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11 pages, 295 KiB

Open AccessArticle

A Survey on Valdivia Open Question on Nikodým Sets

by Salvador López-Alfonso, Manuel López-Pellicer, Santiago Moll-López and Luis M. Sánchez-Ruiz

Mathematics 2022, 10(15), 2660; https://doi.org/10.3390/math10152660 - 28 Jul 2022

Viewed by 1211

Let

A

be an algebra of subsets of a set

Ω

and

b a (A)

the Banach space of bounded finitely additive scalar-valued measures on

A

endowed with the variation norm. A subset

B

A

is a Nikodým set for [...] Read more.

Let

A

be an algebra of subsets of a set

Ω

and

b a (A)

the Banach space of bounded finitely additive scalar-valued measures on

A

endowed with the variation norm. A subset

B

A

is a Nikodým set for

b a (A)

if each countable

B

-pointwise bounded subset M of

b a (A)

is norm bounded. A subset

B

A

is a Grothendieck set for

b a (A)

if for each bounded sequence

{\{μ_{n}\}}_{n = 1}^{\infty}

b a (A)

the

B

-pointwise convergence on

b a (A)

implies its

b a {(A)}^{*}

-pointwise convergence on

b a (A)

. A subset

B

of an algebra

A

is a strong-Nikodým (Grothendieck) set for

b a (A)

if in each increasing covering

{B_{n} : n \in N}

B

there exists

B_{m}

which is a Nikodým (Grothendieck) set for

b a (A)

. The answer of the following open question for an algebra

A

of subsets of a set

Ω

, proposed by Valdivia in 2013, has not yet been found: Is it true that if

A

is a Nikodým set for

b a (A)

then

A

is a strong Nikodým set for

b a (A)

? In this paper we surveyed some results related to this Valdivia’s open question, as well as the corresponding problem for strong Grothendieck sets. The new Propositions 1 and 3 provide more simplified proofs, particularly in their application to Theorems 1 and 2, which were the main results surveyed. Moreover, the proofs of almost all other propositions are wholly or partially original. Full article

17 pages, 337 KiB

Open AccessArticle

On Four Classical Measure Theorems

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

Mathematics 2021, 9(5), 526; https://doi.org/10.3390/math9050526 - 3 Mar 2021

Cited by 3 | Viewed by 1951

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 [...] Read more.

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)

(s V H S)

] if every increasing covering

{B_{n} : n \in N}

B

contains a set

B_{p}

with property

(N)

[

(G)

(V H S)

], and

B

has property

(w N)

[

(w G)

(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}

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)

(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)

. Full article

(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2020)

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