On the setwise convergence of sequences of measures

Annali di Matematica Pura ed Applicata (1923 -)

In this paper convergence theorems for sequences of scalar, vector and multivalued Pettis integrable functions on a topological measure space are proved for varying measures vaguely convergent.

Statistics & Probability Letters

We propose a new approach to vague convergence of measures based on the general theory of boundedness due to Hu (1966). The article explains how this connects and unifies several frequently used types of vague convergence from the literature. Such an approach allows one to translate already developed results from one type of vague convergence to another. We further analyze the corresponding notion of vague topology and give a new and useful characterization of convergence in distribution of random measures in this topology.

PROCEEDINGS OF THE 45TH INTERNATIONAL CONFERENCE ON APPLICATION OF MATHEMATICS IN ENGINEERING AND ECONOMICS (AMEE’19)

Let α = {a n } be a sequence of real numbers and M ⊆ R. For a positive integer n and the subset M, let the counting function # (M; n; α) be defined as the number of terms a k , 1 ≤ k ≤ n for which a k ∈ M. Define further a "measure" µ (M) as a limit µ (M) = lim n→∞ # (M; n; α) n. It is well known that α is an uniformly distributed in the unit interval [0, 1] one obtain Jordan measure. On the other hand the sequenceα = { (−1) n n } generates Dirac δ-measure etc. In the second part in this note we introduce uniformly distributed sequences of sets. Some consequences are obtained.

. The hypo-convergence of upper semicontinuous functions provides a natural framework for the study of the convergence of probability measures. This approach also yields some further characterizations of weak convergence and equi-tightness. Keywords: Narrow (weak, weak*) convergence, epi-convergence Date: April 2, 1987 Printed: April 13, 1999 * This research was supported in part by MPI, Projects: "Calcolo Stocastico e Sistemi Dinamici Statistici" and "Modelli Probabilistic" 1984, and by the National Science Foundation. 1 1. ABOUT CONTINUITY AND MEASURABILITY A probabilistic structure ---a space of possible events, a sigma-field of (observable) subcollections of events, and a probability measure defined on this sigma-field--- does not have a built-in topological structure. This is the source of many technical difficulties in the development of Probability Theory, in particular in the theory of stochastic processes. Much progress was made, in reconciling the measu...

1998

We consider a set of probability measures on a locally compact separable metric space. It is shown that a necessary and sufficient condition for (relative) sequential compactness of in various weak topologies (among which the vague, weak and setwise topologies) has the same simple form; i.e. a uniform principle has to hold in. We also extend this uniform principle to some Köthe function spaces.

Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 1999

Note pr&entke par Paul MALLIAC'IN. 0764~4442/99/0328 1 OS5 0 AcadCmie des SciencesElsevier, Paris

2009

We establish some convergence theorems for sequences of totally-measurable functions with respect to a submeasure of finite variation. We also present relationships among different types of convergences such as convergence in submeasure, almost uniformly convergence, convergence in L p spaces.

Mathematical and Computer Modelling, 2009

A real-valued finitely additive measure µ on N is said to be a measure of statistical type provided µ(k) = 0 for all singletons {k}. Applying the classical representation theorem of finitely additive measures with totally bounded variation, we first present a short proof of the representation theorem of statistical measures. As its application, we show that every kind of statistical convergence is just a type of measure convergence with respect to a specific class of statistical measures.

2018

We propose a notion of convergence of measures with intention of generalizing and unifying several frequently used types of vague convergence. We explain that by general theory of boundedness due to Hu [Hu66], in Polish spaces, this notion of convergence can be always formulated as follows: μn v −→ μ if ∫ fdμn → ∫ fdμ for all continuous bounded functions f with support bounded in some suitably chosen metric. This connects all the related types of vague convergence with the framework of Daley and Vere-Jones [DVJ03] and Kallenberg [Kal17]. In the rest of the note we discuss the vague topology and sufficient conditions for the corresponding notion of convergence in distribution, complementing the theory developed in those two references.

Annals of the Alexandru Ioan Cuza University - Mathematics, 2012

In this paper, we study different types of pseudo-convergences of sequences of measurable functions with respect to set-valued non-additive monotonic set functions and we establish some pseudo-versions of Egoroff theorem in the set-valued case. We also characterize important structural properties of monotone multimeasures.