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Concept
Ba space
Summary
In mathematics, the ba space of an algebra of sets is the Banach space consisting of all bounded and finitely additive signed measures on . The norm is defined as the variation, that is
If Σ is a sigma-algebra, then the space is defined as the subset of consisting of countably additive measures. The notation ba is a mnemonic for bounded additive and ca is short for countably additive.
If X is a topological space, and Σ is the sigma-algebra of Borel sets in X, then is the subspace of consisting of all regular Borel measures on X.
All three spaces are complete (they are Banach spaces) with respect to the same norm defined by the total variation, and thus is a closed subset of , and is a closed set of for Σ the algebra of Borel sets on X. The space of simple functions on is dense in .
The ba space of the power set of the natural numbers, ba(2N), is often denoted as simply and is isomorphic to the dual space of the l∞ space.
Let B(Σ) be the space of bounded Σ-measurable functions, equipped with the uniform norm. Then ba(Σ) = B(Σ)* is the continuous dual space of B(Σ). This is due to Hildebrandt and Fichtenholtz & Kantorovich. This is a kind of Riesz representation theorem which allows for a measure to be represented as a linear functional on measurable functions. In particular, this isomorphism allows one to define the integral with respect to a finitely additive measure (note that the usual Lebesgue integral requires countable additivity). This is due to Dunford & Schwartz, and is often used to define the integral with respect to vector measures, and especially vector-valued Radon measures.
The topological duality ba(Σ) = B(Σ)* is easy to see. There is an obvious algebraic duality between the vector space of all finitely additive measures σ on Σ and the vector space of simple functions (). It is easy to check that the linear form induced by σ is continuous in the sup-norm if σ is bounded, and the result follows since a linear form on the dense subspace of simple functions extends to an element of B(Σ)* if it is continuous in the sup-norm.
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