NDLI: On Four Classical Measure Theorems

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On Four Classical Measure Theorems

Content Provider	MDPI
Author	Salvador, López-Alfonso Manuel, López-Pellicer Santiago, Moll-López
Copyright Year	2021
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 $b a (A)$ is bounded in $b a (A)$ , where $b a (A)$ is the Banach space of real or complex bounded finitely additive measures defined on $A$ endowed with the variation norm. $B$ has property $(G)$ [ $(V H S)$ ] if for each bounded sequence [if for each sequence] in $b a (A)$ the $B$ -pointwise convergence implies its weak convergence. $B$ has property $(s N)$ [ $(s G)$ or $(s V H S)$ ] if every increasing covering ${B_{n} : n \in N}$ of $B$ contains a set $B_{p}$ with property $(N)$ [ $(G)$ or $(V H S)$ ], and $B$ has property $(w N)$ [ $(w G)$ or $(w V H S)$ ] if every increasing web ${B_{n_{1} n_{2} \dots n_{m}} : n_{i} \in N, 1 \leq i \leq m, m \in N}$ of $B$ contains a strand ${B_{p_{1} p_{2} \dots p_{m}} : m \in N}$ formed by elements $B_{p_{1} p_{2} \dots p_{m}}$ with property $(N)$ [ $(G)$ or $(V H S)$ ] for every $m \in N$ . The classical theorems of Nikodým–Grothendieck, Valdivia, Grothendieck and Vitali–Hahn–Saks say, respectively, that every $σ$ -algebra has properties $(N)$ , $(s N)$ , $(G)$ and $(V H S)$ . Valdivia’s theorem was obtained through theorems of barrelled spaces. Recently, it has been proved that every $σ$ -algebra has property $(w N)$ and several applications of this strong Nikodým type property have been provided. In this survey paper we obtain a proof of the property $(w N)$ 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 $(w W H S)$ if and only if $B$ has property $(w N)$ and $A$ has property $(G)$ .
Starting Page	526
e-ISSN	22277390
DOI	10.3390/math9050526
Journal	Mathematics
Issue Number	5
Volume Number	9
Language	English
Publisher	MDPI
Publisher Date	2021-03-03
Access Restriction	Open
Subject Keyword	Mathematics Logic Algebra and Σ-algebra of Subsets Bounded Finitely Additive Scalar Measure Nikodým Strong and Web Nikodým Properties Grothendieck Strong and Web Grothendieck Properties Vitali–hahn–saks Strong and Web Vitali–hahn–saks Properties
Content Type	Text
Resource Type	Article

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