- Ba space
In
mathematics , the ba space of analgebra of sets is theBanach space consisting of all bounded and finitely additive measures on . The norm is defined as the variation, that is harv|Dunford|Schwartz|1958|loc=IV.2.15If Σ is a
sigma-algebra , then the space is defined as the subset of consisting of countably additive measures. harv|Dunford|Schwartz|1958|loc=IV.2.16If "X" is a topological space, and Σ is the sigma-algebra of
Borel set s in "X", then is the subspace of consisting of all regularBorel measure s on "X". harv|Dunford|Schwartz|1958|loc=IV.2.17Properties
All three spaces are complete (they are
Banach space s) 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 ofsimple function s on isdense in .The ba space of the
power set of thenatural number s, "ba"(2N), is often denoted as simply and isisomorphic to thedual 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 thecontinuous dual space of B(Σ). This is due to harvtxt|Hildebrandt|1934 and harvtxt|Fichtenholtz|Kantorovich|1934. This is a kind ofRiesz 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" theintegral 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 tovector measure s harv|Diestel|Uhl|1977|loc=Chapter I, and especially vector-valuedRadon measure s.References
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