no. 2 (1996), pp. 153–185. - Gabriel Debs and Jean Saint Raymond. Compact covering mappings between Borel sets and the size of constructible reals. Transactions of the American Mathematical Society, vol. 356, no. 1 (2004), pp. 73–117. - Volume 10 Issue 3