This document discusses disjointification methods in spaces of integrable functions with respect to a vector measure. Specifically:
1. It introduces two classical disjointification results - Bessaga-Pelczynski and Kadecs-Pelczynski - which allow working with orthogonality notions and analyzing disjoint functions in these spaces.
2. It defines key concepts like Banach function spaces, vector measures, integration with respect to vector measures, and the space L1(m) of integrable functions.
3. The document aims to show how these classical disjointification techniques can be used in analyzing properties of spaces of integrable functions and the structure of their subspaces.
Vector measures and classical disjointification methodsesasancpe
Vector measures and classical disjointification methods.
In this talk we show how the classical disjointification methods (Bessaga-Pelczynski, Kadecs-Pelczynski) can be applied in the setting of the spaces of p-integrable functions with respect to vector measures. These spaces provide in fact a representation of p-convex order continuous Banach lattices with weak unit; the additional tool of the vector valued integral for each function has already shown to be fruitful for the analysis of these spaces. Consequently, our results can be directly extended to a broad class of Banach lattices.
Subespacios de funciones integrables respecto de una medidavectorialye...esasancpe
En esta charla mostramos que la extensio ́n de un operador definido en un espacio de funciones de Banach es equivalente, bajo ciertos requisitos m ́ınimos, a su factorizacio ́natrave ́sdeunsubespaciodefuncionesintegrablesrespectodela medidavectorialmT asociadaaloperadorT.Adema ́s,estesubespacioparticular determina aquella propiedad del operador que se pretende preservar, en el sentido de queconstituyeeldominioo ́ptimo-elmayorsubespacioalquesepuedeextender-,de manera que la extensio ́n conserva esta propiedad. Esto permite un planteamiento abstracto de ciertos aspectos de la teor ́ıa de dominio o ́ptimo, que se ha desarrollado de manera importante en los u ́ltimos tiempos, principalmente en la direccio ́n de caracterizar cua ́l es el mayor espacio de funciones de Banach al que se pueden extender algunos operadores de importancia en ana ́lisis. Adema ́s, permite clasificar laspropiedadesdelosoperadoresqueseconservanenlaextensio ́nenfuncio ́n delsubespaciodeL1(mT)queconstituyesudominioo ́ptimo.
Vector measures and classical disjointification methodsesasancpe
Vector measures and classical disjointification methods.
In this talk we show how the classical disjointification methods (Bessaga-Pelczynski, Kadecs-Pelczynski) can be applied in the setting of the spaces of p-integrable functions with respect to vector measures. These spaces provide in fact a representation of p-convex order continuous Banach lattices with weak unit; the additional tool of the vector valued integral for each function has already shown to be fruitful for the analysis of these spaces. Consequently, our results can be directly extended to a broad class of Banach lattices.
Subespacios de funciones integrables respecto de una medidavectorialye...esasancpe
En esta charla mostramos que la extensio ́n de un operador definido en un espacio de funciones de Banach es equivalente, bajo ciertos requisitos m ́ınimos, a su factorizacio ́natrave ́sdeunsubespaciodefuncionesintegrablesrespectodela medidavectorialmT asociadaaloperadorT.Adema ́s,estesubespacioparticular determina aquella propiedad del operador que se pretende preservar, en el sentido de queconstituyeeldominioo ́ptimo-elmayorsubespacioalquesepuedeextender-,de manera que la extensio ́n conserva esta propiedad. Esto permite un planteamiento abstracto de ciertos aspectos de la teor ́ıa de dominio o ́ptimo, que se ha desarrollado de manera importante en los u ́ltimos tiempos, principalmente en la direccio ́n de caracterizar cua ́l es el mayor espacio de funciones de Banach al que se pueden extender algunos operadores de importancia en ana ́lisis. Adema ́s, permite clasificar laspropiedadesdelosoperadoresqueseconservanenlaextensio ́nenfuncio ́n delsubespaciodeL1(mT)queconstituyesudominioo ́ptimo.
Convex Analysis and Duality (based on "Functional Analysis and Optimization" ...Katsuya Ito
In this presentation, we explain the monograph ”Functional Analysis and Optimization” by Kazufumi Ito
https://kito.wordpress.ncsu.edu/files/2018/04/funa3.pdf
Our goal in this presentation is to
-Understand the basic notions of functional analysis
lower-semicontinuous, subdifferential, conjugate functional
- Understand the formulation of duality problem
primal (P), perturbed (Py), and dual (P∗) problem
-Understand the primal-dual relationships
inf(P)≤sup(P∗), inf(P) = sup(P∗), inf supL≤sup inf L
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
On Extendable Sets in the Reals (R) With Application to the Lyapunov Stabilit...BRNSS Publication Hub
This work produces the authors’ own concept for the definition of extension on R alongside a basic result he tagged the basic extension fact for R. This was continued with the review of existing definitions and theorems on extension prominent among which are the Urysohn’s lemma and the Tietze extension theorem which we exhaustively discussed, and in conclusion, this was applied extensively in resolving proofs of some important results bordering on the comparison principle of Lyapunov stability theory in ordinary differential equation. To start this work, an introduction to the concept of real numbers was reviewed as a definition on which this work was founded.
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Several quantum optical models, such as, coherent states, cat states and squeezed states are constructed in a noncommutative space arising from the generalised uncertainty relation. We explore some advantages of utilising noncommutative models by comparing the nonclassicality and entanglement properties with that of the usual quantum mechanical systems.
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When data are missing at random (MAR), complete-case analysis with the full-data estimating equation is in general not valid. To correct the bias, we can employ the inverse probability weighting (IPW) technique on the complete cases. This requires modeling the missing pattern on the observed data (call it the $\pi$ model). The resulting IPW estimator, however, ignores information contained in cases with missing components, and is thus statistically inefficient. Efficiency can be improved by modifying the estimating equation along the lines of the semiparametric efficiency theory of Bickel et al. (1993). This modification usually requires modeling the distribution of the missing component on the observed ones (call it the $\mu$ model). Hence, when both the $\pi$ and the $\mu$ models are correct, the modified estimator is valid and is more efficient than the IPW one. In addition, the modified estimator is "doubly robust" in the sense that it is valid when either the $\pi$ model or the $\mu$ model is correct.
Essential materials of the slides are extracted from the book "Semiparametric Theory and Missing Data" (Tsiatis, 2006). The slides were originally presented in the class BIOS 773 Statistical Analysis with Missing Data in Spring 2013 at UNC Chapel Hill as a final project.
