View source for Ba space

{{Short description|Class of Banach spaces}}
{{lowercase|ba space}}
{{Use shortened footnotes|date=May 2021}}
In [[mathematics]], the '''ba space''' <math>ba(\Sigma)</math> of an [[ field of sets|algebra of sets]] <math>\Sigma</math> is the [[Banach space]] consisting of all [[bounded measure|bounded]] and finitely additive [[signed measure]]s on <math>\Sigma</math>. The norm is defined as the [[measure variation|variation]], that is <math>\|\nu\|=|\nu|(X).</math>{{sfn|Dunford|Schwartz|1958|loc=IV.2.15}}

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 [[sigma-additive|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 set]]s in ''X'', then <math>rca(X)</math> is the subspace of <math>ca(\Sigma)</math> consisting of all [[regular measure|regular]] [[Borel measure]]s on ''X''.{{sfn|Dunford|Schwartz|1958|loc=IV.2.17}}

== Properties ==
All three spaces are complete (they are [[Banach space]]s) 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 function]]s on <math>\Sigma</math> is [[Dense set|dense]] in <math>ba(\Sigma)</math>.

The ba space of the [[power set]] of the [[natural number]]s, ''ba''(2<sup>'''N'''</sup>), is often denoted as simply <math>ba</math> and is [[isomorphic]] to the [[dual space]] of the [[lp space|ℓ<sup>∞</sup> 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 measure]]s,{{r|DiestelUhl1977_ChptI}} and especially vector-valued [[Radon measure]]s.

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 function]]s (<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''<sup>∞</sup>(''μ'') ===

If Σ is a [[sigma-algebra]] and ''μ'' is a [[sigma-additive]] positive measure on Σ then the [[Lp space]] ''L''<sup>∞</sup>(''μ'') endowed with the [[essential supremum]] norm is by definition the [[Quotient space (topology)|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''<sup>∞</sup>(''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''<sup>∞</sup>(''μ'') is in turn dual to ''L''<sup>1</sup>(''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.

== See also ==
* [[List of Banach spaces]]

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

{{reflist|refs=
<ref name=Hildebrandt1934>{{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}}</ref>

<ref name=FichtenholtzKantorovich1934>{{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}}</ref>

<ref name=DiestelUhl1977_ChptI>{{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}}</ref>
}}

==Further reading==
* {{cite book |last=Diestel |first=Joseph |date=1984 |title=Sequences and series in Banach spaces |publisher=Springer-Verlag |isbn=0-387-90859-5 |oclc=9556781 |url-access=registration |url=https://archive.org/details/sequencesseriesi0000dies}}
* {{cite journal |last1=Yosida |first1=K. |last2=Hewitt |first2=E. |date=1952 |title=Finitely additive measures |journal=Transactions of the American Mathematical Society |volume=72 |issue=1 |pages=46–66 |doi=10.2307/1990654 |jstor=1990654 |doi-access=free}}
* {{cite book |last1=Kantorovitch |first1=Leonid V. |last2=Akilov |first2=Gleb P. |title=Functional Analysis |date=1982 |publisher=Pergamon |isbn=978-0-08-023036-8 |doi=10.1016/C2013-0-03044-7}}

{{Functional analysis}}

[[Category:Measure theory]]
[[Category:Banach spaces]]