Borel Action Research Articles

Topology and descriptive set theory

This paper consists essentially of the text of a series of four lectures given by the author in the Summer Conference on General Topology and Applications, Amsterdam, August 1994. Instead of attempting to give a general survey of the interrelationships between the two subjects mentioned in the title, which would be an enormous and hopeless task, we chose to illustrate them in a specific context, that of the study of Borel actions of Polish groups and Borel equivalence relations. This is a rapidly growing area of research of much current interest, which has interesting connections not only with topology and set theory (which are emphasized here), but also to ergodic theory, group representations, operator algebras and logic (particularly model theory and recursion theory). There are four parts, corresponding roughly to each one of the lectures. The first contains a brief review of some fundamental facts from descriptive set theory. In the second we discuss Polish groups, and in the third the basic theory of their Borel actions. The last part concentrates on Borel equivalence relations. The exposition is essentially self-contained, but proofs, when included at all, are often given in the barest outline.

Discrete type shock semi-markov decision processes with borel state space

This paper investigates discrete type shock semi-Markov decision processes (SMDP for short) with Borel state and action space. The discrete type shock SMDP describes a system which behaves like a discrete type SMDP, except that the system is subject to random shocks from its environment. Following each shock, an instantaneous state transition occurs and the parameters of the SMDP are changed. After presenting the model, we transform the discrete type shock SMDP into an equivalent discrete time Markov decision process under the condition that one of the assumptions P, N, D, holds. So the most results from discrete time Markov decision processes can be generalized directly to hold for the discrete type shock SMDP.

Countable sections for locally compact group actions. II

In this paper we study the structure of the orbit equivalence relation induced by a Borel action of a second countable locally compact group on a standard Borel space.

Amenable versus hyperfinite Borel equivalence relations

LetXbe a standard Borel space (i.e., a Polish space with the associated Borel structure), and letEbe acountableBorel equivalence relation onX, i.e., a Borel equivalence relationEfor which every equivalence class [x]Eis countable. By a result of Feldman-Moore [FM],Eis induced by the orbits of a Borel action of a countable groupGonX.The structure of general countable Borel equivalence relations is very little understood. However, a lot is known for the particularly important subclass consisting of hyperfinite relations. A countable Borel equivalence relation is calledhyperfiniteif it is induced by a Borel ℤ-action, i.e., by the orbits of a single Borel automorphism. Such relations are studied and classified in [DJK] (see also the references contained therein). It is shown in Ornstein-Weiss [OW] and Connes-Feldman-Weiss [CFW] that for every Borel equivalence relationEinduced by a Borel action of a countable amenable groupGonXand for every (Borel) probability measure μ onX, there is a Borel invariant setY⊆Xwith μ(Y) = 1 such thatE↾Y(= the restriction ofEtoY) is hyperfinite. (Recall that a countable group G isamenableif it carries a finitely additive translation invariant probability measure defined on all its subsets.) Motivated by this result, Weiss [W2] raised the question of whether everyEinduced by a Borel action of a countable amenable group is hyperfinite. Later on Weiss (personal communication) showed that this is true forG= ℤn. However, the problem is still open even for abelianG. Our main purpose here is to provide a weaker affirmative answer for general amenableG(and more—see below). We need a definition first. Given two standard Borel spacesX, Y, auniversally measurableisomorphism betweenXandYis a bijection ƒ:X→Ysuch that both ƒ, ƒ-1are universally measurable. (As usual, a mapg:Z→W, withZandWstandard Borel spaces, is calleduniversally measurableif it is μ-measurable for every probability measure μ onZ.) Notice now that to assert that a countable Borel equivalence relation onXis hyperfinite is trivially equivalent to saying that there is a standard Borel spaceYand a hyperfinite Borel equivalence relationFonY, which isBorelisomorphic toE, i.e., there is a Borel bijection ƒ:X→YwithxEy⇔ ƒ(x)Fƒ(y). We have the following theorem.

Borel actions of Polish groups

We show that a Borel action of a Polish group on a standard Borel space is Borel isomorphic to a continuous action of the group on a Polish space, and we apply this result to three aspects of the theory of Borel actions of Polish groups: universal actions, invariant probability measures, and the Topological Vaught Conjecture. We establish the existence of universal actions for any given Polish group, extending a result of Mackey and Varadarajan for the locally compact case. We prove an analog of Tarski’s theorem on paradoxical decompositions by showing that the existence of an invariant Borel probability measure is equivalent to the nonexistence of paradoxical decompositions with countably many Borel pieces. We show that various natural versions of the Topological Vaught Conjecture are equivalent with each other and, in the case of the group of permutations of N {\mathbb {N}} , with the model-theoretic Vaught Conjecture for infinitary logic; this depends on our identification of the universal action for that group.

