scholarly journals On Null-Continuity of Monotone Measures

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 205 ◽  
Author(s):  
Jun Li
Keyword(s):  
Structural Characteristic ◽  
Choquet Integral ◽  
Measurable Space ◽  
Continuity Condition ◽  
Necessary Condition ◽  
Structure Characteristic ◽  
Sugeno Integral ◽  
Compact Spaces ◽  
Basic Properties ◽  
Measurable Functions

The null-continuity of monotone measures is a weaker condition than continuity from below and possesses many special properties. This paper further studies this structure characteristic of monotone measures. Some basic properties of null-continuity are shown and the characteristic of null-continuity is described by using convergence of sequence of measurable functions. It is shown that the null-continuity is a necessary condition that the classical Riesz’s theorem remains valid for monotone measures. When considered measurable space ( X , A ) is S-compact, the null-continuity condition is also sufficient for Riesz’s theorem. By means of the equivalence of null-continuity and property (S) of monotone measures, a version of Egoroff’s theorem for monotone measures on S-compact spaces is also presented. We also study the Sugeno integral and the Choquet integral by using null-continuity and generalize some previous results. We show that the monotone measures defined by the Sugeno integral (or the Choquet integral) preserve structural characteristic of null-continuity of the original monotone measures.

scholarly journals On t-Neighbourhoods in Trigonometric Topological Spaces

2021 ◽  
Vol 12 (1S) ◽  
pp. 655-658
Author(s):  
S. Malathi, Et. al.
Keyword(s):  
Necessary Condition ◽  
Topological Spaces ◽  
Basic Properties ◽  
New Type ◽  
The Relationship

In this paper we introduce a new type of neighbourhoods, namely, t-neighbourhoods in trigonometric topological spaces and study their basic properties. Also, we discuss the relationship between neighbourhoods and t-neighbourhoods. Further, we give the necessary condition for t-neighbourhoods in trigonometric topological spaces.  .

scholarly journals Ideals in rings and intermediate rings of measurable functions

2019 ◽  
Vol 19 (02) ◽  
pp. 2050038
Author(s):  
Sudip Kumar Acharyya ◽  
Sagarmoy Bag ◽  
Joshua Sack
Keyword(s):  
Measurable Space ◽  
Zero Sets ◽  
Measurable Functions ◽  
Maximal Ideals

The set of all maximal ideals of the ring [Formula: see text] of real valued measurable functions on a measurable space [Formula: see text] equipped with the hull-kernel topology is shown to be homeomorphic to the set [Formula: see text] of all ultrafilters of measurable sets on [Formula: see text] with the Stone-topology. This yields a complete description of the maximal ideals of [Formula: see text] in terms of the points of [Formula: see text]. It is further shown that the structure spaces of all the intermediate subrings of [Formula: see text] containing the bounded measurable functions are one and the same and are compact Hausdorff zero-dimensional spaces. It is observed that when [Formula: see text] is a [Formula: see text]-space, then [Formula: see text], where [Formula: see text] is the [Formula: see text]-algebra consisting of the zero-sets of [Formula: see text].

scholarly journals Lacunary Statistical Convergence in Measure for Double Sequences of Fuzzy Valued Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Ömer Kişi
Keyword(s):  
Statistical Convergence ◽  
Double Sequence ◽  
Finite Measure ◽  
Measurable Space ◽  
Double Sequences ◽  
Convergence In Measure ◽  
Measurable Functions ◽  
Lacunary Statistical Convergence ◽  
Convergence Of Sequences ◽  
Sequences Of Fuzzy Numbers

Based on the concept of lacunary statistical convergence of sequences of fuzzy numbers, the lacunary statistical convergence, uniformly lacunary statistical convergence, and equi-lacunary statistical convergence of double sequences of fuzzy-valued functions are defined and investigated in this paper. The relationship among lacunary statistical convergence, uniformly lacunary statistical convergence, equi-lacunary statistical convergence of double sequences of fuzzy-valued functions, and their representations of sequences of α -level cuts are discussed. In addition, we obtain the lacunary statistical form of Egorov’s theorem for double sequences of fuzzy-valued measurable functions in a finite measurable space. Finally, the lacunary statistical convergence in measure for double sequences of fuzzy-valued measurable functions is examined, and it is proved that the inner and outer lacunary statistical convergence in measure are equivalent in a finite measure set for a double sequence of fuzzy-valued measurable functions.