Vector measures and classical disjointification methods...esasancpe
In this talk we show how the classical disjointification methods (Bessaga-Pelczynski, Kadecs-Pelczynski) can be applied in the setting of the spaces of p-integrable functions with respect to vector measures. These spaces provide in fact a representation of p-convex order continuous Banach lattices with weak unit; the additional tool of the vector valued integral for each function has already shown to be fruitful for the analysis of these spaces. Consequently, our results can be directly extended to a broad class of Banach lattices.
Convex Analysis and Duality (based on "Functional Analysis and Optimization" ...Katsuya Ito
In this presentation, we explain the monograph ”Functional Analysis and Optimization” by Kazufumi Ito
https://kito.wordpress.ncsu.edu/files/2018/04/funa3.pdf
Our goal in this presentation is to
-Understand the basic notions of functional analysis
lower-semicontinuous, subdifferential, conjugate functional
- Understand the formulation of duality problem
primal (P), perturbed (Py), and dual (P∗) problem
-Understand the primal-dual relationships
inf(P)≤sup(P∗), inf(P) = sup(P∗), inf supL≤sup inf L
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
On Extendable Sets in the Reals (R) With Application to the Lyapunov Stabilit...BRNSS Publication Hub
This work produces the authors’ own concept for the definition of extension on R alongside a basic result he tagged the basic extension fact for R. This was continued with the review of existing definitions and theorems on extension prominent among which are the Urysohn’s lemma and the Tietze extension theorem which we exhaustively discussed, and in conclusion, this was applied extensively in resolving proofs of some important results bordering on the comparison principle of Lyapunov stability theory in ordinary differential equation. To start this work, an introduction to the concept of real numbers was reviewed as a definition on which this work was founded.
Quantum optical models in noncommutative spacesSanjib Dey
Several quantum optical models, such as, coherent states, cat states and squeezed states are constructed in a noncommutative space arising from the generalised uncertainty relation. We explore some advantages of utilising noncommutative models by comparing the nonclassicality and entanglement properties with that of the usual quantum mechanical systems.
Double Robustness: Theory and Applications with Missing DataLu Mao
When data are missing at random (MAR), complete-case analysis with the full-data estimating equation is in general not valid. To correct the bias, we can employ the inverse probability weighting (IPW) technique on the complete cases. This requires modeling the missing pattern on the observed data (call it the $\pi$ model). The resulting IPW estimator, however, ignores information contained in cases with missing components, and is thus statistically inefficient. Efficiency can be improved by modifying the estimating equation along the lines of the semiparametric efficiency theory of Bickel et al. (1993). This modification usually requires modeling the distribution of the missing component on the observed ones (call it the $\mu$ model). Hence, when both the $\pi$ and the $\mu$ models are correct, the modified estimator is valid and is more efficient than the IPW one. In addition, the modified estimator is "doubly robust" in the sense that it is valid when either the $\pi$ model or the $\mu$ model is correct.
Essential materials of the slides are extracted from the book "Semiparametric Theory and Missing Data" (Tsiatis, 2006). The slides were originally presented in the class BIOS 773 Statistical Analysis with Missing Data in Spring 2013 at UNC Chapel Hill as a final project.
Vector measures and classical disjointification methods...esasancpe
In this talk we show how the classical disjointification methods (Bessaga-Pelczynski, Kadecs-Pelczynski) can be applied in the setting of the spaces of p-integrable functions with respect to vector measures. These spaces provide in fact a representation of p-convex order continuous Banach lattices with weak unit; the additional tool of the vector valued integral for each function has already shown to be fruitful for the analysis of these spaces. Consequently, our results can be directly extended to a broad class of Banach lattices.
Compactness and homogeneous maps on Banach function spacesesasancpe
The aim of this talk is to show how compact operators between Banach function spaces can be approximated by means of homogeneous maps. After explaining the special characterizations of compact and weakly compact sets that are known for Banach function spaces, we develop a factorization method for describing lattice and topological properties of homogeneous maps by approximating with “simple” homogeneous maps. No approximation properties for the spaces involved are needed. A canonical homogeneous map that is defined as φp(f ) := |f |1/p · ∥f ∥1/p′ from a Banach function space X into its p-th power X[p] plays a meaningful role.
This work has its roots in some classical descriptions of weakly compact subsets of Banach spaces (Grothendieck, Fremlin,...), but particular Banach lattice tools (p-convexification, Maurey-Rosenthal type theorems) are also required.
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Disjointification methods in spaces of integrable functions with respect to a vector measure
1. Disjointification methods in spaces of integrable functions
with respect to a vector measure
E. A. S´anchez P´erez (joint work with E. Jim´enez)
Instituto Universitario de Matem´atica Pura y Aplicada
(I.U.M.P.A.).
Departamento de Matem´atica Aplicada.
Universidad Polit´ecnica de Valencia.
Camino de Vera S/N, 46022, Valencia.
Integration, Vector Measures and Related Topics V
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
2. Motivation
Several procedures regarding topological and geometric properties
of spaces of integrable functions with respect to a vector measure
and the structure of their subspaces have their roots in classical
disjointification methods (Bessaga-Pelczynski, Kadecs-Pelczynski).
We show how these techniques can be used in the setting of the
applications of the vector valued integration to the theory of
operators on Banach function spaces.
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
3. Let m : Σ → E be a vector measure. Consider the space L2(m).
1 The first result (Bessaga-Pelczynski) allows to work with orthogonality notions in
the range space: orthogonal integrals.
2 The second one (Kadecs-Pelczynski) provides the tools for analyzing disjoint
functions in L2(m).
3 Combining both results: structure of subspaces in L2(m).
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
4. Let m : Σ → E be a vector measure. Consider the space L2(m).
1 The first result (Bessaga-Pelczynski) allows to work with orthogonality notions in
the range space: orthogonal integrals.
2 The second one (Kadecs-Pelczynski) provides the tools for analyzing disjoint
functions in L2(m).
3 Combining both results: structure of subspaces in L2(m).
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
5. Let m : Σ → E be a vector measure. Consider the space L2(m).
1 The first result (Bessaga-Pelczynski) allows to work with orthogonality notions in
the range space: orthogonal integrals.
2 The second one (Kadecs-Pelczynski) provides the tools for analyzing disjoint
functions in L2(m).
