- Ba space
In
mathematics , the ba space ba(Sigma) of analgebra of sets Sigma is theBanach 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.2.15If 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.2.16If "X" is a topological space, and Σ is the sigma-algebra of
Borel set s in "X", then rca(X) is the subspace of ca(Sigma) 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 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 ofsimple function s on Sigma isdense in ba(Sigma).The ba space of the
power set of thenatural number s, "ba"(2N), is often denoted as simply ba 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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