Ba space

Ba space

In mathematics, the ba space ba(Sigma) of an algebra of sets Sigma is the Banach space consisting of all bounded and finitely additive measures on Sigma. The norm is defined as the variation, that is | u|=| u|(X). harv|Dunford|Schwartz|1958|loc=IV.2.15

If Σ is a sigma-algebra, then the space ca(Sigma) is defined as the subset of ba(Sigma) consisting of countably additive measures. harv|Dunford|Schwartz|1958|loc=IV.2.16

If "X" is a topological space, and Σ is the sigma-algebra of Borel sets in "X", then rca(X) is the subspace of ca(Sigma) consisting of all regular Borel measures on "X". harv|Dunford|Schwartz|1958|loc=IV.2.17

Properties

All three spaces are complete (they are Banach spaces) with respect to the same norm defined by the total variation, and thus ca(Sigma) is a closed subset of ba(Sigma), and rca(X) is a closed set of ca(Sigma) for Σ the algebra of Borel sets on "X". The space of simple functions on Sigma is dense in ba(Sigma).

The ba space of the power set of the natural numbers, "ba"(2N), is often denoted as simply ba and is isomorphic to the dual space of the 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 harvtxt|Hildebrandt|1934 and harvtxt|Fichtenholtz|Kantorovich|1934. 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 harvtxt|Dunford|Schwartz|1958, and is often used to define the integral with respect to vector measures harv|Diestel|Uhl|1977|loc=Chapter I, and especially vector-valued Radon measures.

References

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