Semicontinuous limits of nets of continuous functions

In this paper we present a topology on the space of real-valued functions defined on a functionally Hausdorff space $X$ that is finer than the topology of pointwise convergence and for which (1) the closure of the set of continuous functions $\mathcal{C }(X)$ is the set of upper semicontinuous functions on $X$ , and (2) the pointwise convergence of a net in $\mathcal{C }(X)$ to an upper semicontinuous limit automatically ensures convergence in this finer topology.     

Approximation from the exterior of multifunctions with connected values

We approximate an upper semicontinuous multifunctionF(·) from a metric spaceT into the compact, connected subsets of the Euclidean spaceR ^P by means of a decreasing sequence of multifunctions which are locally Lipschitzian with respect to the Hausdorff distance.Work partially supported by GNAFA-CNR and partially by MURST.     

A representation of random upper semicontinuous functions

In this paper a representation of random upper semicontinuous functions in terms of -valued random elements is stated. This fact allows us to consider for the first time a complete and separable metric, the Skorohod one, on a wide class of upper semicontinuous functions. Finally, different relevant concepts of measurability for random upper semicontinuous functions are studied and the relationships between them are analyzed.

     

Compact supported endographs and fuzzy sets

A metric is defined on a space of functions from a locally compact metric space X into the unit interval I in terms of the Hausdorff metric distance between their compact supported endographs in X × I. Convergence in this metric is shown to be equivalent to the conjunction of the Hausdorff metric convergence of supports in X and two conditions involving numerical values of the functions. The space of nonempty compact subsets of X with the Hausdorff metric is imbedded in the above function space by the characteristic function on subsets of X. Applications of these results to fuzzy subsets of X and fuzzy dynamical systems on X are indicated.     

Some new results on stability of Ekeland's ?-variational principle

In this paper, we give an example and point out that ?-solutions of Ekeland's variational principle are not always lower semicontinuous in infinite-dimensional Banach spaces, even with respect to the uniform metric. Further, the example shows that the ?-solutions need not be almost lower semicontinuous when the convergence of sequence of functions is weakened to Painlevé-Kuratowski epigraphical convergence. To provide some results of stability, we prove the almost lower semicontinuity of ?-solutions in a general framework.     

Generalized Dini theorems for nets of functions on arbitrary sets

We characterize the uniform convergence of pointwise monotonic nets of bounded real functions defined on arbitrary sets, without any particular structure. The resulting condition trivially holds for the classical Dini theorem. Our vector-valued Dini-type theorem characterizes the uniform convergence of pointwise monotonic nets of functions with relatively compact range in Hausdorff topological ordered vector spaces. As a consequence, for such nets of continuous functions on a compact space, we get the equivalence between the pointwise and the uniform convergence. When the codomain is locally convex, we also get the equivalence between the uniform convergence and the weak-pointwise convergence; this also merges the Dini-Weston theorem on the convergence of monotonic nets from Hausdorff locally convex ordered spaces. Most of our results are free of any structural requirements on the common domain and put compactness in the right place: the range of the functions.     

COHOMOLOGICAL DIMENSION OF UNIFORM SPACES

Abstract

In this paper we obtain classification and extension theorems for uniform spaces, using the ?ech cohomology theory based on the finite uniform coverings, and study the associated cohomological dimension theory. In particular, we extend results for the cohomological dimension theory on compact Hausdorff spaces or compact metric spaces to those for our cohomological dimension theory on uniform spaces.     

The weak convergence of uniform measures

This paper presents a necessary and sufficient condition for the weak convergence of uniform measures on an arbitrary Hausdorff uniform space in terms of their projections in metric spaces. This result was inspired by and extends a result of Bartoszynski which characterizes the weak convergence of countably additive measures on C[0,1] in terms of their projections in finite dimensional spaces.     

