On Four Classical Measure Theorems

Salvador López-Alfonso,Santiago Moll-López,Manuel López-Pellicer

doi:10.3390/math9050526

Salvador López-Alfonso, Santiago Moll-López + Show 1 more

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https://doi.org/10.3390/math9050526

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Journal: Mathematics	Publication Date: Mar 3, 2021
Citations: 2	License type: CC BY 4.0

Affiliation: Universitat Politècnica de València

Abstract

A subset B of an algebra A of subsets of a set Ω has property (N) if each B-pointwise bounded sequence of the Banach space ba(A) is bounded in ba(A), where ba(A) is the Banach space of real or complex bounded finitely additive measures defined on A endowed with the variation norm. B has property (G) [(VHS)] if for each bounded sequence [if for each sequence] in ba(A) the B-pointwise convergence implies its weak convergence. B has property (sN) [(sG) or (sVHS)] if every increasing covering {Bn:n∈N} of B contains a set Bp with property (N) [(G) or (VHS)], and B has property (wN) [(wG) or (wVHS)] if every increasing web {Bn1n2⋯nm:ni∈N,1≤i≤m,m∈N} of B contains a strand {Bp1p2⋯pm:m∈N} formed by elements Bp1p2⋯pm with property (N) [(G) or (VHS)] for every m∈N. The classical theorems of Nikodým–Grothendieck, Valdivia, Grothendieck and Vitali–Hahn–Saks say, respectively, that every σ-algebra has properties (N), (sN), (G) and (VHS). Valdivia’s theorem was obtained through theorems of barrelled spaces. Recently, it has been proved that every σ-algebra has property (wN) and several applications of this strong Nikodým type property have been provided. In this survey paper we obtain a proof of the property (wN) of a σ-algebra independent of the theory of locally convex barrelled spaces which depends on elementary basic results of Measure theory and Banach space theory. Moreover we prove that a subset B of an algebra A has property (wWHS) if and only if B has property (wN) and A has property (G).

Highlights

In this paper A and S denote, respectively, an algebra and a σ-algebra of subsets of a set Ω
The converse follows from the observation that if (Bn : n ∈ N) contains a set Bq with property ( N ) and A has property ( G ), by Proposition 6, Bq has property (V HS), so
The converse follows from the observation that if A has property ( G ) and for the increasing web {Bn1 n2 ···nm : ni ∈ N, 1 ≤ i ≤ m, m ∈ N} there exists a sequence

Summary

Introduction

In this paper A and S denote, respectively, an algebra and a σ-algebra of subsets of a set Ω. Let us recall (see [6,7]) ̇ that a subset C of the closed dual unit ball BE∗ of a Banach space E is a Rainwater set for E if for every bounded sequence { xn : n ∈ N} the conditions lim f ( xn ) = 0, for every f ∈ C n→∞. In [19], Problem 2, it was proposed to prove that every σ-algebra has property (wN ) using basic results of Measure theory and Banach space theory. In a class of algebras where property ( N ) implies property (sN ) we will have that property (V HS) imply property (sV HS)

Preliminary Results

A Proof of the Web Nikodým Property of σ-Algebras

Conclusions