3 Combining both results: structure of subspaces in L2(m).
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
6. Let m : Σ → E be a vector measure. Consider the space L2(m).
1 The first result (Bessaga-Pelczynski) allows to work with orthogonality notions in
the range space: orthogonal integrals.
2 The second one (Kadecs-Pelczynski) provides the tools for analyzing disjoint
functions in L2(m).
3 Combining both results: structure of subspaces in L2(m).
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
7. DEFINITIONS: Banach function spaces
(Ω,Σ,µ) be a finite measure space.
L0(µ) space of all (classes of) measurable real functions on Ω.
A Banach function space (briefly B.f.s.) is a Banach space X ⊂ L0(µ) of locally
integrable functions with norm · X such that if f ∈ L0(µ), g ∈ X and |f | ≤ |g|
µ-a.e. then f ∈ X and f X ≤ g X .
A B.f.s. X has the Fatou property if for every sequence (fn) ⊂ X such that
0 ≤ fn ↑ f µ-a.e. and supn fn X < ∞, it follows that f ∈ X and fn X ↑ f X .
We will say that X is order continuous if for every f ,fn ∈ X such that 0 ≤ fn ↑ f
µ-a.e., we have that fn → f in norm.
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
8. DEFINITIONS: Banach function spaces
(Ω,Σ,µ) be a finite measure space.
L0(µ) space of all (classes of) measurable real functions on Ω.
A Banach function space (briefly B.f.s.) is a Banach space X ⊂ L0(µ) of locally
integrable functions with norm · X such that if f ∈ L0(µ), g ∈ X and |f | ≤ |g|
µ-a.e. then f ∈ X and f X ≤ g X .
A B.f.s. X has the Fatou property if for every sequence (fn) ⊂ X such that
0 ≤ fn ↑ f µ-a.e. and supn fn X < ∞, it follows that f ∈ X and fn X ↑ f X .
We will say that X is order continuous if for every f ,fn ∈ X such that 0 ≤ fn ↑ f
µ-a.e., we have that fn → f in norm.
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
9. DEFINITIONS: Banach function spaces
(Ω,Σ,µ) be a finite measure space.
L0(µ) space of all (classes of) measurable real functions on Ω.
A Banach function space (briefly B.f.s.) is a Banach space X ⊂ L0(µ) of locally
integrable functions with norm · X such that if f ∈ L0(µ), g ∈ X and |f | ≤ |g|
µ-a.e. then f ∈ X and f X ≤ g X .
A B.f.s. X has the Fatou property if for every sequence (fn) ⊂ X such that
0 ≤ fn ↑ f µ-a.e. and supn fn X < ∞, it follows that f ∈ X and fn X ↑ f X .
We will say that X is order continuous if for every f ,fn ∈ X such that 0 ≤ fn ↑ f
µ-a.e., we have that fn → f in norm.
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
10. DEFINITIONS: Banach function spaces
(Ω,Σ,µ) be a finite measure space.
L0(µ) space of all (classes of) measurable real functions on Ω.
A Banach function space (briefly B.f.s.) is a Banach space X ⊂ L0(µ) of locally
integrable functions with norm · X such that if f ∈ L0(µ), g ∈ X and |f | ≤ |g|
µ-a.e. then f ∈ X and f X ≤ g X .
A B.f.s. X has the Fatou property if for every sequence (fn) ⊂ X such that
0 ≤ fn ↑ f µ-a.e. and supn fn X < ∞, it follows that f ∈ X and fn X ↑ f X .
We will say that X is order continuous if for every f ,fn ∈ X such that 0 ≤ fn ↑ f
µ-a.e., we have that fn → f in norm.
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
11. DEFINITIONS: Banach function spaces
(Ω,Σ,µ) be a finite measure space.
L0(µ) space of all (classes of) measurable real functions on Ω.
A Banach function space (briefly B.f.s.) is a Banach space X ⊂ L0(µ) of locally
integrable functions with norm · X such that if f ∈ L0(µ), g ∈ X and |f | ≤ |g|
µ-a.e. then f ∈ X and f X ≤ g X .
A B.f.s. X has the Fatou property if for every sequence (fn) ⊂ X such that
0 ≤ fn ↑ f µ-a.e. and supn fn X < ∞, it follows that f ∈ X and fn X ↑ f X .
We will say that X is order continuous if for every f ,fn ∈ X such that 0 ≤ fn ↑ f
µ-a.e., we have that fn → f in norm.
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
12. DEFINITIONS: Banach function spaces
(Ω,Σ,µ) be a finite measure space.
L0(µ) space of all (classes of) measurable real functions on Ω.
A Banach function space (briefly B.f.s.) is a Banach space X ⊂ L0(µ) of locally
integrable functions with norm · X such that if f ∈ L0(µ), g ∈ X and |f | ≤ |g|
µ-a.e. then f ∈ X and f X ≤ g X .
A B.f.s. X has the Fatou property if for every sequence (fn) ⊂ X such that
0 ≤ fn ↑ f µ-a.e. and supn fn X < ∞, it follows that f ∈ X and fn X ↑ f X .
We will say that X is order continuous if for every f ,fn ∈ X such that 0 ≤ fn ↑ f
µ-a.e., we have that fn → f in norm.
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
13. DEFINITIONS: Vector measures and integration
Let m : Σ → E be a vector measure, that is, a countably additive set function,
where E is a real Banach space.
A set A ∈ Σ is m-null if m(B) = 0 for every B ∈ Σ with B ⊂ A. For each x∗ in the
topological dual E∗ of E, we denote by |x∗m| the variation of the real measure
x∗m given by the composition of m with x∗. There exists x∗
0 ∈ E∗ such that |x∗
0 m|
has the same null sets as m. We will call |x∗
0 m| a Rybakov control measure for m.
A measurable function f : Ω → R is integrable with respect to m if
(i) |f |d|x∗
m| < ∞ for all x∗
∈ E∗
.
(ii) For each A ∈ Σ, there exists xA ∈ E such that
x∗
(xA) =
A
f dx∗
m, for all x∗
∈ E.
The element xA will be written as A f dm.
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
14. DEFINITIONS: Vector measures and integration
Let m : Σ → E be a vector measure, that is, a countably additive set function,
where E is a real Banach space.