Countable sections for locally compact group actions

AbstractIt has been shown by J. Feldman, P. Hahn and C. C. Moore that every non-singular action of a second countable locally compact group has a countable (in fact so-called lacunary) complete measurable section. This is extended here to the purely Borel theoretic category, consisting of a Borel action of such a group on an analytic Borel space (without any measure). Characterizations of when an arbitrary Borel equivalence relation admits a countable complete Borel section are also established.

Recursive adaptive control of Markov decision processes with the average reward criterion

We are concerned with Markov decision processes with Borel state and action spaces; the transition law and the reward function depend on anunknown parameter. In this framework, we study therecursive adaptive nonstationary value iteration policy, which is proved to be optimal under thesame conditions usually imposed to obtain the optimality of other well-knownnonrecursive adaptive policies. The results are illustrated by showing the existence of optimal adaptive policies for a class of additive-noise systems with unknown noise distribution.

Existence of Optimal Strategies in Zero-Sum Nonstationary Stochastic Games with Lack of Information on Both Sides

This paper studies a zero-sum discrete-time stochastic game model with Borel state and action spaces. The law of motion of the system in the model is assumed to be nonstationary. Following M. Schal, at each stage of the game every player is assumed to know the sequence of states occurring up to this stage, but has no explicit information about his opponent’s previous decisions. Under certain semicontinuity and compactness conditions, the existence of a value is proved for such a game and the existence of optimal ($\varepsilon $-optimal) universally measurable strategies for the minimizes (maximizes). This essentially improves a result of Schal on this subject.

On extending actions

Consider a Polish topological group G G acting via J J on a substandard (= countably generated) Borel space. Theorem 1. Any such "Borel action" can be extended to a Borel action J ′ : G × X ′ → X ′ J’:G \times X’ \to X’ where X ′ X’ is coanalytic. (Theorem 3 gives an analogue for continuous actions.) Corollary 2. The result "in any Borel action, orbits are Borel" implies the (well-known) result "all such orbits are absolutely Borel".

An Equivalence Relation that is not Freely Generated

In this paper it is shown that there exists a Borel equivalence relation with countable equivalence classes that is not generated by a free Borel action of a countable discrete group.

An equivalence relation that is not freely generated

In this paper it is shown that there exists a Borel equivalence relation with countable equivalence classes that is not generated by a free Borel action of a countable discrete group.

Measurable group actions are essentially Borel actions

If a locally compact groupG acts on a Lebesgue probability space (X, λ), it is natural to consider these conditions: (a) each group element preserves the class of λ, and (b) the action function is measurable. The latter is a weakening of the requirement that the action be Borel, providedX has a particular Borel structure as well as the σ-algebra of measurable sets. In this paper, we give an example showing that such an action need not be Borel relative to the given Borel structure, and prove that there is always a conull invariant subset and a new standard Borel structure on that subset for which the action is Borel. This is the meaning of the title.

Borel selectors for separated quotients

This work was stimulated by a recent paper of Burgess [1] in which some techniques from Vaught [11] were used to obtain a Borel selector in the case A = Y of Theorem A. The selector in [1] is obtained via an application of Suslin's theorem, so its complexity is very hard to estimate. We will use Vaught's method in a different way to provide a more direct (and simpler) construction in which the complexity of the selector is apparent. Burgess's result extends a theorem of E. Effros [2] on locally compact groups. Theorem A will be proved as a special case of a more general result (3.2) about Borel actions. The existence of a Borel selector for noncontinuous actions appears to be new in all cases. The existence of a Borel selector (of unknown complexity) under the hypothesis of Theorem B is a recent result of S. M. Srivastava [10]. The special case, 7 = 1, was previously established by the present author in [8]. These results extend earlier work by Kallman and Mauldin [3] and Kuratowski and Maitra [5]. Theorem B is a

On the Measurability of Orbits in Borel Actions

We replace measure with category in an argument of G. W. Mackey to characterize closed subgroups H of a totally nonmeager, 2nd countable topological group G in terms of the quotient Borel structure G/H. As a corollary, we obtain an improved version of a theorem of C. Ryll-Nardzewski on the Borel measurability of orbits in continuous actions by Polish groups.

On the measurability of orbits in Borel actions

We replace measure with category in an argument of G. W. Mackey to characterize closed subgroups H of a totally nonmeager, 2nd countable topological group G in terms of the quotient Borel structure G/H. As a corollary, we obtain an improved version of a theorem of C. Ryll-Nardzewski on the Borel measurability of orbits in continuous actions by Polish groups.