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

References:
  1. J. Diestel, Sequences and Series in Banach Spaces, Number 92 in Graduate Texts in Mathematics. Springer-Verlag, New York, 1984.
  2. J. Diestel and J. J. Uhl, Vector Measures, Number 15 in Mathematical Surveys and Monographs, American Mathematical Society, Providence, 1977.
  3. J. C. Ferrando, Strong barrelledness properties in certain  l_{0}^{\infty }(\mathcal{A})  spaces, J. Math. Anal. Appl. 190, 194-202, 1995.
  4. J. C. Ferrando, S. López-Alfonso and M. López-Pellicer, On Nikodým and Rainwater sets for  ba(R)  and a problem of M. Valdivia, Filomat 33 (8), 2409-2416, 2019.
  5. J. C. Ferrando and M. López-Pellicer, Strong barrelledness properties in  l_{0}^{\infty }(x,\mathcal{A})  and bounded finite additive measures, J. Math. Anal. Appl. 287, 727-736, 1990.
  6. J. C. Ferrando, M. López-Pellicer and L. M. Sánchez Ruiz, Metrizable Barrelled Spaces, Number 332 in Pitman Research Notes in Mathematics Series, Longman, Harlow, 1995.
  7. J. C. Ferrando and L. M. Sánchez Ruiz, A survey on recent advances on the Nikodým boundedness theorem and spaces of simple functions, Rocky Mount. J. Math. 34, 139-172, 2004.
  8. J. Kakol and M. López-Pellicer, On Valdivia strong version of Nikodým boundedness property, J. Math. Anal. Appl. 446, 1-17, 2017.
  9. G. Köthe, Topological Vector Spaces I and II, Springer, Berlin, 1979.
  10. S. López-Alfonso, On Schachermayer and Valdivia results in algebras of Jordan measurable sets, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 110, 799-808, 2016.
  11. S. López-Alfonso, Vitali–Hahn–Saks property in coverings of sets algebras, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 115, Paper No 17, 2021.
  12. S. López-Alfonso, J. Mas and S. Moll, Nikodým boundedness property for webs in σ-algebras, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 110, 711-722, 2016.
  13. S. López-Alfonso and S. Moll, The uniform bounded deciding property and the separable quotient problem, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113, 1223-1230, 2019.
  14. M. López-Pellicer, Webs and bounded finitely additive measures, J. Math. Anal. Appl. 210, 257-267, 1997.
  15. P. Pérez Carreras and J. Bonet, Barrelled Locally Convex Spaces, Number 131 in North-Holland Mathematics Studies, Notas de Matemática, North-Holland Publishing Co., Amsterdam, 1987.
  16. W. Schachermayer, On some classical measure-theoretic theorems for non-sigma-complete Boolean algebras, Dissertationes Math. (Rozprawy Mat.) 214, 33 pp., 1982.
  17. M. Valdivia, On certain barrelled normed spaces, Ann. Inst. Fourier (Grenoble) 29, 39-56, 1979.
  18. M. Valdivia, On Nikodým boundedness property, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 107, 355-372, 2013.