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In mathematics, the ba space <math>ba(\Sigma)<math> of a sigma-algebra <math>\Sigma<math> is the Banach space consisting of all bounded and finitely additive measures on <math>\Sigma<math>. The norm is defined as the variation, that is <math>\|\nu\|=|\nu|(X).<math>
The space <math>ca(\Sigma)<math> is defined as the subset of <math>ba(\Sigma)<math> consisting of sigma-additive measures.
Properties
Both spaces are complete, and thus <math>ca(\Sigma)<math> is a closed subset of <math>ba(\Sigma)<math>.
The space of simple functions on <math>\Sigma<math> is dense in <math>ba(\Sigma)<math>.
When <math>\Sigma<math> is the power set of the natural numbers, <math>ba(\Sigma)<math> is denoted as <math>ba<math>; it is isomorphic to the dual space of <math>l^\infty<math>.
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