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 be the Banach space of the real (or complex) finitely additive measures of bounded variation defined on an algebra
of subsets of
and endowed with the variation norm. A subset
of
is a Nikodým set for
if each
-pointwise bounded subset
of
is uniformly bounded on
and
is a strong Nikodým set for
if each increasing covering
of
contains a
which is a Nikodým set for
. If, additionally, the Nikodým subset
verifies that the sequential
-pointwise convergence in
implies weak convergence then
has the Vitali-Hahn-Saks property,
in brief, and
has the strong
property if for each increasing covering
of
there exists
that has
property.
Motivated by Valdivia result that every -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
property implies the strong
property.
Keywords: Bounded set, Algebra and -algebra of subsets, Bounded finitely additive scalar measure, Nikodým and strong Nikodým property, Vitali-Hahn-Saks and strong Vitali-Hahn-Saks property
- J. Diestel, Sequences and Series in Banach Spaces, Number 92 in Graduate Texts in Mathematics. Springer-Verlag, New York, 1984.
- J. Diestel and J. J. Uhl, Vector Measures, Number 15 in Mathematical Surveys and Monographs, American Mathematical Society, Providence, 1977.
- J. C. Ferrando, Strong barrelledness properties in certain
spaces, J. Math. Anal. Appl. 190, 194-202, 1995.
- J. C. Ferrando, S. López-Alfonso and M. López-Pellicer, On Nikodým and Rainwater sets for
and a problem of M. Valdivia, Filomat 33 (8), 2409-2416, 2019.
- J. C. Ferrando and M. López-Pellicer, Strong barrelledness properties in
and bounded finite additive measures, J. Math. Anal. Appl. 287, 727-736, 1990.
- 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.
- 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.
- J. Kakol and M. López-Pellicer, On Valdivia strong version of Nikodým boundedness property, J. Math. Anal. Appl. 446, 1-17, 2017.
- G. Köthe, Topological Vector Spaces I and II, Springer, Berlin, 1979.
- 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.
- 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.
- 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.
- 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.
- M. López-Pellicer, Webs and bounded finitely additive measures, J. Math. Anal. Appl. 210, 257-267, 1997.
- 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.
- W. Schachermayer, On some classical measure-theoretic theorems for non-sigma-complete Boolean algebras, Dissertationes Math. (Rozprawy Mat.) 214, 33 pp., 1982.
- M. Valdivia, On certain barrelled normed spaces, Ann. Inst. Fourier (Grenoble) 29, 39-56, 1979.
- M. Valdivia, On Nikodým boundedness property, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 107, 355-372, 2013.