Related papers: Weak convergence of probability measures: a topolo…

This expository note aims at illustrating weak convergence of probability measures from a broader view than a previously published paper. Though the results are standard for functional analysts, this approach is rarely known by…

This paper deals with three major types of convergence of probability measures on metric spaces: weak convergence, setwise converges, and convergence in the total variation. First, it describes and compares necessary and sufficient…

Necessary and sufficient conditions for weak and vague convergence of measures are important for a diverse host of applications. This paper aims to give a comprehensive description of the relationship between the two modes of convergence…

Let E be a locally compact second countable Hausdorff space and F the pertaining family of all closed sets. We endow F respectively with the Fell-topology, the upper Fell topology or the upper Vietoris-topology and investigate weak…

We introduce a notion of weak convergence in arbitrary metric spaces. Metric functionals are key in our analysis: weak convergence of sequences in a given metric space is tested against all the metric functionals defined on said space. When…

Topological measures and deficient topological measures are defined on open and closed subsets of a topological space, generalize regular Borel measures, and correspond to (non-linear in general) functionals that are linear on singly…

Finite metric spaces are the object of study in many data analysis problems. We examine the concept of weak isometry between finite metric spaces, in order to analyse properties of the spaces that are invariant under strictly increasing…

We introduce a notion of vague convergence for random marked metric measure spaces. Our main result shows that convergence of the moments of order $k \ge 1$ of a random marked metric measure space is sufficient to obtain its vague…

We analyze weak convergence on CAT(0) spaces and the existence and properties of corresponding weak topologies.

Weak values and measurements have been proposed as means to achieve dramatic enhancements in metrology based on the greatly increased range of possible measurement outcomes. Unfortunately, the very large values of measurement outcomes occur…

It is known that the space of boundedly finite integer-valued measures on a complete separable metric space becomes itself a complete separable metric space when endowed with the weak-hash metric. It is also known that convergence under…

We investigate the power of weak measurements in the framework of quantum state discrimination. First, we define and analyze the notion of weak consecutive measurements. Our main result is a convergence theorem whereby we demonstrate when…

A weak measurement consists in coupling a system to a probe in such a way that constructive interference generates a large output. So far, only the average output of the probe and its variance were studied. Here, the characteristic function…

In the past decades, weak convergence theory for stochastic processes has become a standard tool for analyzing the asymptotic properties of various statistics. Routinely, weak convergence is considered in the space of bounded functions…

In this paper we introduce a metrics on the space of idempotent probability measures on a given compactum, which extends the metrics on the compactum. It is proven the introduced metrics generates the pointwise convergence topology on the…

We discuss some basic properties of polar convergence in metric spaces. Polar convergence is closely connected with the notion of Delta-convergence of T.C. Lim known for several years. Possible existence of a topology which induces polar…

This paper is concerned with relationships of weakly mixing, topologically weakly mixing, and sensitivity for non-autonomous discrete systems. It is shown that weakly mixing implies topologically weakly mixing and sensitivity for measurable…

Spaces of quasi-invariant measures supplied with different topologies are studied. Their embeddings, projective decompositions, conditions for their metrizability are investigated. Theorems about convergence of nets of quasi-invariant…

A weak measurement on a system is made by coupling a pointer weakly to the system and then measuring the position of the pointer. If the initial wavefunction for the pointer is real, the mean displacement of the pointer is proportional to…

A general approach to the measurement of an observable with pre- and post-selection is presented. The limit of weak measurement is studied in detail, and it is shown that the phase of the probe, including a Hamiltonian contribution to it,…