A set A ∈ Σ is m-null if m(B) = 0 for every B ∈ Σ with B ⊂ A. For each x∗ in the
topological dual E∗ of E, we denote by |x∗m| the variation of the real measure
x∗m given by the composition of m with x∗. There exists x∗
0 ∈ E∗ such that |x∗
0 m|
has the same null sets as m. We will call |x∗
0 m| a Rybakov control measure for m.
A measurable function f : Ω → R is integrable with respect to m if
(i) |f |d|x∗
m| < ∞ for all x∗
∈ E∗
.
(ii) For each A ∈ Σ, there exists xA ∈ E such that
x∗
(xA) =
A
f dx∗
m, for all x∗
∈ E.
The element xA will be written as A f dm.
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
15. DEFINITIONS: Vector measures and integration
Let m : Σ → E be a vector measure, that is, a countably additive set function,
where E is a real Banach space.
A set A ∈ Σ is m-null if m(B) = 0 for every B ∈ Σ with B ⊂ A. For each x∗ in the
topological dual E∗ of E, we denote by |x∗m| the variation of the real measure
x∗m given by the composition of m with x∗. There exists x∗
0 ∈ E∗ such that |x∗
0 m|
has the same null sets as m. We will call |x∗
0 m| a Rybakov control measure for m.
A measurable function f : Ω → R is integrable with respect to m if
(i) |f |d|x∗
m| < ∞ for all x∗
∈ E∗
.
(ii) For each A ∈ Σ, there exists xA ∈ E such that
x∗
(xA) =
A
f dx∗
m, for all x∗
∈ E.
The element xA will be written as A f dm.
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
16. DEFINITIONS: Vector measures and integration
Let m : Σ → E be a vector measure, that is, a countably additive set function,
where E is a real Banach space.
A set A ∈ Σ is m-null if m(B) = 0 for every B ∈ Σ with B ⊂ A. For each x∗ in the
topological dual E∗ of E, we denote by |x∗m| the variation of the real measure
x∗m given by the composition of m with x∗. There exists x∗
0 ∈ E∗ such that |x∗
0 m|
has the same null sets as m. We will call |x∗
0 m| a Rybakov control measure for m.
A measurable function f : Ω → R is integrable with respect to m if
(i) |f |d|x∗
m| < ∞ for all x∗
∈ E∗
.
(ii) For each A ∈ Σ, there exists xA ∈ E such that
x∗
(xA) =
A
f dx∗
m, for all x∗
∈ E.
The element xA will be written as A f dm.
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
17. DEFINITIONS: Vector measures and integration
Let m : Σ → E be a vector measure, that is, a countably additive set function,
where E is a real Banach space.
A set A ∈ Σ is m-null if m(B) = 0 for every B ∈ Σ with B ⊂ A. For each x∗ in the
topological dual E∗ of E, we denote by |x∗m| the variation of the real measure
x∗m given by the composition of m with x∗. There exists x∗
0 ∈ E∗ such that |x∗
0 m|
has the same null sets as m. We will call |x∗
0 m| a Rybakov control measure for m.
A measurable function f : Ω → R is integrable with respect to m if
(i) |f |d|x∗
m| < ∞ for all x∗
∈ E∗
.
(ii) For each A ∈ Σ, there exists xA ∈ E such that
x∗
(xA) =
A
f dx∗
m, for all x∗
∈ E.
The element xA will be written as A f dm.
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
18. DEFINITIONS: Spaces of integrable functions
Denote by L1(m) the space of integrable functions with respect to m, where
functions which are equal m-a.e. are identified.
The space L1(m) is a Banach space endowed with the norm
f m = sup
x∗∈BE∗
|f |d|x∗
m|.
Note that L∞(|x∗
0 m|) ⊂ L1(m). In particular every measure of the type |x∗m| is
finite as |x∗m|(Ω) ≤ x∗ · χΩ m.
Given f ∈ L1(m), the set function mf : Σ → E given by mf (A) = A f dm for all
A ∈ Σ is a vector measure. Moreover, g ∈ L1(mf ) if and only if gf ∈ L1(m) and in
this case g dmf = gf dm.
The space of square integrable functions L2(m) is defined by the set of measurable
functions f such that f 2 ∈ L1(m). It is a Banach function space with the norm
f := f 2 1/2
L1(m)
.
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
19. DEFINITIONS: Spaces of integrable functions
Denote by L1(m) the space of integrable functions with respect to m, where
functions which are equal m-a.e. are identified.
The space L1(m) is a Banach space endowed with the norm
f m = sup
x∗∈BE∗
|f |d|x∗
m|.
Note that L∞(|x∗
0 m|) ⊂ L1(m). In particular every measure of the type |x∗m| is
finite as |x∗m|(Ω) ≤ x∗ · χΩ m.
Given f ∈ L1(m), the set function mf : Σ → E given by mf (A) = A f dm for all
A ∈ Σ is a vector measure. Moreover, g ∈ L1(mf ) if and only if gf ∈ L1(m) and in
this case g dmf = gf dm.
The space of square integrable functions L2(m) is defined by the set of measurable
functions f such that f 2 ∈ L1(m). It is a Banach function space with the norm
f := f 2 1/2
L1(m)
.
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
20. DEFINITIONS: Spaces of integrable functions
Denote by L1(m) the space of integrable functions with respect to m, where
functions which are equal m-a.e. are identified.
The space L1(m) is a Banach space endowed with the norm
f m = sup
x∗∈BE∗
|f |d|x∗
m|.
Note that L∞(|x∗
0 m|) ⊂ L1(m). In particular every measure of the type |x∗m| is
finite as |x∗m|(Ω) ≤ x∗ · χΩ m.
Given f ∈ L1(m), the set function mf : Σ → E given by mf (A) = A f dm for all
A ∈ Σ is a vector measure. Moreover, g ∈ L1(mf ) if and only if gf ∈ L1(m) and in
this case g dmf = gf dm.
The space of square integrable functions L2(m) is defined by the set of measurable
functions f such that f 2 ∈ L1(m). It is a Banach function space with the norm
f := f 2 1/2
L1(m)
.
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
21. DEFINITIONS: Spaces of integrable functions
Denote by L1(m) the space of integrable functions with respect to m, where
functions which are equal m-a.e. are identified.
The space L1(m) is a Banach space endowed with the norm
f m = sup
x∗∈BE∗
|f |d|x∗
m|.
