scholarly journals Nonstandard Methods in Measure Theory

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Grigore Ciurea
Keyword(s):  
Banach Spaces ◽  
Nonstandard Analysis ◽  
Measure Theory ◽  
New Approach ◽  
Vector Measures ◽  
Measure Spaces

Ideas and techniques from standard and nonstandard theories of measure spaces and Banach spaces are brought together to give a new approach to the study of the extension of vector measures. Applications of our results lead to simple new proofs for theorems of classical measure theory. The novelty lies in the use of the principle of extension by continuity (for which we give a nonstandard proof) to obtain in an unified way some notable theorems which have been obtained by Fox, Brooks, Ohba, Diestel, and others. The methods of proof are quite different from those used by previous authors, and most of them are realized by means of nonstandard analysis.

scholarly journals On copies of the null sequence Banach space in some vector measure spaces

1999 ◽  
Vol 59 (3) ◽  
pp. 443-447
Author(s):  
J.C. Ferrando ◽  
J.M. Amigó
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Vector Measure ◽  
Null Sequence ◽  
Vector Measures ◽  
Measure Spaces ◽  
Additive Vector

In this note we extend a result of Drewnowski concerning copies of C0 in the Banach space of all countably additive vector measures and study some properties of complemented copies of C0 in several Banach spaces of vector measures.

scholarly journals Pseudo almost automorphic solutions of class r in α-norm under the light of measure theory

2020 ◽  
Vol 7 (1) ◽  
pp. 81-101
Author(s):  
Issa Zabsonre ◽  
Djendode Mbainadji
Keyword(s):  
Phase Space ◽  
Spectral Decomposition ◽  
Measure Theory ◽  
New Approach ◽  
Almost Automorphic ◽  
Almost Automorphic Functions ◽  
Pseudo Almost Automorphic ◽  
Automorphic Functions

AbstractUsing the spectral decomposition of the phase space developed in Adimy and co-authors, we present a new approach to study weighted pseudo almost automorphic functions in the α-norm using the measure theory.

scholarly journals An Elementary Proof of Jin's Theorem with a Bound

10.37236/3846 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Mauro Di Nasso
Keyword(s):  
Ergodic Theory ◽  
Elementary Proof ◽  
Nonstandard Analysis ◽  
Measure Theory ◽  
Difference Sets ◽  
Explicit Bound ◽  
Finite Set

We present a short and self-contained proof of Jin's theorem about the piecewise syndeticity of difference sets which is entirely elementary, in the sense that no use is made of nonstandard analysis, ergodic theory, measure theory, ultrafilters, or other advanced tools. An explicit bound to the number of shifts that are needed to cover a thick set is provided. Precisely, we prove the following: If $A$ and $B$ are sets of integers having positive upper Banach densities $a$ and $b$ respectively, then there exists a finite set $F$ of cardinality at most $1/ab$ such that $(A-B)+F$ covers arbitrarily long intervals.

scholarly journals Remarks on the semivariation of vector measures with respect to Banach spaces.

2007 ◽  
Vol 75 (3) ◽  
pp. 469-480 ◽  
Author(s):  
Oscar Blasco
Keyword(s):  
Banach Spaces ◽  
Vector Measures ◽  
Tensor Norm

Suppose that and . It is shown that any Lp(µ)-valued measure has finite L2(v)-semivariation with respect to the tensor norm for 1 ≤ p < ∞ and finite Lq(v)-semivariation with respect to the tensor norm whenever either q = 2 and 1 ≤ p ≤ 2 or q > max{p, 2}. However there exist measures with infinite Lq-semivariation with respect to the tensor norm for any 1 ≤ q < 2. It is also shown that the measure m (A) = χA has infinite Lq-semivariation with respect to the tensor norm if q < p.

scholarly journals Nonlinear sequential fractional differential equations in partially ordered spaces

Filomat ◽  
2018 ◽  
Vol 32 (13) ◽  
pp. 4577-4586 ◽  
Author(s):  
Hossein Fazli ◽  
Juan Nieto
Keyword(s):  
Differential Equations ◽  
Banach Spaces ◽  
Fractional Differential Equations ◽  
Fixed Point Theorems ◽  
New Approach ◽  
Partially Ordered ◽  
Fractional Differential ◽  
Partially Ordered Spaces ◽  
Sequential Fractional Differential Equations ◽  
Ordered Banach Spaces

In this paper, some new partially ordered Banach spaces are introduced. Based on those new partially ordered Banach spaces and applying some fixed point theorems, we present a new approach to the theory of nonlinear sequential fractional differential equations. An example illustrating our approach is also discussed.

Vector Integration and Stochastic Integration in Banach Spaces

2018 ◽  
Author(s):  
Nicolae Dinculeanu
Keyword(s):  
Banach Spaces ◽  
Positive Measure ◽  
Integration Theory ◽  
Stochastic Integration ◽  
Bochner Integral ◽  
Finite Variation ◽  
Vector Measures ◽  
Vector Integration ◽  
Step Functions ◽  
Integrable Variation

This article deals with vector integration and stochastic integration in Banach spaces. In particular, it considers the theory of integration with respect to vector measures with finite semivariation and its applications. This theory reduces to integration with respect to vector measures with finite variation which, in turn, reduces to the Bochner integral with respect to a positive measure. The article describes the four stages in the development of integration theory. It first provides an overview of the relevant notation for Banach spaces, measurable functions, the integral of step functions, and measurability with respect to a positive measure before discussing the Bochner integral. It then examines integration with respect to measures with finite variation, semivariation of vector measures, integration with respect to a measure with finite semivariation, and stochastic integrals. It also reviews processes with integrable variation or integrable semivariation and concludes with an analysis of martingales.

A new approach to measures of noncompactness of Banach spaces

2018 ◽  
Vol 240 (1) ◽  
pp. 21-45 ◽  
Author(s):  
Lixin Cheng ◽  
Qingjin Cheng ◽  
Qinrui Shen ◽  
Kun Tu ◽  
Wen Zhang
Keyword(s):  
Banach Spaces ◽  
Measures Of Noncompactness ◽  
New Approach
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