Mutual absolute continuity of sets of measures

Abstract

A theorem slightly stronger than the following is proved: If <italic>K</italic> is a convex set of (signed) measures that are absolutely continuous with respect to some fixed positive sigma-finite measure, then the subset consisting of those measures in <italic>K</italic> with respect to which all measures in <italic>K</italic> are absolutely continuous is the complement of a set of first category in any topology finer than the norm topology of measures. This implies, e.g., that any Banach-space-valued measure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu"> <mml:semantics> <mml:mi>μ<!-- μ --></mml:mi> <mml:annotation encoding="application/x-tex">\mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is absolutely continuous with respect to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue mathematical left-angle mu left-parenthesis dot right-parenthesis comma x prime mathematical right-angle EndAbsoluteValue"> <mml:semantics> <mml:mrow> <mml:mo>|</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo fence="false" stretchy="false">⟨<!-- ⟨ --></mml:mo> <mml:mi>μ<!-- μ --></mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mo>⋅<!-- ⋅ --></mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo fence="false" stretchy="false">⟩<!-- ⟩ --></mml:mo> </mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\left | {\langle \mu ( \cdot ),x’\rangle } \right |</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for a norm-dense <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Subscript delta"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>δ<!-- δ --></mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{G_\delta }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of elements <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x prime"> <mml:semantics> <mml:msup> <mml:mi>x</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">x’</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the dual of the Banach space.

Locations

• Proceedings of the American Mathematical Society - View - PDF