Note that L∞(|x∗
0 m|) ⊂ L1(m). In particular every measure of the type |x∗m| is
finite as |x∗m|(Ω) ≤ x∗ · χΩ m.
Given f ∈ L1(m), the set function mf : Σ → E given by mf (A) = A f dm for all
A ∈ Σ is a vector measure. Moreover, g ∈ L1(mf ) if and only if gf ∈ L1(m) and in
this case g dmf = gf dm.
The space of square integrable functions L2(m) is defined by the set of measurable
functions f such that f 2 ∈ L1(m). It is a Banach function space with the norm
f := f 2 1/2
L1(m)
.
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
22. DEFINITIONS: Spaces of integrable functions
Denote by L1(m) the space of integrable functions with respect to m, where
functions which are equal m-a.e. are identified.
The space L1(m) is a Banach space endowed with the norm
f m = sup
x∗∈BE∗
|f |d|x∗
m|.
Note that L∞(|x∗
0 m|) ⊂ L1(m). In particular every measure of the type |x∗m| is
finite as |x∗m|(Ω) ≤ x∗ · χΩ m.
Given f ∈ L1(m), the set function mf : Σ → E given by mf (A) = A f dm for all
A ∈ Σ is a vector measure. Moreover, g ∈ L1(mf ) if and only if gf ∈ L1(m) and in
this case g dmf = gf dm.
The space of square integrable functions L2(m) is defined by the set of measurable
functions f such that f 2 ∈ L1(m). It is a Banach function space with the norm
f := f 2 1/2
L1(m)
.
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
23. DEFINITIONS: Spaces of integrable functions
The space of square integrable functions L2(m) is defined by the set of measurable
functions f such that f 2 ∈ L1(m). It is a Banach function space with the norm
f := f 2 1/2
L1(m)
.
L2(m)·L2(m) ⊆ L1(m), and
f ·gdm ≤ f L2(m) · g L2(m).
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
24. DEFINITIONS: Spaces of integrable functions
The space of square integrable functions L2(m) is defined by the set of measurable
functions f such that f 2 ∈ L1(m). It is a Banach function space with the norm
f := f 2 1/2
L1(m)
.
L2(m)·L2(m) ⊆ L1(m), and
f ·gdm ≤ f L2(m) · g L2(m).
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
25. Disjointification procedures:
Bessaga-Pelczynski Selection Principle. If {xn}∞
n=1 is a basis of the Banach space
X and {x∗
n }∞
n=1 is the sequence of coefficient functionals, if we take a normalized
sequence {yn}∞
n=1 that is weakly null, then {yn}∞
n=1 admits a basic subsequence
that is equivalent to a block basic sequence of {xn}∞
n=1
Kadec-Pelczynski Disjointification Procedure / Dichotomy.
M. I. Kadec and A. Pelczynski, Bases, lacunary sequences, and complemented
subspaces in the spaces Lp , Studia Math. (1962)
THEOREM 4.1. (Figiel-Johnson-Tzafriri, 1975)
Let L be a σ-complete and σ-order continuous Banach lattice. Suppose X is a
subspace of L. Either X is isomorphic to a subspace of L1(µ) for some measure µ or
there is a sequence (xi ) of unit vectors in X and a disjointly supported sequence (ei )
in L with xi −ei → 0. Consequently, every subspace of L contains an unconditional
basic sequence.
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
26. Disjointification procedures:
Bessaga-Pelczynski Selection Principle. If {xn}∞
n=1 is a basis of the Banach space
X and {x∗
n }∞
n=1 is the sequence of coefficient functionals, if we take a normalized
sequence {yn}∞
n=1 that is weakly null, then {yn}∞
n=1 admits a basic subsequence
that is equivalent to a block basic sequence of {xn}∞
n=1
Kadec-Pelczynski Disjointification Procedure / Dichotomy.
M. I. Kadec and A. Pelczynski, Bases, lacunary sequences, and complemented
subspaces in the spaces Lp , Studia Math. (1962)
THEOREM 4.1. (Figiel-Johnson-Tzafriri, 1975)
Let L be a σ-complete and σ-order continuous Banach lattice. Suppose X is a
subspace of L. Either X is isomorphic to a subspace of L1(µ) for some measure µ or
there is a sequence (xi ) of unit vectors in X and a disjointly supported sequence (ei )
in L with xi −ei → 0. Consequently, every subspace of L contains an unconditional
basic sequence.
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
27. Disjointification procedures:
Bessaga-Pelczynski Selection Principle. If {xn}∞
n=1 is a basis of the Banach space
X and {x∗
n }∞
n=1 is the sequence of coefficient functionals, if we take a normalized
sequence {yn}∞
n=1 that is weakly null, then {yn}∞
n=1 admits a basic subsequence
that is equivalent to a block basic sequence of {xn}∞
n=1
Kadec-Pelczynski Disjointification Procedure / Dichotomy.
M. I. Kadec and A. Pelczynski, Bases, lacunary sequences, and complemented
subspaces in the spaces Lp , Studia Math. (1962)
THEOREM 4.1. (Figiel-Johnson-Tzafriri, 1975)
Let L be a σ-complete and σ-order continuous Banach lattice. Suppose X is a
subspace of L. Either X is isomorphic to a subspace of L1(µ) for some measure µ or
there is a sequence (xi ) of unit vectors in X and a disjointly supported sequence (ei )
in L with xi −ei → 0. Consequently, every subspace of L contains an unconditional
basic sequence.
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
28. Disjointification procedures:
Bessaga-Pelczynski Selection Principle. If {xn}∞
n=1 is a basis of the Banach space
X and {x∗
n }∞
n=1 is the sequence of coefficient functionals, if we take a normalized
sequence {yn}∞
n=1 that is weakly null, then {yn}∞
n=1 admits a basic subsequence
that is equivalent to a block basic sequence of {xn}∞
n=1
Kadec-Pelczynski Disjointification Procedure / Dichotomy.
M. I. Kadec and A. Pelczynski, Bases, lacunary sequences, and complemented
subspaces in the spaces Lp , Studia Math. (1962)
THEOREM 4.1. (Figiel-Johnson-Tzafriri, 1975)
Let L be a σ-complete and σ-order continuous Banach lattice. Suppose X is a
subspace of L. Either X is isomorphic to a subspace of L1(µ) for some measure µ or
there is a sequence (xi ) of unit vectors in X and a disjointly supported sequence (ei )
in L with xi −ei → 0. Consequently, every subspace of L contains an unconditional
basic sequence.
