ba space

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 $\|\nu\|=|\nu|(X).$ {{sfn|Dunford|Schwartz|1958|loc=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.{{sfn|Dunford|Schwartz|1958|loc=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.{{sfn|Dunford|Schwartz|1958|loc=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(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{{r|Hildebrandt1934}} and Fichtenholtz & Kantorovich.{{r|FichtenholtzKantorovich1934}} 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,{{sfn|Dunford|Schwartz|1958}} and is often used to define the integral with respect to vector measures,{{r|DiestelUhl1977_ChptI}} 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''<sup>∞</sup>(''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

{{cite book |last1=Dunford |first1=N. |last2=Schwartz |first2=J.T. |date=1958 |title=Linear operators, Part I |publisher=Wiley-Interscience

}}

{{reflist|refs=

{{cite journal |last=Hildebrandt |first=T.H. |date=1934 |title=On bounded functional operations |journal=Transactions of the American Mathematical Society |volume=36 |issue=4 |pages=868–875 |doi=10.2307/1989829 |jstor=1989829 |doi-access=free}}

{{cite journal |last1=Fichtenholz |first1=G. |last2=Kantorovich |first2=L.V. |date=1934 |title=Sur les opérations linéaires dans l'espace des fonctions bornées |journal=Studia Mathematica |volume=5 |pages=69–98 |doi=10.4064/sm-5-1-69-98 |doi-access=free}}

{{cite book |last1=Diestel |first1=J. |last2=Uhl |first2=J.J. |date=1977 |title=Vector measures |series=Mathematical Surveys |volume=15 |publisher=American Mathematical Society |at=Chapter I}}

}}

ba space

Properties

= Dual of B(Σ) =

= Dual of ''L''<sup>∞</sup>(''μ'') =

See also

References

Further reading