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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 $\|\nu \|=|\nu |(X).$ ^[1]

If Σ is a sigma-algebra, then the space $ca(\Sigma )$ is defined as the subset of $ba(\Sigma )$ consisting of countably additive measures.^[2] 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.^[3]

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.

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^[4] and Fichtenholtz & Kantorovich.^[5] 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,^[6] and is often used to define the integral with respect to vector measures,^[7] 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 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.

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

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

^ Dunford & Schwartz 1958, IV.2.15.
^ Dunford & Schwartz 1958, IV.2.16.
^ Dunford & Schwartz 1958, IV.2.17.
^ 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.
^ Dunford & Schwartz 1958.
^ Diestel, J.; Uhl, J.J. (1977). Vector measures. Mathematical Surveys. Vol. 15. American Mathematical Society. Chapter I.

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Properties

Dual of B(Σ)

Dual of L∞(μ)

References

Further reading

Dual of L^∞(μ)