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
29. Kadec-Pelczynski Disjointification Procedure for sequences.
Let X(µ) be an order continuous Banach function space over a finite measure µ with
a weak unit (this implies X(µ) → L1(µ)). Consider a normalized sequence {xn}∞
n=1 in
X(µ). Then
(1) either { xn L1(µ)}∞
n=1 is bounded away from zero,
(2) or there exist a subsequence {xnk
}∞
k=1 and a disjoint sequence {zk }∞
k=1 in X(µ)
such that zk −xnk
→k 0.
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
30. Kadec-Pelczynski Disjointification Procedure for sequences.
Let X(µ) be an order continuous Banach function space over a finite measure µ with
a weak unit (this implies X(µ) → L1(µ)). Consider a normalized sequence {xn}∞
n=1 in
X(µ). Then
(1) either { xn L1(µ)}∞
n=1 is bounded away from zero,
(2) or there exist a subsequence {xnk
}∞
k=1 and a disjoint sequence {zk }∞
k=1 in X(µ)
such that zk −xnk
→k 0.
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
31. Kadec-Pelczynski Disjointification Procedure for sequences.
Let X(µ) be an order continuous Banach function space over a finite measure µ with
a weak unit (this implies X(µ) → L1(µ)). Consider a normalized sequence {xn}∞
n=1 in
X(µ). Then
(1) either { xn L1(µ)}∞
n=1 is bounded away from zero,
(2) or there exist a subsequence {xnk
}∞
k=1 and a disjoint sequence {zk }∞
k=1 in X(µ)
such that zk −xnk
→k 0.
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
32. m-orthonormal sequences in L2(m):
Definition.
A sequence {fi }∞
i=1 in L2(m) is called m-orthogonal if fi fj dm = δi,j ki for positive
constants ki . If fi L2(m) = 1 for all i ∈ N, it is called m-orthonormal.
Definition.
Let m : Σ −→ 2 be a vector measure. We say that {fi }∞
i=1 ⊂ L2(m) is a strongly
m-orthogonal sequence if fi fj dm = δi,j ei ki for an orthonormal sequence {ei }∞
i=1 in 2
and for ki > 0. If ki = 1 for every i ∈ N, we say that it is a strongly m-orthonormal
sequence.
Definition. A function f ∈ L2(m) is normed by the integral if f L2(m) = Ω f 2 dm 1/2.
This happens for all the functions in L2(m) if the measure m is positive.
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
33. m-orthonormal sequences in L2(m):
Definition.
A sequence {fi }∞
i=1 in L2(m) is called m-orthogonal if fi fj dm = δi,j ki for positive
constants ki . If fi L2(m) = 1 for all i ∈ N, it is called m-orthonormal.
Definition.
Let m : Σ −→ 2 be a vector measure. We say that {fi }∞
i=1 ⊂ L2(m) is a strongly
m-orthogonal sequence if fi fj dm = δi,j ei ki for an orthonormal sequence {ei }∞
i=1 in 2
and for ki > 0. If ki = 1 for every i ∈ N, we say that it is a strongly m-orthonormal
sequence.
Definition. A function f ∈ L2(m) is normed by the integral if f L2(m) = Ω f 2 dm 1/2.
This happens for all the functions in L2(m) if the measure m is positive.
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
34. m-orthonormal sequences in L2(m):
Definition.
A sequence {fi }∞
i=1 in L2(m) is called m-orthogonal if fi fj dm = δi,j ki for positive
constants ki . If fi L2(m) = 1 for all i ∈ N, it is called m-orthonormal.
Definition.
Let m : Σ −→ 2 be a vector measure. We say that {fi }∞
i=1 ⊂ L2(m) is a strongly
m-orthogonal sequence if fi fj dm = δi,j ei ki for an orthonormal sequence {ei }∞
i=1 in 2
and for ki > 0. If ki = 1 for every i ∈ N, we say that it is a strongly m-orthonormal
sequence.
Definition. A function f ∈ L2(m) is normed by the integral if f L2(m) = Ω f 2 dm 1/2.
This happens for all the functions in L2(m) if the measure m is positive.
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
35. m-orthonormal sequences in L2(m):
Definition.
A sequence {fi }∞
i=1 in L2(m) is called m-orthogonal if fi fj dm = δi,j ki for positive
constants ki . If fi L2(m) = 1 for all i ∈ N, it is called m-orthonormal.
Definition.
Let m : Σ −→ 2 be a vector measure. We say that {fi }∞
i=1 ⊂ L2(m) is a strongly
m-orthogonal sequence if fi fj dm = δi,j ei ki for an orthonormal sequence {ei }∞
i=1 in 2
and for ki > 0. If ki = 1 for every i ∈ N, we say that it is a strongly m-orthonormal
sequence.
Definition. A function f ∈ L2(m) is normed by the integral if f L2(m) = Ω f 2 dm 1/2.
This happens for all the functions in L2(m) if the measure m is positive.
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
36. Example.
Let ([0,∞),Σ,µ) be Lebesgue measure space. Let rk (x) := sign{sin(2k−1x)} be the
Rademacher function of period 2π defined at the interval Ek = [2(k −1)π,2kπ], k ∈ N.
Consider the vector measure m : Σ → 2 given by m(A) := ∑∞
k=1
−1
2k ( A∩Ek
rk dµ)ek ∈ 2,
A ∈ Σ.
Note that if f ∈ L2(m) then [0,∞) fdm = ( −1
2k Ek
frk dµ)k ∈ 2. Consider the sequence of
functions
f1(x) = sin(x)· χ[π,2π](x)
f2(x) = sin(2x)· χ[0,2π](x)+ χ[ 7
2 π,4π](x)
f3(x) = sin(4x)· χ[0,4π](x)+ χ[ 23
4 π,6π](x)
...
fn(x) = sin(2n−1
x)· χ[0,2(n−1)π](x)+ χ[(2n− 2
2n )π,2nπ](x) , n ≥ 2.
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
37. Example.
Let ([0,∞),Σ,µ) be Lebesgue measure space. Let rk (x) := sign{sin(2k−1x)} be the
Rademacher function of period 2π defined at the interval Ek = [2(k −1)π,2kπ], k ∈ N.
Consider the vector measure m : Σ → 2 given by m(A) := ∑∞
k=1
−1
2k ( A∩Ek
rk dµ)ek ∈ 2,
A ∈ Σ.