Random capacities and their distributions

Summary We formalize the notion of an increasing and outer continuous random process, indexed by a class of compact sets, that maps the empty set on zero. Existence and convergence theorems for distributions of such processes are proved. These results generalize or are similar to those known in the special cases of random measures, random (closed) sets and random (upper) semicontinuous functions. For the latter processes infinite divisibility under the maximum is introduced and characterized. Our result generalizes known characterizations of infinite divisibility for random sets and max-infinite divisibility for random vectors. Also discussed is the convergence in distribution of the row-vise maxima of a null-array of random semicontinuous functions.Research supported by the Swedish Natural Science Research Council     

Laws of Large Numbers for Exchangeable Random Sets in Kuratowski-Mosco Sense

Abstract

Most of the results for laws of large numbers based on Banach space valued random sets assume that the sets are independent and identically distributed (IID) and compact, in which Rådström embedding or the refined method for collection of compact and convex subsets of a Banach space plays an important role. In this paper, exchangeability among random sets as a dependency, instead of IID, is assumed in obtaining strong laws of large numbers, since some kind of dependency of random variables may be often required for many statistical analyses. Also, the Hausdorff convergence usually used is replaced by another topology, Kuratowski-Mosco convergence. Thus, we prove strong laws of large numbers for exchangeable random sets in Kuratowski-Mosco convergence, without assuming the sets are compact, which is weaker than Hausdorff sense.     

Some topologies on the spaces of USCO maps and densely continuous forms

We study the completeness of three (metrizable) uniformities on the sets D(X, Y) and U(X, Y) of densely continuous forms and USCO maps from X to Y: the uniformity of uniform convergence on bounded sets, the Hausdorff metric uniformity and the uniformity U _B . We also prove that if X is a nondiscrete space, then the Hausdorff metric on real-valued densely continuous forms D(X, ?) (identified with their graphs) is not complete. The key to guarantee completeness of closed subsets of D(X, Y) equipped with the Hausdorff metric is dense equicontinuity introduced by Hammer and McCoy in [7].     

On visual distances for spectrum-type functional data

A functional distance \({\mathbb H}\), based on the Hausdorff metric between the function hypographs, is proposed for the space \({\mathcal E}\) of non-negative real upper semicontinuous functions on a compact interval. The main goal of the paper is to show that the space \(({\mathcal E},{\mathbb H})\) is particularly suitable in some statistical problems with functional data which involve functions with very wiggly graphs and narrow, sharp peaks. A typical example is given by spectrograms, either obtained by magnetic resonance or by mass spectrometry. On the theoretical side, we show that \(({\mathcal E},{\mathbb H})\) is a complete, separable locally compact space and that the \({\mathbb H}\)-convergence of a sequence of functions implies the convergence of the respective maximum values of these functions. The probabilistic and statistical implications of these results are discussed, in particular regarding the consistency of k-NN classifiers for supervised classification problems with functional data in \({\mathbb H}\). On the practical side, we provide the results of a small simulation study and check also the performance of our method in two real data problems of supervised classification involving mass spectra.     

Cartesian-closed coreflective subcategories of uniform spaces generated by classes of metric spaces

The main result, in Theorem 3, is that in the category Unif of Hausdorff uniform spaces and uniformly continuous maps, the coreflective hulls of the following classes are cartesian-closed: all metric spaces having no infinite uniform partition, all connected metric spaces, all bounded metric spaces, and all injective metric spaces.Furthermore, Theorems 1 and 4 imply that if $\text{C}$ is any coreflective, cartesian-closed subcategory of Unif in which enough function space structures are finer than the uniformity of uniform convergence (as in the above examples), then either (1) $\text{C}$ is a subclass of the locally fine spaces, or (2) $\text{C}$ contains all injective metric spaces and $\text{C}$ is a subclass of the coreflective hull of all uniform spaces having no infinite uniform partition.     

