Inmathematics, theba space${\displaystyle ba(\mathrm{\Sigma })}$ of analgebra of sets${\displaystyle \mathrm{\Sigma }}$ is theBanach space consisting of allbounded and finitely additivesigned measures on${\displaystyle \mathrm{\Sigma }}$. The norm is defined as thevariation, that is${\displaystyle \Vert \nu \Vert =|\nu |(X).}$^[1]

If Σ is asigma-algebra, then the space${\displaystyle ca(\mathrm{\Sigma })}$ is defined as the subset of${\displaystyle ba(\mathrm{\Sigma })}$ consisting ofcountably additive measures.^[2] The notationba is amnemonic forbounded additive andca is short forcountably additive.

IfX is atopological space, and Σ is the sigma-algebra ofBorel sets inX, then${\displaystyle rca(X)}$ is the subspace of${\displaystyle ca(\mathrm{\Sigma })}$ consisting of allregularBorel measures onX.^[3]

All three spaces are complete (they areBanach spaces) with respect to the same norm defined by the total variation, and thus${\displaystyle ca(\mathrm{\Sigma })}$ is a closed subset of${\displaystyle ba(\mathrm{\Sigma })}$, and${\displaystyle rca(X)}$ is a closed set of${\displaystyle ca(\mathrm{\Sigma })}$ for Σ the algebra of Borel sets onX. The space ofsimple functions on${\displaystyle \mathrm{\Sigma }}$ isdense in${\displaystyle ba(\mathrm{\Sigma })}$.

The ba space of thepower set of thenatural numbers,ba(2^N), is often denoted as simply${\displaystyle ba}$ and isisomorphic to thedual space of theℓ^∞ space.

Let B(Σ) be the space of bounded Σ-measurable functions, equipped with theuniform norm. Thenba(Σ) = B(Σ)* is thecontinuous dual space of B(Σ). This is due to Hildebrandt^[4] and Fichtenholtz & Kantorovich.^[5] 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 todefine theintegral with respect to a finitely additive measure (note that the usual Lebesgue integral requirescountable additivity). This is due to Dunford & Schwartz,^[6] and is often used to define the integral with respect tovector measures,^[7] and especially vector-valuedRadon measures.

The topological dualityba(Σ) = B(Σ)* is easy to see. There is an obviousalgebraic duality between the vector space ofall finitely additive measures σ on Σ and the vector space ofsimple functions (${\displaystyle \mu (A)=\zeta \left(1_A\right)}$). 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.

If Σ is asigma-algebra andμ is asigma-additive positive measure on Σ then theLp spaceL^∞(μ) endowed with theessential supremum norm is by definition thequotient space of B(Σ) by the closed subspace of boundedμ-null functions:

${\displaystyle N_{\mu }:=\{f\in B(\mathrm{\Sigma }):f=0\mu \text{-almost everywhere}\}.}$

The dual Banach spaceL^∞(μ)* is thus isomorphic to

${\displaystyle N_{\mu }^{\perp }=\{\sigma \in ba(\mathrm{\Sigma }):\mu (A)=0\Rightarrow \sigma (A)=0\text{ for any }A\in \mathrm{\Sigma }\},}$

i.e. the space offinitely additive signed measures onΣ that areabsolutely continuous with respect toμ (μ-a.c. for short).

When the measure space is furthermoresigma-finite thenL^∞(μ) is in turn dual toL¹(μ), which by theRadon–Nikodym theorem is identified with the set of allcountably additiveμ-a.c. measures.In other words, the inclusion in the bidual

${\displaystyle L^1(\mu )\subset L^1(\mu {)}^{\ast \ast }=L^{\mathrm{\infty }}(\mu {)}^{\ast }}$

is isomorphic to the inclusion of the space of countably additiveμ-a.c. bounded measures inside the space of all finitely additiveμ-a.c. bounded measures.

• List of Banach spaces

• Dunford, N.; Schwartz, J.T. (1958).Linear operators, Part I. Wiley-Interscience.

• Diestel, Joseph (1984).Sequences and series in Banach spaces. Springer-Verlag.ISBN 0-387-90859-5.OCLC 9556781.
• Yosida, K.; Hewitt, E. (1952)."Finitely additive measures".Transactions of the American Mathematical Society.72 (1):46–66.doi:10.2307/1990654.JSTOR 1990654.
• Kantorovitch, Leonid V.; Akilov, Gleb P. (1982).Functional Analysis. Pergamon.doi:10.1016/C2013-0-03044-7.ISBN 978-0-08-023036-8.

• Measure theory
• Banach spaces

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• Use shortened footnotes from May 2021