On Uniform Approximations of Abstract Functions

1967 ◽  
Vol 10 (1) ◽  
pp. 99-108
Author(s):  
Elias Zakon
Keyword(s):  
Metric Space ◽  
Real Function ◽  
Present Note ◽  
Uniform Limit ◽  
Uniform Approximations ◽  
Range Of Values ◽  
Separable Metric Space ◽  
Sequence Of Functions

As is well known, every real function is the pointwise (uniform) limit of a sequence of functions with a finite (countable) range of values. Monna [5] and Kvačko [4] suggested some extensions of this theorem to functions with values in a separable metric space. In the present note we give some further generalizations, with an emphasis on uniform approximations which have many applications in the generalized theory of measure and integration. In particular, we consider measurable abstract functions (mappings).

scholarly journals Quasi-uniform and uniform convergence of Riemann and Riemann-type integrable functions with values in a Banach space

Filomat ◽  
2020 ◽  
Vol 34 (6) ◽  
pp. 1899-1913
Author(s):  
Pratikshan Mondal ◽  
Lakshmi Dey ◽  
Ali Jaker
Keyword(s):  
Banach Space ◽  
Topological Space ◽  
Uniform Convergence ◽  
Sequence Space ◽  
Metric Space ◽  
Uniform Limit ◽  
Bounded Interval ◽  
Integrable Functions ◽  
Different Types ◽  
Sequence Of Functions

In this article, we study quasi-uniform and uniform convergence of nets and sequences of different types of functions defined on a topological space, in particular, on a closed bounded interval of R, with values in a metric space and in some cases in a Banach space. We show that boundedness and continuity are inherited to the quasi-uniform limit, and integrability is inherited to the uniform limit of a net of functions. Given a sequence of functions, we construct functions with values in a sequence space and consequently we infer some important properties of such functions. Finally, we study convergence of partially equi-regulated* nets of functions which is shown to be a generalized notion of exhaustiveness.

scholarly journals On Iterated Limits of Measurable Mappings

1965 ◽  
Vol 8 (1) ◽  
pp. 83-91
Author(s):  
Elias Zakon
Keyword(s):  
Metric Space ◽  
Present Note ◽  
Separable Metric Space ◽  
Measurable Mappings

Egoroff' s theorem [1] was extended by Kvačko [3] to functions with values in a separable metric space; and, as is easily seen, this result applies also to separable pseudometric spaces. In the present note we shall use this theorem to obtain some propositions on iterated limits, which, despite their simplicity, seem not yet to be known in the proposed generality.

scholarly journals On limit theorems for random fields

2009 ◽  
Vol 50 ◽  
Author(s):  
Rimas Banys
Keyword(s):  
Metric Space ◽  
Random Fields ◽  
Limit Theorems ◽  
Characteristic Property ◽  
Complete Separable Metric Space ◽  
Separable Metric Space

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.

scholarly journals DIMENSIONS IN A SEPARABLE METRIC SPACE

1995 ◽  
Vol 49 (1) ◽  
pp. 143-162 ◽  
Author(s):  
Masakazu TAMASHIRO
Keyword(s):  
Metric Space ◽  
Separable Metric Space

Approximation Par des Fonctions Lipschitziennes et Critere de Convergence Etroite D'une Suite de Probabilites

1984 ◽  
Vol 27 (4) ◽  
pp. 514-516 ◽  
Author(s):  
I. Rihaoui
Keyword(s):  
Weak Convergence ◽  
Metric Space ◽  
Bounded Function ◽  
Uniform Limit ◽  
Bounded Functions ◽  
New Criterion ◽  
Uniformly Continuous

AbstractIn this paper, we prove that a real valued bounded function, defined on a metric space and uniformly continuous is the uniform limit of a sequence of Lipschitzian bounded functions.As a consequence, a new criterion for the weak convergence of probabilities is given.

scholarly journals Super-extremal processes and the argmax process

1994 ◽  
Vol 31 (04) ◽  
pp. 958-978 ◽  
Author(s):  
Sidney I. Resnick ◽  
Rishin Roy
Keyword(s):  
Metric Space ◽  
Semicontinuous Function ◽  
Closed Set ◽  
Continuous Choice ◽  
Extremal Processes ◽  
Upper Semicontinuous Function ◽  
Classical Vector ◽  
Vector Valued ◽  
Path Properties ◽  
Separable Metric Space

In this paper, we develop the probabilistic foundations of the dynamic continuous choice problem. The underlying choice set is a compact metric space E such as the unit interval or the unit square. At each time point t, utilities for alternatives are given by a random function . To achieve a model of dynamic continuous choice, the theory of classical vector-valued extremal processes is extended to super-extremal processes Y = {Yt, t > 0}. At any t > 0, Y t is a random upper semicontinuous function on a locally compact, separable, metric space E. General path properties of Y are discussed and it is shown that Y is Markov with state-space US(E). For each t > 0, Y t is associated. For a compact metric E, we consider the argmax process M = {Mt, t > 0}, where . In the dynamic continuous choice application, the argmax process M represents the evolution of the set of random utility maximizing alternatives. M is a closed set-valued random process, and its path properties are investigated.

Uniform Embeddings into Hilbert Space and a Question of Gromov

2002 ◽  
Vol 45 (1) ◽  
pp. 60-70 ◽  
Author(s):  
A. N. Dranishnikov ◽  
G. Gong ◽  
V. Lafforgue ◽  
G. Yu
Keyword(s):  
Hilbert Space ◽  
Metric Space ◽  
Uniform Embedding ◽  
Separable Metric Space

AbstractGromov introduced the concept of uniform embedding into Hilbert space and asked if every separable metric space admits a uniform embedding into Hilbert space. In this paper, we study uniform embedding into Hilbert space and answer Gromov’s question negatively.

scholarly journals Note on limits of simply continuous and cliquish functions

1994 ◽  
Vol 17 (3) ◽  
pp. 447-450 ◽  
Author(s):  
Janina Ewert
Keyword(s):  
Metric Space ◽  
Continuous Functions ◽  
Baire Space ◽  
Baire Property ◽  
Baire Class ◽  
Class 1 ◽  
Pointwise Limit ◽  
Separable Metric Space

The main result of this paper is that any functionfdefined on a perfect Baire space(X,T)with values in a separable metric spaceYis cliquish (has the Baire property) iff it is a uniform (pointwise) limit of sequence{fn:n≥1}of simply continuous functions. This result is obtained by a change of a topology onXand showing that a functionf:(X,T)→Yis cliquish (has the Baire property) iff it is of the Baire class 1 (class 2) with respect to the new topology.

scholarly journals A Universal Separable Diversity

2017 ◽  
Vol 5 (1) ◽  
pp. 138-151 ◽  
Author(s):  
David Bryant ◽  
André Nies ◽  
Paul Tupper
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Complete Metric Space ◽  
Urysohn Space ◽  
Fundamental Properties ◽  
Finite Set ◽  
Whole Space ◽  
Set Of Points ◽  
Separable Metric Space

AbstractThe Urysohn space is a separable complete metric space with two fundamental properties: (a) universality: every separable metric space can be isometrically embedded in it; (b) ultrahomogeneity: every finite isometry between two finite subspaces can be extended to an auto-isometry of the whole space. The Urysohn space is uniquely determined up to isometry within separable metric spaces by these two properties. We introduce an analogue of the Urysohn space for diversities, a recently developed variant of the concept of a metric space. In a diversity any finite set of points is assigned a non-negative value, extending the notion of a metric which only applies to unordered pairs of points.We construct the unique separable complete diversity that it is ultrahomogeneous and universal with respect to separable diversities.

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