ba space

<h2 id="properties">Properties</h2>
All three spaces are complete (they are <a href="/facts/Banach_space/sDEIk7EC">Banach spaces</a>) with respect to the same norm defined by the total variation, and thus 
 
 
 
 c
 a
 (
 Σ
 )
 
 
 {\displaystyle ca(\Sigma )}
 
 is a closed subset of 
 
 
 
 b
 a
 (
 Σ
 )
 
 
 {\displaystyle ba(\Sigma )}
 
, and 
 
 
 
 r
 c
 a
 (
 X
 )
 
 
 {\displaystyle rca(X)}
 
 is a closed set of 
 
 
 
 c
 a
 (
 Σ
 )
 
 
 {\displaystyle ca(\Sigma )}
 
 for Σ the algebra of Borel sets on X. The space of <a href="/facts/Simple_function/hWAJkmDQ">simple functions</a> on 
 
 
 
 Σ
 
 
 {\displaystyle \Sigma }
 
 is <a href="/facts/Dense_set/nPT9Yxao">dense</a> in 
 
 
 
 b
 a
 (
 Σ
 )
 
 
 {\displaystyle ba(\Sigma )}
 
.
The ba space of the <a href="/facts/Power_set/FVuf8PDD">power set</a> of the <a href="/facts/Natural_number/u65og8JE">natural numbers</a>, ba(2N), is often denoted as simply 
 
 
 
 b
 a
 
 
 {\displaystyle ba}
 
 and is <a href="/facts/Isomorphic/pj8KgkxU">isomorphic</a> to the <a href="/facts/Dual_space/4Uw4Knz1">dual space</a> of the <a href="/facts/Lp_space/gbad0Rv1">ℓ∞ space</a>.

<h3>Dual of B(Σ)</h3>
Let B(Σ) be the space of bounded Σ-measurable functions, equipped with the <a href="/facts/Uniform_norm/5yIkjNE3">uniform norm</a>. Then ba(Σ) = B(Σ)* is the <a href="/facts/Continuous_dual_space/4Uw4Knz1">continuous dual space</a> of B(Σ). This is due to Hildebrandt<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> and Fichtenholtz & Kantorovich.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> This is a kind of <a href="/facts/Riesz_representation_theorem/9CjKDUII">Riesz representation theorem</a> which allows for a measure to be represented as a linear functional on measurable functions. In particular, this isomorphism allows one to define the <a href="/facts/Integral/V7lyV997">integral</a> with respect to a finitely additive measure (note that the usual Lebesgue integral requires countable additivity). This is due to Dunford & Schwartz,<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> and is often used to define the integral with respect to <a href="/facts/Vector_measure/pQXUI3mv">vector measures</a>,<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> and especially vector-valued <a href="/facts/Radon_measure/21adq3XN">Radon measures</a>.
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 <a href="/facts/Simple_function/hWAJkmDQ">simple functions</a> (
 
 
 
 μ
 (
 A
 )
 =
 ζ
 
 (
 
 1
 
 A
 
 
 )
 
 
 
 {\displaystyle \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.

<h3>Dual of L∞(μ)</h3>
If Σ is a <a href="/facts/Sigma-algebra/8LJINLNc">sigma-algebra</a> and μ is a <a href="/facts/Sigma-additive/gpgoKIeU">sigma-additive</a> positive measure on Σ then the <a href="/facts/Lp_space/gbad0Rv1">Lp space</a> L∞(μ) endowed with the <a href="/facts/Essential_supremum/pN5r06dj">essential supremum</a> norm is by definition the <a href="/facts/Quotient_space_(topology)/muE0qQVC">quotient space</a> of B(Σ) by the closed subspace of bounded μ-null functions:

N
          
            μ
          
        
        :=
        {
        f
        ∈
        B
        (
        Σ
        )
        :
        f
        =
        0
         
        μ
        
          -almost everywhere
        
        }
        .
      
