Nikodym boundedness property for webs in sigma-algebras

[EN] A subset B of an algebra A of subsets of Omega is said to have property N if a B-pointwise bounded subset M of ba(A ) is uniformly bounded on A , where ba(A ) is the Banach space of the real (or complex) finitely additive measures of bounded variation defined on A with the norm variation. Moreover B is said to have property sN if for each increasing countable covering (B_m)_m of B there exists B_n which has property N and B is said to have property wN if given the increasing countable coverings (B_m_1 )_m_1 of B and (B_m_1m_2...m_pm_(p+1) )_m_(p+1) of B_m_1m_2...m_p , for each p,m_i &#8712; N, 1<= i <= p + 1, there exists a sequence (n_i )_i such that each B_n_1n_2...n_r , r &#8712; N, has property N. For a &#963;-algebra S of subsets of Omega it has been proved that S has property N (Nikodym Grothendieck), property sN (Valdivia) and property w(sN) (Kakol López-Pellicer). We give a proof of property wN for a &#963;-algebra S which is independent of properties N and sN. This result and the equivalence of properties wN and w2N enable us to give some applications to localization of bounded additive vector measures.This work was supported for the second named author by the Spanish Ministerio de Economía y Competitividad under Grant MTM2014-58159-PLópez Alfonso, S.; Mas Marí, J.; Moll López, SE. (2016). Nikodym boundedness property for webs in sigma-algebras. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 110(2):711-722. https://doi.org/10.1007/s13398-015-0260-47117221102Diestel, J.: Sequences and Series in Banach Spaces. Springer, New York (1984)Diestel, J., Uhl, J.J.: Vector Measures. Mathematical Surveys and Monographs, vol. 15. American Mathematical Society, Providence (1977)Dieudonné, J.: Sur la convergence de suites de measures de Radon. An. Acad. Brasi. Ciên. 23, 277–282 (1951)Ferrando, J.C.: Strong barrelledness properties in certain $$l_{0}^{\infty }({{\fancyscript {A}}} )$$ l 0 ∞ ( A ) spaces. J. Math. Anal. Appl. 190, 194–202 (1995)Ferrando, J.C., López-Pellicer, M.: Strong barrelledness properties in $$l_{0}^{\infty }(X,{\cal A})$$ l 0 ∞ ( X , A ) and bounded finite additive measures. Math. Ann. 287, 727–736 (1990)Kakol, J., López-Pellicer, M.: On Valdivia strong version of Nikodym boundedness property, preprintKöthe, G.: Topological Vector Spaces I and II. Springer, Berlin (1979)López-Pellicer, M.: Webs and bounded finitely additive measures. J. Math. Anal. Appl. 210, 257–267 (1997)Nikodym, O.M.: Sur les familles bornées de fonctions parfaitement additives d’ensembles abstrait. Monatsh. Math. U. Phys. 40, 418–426 (1933)Schachermayer, W.: On some classical measure-theoretic theorems for non-sigma-complete Boolean algebras. Dissertationes Math. (Rozprawy Mat.) 214, 33 pp., 1982Valdivia, M.: On the closed graph theorem. Collect. Math. 22, 51–72 (1971)Valdivia, M.: On certain barrelled normed spaces. Ann. Inst. Fourier (Grenoble) 29, 39–56 (1979)Valdivia, M.: On Nikodym boundedness property, RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 107, 355–372, 201