Random Steiner symmetrizations of sets and functions

In this article we prove almost sure convergence, in the L ₁ distance, of sequences of random Steiner symmetrizations of measurable sets having finite measure to the ball having the same measure. From this result we deduce analogous statements concerning the almost sure convergence to the spherical symmetrization of random Steiner symmetrizations of non negative L _p functions in the natural norm and uniform convergence of non negative continuous functions with bounded support. The latter result is finally used to prove that sequences of random symmetrizations of a compact set converge almost surely in the Hausdorff distance to the ball having the same measure, providing another proof of Mani-Levitska’s conjecture besides the one given in 2006 by Van Schaftingen (Topol Methods Nonlinear Anal 28(1): 61–85, 2006).     

A characterization of relatively compact sets of fuzzy sets

A purely topological characterization of relatively compact sets is given for the metric space (K(Y),D)$(\mathbb{K}(Y),D)$ of upper semicontinuous, compact-supported, normal fuzzy subsets of a metric space Y$Y$. The considered metric D$D$ is that of the distance between fuzzy subsets, which is the supremum of the Hausdorff distances of the corresponding level sets. In the given proof the compactness of a variational convergence which was introduced by De Giorgi and Franzoni is fundamental.     

LOCAL COMPACTNESS IN SEMI-UNIFORM CONVERGENCE SPACES

Abstract

Local compactness is studied in the highly convenient setting of semi-uniform convergence spaces which form a common generalization of (symmetric) limit spaces (and thus of symmetric topological spaces) as well as of uniform limit spaces (and thus of uniform spaces). It turns out that it leads to a cartesian closed topological category and, in contrast to the situation for topological spaces, the local compact spaces are exactly the compactly generated spaces. Furthermore, a one-point Hausdorff compactification for noncompact locally compact Hausdorff convergence spaces is considered.¹     

Convergence theorems for adapted sequences of random sets

We consider random sets with values in a separable Banach space. We study set-valued amarts, L¹-amarts, uniform amarts and submartingales. For all these classes of random sets, we prove convergence theorems in all main modes of set convergence (weak, Wijsman, Mosco, and Hausdorff). We also prove new convergence theorems for vector-valued subpramarts and pramarts.     

Minimizing Irregular Convex Functions: Ulam Stability for Approximate Minima

The main concern of this article is to study Ulam stability of the set of ε-approximate minima of a proper lower semicontinuous convex function bounded below on a real normed space X, when the objective function is subjected to small perturbations (in the sense of Attouch & Wets). More precisely, we characterize the class all proper lower semicontinuous convex functions bounded below such that the set-valued application which assigns to each function the set of its ε-approximate minima is Hausdorff upper semi-continuous for the Attouch–Wets topology when the set $\mathcal{C}(X)$ of all the closed and nonempty convex subsets of X is equipped with the Hausdorff topology. We prove that a proper lower semicontinuous convex function bounded below has Ulam-stable ε-approximate minima if and only if the boundary of any of its sublevel sets is bounded.     

On Convergence to Infinity

 By a metric mode of convergence to infinity in a regular Hausdorff space X, we mean a sequence of closed subsets of X with and , and a sequence (or net) in X is convergent to infinity with respect to provided for each contains eventually. Modulo a natural equivalence relation, these correspond to one-point extensions of the space with a countable base at the ideal point, and in the metrizable setting, they correspond to metric boundedness structures for the space. In this article, we study the interplay between these objects and certain continuous functions that may determine the metric mode of convergence to infinity, called forcing functions. Falling out of our results is a simple proof that each noncompact metrizable space admits uncountably many distinct metric uniformities. (Received 2 March 1999)     

A new look at some classical theorems on continuous functions on normal spaces

A sufficient condition for the strict insertion of a continuous function between two comparable upper and lower semicontinuous functions on a normal space is given. Among immediate corollaries are the classical insertion theorems of Michael and Dowker. Our insertion lemma also provides purely topological proofs of some standard results on closed subsets of normal spaces which normally depend upon uniform convergence of series of continuous functions. We also establish a Tietze-type extension theorem characterizing closed G _δ -sets in a normal space. This research was supported by the Ministry of Education and Science of Spain and FEDER under grant MTM2006-14925-C02-02. The first named author also acknowledges financial support from the University of the Basque Country under grant UPV05/101.