Ba space
Template:Short description Template:Lowercase Template:Use shortened footnotes In mathematics, the ba space <math>ba(\Sigma)</math> of an algebra of sets <math>\Sigma</math> is the Banach space consisting of all bounded and finitely additive signed measures on <math>\Sigma</math>. The norm is defined as the variation, that is <math>\|\nu\|=|\nu|(X).</math>Template:Sfn
If Σ is a sigma-algebra, then the space <math>ca(\Sigma)</math> is defined as the subset of <math>ba(\Sigma)</math> consisting of countably additive measures.Template:Sfn 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>rca(X)</math> is the subspace of <math>ca(\Sigma)</math> consisting of all regular Borel measures on X.Template:Sfn
PropertiesEdit
All three spaces are complete (they are Banach spaces) with respect to the same norm defined by the total variation, and thus <math>ca(\Sigma)</math> is a closed subset of <math>ba(\Sigma)</math>, and <math>rca(X)</math> is a closed set of <math>ca(\Sigma)</math> for Σ the algebra of Borel sets on X. The space of simple functions on <math>\Sigma</math> is dense in <math>ba(\Sigma)</math>.
The ba space of the power set of the natural numbers, ba(2N), is often denoted as simply <math>ba</math> and is isomorphic to the dual space of the ℓ∞ space.
Dual of B(Σ)Edit
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 HildebrandtTemplate:R and Fichtenholtz & Kantorovich.Template:R 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,Template:Sfn and is often used to define the integral with respect to vector measures,Template:R 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>\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)Edit
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>N_\mu:=\{f\in B(\Sigma) : f = 0 \ \mu\text{-almost everywhere} \}.</math>
The dual Banach space L∞(μ)* is thus isomorphic to
- <math>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>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.