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"(2^N), is often denoted as simply $ba$ 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 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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