Abstract
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Manuel López-Pellicer to go tu the abstract, or directly go to
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ABSTRACT
Nikodym proved that if A is a sigma-algebra of subsets of a set and H is a subset of bounded real (or complex) measures defined on A pointwise bounded in each set of the sigma-algebra A, then H is a bounded subset of the Banach space ba(A) of all bounded additive real (or complex) measures defined on A endowed with the variation norm. The extension of this result to a subset H of ba(A) of bounded additive measures on A it is known as the Nikodym-Grothendieck's theorem. Nevertheless, this result does not hold in general for an algebra A of subsets of a set. This fact motivated Schachermayer to distinguish a family B of an algebra A of subsets of a set for which both conditions H contained in ba(A) and H pointwise bounded in each element of B imply H is a bounded subset of ba(A); it is said then that B has property N.
Valdivia proved in 2013 that if (A_n)_n is an increasing covering of the sigma-algebra A, there exists a positive integer p such that A_p has property N. The main our result states that if W={A_(n_1)_(n_2)_..._(n_p):p, n_1, n_2, ..., n_p in {1,2,...}} is an increasing web in a sigma-algebra A, there exists a sequence (m_i)_i such that each A_(m_1)_(m_2)..._(m_i), i in {1,2,...}, has property N. In particular, if A_(n_1)_(n_2)_..._(n_p)=A_(n_1) for each p, n_1, n_2, ..., n_p in {1,2,...}, we reach Valdivia's result.
Applications to bounded vector additive measures are provided
