ON THE WEAK-HASH METRIC FOR BOUNDEDLY FINITE INTEGER-VALUED MEASURES

It is known that the space of boundedly finite integer-valued measures on a complete separable metric space becomes a complete separable metric space when endowed with the weak-hash metric. It is also known that convergence under this topology can be characterised in a way that is similar to the weak convergence of totally finite measures. However, the original proofs of these two fundamental results assume that a certain term is monotonic, which is not the case as we show by a counterexample. We clarify these original proofs by addressing the parts that rely on this assumption and finding alternative arguments.

On computability and disintegration

We show that the disintegration operator on a complete separable metric space along a projection map, restricted to measures for which there is a unique continuous disintegration, is strongly Weihrauch equivalent to the limit operator Lim. When a measure does not have a unique continuous disintegration, we may still obtain a disintegration when some basis of continuity sets has the Vitali covering property with respect to the measure; the disintegration, however, may depend on the choice of sets. We show that, when the basis is computable, the resulting disintegration is strongly Weihrauch reducible to Lim, and further exhibit a single distribution realizing this upper bound.

On limit theorems for random fields

A complete separable metric space of functions defined on the positive quadrant of the plane is constructed. The characteristic property of these functions is that at every point x there exist two lines intersecting at this point such that limits limy→x f (y) exist when y approaches x along any path not intersecting these lines. A criterion of compactness of subsets of this space is obtained.

REVERSE MATHEMATICS OF MF SPACES

This paper gives a formalization of general topology in second-order arithmetic using countably based MF spaces. This formalization is used to study the reverse mathematics of general topology. For each poset P we let MF (P) denote the set of maximal filters on P endowed with the topology generated by {Np | p ∈ P}, where Np = {F ∈ MF (P) | p ∈ F}. We define a countably based MF space to be a space of the form MF (P) for some countable poset P. The class of countably based MF spaces includes all complete separable metric spaces as well as many nonmetrizable spaces. The following reverse mathematics results are obtained. The proposition that every nonempty Gδ subset of a countably based MF space is homeomorphic to a countably based MF space is equivalent to [Formula: see text] over ACA0. The proposition that every uncountable closed subset of a countably based MF space contains a perfect set is equivalent over [Formula: see text] to the proposition that [Formula: see text] is countable for all A ⊆ ℕ. The proposition that every regular countably based MF space is homeomorphic to a complete separable metric space is equivalent to [Formula: see text] over [Formula: see text].

Reverse Mathematics and Π12 Comprehension

We initiate the reverse mathematics of general topology. We show that a certain metrization theorem is equivalent to Π12 comprehension. An MF space is defined to be a topological space of the form MF(P) with the topology generated by {Np ∣ p ϵ P}. Here P is a poset, MF(P) is the set of maximal filters on P, and Np = {F ϵ MF(P) ∣ p ϵ F }. If the poset P is countable, the space MF(P) is said to be countably based. The class of countably based MF spaces can be defined and discussed within the subsystem ACA0 of second order arithmetic. One can prove within ACA0 that every complete separable metric space is homeomorphic to a countably based MF space which is regular. We show that the converse statement, “every countably based MF space which is regular is homeomorphic to a complete separable metric space,” is equivalent to . The equivalence is proved in the weaker system . This is the first example of a theorem of core mathematics which is provable in second order arithmetic and implies Π12 comprehension.

On constructing completions

The Dedekind cuts in an ordered set form a set in the sense of constructive Zermelo–Fraenkel set theory. We deduce this statement from the principle of refinement, which we distill before from the axiom of fullness. Together with exponentiation, refinement is equivalent to fullness. None of the defining properties of an ordering is needed, and only refinement for two–element coverings is used.In particular, the Dedekind reals form a set: whence we have also refined an earlier result by Aczel and Rathjen, who invoked the full form of fullness. To further generalise this, we look at Richman's method to complete an arbitrary metric space without sequences, which he designed to avoid countable choice. The completion of a separable metric space turns out to be a set even if the original space is a proper class: in particular, every complete separable metric space automatically is a set.

Attractors of iterated function systems and Markov operators

This paper contains a review of results concerning “generalized” attractors for a large class of iterated function systems{wi:i∈I}acting on a complete separable metric space. This generalization, which originates in the Banach contraction principle, allows us to consider a new class of sets, which we call semi-attractors (or semifractals). These sets have many interesting properties. Moreover, we give some fixed-point results for Markov operators acting on the space of Borel measures, and we show some relations between semi-attractors and supports of invariant measures for such Markov operators. Finally, we also show some relations between multifunctions and transition functions appearing in the theory of Markov operators.

Asymptotic values of fine continuous functions

The set of asymptotic values of a continuous function on the open unit disc in ℝ2 forms an analytic set, in the sense of being a continuous image of a Polish space (complete, separable metric space). This was proved in [9] by J. E. McMillan, who had earlier given versions of this result for holomorphic and meromorphic functions. We extend his method to the case of a function on the open unit ball of ℝn which is continuous merely in the fine topology, the coarsest topology making all subharmonic functions continuous. In particular, we use a version of McMillan's ingenious metric on a certain space of equivalence classes of asymptotic paths. McMillan also proved in [9] that the set of point asymptotic values of a continuous function in the unit disc forms an analytic set. We use a modification of the McMillan metric to extend this result to fine continuous functions in the unit ball and deduce that the set of boundary points of the unit ball at which the function has an asymptotic value forms an analytic set.

Bisimulation for Labelled Markov Processes

In this paper we introduce a new class of labelled transition systems<br />- Labelled Markov Processes - and define bisimulation for them.<br />Labelled Markov processes are probabilistic labelled transition systems<br />where the state space is not necessarily discrete, it could be the<br />reals, for example. We assume that it is a Polish space (the underlying<br />topological space for a complete separable metric space). The mathematical<br /> theory of such systems is completely new from the point of<br />view of the extant literature on probabilistic process algebra; of course,<br />it uses classical ideas from measure theory and Markov process theory.<br />The notion of bisimulation builds on the ideas of Larsen and Skou and<br />of Joyal, Nielsen and Winskel. The main result that we prove is that<br />a notion of bisimulation for Markov processes on Polish spaces, which<br />extends the Larsen-Skou denition for discrete systems, is indeed an<br />equivalence relation. This turns out to be a rather hard mathematical<br />result which, as far as we know, embodies a new result in pure probability<br />theory. This work heavily uses continuous mathematics which<br />is becoming an important part of work on hybrid systems.