| Summary: | We call a subset <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">M</mi> </semantics> </math> </inline-formula> of an algebra of sets <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">A</mi> </semantics> </math> </inline-formula> a <i>Grothendieck set</i> for the Banach space <inline-formula> <math display="inline"> <semantics> <mrow> <mi>b</mi> <mi>a</mi> <mo>(</mo> <mi mathvariant="script">A</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> of bounded finitely additive scalar-valued measures on <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">A</mi> </semantics> </math> </inline-formula> equipped with the variation norm if each sequence <inline-formula> <math display="inline"> <semantics> <msubsup> <mfenced separators="" open="{" close="}"> <msub> <mi>μ</mi> <mi>n</mi> </msub> </mfenced> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mo>∞</mo> </msubsup> </semantics> </math> </inline-formula> in <inline-formula> <math display="inline"> <semantics> <mrow> <mi>b</mi> <mi>a</mi> <mo>(</mo> <mi mathvariant="script">A</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> which is pointwise convergent on <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">M</mi> </semantics> </math> </inline-formula> is weakly convergent in <inline-formula> <math display="inline"> <semantics> <mrow> <mi>b</mi> <mi>a</mi> <mo>(</mo> <mi mathvariant="script">A</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, i.e., if there is <inline-formula> <math display="inline"> <semantics> <mrow> <mi>μ</mi> <mo>∈</mo> <mi>b</mi> <mi>a</mi> <mfenced open="(" close=")"> <mi mathvariant="script">A</mi> </mfenced> </mrow> </semantics> </math> </inline-formula> such that <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>μ</mi> <mi>n</mi> </msub> <mfenced open="(" close=")"> <mi>A</mi> </mfenced> <mo>→</mo> <mi>μ</mi> <mfenced open="(" close=")"> <mi>A</mi> </mfenced> </mrow> </semantics> </math> </inline-formula> for every <inline-formula> <math display="inline"> <semantics> <mrow> <mi>A</mi> <mo>∈</mo> <mi mathvariant="script">M</mi> </mrow> </semantics> </math> </inline-formula> then <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>μ</mi> <mi>n</mi> </msub> <mo>→</mo> <mi>μ</mi> </mrow> </semantics> </math> </inline-formula> weakly in <inline-formula> <math display="inline"> <semantics> <mrow> <mi>b</mi> <mi>a</mi> <mo>(</mo> <mi mathvariant="script">A</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>. A subset <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">M</mi> </semantics> </math> </inline-formula> of an algebra of sets <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">A</mi> </semantics> </math> </inline-formula> is called a <i>Nikodým set</i> for <inline-formula> <math display="inline"> <semantics> <mrow> <mi>b</mi> <mi>a</mi> <mo>(</mo> <mi mathvariant="script">A</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> if each sequence <inline-formula> <math display="inline"> <semantics> <msubsup> <mfenced separators="" open="{" close="}"> <msub> <mi>μ</mi> <mi>n</mi> </msub> </mfenced> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mo>∞</mo> </msubsup> </semantics> </math> </inline-formula> in <inline-formula> <math display="inline"> <semantics> <mrow> <mi>b</mi> <mi>a</mi> <mo>(</mo> <mi mathvariant="script">A</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> which is pointwise bounded on <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">M</mi> </semantics> </math> </inline-formula> is bounded in <inline-formula> <math display="inline"> <semantics> <mrow> <mi>b</mi> <mi>a</mi> <mo>(</mo> <mi mathvariant="script">A</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>. We prove that if <inline-formula> <math display="inline"> <semantics> <mi mathvariant="sans-serif">Σ</mi> </semantics> </math> </inline-formula> is a <inline-formula> <math display="inline"> <semantics> <mi>σ</mi> </semantics> </math> </inline-formula>-algebra of subsets of a set <inline-formula> <math display="inline"> <semantics> <mi mathvariant="sans-serif">Ω</mi> </semantics> </math> </inline-formula> which is covered by an increasing sequence <inline-formula> <math display="inline"> <semantics> <mfenced separators="" open="{" close="}"> <msub> <mi mathvariant="sans-serif">Σ</mi> <mi>n</mi> </msub> <mo>:</mo> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mfenced> </semantics> </math> </inline-formula> of subsets of <inline-formula> <math display="inline"> <semantics> <mi mathvariant="sans-serif">Σ</mi> </semantics> </math> </inline-formula> there exists <inline-formula> <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </semantics> </math> </inline-formula> such that <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="sans-serif">Σ</mi> <mi>p</mi> </msub> </semantics> </math> </inline-formula> is a Grothendieck set for <inline-formula> <math display="inline"> <semantics> <mrow> <mi>b</mi> <mi>a</mi> <mo>(</mo> <mi mathvariant="script">A</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>. This statement is the exact counterpart for Grothendieck sets of a classic result of Valdivia asserting that if a <inline-formula> <math display="inline"> <semantics> <mi>σ</mi> </semantics> </math> </inline-formula>-algebra <inline-formula> <math display="inline"> <semantics> <mi mathvariant="sans-serif">Σ</mi> </semantics> </math> </inline-formula> is covered by an increasing sequence <inline-formula> <math display="inline"> <semantics> <mfenced separators="" open="{" close="}"> <msub> <mi mathvariant="sans-serif">Σ</mi> <mi>n</mi> </msub> <mo>:</mo> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mfenced> </semantics> </math> </inline-formula> of subsets, there is <inline-formula> <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </semantics> </math> </inline-formula> such that <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="sans-serif">Σ</mi> <mi>p</mi> </msub> </semantics> </math> </inline-formula> is a Nikodým set for <inline-formula> <math display="inline"> <semantics> <mrow> <mi>b</mi> <mi>a</mi> <mfenced open="(" close=")"> <mi mathvariant="sans-serif">Σ</mi> </mfenced> </mrow> </semantics> </math> </inline-formula>. This also refines the Grothendieck result stating that for each <inline-formula> <math display="inline"> <semantics> <mi>σ</mi> </semantics> </math> </inline-formula>-algebra <inline-formula> <math display="inline"> <semantics> <mi mathvariant="sans-serif">Σ</mi> </semantics> </math> </inline-formula> the Banach space <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>ℓ</mi> <mo>∞</mo> </msub> <mfenced open="(" close=")"> <mi mathvariant="sans-serif">Σ</mi> </mfenced> </mrow> </semantics> </math> </inline-formula> is a Grothendieck space. Some applications to classic Banach space theory are given.
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