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23 pages, 359 KiB

Open AccessArticle

Hausdorff Outer Measures and the Representation of Coherent Upper Conditional Previsions by the Countably Additive Möbius Transform

by Serena Doria

Fractal Fract. 2025, 9(8), 496; https://doi.org/10.3390/fractalfract9080496 - 29 Jul 2025

Viewed by 234

Abstract

This paper explores coherent upper conditional previsions, a class of nonlinear functionals that generalize expectations while preserving consistency properties. The study focuses on their integral representation using the countably additive Möbius transform, which is possible if coherent upper previsions are defined with respect to a monotone set function of bounded variation. In this work, we prove that an integral representation with respect to a countably additive measure is also possible, on the Borel

σ

-algebra, even when the coherent upper prevision is defined by the Choquet integral with respect to a Hausdorff measure, which is not of bounded variation. It occurs since Hausdorff outer measures are metric measures, and therefore every Borel set is measurable with respect to them. Furthermore, when the conditioning event has a Hausdorff measure in its own Hausdorff dimension equal to zero or infinity, coherent conditional probability is defined via the countably additive Möbius transform of a monotone set function of bounded variation. The paper demonstrates the continuity of coherent conditional previsions induced by Hausdorff measures. Full article

(This article belongs to the Special Issue Fixed Point Theory and Fractals)

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16 pages, 304 KiB

Open AccessArticle

On the Characterizations of Some Strongly Bounded Operators on C(K, X) Spaces

by Ioana Ghenciu

Axioms 2025, 14(8), 558; https://doi.org/10.3390/axioms14080558 - 23 Jul 2025

Viewed by 118

Abstract

Suppose X and Y are Banach spaces, K is a compact Hausdorff space, and

C (K, X)

is the Banach space of all continuous X-valued functions (with the supremum norm). We will study some strongly bounded operators [...] Read more.

Suppose X and Y are Banach spaces, K is a compact Hausdorff space, and

C (K, X)

is the Banach space of all continuous X-valued functions (with the supremum norm). We will study some strongly bounded operators

T : C (K, X) \to Y

with representing measures

m : Σ \to L (X, Y)

, where

L (X, Y)

is the Banach space of all operators

T : X \to Y

and

Σ

is the

σ

-algebra of Borel subsets of K. The classes of operators that we will discuss are the Grothendieck, p-limited, p-compact, limited, operators with completely continuous, unconditionally converging, and p-converging adjoints, compact, and absolutely summing. We give a characterization of the limited operators (resp. operators with completely continuous, unconditionally converging, p-convergent adjoints) in terms of their representing measures. Full article

16 pages, 313 KiB

Open AccessArticle

Riemann Integral on Fractal Structures

by José Fulgencio Gálvez-Rodríguez, Cristina Martín-Aguado and Miguel Ángel Sánchez-Granero

Mathematics 2024, 12(2), 310; https://doi.org/10.3390/math12020310 - 17 Jan 2024

Viewed by 1551

Abstract

In this work we start developing a Riemann-type integration theory on spaces which are equipped with a fractal structure. These topological structures have a recursive nature, which allows us to guarantee a good approximation to the true value of a certain integral with [...] Read more.

σ

-algebra of the space. We give the notion of Darboux sums and lower and upper Riemann integrals of a bounded function when given a measure and a fractal structure. Furthermore, we give the notion of a Riemann-integrable function in this context and prove that each

μ

-measurable function is Riemann-integrable with respect to

μ

. Moreover, if

μ

is the Lebesgue measure, then the Lebesgue integral on a bounded set of

R^{n}

meets the Riemann integral with respect to the Lebesgue measure in the context of measures and fractal structures. Finally, we give some examples showing that we can calculate improper integrals and integrals on fractal sets. Full article

(This article belongs to the Special Issue New Advances in Fuzzy Metric Spaces, Soft Metric Spaces, and Other Related Structures, 2nd Edition)

