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In mathematics, the ba space $$ba(\Sigma)$$ of an algebra of sets $$\Sigma$$ is the Banach space consisting of all bounded and finitely additive signed measures on $$\Sigma$$ . The norm is defined as the variation, that is $$\|\nu\|=|\nu|(X)$$ . (Dunford & Schwartz 1958, 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. (Dunford & Schwartz 1958, IV.2.16) The notation ba is a mnemonic for bounded additive and ca is short for countably additive.

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. (Dunford & Schwartz 1958, 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 ℓ∞ space.

Dual of B(Σ)

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 Hildebrandt (1934) and 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 Dunford & Schwartz (1958), and is often used to define the integral with respect to vector measures (Diestel & Uhl 1977, Chapter I), and especially vector-valued Radon measures.

The topological duality ba(Σ) = B(Σ)* is easy to see. There is an obvious algebraic duality between the vector space of all finitely additive measures σ on Σ and the vector space of simple functions $$\mu(A)=\zeta\left(1_A\right))$$. It is easy to check that the linear form induced by σ is continuous in the sup-norm iff 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)* iff it is continuous in the sup-norm.

Dual of L∞(μ)

If Σ is a sigma-algebra and μ is a sigma-additive positive measure on Σ then the Lp space L∞(μ) endowed with the essential supremum norm is by definition the quotient space of B(Σ) by the closed subspace of bounded μ-null functions:

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

The dual Banach space L∞(μ)* is thus isomorphic to

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

i.e. the space of finitely additive signed measures on Σ that are absolutely continuous with respect to μ (μ-a.c. for short).

When the measure space is furthermore sigma-finite then L∞(μ) is in turn dual to L1(μ), which by the Radon–Nikodym theorem is identified with the set of all countably additive μ-a.c. measures. In other words, the inclusion in the bidual

$$L^1(\mu)\subset L^1(\mu)^{**}=L^{\infty}(\mu)^*$$

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

Diestel, Joseph (1984), Sequences and series in Banach spaces, Springer-Verlag, ISBN 0-387-90859-5, OCLC 9556781.
Diestel, J.; Uhl, J.J. (1977), Vector measures, Mathematical Surveys, 15, American Mathematical Society.
Dunford, N.; Schwartz, J.T. (1958), Linear operators, Part I, Wiley-Interscience.
Hildebrandt, T.H. (1934), "On bounded functional operations", Transactions of the American Mathematical Society, 36 (4): 868–875, doi:10.2307/1989829, JSTOR 1989829.
Fichtenholz, G; Kantorovich, L.V. (1934), "Sur les opérations linéaires dans l'espace des fonctions bornées", Studia Mathematica, 5: 69–98, doi:10.4064/sm-5-1-69-98.
Yosida, K; Hewitt, E (1952), "Finitely additive measures", Transactions of the American Mathematical Society, 72 (1): 46–66, doi:10.2307/1990654, JSTOR 1990654.

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