Note that if f ∈ L2(m) then [0,∞) fdm = ( −1
2k Ek
frk dµ)k ∈ 2. Consider the sequence of
functions
f1(x) = sin(x)· χ[π,2π](x)
f2(x) = sin(2x)· χ[0,2π](x)+ χ[ 7
2 π,4π](x)
f3(x) = sin(4x)· χ[0,4π](x)+ χ[ 23
4 π,6π](x)
...
fn(x) = sin(2n−1
x)· χ[0,2(n−1)π](x)+ χ[(2n− 2
2n )π,2nπ](x) , n ≥ 2.
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
38. Proposition.
Let {gn}∞
n=1 be a normalized sequence in L2(m). Suppose that there exists a Rybakov
measure µ = | m,x0 | for m such that { gn L1(µ)}∞
n=1 is not bounded away from zero.
Then there are a subsequence {gnk
}∞
k=1 of {gn}∞
n=1 and an m-orthonormal sequence
{fk }∞
k=1 such that gnk
−fk L2(m) →k 0.
Theorem.
Let us consider a vector measure m : Σ → 2 and an m-orthonormal sequence {fn}∞
n=1
of functions in L2(m) that are normed by the integrals. Let {en}∞
n=1 be the canonical
basis of 2. If l´ımn f 2
n dm,ek = 0 for every k ∈ N, then there exists a subsequence
{fnk
}∞
k=1 of {fn}∞
n=1 and a vector measure m∗ : Σ → 2 such that {fnk
}∞
k=1 is strongly
m∗-orthonormal.
Moreover, m∗ can be chosen to be as m∗ = φ ◦m for some Banach space isomorphism
φ from 2 onto 2, and so L2(m) = L2(m∗).
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
39. Proposition.
Let {gn}∞
n=1 be a normalized sequence in L2(m). Suppose that there exists a Rybakov
measure µ = | m,x0 | for m such that { gn L1(µ)}∞
n=1 is not bounded away from zero.
Then there are a subsequence {gnk
}∞
k=1 of {gn}∞
n=1 and an m-orthonormal sequence
{fk }∞
k=1 such that gnk
−fk L2(m) →k 0.
Theorem.
Let us consider a vector measure m : Σ → 2 and an m-orthonormal sequence {fn}∞
n=1
of functions in L2(m) that are normed by the integrals. Let {en}∞
n=1 be the canonical
basis of 2. If l´ımn f 2
n dm,ek = 0 for every k ∈ N, then there exists a subsequence
{fnk
}∞
k=1 of {fn}∞
n=1 and a vector measure m∗ : Σ → 2 such that {fnk
}∞
k=1 is strongly
m∗-orthonormal.
Moreover, m∗ can be chosen to be as m∗ = φ ◦m for some Banach space isomorphism
φ from 2 onto 2, and so L2(m) = L2(m∗).
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
40. Proposition.
Let {gn}∞
n=1 be a normalized sequence in L2(m). Suppose that there exists a Rybakov
measure µ = | m,x0 | for m such that { gn L1(µ)}∞
n=1 is not bounded away from zero.
Then there are a subsequence {gnk
}∞
k=1 of {gn}∞
n=1 and an m-orthonormal sequence
{fk }∞
k=1 such that gnk
−fk L2(m) →k 0.
Theorem.
Let us consider a vector measure m : Σ → 2 and an m-orthonormal sequence {fn}∞
n=1
of functions in L2(m) that are normed by the integrals. Let {en}∞
n=1 be the canonical
basis of 2. If l´ımn f 2
n dm,ek = 0 for every k ∈ N, then there exists a subsequence
{fnk
}∞
k=1 of {fn}∞
n=1 and a vector measure m∗ : Σ → 2 such that {fnk
}∞
k=1 is strongly
m∗-orthonormal.
Moreover, m∗ can be chosen to be as m∗ = φ ◦m for some Banach space isomorphism
φ from 2 onto 2, and so L2(m) = L2(m∗).
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
41. Corollary.
Let m : Σ → 2 be a countably additive vector measure. Let {gn}∞
n=1 be a normalized
sequence of functions in L2(m) that are normed by the integrals. Suppose that there
exists a Rybakov measure µ = | m,x0 | for m such that { gn L1(µ)}∞
n=1 is not bounded
away from zero.
If l´ımn g2
n dm,ek = 0 for every k ∈ N, then there is a (disjoint) sequence {fk }∞
k=1
such that
(1) l´ımk gnk
−fk L2(m) = 0 for a given subsequence {gnk
}∞
k=1 of {gn}∞
n=1, and
(2) it is strongly m∗-orthonormal for a certain Hilbert space valued vector measure
m∗ defined as in the theorem that satisfies that L2(m) = L2(m∗).
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
42. Corollary.
Let m : Σ → 2 be a countably additive vector measure. Let {gn}∞
n=1 be a normalized
sequence of functions in L2(m) that are normed by the integrals. Suppose that there
exists a Rybakov measure µ = | m,x0 | for m such that { gn L1(µ)}∞
n=1 is not bounded
away from zero.
If l´ımn g2
n dm,ek = 0 for every k ∈ N, then there is a (disjoint) sequence {fk }∞
k=1
such that
(1) l´ımk gnk
−fk L2(m) = 0 for a given subsequence {gnk
}∞
k=1 of {gn}∞
n=1, and
(2) it is strongly m∗-orthonormal for a certain Hilbert space valued vector measure
m∗ defined as in the theorem that satisfies that L2(m) = L2(m∗).
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
43. Corollary.
Let m : Σ → 2 be a countably additive vector measure. Let {gn}∞
n=1 be a normalized
sequence of functions in L2(m) that are normed by the integrals. Suppose that there
exists a Rybakov measure µ = | m,x0 | for m such that { gn L1(µ)}∞
n=1 is not bounded
away from zero.
If l´ımn g2
n dm,ek = 0 for every k ∈ N, then there is a (disjoint) sequence {fk }∞
k=1
such that
(1) l´ımk gnk
−fk L2(m) = 0 for a given subsequence {gnk
}∞
k=1 of {gn}∞
n=1, and
(2) it is strongly m∗-orthonormal for a certain Hilbert space valued vector measure
m∗ defined as in the theorem that satisfies that L2(m) = L2(m∗).
