scholarly journals Ultrabornological Bochner integrable function spaces

1993 ◽  
Vol 47 (1) ◽  
pp. 119-126
Author(s):  
J.C. Ferrando
Keyword(s):  
Normed Space ◽  
Measure Space ◽  
Integrable Function ◽  
Finite Measure ◽  
Function Spaces ◽  
Nikodym Property ◽  
Finite Measure Space ◽  
Bochner Integrable Function

If (Ω, Σ, μ) is a finite measure space and X is a normed space such that X* has the Radon-Nikodym property with respect to μ, we show first that each space Lp(μ, x), 1 < p < ∞, is ultrabornological whenever μ is atomless. When μ is arbitrary, we prove later on that the space Lp(μ, X) is ultrabornological if X* has the Radon-Nikodym property with respect to μ and X is itself an ultrabornological space.

On the Integrability of the Ergodic Hilbert Transform for A Class of Functions with Equal Absolute Values

1998 ◽  
Vol 5 (2) ◽  
pp. 101-106
Author(s):  
L. Ephremidze
Keyword(s):  
Hilbert Transform ◽  
Measure Space ◽  
Integrable Function ◽  
Finite Measure ◽  
Ergodic Transformation ◽  
Finite Measure Space ◽  
Class Of Functions

Abstract It is proved that for an arbitrary non-atomic finite measure space with a measure-preserving ergodic transformation there exists an integrable function f such that the ergodic Hilbert transform of any function equal in absolute values to f is non-integrable.

Rearrangement Inequalities

1972 ◽  
Vol 24 (5) ◽  
pp. 930-943 ◽  
Author(s):  
Peter W. Day
Keyword(s):  
Measure Space ◽  
Finite Measure ◽  
Function Spaces ◽  
Banach Function Spaces ◽  
Measurable Functions ◽  
Finite Measure Space ◽  
Rearrangement Inequalities

In recent years a number of inequalities have appeared which involve rearrangements of vectors in Rn and of measurable functions on a finite measure space. These inequalities are not only interesting in themselves, but also are important in investigations involving rearrangement invariant Banach function spaces and interpolation theorems for these spaces [2; 8; 9].

Lacunary ideal convergence in measure for sequences of fuzzy valued functions

2021 ◽  
Vol 40 (3) ◽  
pp. 5517-5526
Author(s):  
Ömer Kişi
Keyword(s):  
Measure Space ◽  
Finite Measure ◽  
Ideal Convergence ◽  
Convergence In Measure ◽  
Ideal Form ◽  
Measurable Functions ◽  
Finite Measure Space ◽  
Convergence Of Sequences

We investigate the concepts of pointwise and uniform I θ -convergence and type of convergence lying between mentioned convergence methods, that is, equi-ideally lacunary convergence of sequences of fuzzy valued functions and acquire several results. We give the lacunary ideal form of Egorov’s theorem for sequences of fuzzy valued measurable functions defined on a finite measure space ( X , M , μ ) . We also introduce the concept of I θ -convergence in measure for sequences of fuzzy valued functions and proved some significant results.

On Unsymmetric Dirichlet Forms

1973 ◽  
Vol 25 (2) ◽  
pp. 252-260 ◽  
Author(s):  
Joanne Elliott
Keyword(s):  
Compact Space ◽  
Measure Space ◽  
Closed Subspace ◽  
Finite Measure ◽  
Dirichlet Forms ◽  
Locally Compact ◽  
Locally Compact Space ◽  
Finite Measure Space

Let F be a linear, but not necessarily closed, subspace of L2[X, dm], where (X,,m) is a σ-finite measure space with the Borel subsets of the locally compact space X. If u and v are measureable functions, then v is called a normalized contraction of u if and Assume that F is stable under normalized contractions, that is, if u ∈ F and v is a normalized contraction of u, then v ∈ F.

