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.Abstract A subset B of an algebra A of subsets of Ω is said to have the 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 the property sN if for each increasing countable covering (B m ) m of B there exists B n which has the 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 1 m 2 ...m p m p+1 ) m p+1 of B m 1 m 2 ...m p , for each p, m i ∈ N, 1 i p + 1, there exists a sequence (n i ) i such that each B n 1 n 2 ...n r , r ∈ N, has property N. For a σ -algebra S of subsets of Ω 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 σ -algebra S which is independent of properties N and sN. This result and the equivalence of properties wN and w 2 N enable us to give some applications to localization of bounded additive vector measures.Let Ω be a set and A a set-algebra of subsets of Ω. If B is a subset of A then L(B) is the normed space of the real or complex linear hull of the set of characteristics functions {e C : C ∈ B} endowed with the supremum norm · . The dual of L(A ) with the dual norm is named L(A ) and it is isometric to the Banach space ba(A ) of finitely additive measures on A with bounded variation provided with the variation norm, i.e., | · | := | · | (Ω), being the isometry the map Θ : ba(A ) → L(A ) such that, for each µ ∈ ba(A ), Θ(µ) is the linear form named also by µ and defined by µ(e C ) := µ(C), for each C ∈ A , [2, Chpater 1]. A norm in L(A ) equivalent to the supremum norm is defined by the Minkowski functional of absco({e C : C ∈ A }) ([12, Propositions 1 and 2]), which dual norm is the A -supremum norm, i.e., µ := sup{|µ(C)| : C ∈ A }, µ ∈ ba(A ).In this paper duality is referred to the dual pair L(A ), ba(A ) and we follow notations of [7]. Then the weak * dual of a locally convex space E is (E , τ s (E)), whence the topology τ s (L(A )) is the topology τ s (A ) of pointwise convergence in the elements of A , the cardinal of a set C is denoted by |C|, N is the set {1, 2, . . .} of positive integers, the closure of a set is marked by an overline, the convex (absolutely convex) hull of a subset M of a topological vector space is represented by co(M) (absco(M)) and absco(M) = co(∪{rM : |r| = 1}). A subset B of a set-algebra A has the Nikodym property, property N in brief, if each B-pointwise bounded subset M of ba(A ) is bounded in ba(A ) (see [10, Definition 2.4] or [13, Definition 1]). If B has property N the polar set {e C : C ∈ B} • is bounded in ba(A ), hence the bipolar set {e C : C ∈ B} •• = absco{e C : C ∈ B} is ...

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If R is a ring of subsets of a set Ω and ba (R) is the Banach space of bounded finitely additive measures defined on R equipped with the supremum norm, a subfamily ∆ of R is called a Nikodým set for ba (R) if each set {µ α : α ∈ Λ} in ba (R) which is pointwise bounded on ∆ is norm-bounded in ba (R). If the whole ring R is a Nikodým set, R is said to have property (N), which means that R satisfies the Nikodým-Grothendieck boundedness theorem. In this paper we find a class of rings with property (N) that fail Grothendieck's property (G) and prove that a ring R has property (G) if and only if the set of the evaluations on the sets of R is a so-called Rainwater set for ba(R). Recalling that R is called a (wN)-ring if each increasing web in R contains a strand consisting of Nikodým sets, we also give a partial solution to a question raised by Valdivia by providing a class of rings without property (G) for which the relation (N) ⇔ (wN) holds.

We call a subset M of an algebra of sets A a Grothendieck set for the Banach space b a ( A ) of bounded finitely additive scalar-valued measures on A equipped with the variation norm if each sequence μ n n = 1 ∞ in b a ( A ) which is pointwise convergent on M is weakly convergent in b a ( A ) , i.e., if there is μ ∈ b a A such that μ n A → μ A for every A ∈ M then μ n → μ weakly in b a ( A ) . A subset M of an algebra of sets A is called a Nikodým set for b a ( A ) if each sequence μ n n = 1 ∞ in b a ( A ) which is pointwise bounded on M is bounded in b a ( A ) . We prove that if Σ is a σ -algebra of subsets of a set Ω which is covered by an increasing sequence Σ n : n ∈ N of subsets of Σ there exists p ∈ N such that Σ p is a Grothendieck set for b a ( A ) . This statement is the exact counterpart for Grothendieck sets of a classic result of Valdivia asserting that if a σ -algebra Σ is covered by an increasing sequence Σ n : n ∈ N of subsets, there is p ∈ N such that Σ p is a Nikodým set for b a Σ . This also refines the Grothendieck result stating that for each σ -algebra Σ the Banach space ℓ ∞ Σ is a Grothendieck space. Some applications to classic Banach space theory are given.