On Grothendieck Sets

Juan Carlos Ferrando,Salvador López-Alfonso,Manuel López-Pellicer

doi:10.3390/axioms9010034

Juan Carlos Ferrando, Salvador López-Alfonso + Show 1 more

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https://doi.org/10.3390/axioms9010034

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Abstract

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.

Highlights

With a different terminology, Valdivia showed in [1] that if a σ-algebra Σ of subsets of a setΩ is covered by an increasing sequence {Σn : n ∈ N} of subsets, there is p ∈ N such that Σ p is a Nikodým set for ba (Σ)
Of subsets of Σ there is p ∈ N such that Σ p is a Grothendieck set for ba(A)
This statement is both the exact counterpart for Grothendieck sets of Valdivia’s result for Nikodým sets and a refinement of Grothendieck’s classic result stating that the Banach space∞ (Σ) of bounded scalar-valued Σ-measurable functions defined on Ω equipped with the supremum-norm is a Grothendieck space

Summary

Introduction

Valdivia showed in [1] that if a σ-algebra Σ of subsets of a set. Ω is covered by an increasing sequence {Σn : n ∈ N} of subsets, there is p ∈ N such that Σ p is a Nikodým set for ba (Σ). Of subsets of Σ there is p ∈ N such that Σ p is a Grothendieck set for ba(A) (definitions below) This statement is both the exact counterpart for Grothendieck sets of Valdivia’s result for Nikodým sets and a refinement of Grothendieck’s classic result stating that the Banach space∞ (Σ) of bounded scalar-valued Σ-measurable functions defined on Ω equipped with the supremum-norm is a Grothendieck space. If {Σn : n ∈ N} is an increasing sequence of subsets of Σ covering Σ, there is p ∈ N such that if {μn }∞.

Preliminaries

Application to Banach Spaces