Related papers: Universal generalized functionals and finitely abs…

We study universal approximation of continuous functionals on compact subsets of products of Hilbert spaces. We prove that any such functional can be uniformly approximated by models that first take finitely many continuous linear…

This paper deals with functions that defined in metric spaces and valued in complete paranormed vector spaces or valued in Banach spaces, and obtains some necessary and sufficient conditions for weak convergence of finite measures.

The existence of a Banach limit as a translation invariant positive continuous linear functional on the space of bounded scalar sequences which is equal to 1 at the constant sequence (1,1,...,1,...) is proved in a first course on functional…

We derive Onsager-Machlup functionals for countable product measures on weighted $\ell^p$ subspaces of the sequence space $\mathbb{R}^{\mathbb{N}}$. Each measure in the product is a shifted and scaled copy of a reference probability measure…

Let C(K) be the Banach space of all continuous functions on a given compact space K. We investigate the w*-sequential closure in C(K)* of the set of all finitely supported probabilities on K. We discuss the coincidence of the Baire…

The objective of this paper is to introduce the notion of generalized almost statistical (briefly, GAS) convergence of bounded real sequences, which generalizes the notion of almost convergence as well as statistical convergence of bounded…

With a new proof approach we prove in a more general setting the classical convergence theorem that almost everywhere convergence of measurable functions on a finite measure space implies convergence in measure. Specifically, we generalize…

Uniform measures are defined as the functionals on the space of bounded uniformly continuous functions that are continuous on bounded uniformly equicontinuous sets. If every cardinal has measure zero then every countably additive measure is…

We introduce and study two new relations between function spaces over measure spaces of infinite measure, motivated by the question of establishing compactness. The first relation captures the uniform decay of function (quasi-)norms ``at…

It is shown that a Banach space with locally uniformly convex dual admits an equivalent norm which is itself locally uniformly convex. It follows that on any such space all continuous real-valued functions may be uniformly approximated by…

This paper investigates the approximation of Gaussian random variables in Banach spaces, focusing on the high-probability bounds for the approximation of Gaussian random variables using finitely many observations. We derive non-asymptotic…

In this paper we provide a sufficient condition for a Furstenberg measure generated by a finitely supported measure to be absolutely continuous. Using this, we give a very broad class of examples of absolutely continuous Furstenberg…

It is well known, as follows from the Banach-Steinhaus theorem, that if a sequence $\left\{y_{n}\right\}_{n=1}^{\infty}$ of linear continuous functionals in a Fr\'echet space converges pointwise to a linear functional $Y,$ $Y\left( x\right)…

In this work we study certain invariant measures that can be associated to the time averaged observation of a broad class of dissipative semigroups via the notion of a generalized Banach limit. Consider an arbitrary complete separable…

Recently, the different types of unbounded convergences (uo, un, uaw, uaw*) in Banach lattices were studied. In this paper, we study the continuous functionals with respect to unbounded convergences. We first characterize the continuity of…

If $\Lambda $ is a measure space, $u:\Lambda ^{m}\rightarrow \Bbb{R}$ is a given function and $N\geq m,$ the function $U(x_{1},...,x_{N})=\left( \begin{array}{l} N \\ m \end{array} \right) ^{-1}\sum_{1\leq i_{1}<\cdots <i_{m}\leq…

Let $\xi_0,\xi_1,...$ be independent identically distributed (i.i.d.) random variables such that $\E \log (1+|\xi_0|)<\infty$. We consider random analytic functions of the form $$ G_n(z)=\sum_{k=0}^{\infty} \xi_k f_{k,n} z^k, $$ where…

In this monograph, we construct and study a sigma-finite measure on continuous functions from R_+ to R, strongly related to many probability measures obtained by penalisation of Brownian motion, i.e. as limits of probabilities which are…

A Banach space has the Schur property when every weakly convergent sequence converges in norm. We prove a Schur-like property for measures: if a sequence of finite signed Borel measures on a Polish space is such that it is bounded in total…

Let $\mu$ be a probability measure on a separable Banach space $X$. A subset $U\subset X$ is $\mu$-continuous if $\mu(\partial U)=0$. In the paper the $\mu$-continuity and uniform $\mu$-continuity of convex bodies in $X$, especially of…