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| {{lowercase|ba space}}
| | My name's Gilda Trenwith but everybody calls me Gilda. I'm from France. I'm studying at the university (3rd year) and I play the Banjo for 6 years. Usually I choose music from the famous films ;). <br>I have two sister. I love Fishkeeping, watching movies and Amateur radio.<br><br>My website; Review Bookbyte Buyback ([http://filipinagals.com/blogs_post.php?id=113403 simply click the following internet page]) |
| 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 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> {{harv|Dunford|Schwartz|1958|loc=IV.2.15}}
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| 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]]. {{harv|Dunford|Schwartz|1958|loc=IV.2.16}} The notation ''ba'' is a [[mnemonic]] for ''bounded additive'' and ''ca'' is short for ''countably additive''.
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| 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''. {{harv|Dunford|Schwartz|1958|loc=IV.2.17}}
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| == Properties ==
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| 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>.
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| 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]].
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| === Dual of B(Σ) ===
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| 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 {{harvtxt|Hildebrandt|1934}} and {{harvtxt|Fichtenholtz|Kantorovich|1934}}. 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 {{harvtxt|Dunford|Schwartz|1958}}, and is often used to define the integral with respect to [[vector measure]]s {{harv|Diestel|Uhl|1977|loc=Chapter I}}, and especially vector-valued [[Radon measure]]s.
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| 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 iff 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)* iff it is continuous in the sup-norm.
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| === Dual of ''L''<sup>∞</sup>(''μ'') ===
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| 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]] of B(Σ) by the closed subspace of bounded ''μ''-null functions:
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| :<math>N_\mu:=\{f\in B(\Sigma) : f = 0 \ \mu\text{-almost everywhere} \}.</math>
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| The dual Banach space ''L''<sup>∞</sup>(''μ'')* is thus isomorphic to
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| :<math>N_\mu^\perp=\{\sigma\in ba(\Sigma) : \mu(A)=0\Rightarrow \sigma(A)= 0 \text{ for any }A\in\Sigma\},</math>
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| i.e. the space of [[finitely additive]] signed measures on ''Σ'' that are [[absolutely continuous]] with respect to ''μ'' (''μ''-a.c. for short).
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| 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.
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| In other words the inclusion in the bidual
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| :<math>L^1(\mu)\subset L^1(\mu)^{**}=L^{\infty}(\mu)^*</math>
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| 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.
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| == References ==
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| * {{citation|title=Sequences and series in Banach spaces|first=Joseph|last=Diestel|publisher=Springer-Verlag|year=1984|isbn=0-387-90859-5|oclc=9556781}}.
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| * {{citation|first1=J.|last1=Diestel|first2=J.J.|last2=Uhl|title=Vector measures|publisher=American Mathematical Society|year=1977|series=Mathematical Surveys|volume=15}}.
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| * {{citation|first1=N.|last1=Dunford|first2=J.T.|last2=Schwartz|title=Linear operators, Part I|publisher=Wiley-Interscience|year=1958}}.
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| * {{citation|first=T.H.|last=Hildebrandt|title=On bounded functional operations|journal=Transactions of the American Mathematical Society|volume=36|year=1934|pages=868–875|doi=10.2307/1989829|jstor=1989829|issue=4}}.
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| * {{citation|last1=Fichtenholz|first1=G|last2=Kantorovich|first2=L.V.|year=1934|journal=Studia Mathematica|volume=5|pages=69–98|title=Sur les opérations linéaires dans l'espace des fonctions bornées}}.
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| * {{citation|title=Finitely additive measures|first1=K|last1=Yosida|first2=E|last2=Hewitt|journal=Transactions of the American Mathematical Society|volume=72|pages=46–66|doi=10.2307/1990654|year=1952|issue=1|jstor=1990654}}.
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| [[Category:Measure theory]]
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| [[Category:Banach spaces]]
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My name's Gilda Trenwith but everybody calls me Gilda. I'm from France. I'm studying at the university (3rd year) and I play the Banjo for 6 years. Usually I choose music from the famous films ;).
I have two sister. I love Fishkeeping, watching movies and Amateur radio.
My website; Review Bookbyte Buyback (simply click the following internet page)