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
44. Corollary.
Let m : Σ → 2 be a countably additive vector measure. Let {gn}∞
n=1 be a normalized
sequence of functions in L2(m) that are normed by the integrals. Suppose that there
exists a Rybakov measure µ = | m,x0 | for m such that { gn L1(µ)}∞
n=1 is not bounded
away from zero.
If l´ımn g2
n dm,ek = 0 for every k ∈ N, then there is a (disjoint) sequence {fk }∞
k=1
such that
(1) l´ımk gnk
−fk L2(m) = 0 for a given subsequence {gnk
}∞
k=1 of {gn}∞
n=1, and
(2) it is strongly m∗-orthonormal for a certain Hilbert space valued vector measure
m∗ defined as in the theorem that satisfies that L2(m) = L2(m∗).
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
45. Consequences on the structure of L1(m):
Lemma.
Let m : Σ → 2 be a positive vector measure, and suppose that the bounded sequence
{gn}∞
n=1 in L2(m) satisfies that l´ımn g2
n dm,ek = 0 for all k ∈ N. Then there is a
Rybakov measure µ for m such that l´ımn gn L1(µ) = 0.
Proposition.
Let m : Σ → 2 be a positive (countably additive) vector measure. Let {gn}∞
n=1 be a
normalized sequence in L2(m) such that for every k ∈ N, l´ımn g2
n dm,ek = 0. Then
L2(m) contains a lattice copy of 4. In particular, there is a subsequence {gnk
}∞
k=1 of
{gn}∞
n=1 that is equivalent to the unit vector basis of 4.
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
46. Consequences on the structure of L1(m):
Lemma.
Let m : Σ → 2 be a positive vector measure, and suppose that the bounded sequence
{gn}∞
n=1 in L2(m) satisfies that l´ımn g2
n dm,ek = 0 for all k ∈ N. Then there is a
Rybakov measure µ for m such that l´ımn gn L1(µ) = 0.
Proposition.
Let m : Σ → 2 be a positive (countably additive) vector measure. Let {gn}∞
n=1 be a
normalized sequence in L2(m) such that for every k ∈ N, l´ımn g2
n dm,ek = 0. Then
L2(m) contains a lattice copy of 4. In particular, there is a subsequence {gnk
}∞
k=1 of
{gn}∞
n=1 that is equivalent to the unit vector basis of 4.
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
47. Consequences on the structure of L1(m):
Lemma.
Let m : Σ → 2 be a positive vector measure, and suppose that the bounded sequence
{gn}∞
n=1 in L2(m) satisfies that l´ımn g2
n dm,ek = 0 for all k ∈ N. Then there is a
Rybakov measure µ for m such that l´ımn gn L1(µ) = 0.
Proposition.
Let m : Σ → 2 be a positive (countably additive) vector measure. Let {gn}∞
n=1 be a
normalized sequence in L2(m) such that for every k ∈ N, l´ımn g2
n dm,ek = 0. Then
L2(m) contains a lattice copy of 4. In particular, there is a subsequence {gnk
}∞
k=1 of
{gn}∞
n=1 that is equivalent to the unit vector basis of 4.
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
48. Theorem.
Let m : Σ → 2 be a positive (countably additive) vector measure. Let {gn}∞
n=1 be a
normalized sequence in L2(m) such that for every k ∈ N,
l´ım
n
ek , g2
n dm = 0
for all k ∈ N. Then there is a subsequence {gnk
}∞
k=1 such that {g2
nk
}∞
k=1 generates an
isomorphic copy of 2 in L1(m) that is preserved by the integration map. Moreover,
there is a normalized disjoint sequence {fk }∞
k=1 that is equivalent to the previous one
and {f 2
k }∞
k=1 gives a lattice copy of 2 in L1(m) that is preserved by Im∗ .
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
49. Corollary.
Let m : Σ → 2 be a positive (countably additive) vector measure. The following
assertions are equivalent:
(1) There is a normalized sequence in L2(m) satisfying that l´ımn Ω g2
n dm,ek = 0 for
all the elements of the canonical basis {ek }∞
k=1 of 2.
(2) There is an 2-valued vector measure m∗ = φ ◦m —φ an isomorphism— such that
L2(m) = L2(m∗) and there is a disjoint sequence in L2(m) that is strongly
m∗-orthonormal.
(3) The subspace S that is fixed by the integration map Im satisfies that there are
positive functions hn ∈ S such that { Ω hndm}∞
n=1 is an orthonormal basis for
Im(S).
(4) There is an 2-valued vector measure m∗ defined as m∗ = φ ◦m —φ an
isomorphism— such that L1(m) = L1(m∗) and a subspace S of L1(m) such that
the restriction of Im∗ to S is a lattice isomorphism in 2.
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
50. Theorem.
The following assertions for a positive vector measure m : Σ → 2 are equivalent.
(1) L1(m) contains a lattice copy of 2.
(2) L1(m) has a reflexive infinite dimensional sublattice.
(3) L1(m) has a relatively weakly compact normalized sequence of disjoint functions.
(4) L1(m) contains a weakly null normalized sequence.
(5) There is a vector measure m∗ defined by m∗ = φ ◦m such that integration map
Im∗ fixes a copy of 2.
(6) There is a vector measure m∗ defined as m∗ = φ ◦m that is not disjointly strictly
singular.
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions
51. C. Bessaga and A. Pelczynski, On bases and unconditional convergence of series
in Banach spaces, Studia Math. 17, 151-164 (1958)
C. Bessaga and A. Pelczynski, A generalization of results of R. C. James
concerning absolute bases in Banach spaces, Studia Math. 17, 165-174 (1958)
M. I. Kadec and A. Pelczynski, Bases, lacunary sequences, and complemented
subspaces in the spaces Lp, Studia Math. 21, 161-176 (1962)
T. Figiel, W.B. Johnson and L. Tzafriri, On Banach lattices and spaces having
local unconditional structure, with applications to Lorentz function spaces, J.
Appr. Th. 13, 395-412 (1975)
E. Jim´enez Fern´andez and E.A. S´anchez P´erez. Lattice copies of 2 in L1 of a
vector measure and strongly orthogonal sequences. To be published in J. Func.
Sp. Appl.
E.A. S´anchez P´erez, Vector measure orthonormal functions and best
approximation for the 4-norm, Arch. Math. 80, 177-190 (2003)
E. A. S´anchez P´erez Disjointification methods in spaces of integrable functions