Weak Precompactness in the Space of Vector-Valued Measures of Bounded Variation
Weak Precompactness in the Space of Vector-Valued Measures of Bounded Variation
For a Banach space<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi>X</mml:mi></mml:mrow></mml:math>and a measure space<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Σ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>, let<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mi>M</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>be the space of all<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mrow><mml:mi>X</mml:mi></mml:mrow></mml:math>-valued countably additive measures on<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Σ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>of bounded variation, with the total variation norm. In this paper we give a …