    
    {\displaystyle N_{\mu }:=\{f\in B(\Sigma ):f=0\ \mu {\text{-almost everywhere}}\}.}

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

N
          
            μ
          
          
            ⊥
          
        
        =
        {
        σ
        ∈
        b
        a
        (
        Σ
        )
        :
        μ
        (
        A
        )
        =
        0
        ⇒
        σ
        (
        A
        )
        =
        0
        
           for any 
        
        A
        ∈
        Σ
        }
        ,
      
    
    {\displaystyle 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 <a href="/facts/Finitely_additive/gpgoKIeU">finitely additive</a> signed measures on Σ that are <a href="/facts/Absolutely_continuous/NPKDt9Qn">absolutely continuous</a> with respect to μ (μ-a.c. for short).
When the measure space is furthermore <a href="/facts/Sigma-finite/5PXwoAX7">sigma-finite</a> then L∞(μ) is in turn dual to L1(μ), which by the <a href="/facts/Radon%E2%80%93Nikodym_theorem/uWepNG2B">Radon–Nikodym theorem</a> is identified with the set of all <a href="/facts/Countably_additive/gpgoKIeU">countably additive</a> μ-a.c. measures.
In other words, the inclusion in the bidual

L
          
            1
          
        
        (
        μ
        )
        ⊂
        
          L
          
            1
          
        
        (
        μ
        
          )
          
            ∗
            ∗
          
        
        =
        
          L
          
            ∞
          
        
        (
        μ
        
          )
          
            ∗
          
        
      
    
    {\displaystyle 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.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/List_of_Banach_spaces/6dyURAo0">List of Banach spaces</a></li></ul>

<ul><li>Dunford, N.; Schwartz, J.T. (1958). Linear operators, Part I. Wiley-Interscience.</li></ul>

<h2 id="further-reading">Further reading</h2>
<ul><li>Diestel, Joseph (1984). <a href="https://archive.org/details/sequencesseriesi0000dies">Sequences and series in Banach spaces</a>. Springer-Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-90859-5. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/9556781">9556781</a>.</li>
<li>Yosida, K.; Hewitt, E. (1952). <a href="https://doi.org/10.2307%2F1990654">"Finitely additive measures"</a>. Transactions of the American Mathematical Society. 72 (1): 46–66. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1990654">10.2307/1990654</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1990654">1990654</a>.</li>
<li>Kantorovitch, Leonid V.; Akilov, Gleb P. (1982). Functional Analysis. Pergamon. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2FC2013-0-03044-7">10.1016/C2013-0-03044-7</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-08-023036-8.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Dunford & Schwartz 1958, IV.2.15. - Dunford, N.; Schwartz, J.T. (1958). Linear operators, Part I. Wiley-Interscience. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Dunford & Schwartz 1958, IV.2.16. - Dunford, N.; Schwartz, J.T. (1958). Linear operators, Part I. Wiley-Interscience. <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Dunford & Schwartz 1958, IV.2.17. - Dunford, N.; Schwartz, J.T. (1958). Linear operators, Part I. Wiley-Interscience. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Hildebrandt, T.H. (1934). "On bounded functional operations". Transactions of the American Mathematical Society. 36 (4): 868–875. doi:10.2307/1989829. JSTOR 1989829. <a href="https://doi.org/10.2307%2F1989829" target="_blank">https://doi.org/10.2307%2F1989829</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Fichtenholz, G.; Kantorovich, L.V. (1934). "Sur les opérations linéaires dans l'espace des fonctions bornées". Studia Mathematica. 5: 69–98. doi:10.4064/sm-5-1-69-98. <a href="https://doi.org/10.4064%2Fsm-5-1-69-98" target="_blank">https://doi.org/10.4064%2Fsm-5-1-69-98</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Dunford & Schwartz 1958. - Dunford, N.; Schwartz, J.T. (1958). Linear operators, Part I. Wiley-Interscience. <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Diestel, J.; Uhl, J.J. (1977). Vector measures. Mathematical Surveys. Vol. 15. American Mathematical Society. Chapter I. <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
</ol>

ba space open-in-new

ba space