Elements of probability theory with values in bihyperbolic algebra

2021 ◽  
Vol 34 ◽  
pp. 36-49
Author(s):  
Tamila Kolomiiets
Keyword(s):  
Statistical Physics ◽  
Random Variable ◽  
Measurable Space ◽  
Probabilistic Measure ◽  
Basic Properties ◽  
Bicomplex Numbers ◽  
Zero Divisors ◽  
Special Cases ◽  
Testing Complex ◽  
Significant Probability

In this paper we expand the concept of a really significant probabilistic measure in the case when the measure takes values in the algebra of bihyperbolic numbers. The basic properties of bihyperbolic numbers are given, in particular idempotents, main ideals generated by idempotents, Pierce's decompo\-sition and the set of zero divisors of the algebra of bihyperbolic numbers are determined. We entered the relation of partial order on the set of bihyperbolic numbers, by means of which the bihyperbolic significant modulus is defined and its basic properties are proved. In addition, some bihyperbolic modules can be endowed with a bihyperbolic significant norms that take values in a set of non-negative bihyperbolic numbers. We define $\sigma$-additive functions of sets in a measurable space that take appropriately normalized bihyperbolic values, which we call a bihyperbolic significant probability. It is proved that such a bihyperbolic probability satisfies the basic properties of the classical probability. A representation of the bihyperbolic probability measure is given and its main properties are proved. A bihyperbolically significant random variable is defined on a bihyperbolic probability space, and this variable is a bihyperbolic measurable function in the same space. We proved the criterion of measurability of a function with values in the algebra of bihyperbolic numbers, and the basic properties of bihyperbolic random variables are formulated and proved. Special cases have been studied in which the bihyperbolic probability and the bihyperbolic random variable take values that are zero divisors of bihyperbolic algebra. Although bihyperbolic numbers are less popular than hyperbolic numbers, bicomplex numbers, or quaternions, they have a number of important properties that can be useful, particularly in the study of partial differential equations also in mathematical statistics for testing complex hypotheses, in thermodynamics and statistical physics.

scholarly journals A Multi-Valued Choquet Integral with Respect to a Multisubmeasure

2015 ◽  
Vol 61 (1) ◽  
pp. 129-152 ◽  
Author(s):  
Floarea-Nicoleta Sofian-Boca
Keyword(s):  
Convex Sets ◽  
Choquet Integral ◽  
Order Relation ◽  
Convergence Theorems ◽  
Fuzzy Measures ◽  
Measurable Functions

Abstract Jang, Kim and Kwon introduced a multi-valued Choquet integral for multifunctions with respect to real fuzzy measures and Zhang, Guo and Liu established for this kind of integral some convergence theorems. The aim of this paper is to present another type of set-valued Choquet integral, called by us the Aumann-Choquet integral, for non-negative measurable functions with respect to multisubmeasures taking values in the class of all non-empty,compact and convex sets of ℝ+ on which we use the order relation considered by Guo and Zhang. For this kind of integral, we study some important properties and we prove that if we add some supplementary properties to the multisubmeasure then they are also preserved by the set-valued function defined as Aumann-Choquet integral.

scholarly journals Abstract Daniell-Loomis spaces

1996 ◽  
Vol 53 (1) ◽  
pp. 135-142 ◽  
Author(s):  
M. Díaz Carrillo ◽  
H. Günzler
Keyword(s):  
Integral Extension ◽  
Basic Properties ◽  
General Integral ◽  
Measurable Functions

In [3] for general integral metric q an integral extension of Lebesgue power was discussed. In this paper we introduce the abstract Daniell-Loomis spaces Rp, p real, 0 < p < ∞, of q-measurable functions with finite “p-norm”, and study their basic properties.

To the Mathematical Concept of Thermostatics To the Mathematical Concept of Thermostatics

1982 ◽  
Vol 37 (1) ◽  
pp. 1-5
Author(s):  
A. Grauel
Keyword(s):  
Mathematical Concept ◽  
Necessary Condition ◽  
Second Law ◽  
Field Equations ◽  
Basic Properties ◽  
Integrating Factor

Abstract We discuss the integrated form of the laws of thermostatics by using Stokes theorem on manifolds. We give a new insight to the basic properties of thermostatics, in particular we show that only the second law is a necessary condition to identify the integrating factor with the temperature. Moreover we consider the thermodynamical field equations for discontinuous media and discuss the properties of thermostatics.

scholarly journals The Near Subnormal Weighted Shift and Recursiveness

2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
R. Ben Taher ◽  
M. Rachidi
Keyword(s):  
Finite Order ◽  
Weighted Shift ◽  
Necessary Condition ◽  
Important Class ◽  
Weighted Shifts ◽  
Basic Properties ◽  
Recursive Sequences ◽  
Recursive Relations

We aim at studying the near subnormality of the unilateral weighted shifts, whose moment sequences are defined by linear recursive relations of finite order. Using the basic properties of recursive sequences, we provide a natural necessary condition, that ensure the near subnormality of this important class of weighted shifs. Some related new results are established; moreover, applications and consequences are presented; notably the notion of near subnormal completion weighted shift is implanted and explored.

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