A Packing Problem for Measurable Sets

1967 ◽  
Vol 19 ◽  
pp. 749-756
Author(s):  
D. Sankoff ◽  
D. A. Dawson
Keyword(s):  
Probability Measure ◽  
Measure Space ◽  
Packing Problem ◽  
Finite Measure ◽  
Finite Measure Space ◽  
Probability Measure Space

Given a probability measure space (Ω,,P)consider the followingpacking problem.What is the maximum number,b(K,Λ), of sets which may be chosen fromso that each set has measureKand no two sets have intersection of measure larger than Λ <K?In this paper the packing problem is solved for any non-atomic probability measure space. Rather than obtaining the solution explicitly, however, it is convenient to solve the followingminimal paving problem.In a non-atomic a-finite measure space (Ω,,μ)what is the measure,V(b, K,Λ), of the smallest set which is the union of exactlybsubsets of measureKsuch that no subsets have intersection of measure larger than Λ?

scholarly journals Duals of vector-valued Köthe function spaces

1992 ◽  
Vol 112 (1) ◽  
pp. 165-174 ◽  
Author(s):  
Miguel Florencio ◽  
Pedro J. Paúl ◽  
Carmen Sáez
Keyword(s):  
Function Space ◽  
Normed Space ◽  
Function Spaces ◽  
Integrable Functions ◽  
Nikodym Property ◽  
Köthe Dual ◽  
Vector Valued

AbstractLet Λ be a perfect Köthe function space in the sense of Dieudonné, and Λ× its Köthe-dual. Let E be a normed space. Then the topological dual of the space Λ(E) of Λ-Bochner integrable functions equals the corresponding Λ×(E′) if and only if E′ has the Radon–Nikodým property. We also give some results concerning barrelledness for spaces of this kind.

scholarly journals M-Cohyponormal powers of composition operators

1992 ◽  
Vol 53 (1) ◽  
pp. 9-16
Author(s):  
Satish K. Khurana ◽  
Babu Ram
Keyword(s):  
Composition Operator ◽  
Measure Space ◽  
Composition Operators ◽  
Finite Measure ◽  
Finite Measure Space ◽  
Positive Integers ◽  
Nikodym Derivative

AbstractLet T1, i = 1, 2 be measurable transformations which define bounded composition operators C Ti on L2 of a σ-finite measure space. Let us denote the Radon-Nikodym derivative of with respect to m by hi, i = 1, 2. The main result of this paper is that if and are both M-hyponormal with h1 ≤ M2(h2 o T2) a.e. and h2 ≤ M2(h1 o T1) a.e., then for all positive integers m, n and p, []* is -hyponormal. As a consequence, we see that if is an M-hyponormal composition operator, then is -hyponormal for all positive integers n.

Ratio and Stochastic Ergodic Theorems for Superadditive Processes

1979 ◽  
Vol 31 (2) ◽  
pp. 441-447 ◽  
Author(s):  
Humphrey Fong
Keyword(s):  
Linear Operator ◽  
Measure Space ◽  
Measure Zero ◽  
Positive Linear Operator ◽  
Finite Measure ◽  
Ergodic Theorems ◽  
Finite Measure Space ◽  
Image Position

1. Introduction. Let (X, , m) be a σ-finite measure space and let T be a positive linear operator on L1 = L1(X, , m). T is called Markovian if(1.1)T is called sub-Markovian if(1.2)All sets and functions are assumed measurable; all relations and statements are assumed to hold modulo sets of measure zero.For a sequence of L1+ functions (ƒ0, ƒ1, ƒ2, …), let(ƒn) is called a super additive sequence or process, and (sn) a super additive sum relative to a positive linear operator T on L1 if(1.3)and(1.4)

scholarly journals Classes of operators on vector valued integration spaces

1977 ◽  
Vol 24 (2) ◽  
pp. 129-138 ◽  
Author(s):  
R. J. Fleming ◽  
J. E. Jamison
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Measure Space ◽  
Separable Hilbert Space ◽  
Hermitian Operator ◽  
Finite Measure ◽  
Finite Measure Space ◽  
Image Position ◽  
Vector Valued ◽  
Uniformly Bounded

AbstractLet Lp(Ω, K) denote the Banach space of weakly measurable functions F defined on a finite measure space and taking values in a separable Hilbert space K for which ∥ F ∥p = ( ∫ | F(ω) |p)1/p < + ∞. The bounded Hermitian operators on Lp(Ω, K) (in the sense of Lumer) are shown to be of the form , where B(ω) is a uniformly bounded Hermitian operator valued function on K. This extends the result known for classical Lp spaces. Further, this characterization is utilized to obtain a new proof of Cambern's theorem describing the surjective isometries of Lp(Ω, K). In addition, it is shown that every adjoint abelian operator on Lp(Ω, K) is scalar.

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