Ba space

In mathematics, the ba space [math]\displaystyle{ ba(\Sigma) }[/math] of an algebra of sets [math]\displaystyle{ \Sigma }[/math] is the Banach space consisting of all bounded and finitely additive signed measures on [math]\displaystyle{ \Sigma }[/math]. The norm is defined as the variation, that is [math]\displaystyle{ \|\nu\|=|\nu|(X). }[/math]^[1]

If Σ is a sigma-algebra, then the space [math]\displaystyle{ ca(\Sigma) }[/math] is defined as the subset of [math]\displaystyle{ ba(\Sigma) }[/math] 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 [math]\displaystyle{ rca(X) }[/math] is the subspace of [math]\displaystyle{ ca(\Sigma) }[/math] 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 [math]\displaystyle{ ca(\Sigma) }[/math] is a closed subset of [math]\displaystyle{ ba(\Sigma) }[/math], and [math]\displaystyle{ rca(X) }[/math] is a closed set of [math]\displaystyle{ ca(\Sigma) }[/math] for Σ the algebra of Borel sets on X. The space of simple functions on [math]\displaystyle{ \Sigma }[/math] is dense in [math]\displaystyle{ ba(\Sigma) }[/math].

The ba space of the power set of the natural numbers, ba(2^N), is often denoted as simply [math]\displaystyle{ ba }[/math] 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 ([math]\displaystyle{ \mu(A)=\zeta\left(1_A\right) }[/math]). 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:

[math]\displaystyle{ N_\mu:=\{f\in B(\Sigma) : f = 0 \ \mu\text{-almost everywhere} \}. }[/math]

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

[math]\displaystyle{ N_\mu^\perp=\{\sigma\in ba(\Sigma) : \mu(A)=0\Rightarrow \sigma(A)= 0 \text{ for any }A\in\Sigma\}, }[/math]

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

[math]\displaystyle{ L^1(\mu)\subset L^1(\mu)^{**}=L^{\infty}(\mu)^* }[/math]

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. 

Further reading

• Diestel, Joseph (1984). Sequences and series in Banach spaces. Springer-Verlag. ISBN 0-387-90859-5. OCLC 9556781. https://archive.org/details/sequencesseriesi0000dies. 
• Yosida, K.; Hewitt, E. (1952). "Finitely additive measures". Transactions of the American Mathematical Society 72 (1): 46–66. doi:10.2307/1990654. 
• Kantorovitch, Leonid V.; Akilov, Gleb P. (1982). Functional Analysis. Pergamon. doi:10.1016/C2013-0-03044-7. ISBN 978-0-08-023036-8. 

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