# Article ID: MTJPAM-D-20-00058

## Title: A Survey on Nikodým and Vitali-Hahn-Saks Properties

Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-20-00058; Volume 3 / Issue 3 / Year 2021 (Special Issue), Pages 112-121

Document Type: Research Paper

Author(s):   Salvador López-Alfonso a ,   Manuel López-Pellicer b ,   José Mas c

aDepartamento Construcciones Arquitectónicas, Universitat Politècnica de València, 46022 Valencia, Spain

bProfessor Emeritus, Universitat Politècnica de València and IUMPA, 46022 Valencia, Spain

cUniversitat Politècnica de València, Instituto de Matemática Multidisciplinar, 46022 Valencia, Spain

Received: 31 December 2020, Accepted: 11 January 2021, Published: 25 April 2021.

Corresponding Author: Manuel López-Pellicer (Email address: mlopezpe@mat.upv.es)

Full Text: PDF

Abstract

Let  $ba(\mathcal{A})$  be the Banach space of the real (or complex) finitely additive measures of bounded variation defined on an algebra  $\mathcal{A}$  of subsets of  $\Omega$  and endowed with the variation norm. A subset  $\mathcal{B}$  of  $\mathcal{A}$  is a Nikodým set for  $ba(\mathcal{A})$  if each  $\mathcal{B}$-pointwise bounded subset  $M$  of  $ba (\mathcal{A})$ is uniformly bounded on  $\mathcal{A}$  and  $\mathcal{B}$  is a strong Nikodým set for  $ba (\mathcal{A})$  if each increasing covering  $(\mathcal{B}_{m})_{m=1}^{\infty }$  of  $\mathcal{B}$  contains a  $\mathcal{B}_{n}$  which is a Nikodým set for  $ba (\mathcal{A})$. If, additionally, the Nikodým subset  $\mathcal{B}$  verifies that the sequential  $\mathcal{B}$-pointwise convergence in  $ba (\mathcal{A})$  implies weak convergence then  $\mathcal{B}$  has the Vitali-Hahn-Saks property,  $(VHS)$  in brief, and  $\mathcal{B}$  has the strong  $(VHS)$  property if for each increasing covering  $(\mathcal{B}_{m})_{m=1}^{\infty }$  of  $\mathcal{B}$  there exists  $\mathcal{B}_q$  that has  $(VHS)$  property.
Motivated by Valdivia result that every  $\sigma$-algebra has strong Nikodým property and by his 2013 open question concerning that if Nikodým property in an algebra of subsets implies strong Nikodým property we survey this Valdivia theorem and we get that in a strong Nikodým set the  $(VHS)$  property implies the strong  $(VHS)$  property.

Keywords: Bounded set, Algebra and $\sigma$-algebra of subsets, Bounded finitely additive scalar measure, Nikodým and strong Nikodým property, Vitali-Hahn-Saks and strong Vitali-Hahn-Saks property

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