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15 pages, 352 KiB

Open AccessArticle

Congruence Representations via Soft Ideals in Soft Topological Spaces

by Zanyar A. Ameen and Mesfer H. Alqahtani

Axioms 2023, 12(11), 1015; https://doi.org/10.3390/axioms12111015 - 28 Oct 2023

Cited by 15 | Viewed by 1377

Abstract

This article starts with a study of the congruence of soft sets modulo soft ideals. Different types of soft ideals in soft topological spaces are used to introduce new weak classes of soft open sets. Namely, soft open sets modulo soft nowhere dense sets and soft open sets modulo soft sets of the first category. The basic properties and representations of these classes are established. The class of soft open sets modulo the soft nowhere dense sets forms a soft algebra. Elements in this soft algebra are primarily the soft sets whose soft boundaries are soft nowhere dense sets. The class of soft open sets modulo soft sets of the first category, known as soft sets of the Baire property, is a soft

σ

-algebra. In this work, we mainly focus on the soft

σ

-algebra of soft sets with the Baire property. We show that soft sets with the Baire property can be represented in terms of various natural classes of soft sets in soft topological spaces. In addition, we see that the soft

σ

-algebra of soft sets with the Baire property includes the soft Borel

σ

-algebra. We further show that soft sets with the Baire property in a certain soft topology are equal to soft Borel sets in the cluster soft topology formed by the original one. Full article

(This article belongs to the Special Issue Computational Mathematics and Mathematical Physics)

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13 pages, 311 KiB

Open AccessFeature PaperArticle

On Focal Borel Probability Measures

by Francisco Javier García-Pacheco, Jorge Rivero-Dones and Moisés Villegas-Vallecillos

Mathematics 2022, 10(22), 4365; https://doi.org/10.3390/math10224365 - 20 Nov 2022

Cited by 1 | Viewed by 1803

Abstract

The novel concept of focality is introduced for Borel probability measures on compact Hausdorff topological spaces. We characterize focal Borel probability measures as those Borel probability measures that are strictly positive on every nonempty open subset. We also prove the existence of focal Borel probability measures on compact metric spaces. Lastly, we prove that the set of focal (regular) Borel probability measures is convex but not extremal in the set of all (regular) Borel probability measures. Full article

(This article belongs to the Special Issue Functional Analysis, Topology and Quantum Mechanics II)

15 pages, 285 KiB

Open AccessArticle

Steffensen Type Inequalites for Convex Functions on Borel σ-Algebra

by Ksenija Smoljak Kalamir

Mathematics 2021, 9(24), 3276; https://doi.org/10.3390/math9243276 - 16 Dec 2021

Viewed by 2016

Abstract

In the paper, we prove Steffensen type inequalities for positive finite measures by using functions which are convex in point. Further, we prove Steffensen type inequalities on Borel

σ

-algebra for the function of the form

f / h

which is convex in [...] Read more.

In the paper, we prove Steffensen type inequalities for positive finite measures by using functions which are convex in point. Further, we prove Steffensen type inequalities on Borel

σ

-algebra for the function of the form

f / h

which is convex in point. We conclude the paper by showing that these results also hold for convex functions. Full article

21 pages, 327 KiB

Open AccessArticle

The Distribution Function of a Probability Measure on a Linearly Ordered Topological Space

by José Fulgencio Gálvez-Rodríguez and Miguel Ángel Sánchez-Granero

Mathematics 2019, 7(9), 864; https://doi.org/10.3390/math7090864 - 18 Sep 2019

Cited by 4 | Viewed by 2648

Abstract

In this paper, we describe a theory of a cumulative distribution function on a space with an order from a probability measure defined in this space. This distribution function plays a similar role to that played in the classical case. Moreover, we define its pseudo-inverse and study its properties. Those properties will allow us to generate samples of a distribution and give us the chance to calculate integrals with respect to the related probability measure. Full article

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