ON GRAEV’S THEOREM FOR FREE PRODUCTS OF HAUSDORFF TOPOLOGICAL GROUPS

The paper gives a simple proof of Graev’s theorem (asserting that the free product of Hausdorff topological groups is Hausdorff) for a particular case which includes the countable case of $k_\omega $ -groups and the countable case of Lindelöf P-groups. For this it is shown that in these particular cases the topology of the free product of Hausdorff topological groups coincides with the $X_0$ -topology in the Mal’cev sense, where X is the disjoint union of the topological groups identifying their units.

On the support of extremal martingale measures with given marginals: the countable case

We investigate the supports of extremal martingale measures with prespecified marginals in a two-period setting. First, we establish in full generality the equivalence between the extremality of a given measure Q and the denseness in $L^1(Q)$ of a suitable linear subspace, which can be seen in a financial context as the set of all semistatic trading strategies. Moreover, when the supports of both marginals are countable, we focus on the slightly stronger notion of weak exact predictable representation property (WEP) and provide two combinatorial sufficient conditions, called the ‘2-link property’ and ‘full erasability’, on how the points in the supports are linked to each other for granting extremality. When the support of the first marginal is a finite set, we give a necessary and sufficient condition for the WEP to hold in terms of the new concepts of 2-net and deadlock. Finally, we study the relation between cycles and extremality.

UNIFORM PROCEDURES IN UNCOUNTABLE STRUCTURES

This article contributes to the general program of extending techniques and ideas of effective algebra to computable metric space theory. It is well-known that relative computable categoricity (to be defined) of a computable algebraic structure is equivalent to having a c.e. Scott family with finitely many parameters (e.g., [1]). The first main result of the article extends this characterisation to computable Polish metric spaces. The second main result illustrates that just a slight change of the definitions will give us a new notion of categoricity unseen in the countable case (to be stated formally). The second result also shows that the characterisation of computably categorical closed subspaces of ${\Cal R}^n $ contained in [17] cannot be improved. The third main result extends the characterisation to not necessarily separable structures of cardinality κ using κ-computability.

MEASURABLE PERFECT MATCHINGS FOR ACYCLIC LOCALLY COUNTABLE BOREL GRAPHS

We characterize the structural impediments to the existence of Borel perfect matchings for acyclic locally countable Borel graphs admitting a Borel selection of finitely many ends from their connected components. In particular, this yields the existence of Borel matchings for such graphs of degree at least three. As a corollary, it follows that acyclic locally countable Borel graphs of degree at least three generating μ-hyperfinite equivalence relations admit μ-measurable matchings. We establish the analogous result for Baire measurable matchings in the locally finite case, and provide a counterexample in the locally countable case.

An atomic theory with no prime models

We construct an atomic uncountable theory with no prime models. This contrasts with the countable case.

GENERAL ENTROPY OF GENERAL MEASURES

The concept of entropy is an important part of the theory of additive measures. In this paper, a definition of entropy is introduced for general (not necessarily additive) measures as the infinum of the Shannon entropies of "subordinate" additive measures. Several properties of the general entropy are discussed and proved. Some of the properties require that the measure belongs to the class of so-called "equientropic" general measures introduced and studied in this paper. The definition of general entropy is extended to the countable case for which a sufficient condition of convergence is proved. We introduce a method of "conditional combination" of general measures and prove that in that case the general entropy possesses the "subset independence" property.

An application of ultrapowers to changing cofinality

The problem of changing the cofinality of a measurable cardinal to ω with the help of an iterated ultrapower construction has been introduced in [Bu] and more completely studied in [De]. The aim of this paper is to investigate how the construction above has to be changed to obtain an uncountable cofinality for the (previously) measurable cardinal.A forcing approach of this question has been developed by Magidor in [Ma]. Just as in the countable case with Prikry forcing, it turns out that the needed hypothesis and the models constructed are the same in both techniques. However the ultrapowers yield a solution which may appear as more effective. In particular the sequence used to change the measurable cardinal into a cardinal of cofinality α has the property that for any β < α the restriction to β of this sequence can be used to change the cofinality of the (same) measurable cardinal to β.The result we prove is as follows:Theorem. Assume that α is a limit ordinal, that (Uβ)β<α is a sequence of complete ultrafilters on κ > α in the model N0, andfor B included in α let NB be the ultrapower of N0 by those Uβ which are such that β is in B.