Lebesgue's decomposition theorem

<h2 id="definition">Definition</h2>
<p>The theorem states that if 
  
    
      
        (
        Ω
        ,
        Σ
        )
      
    
    {\displaystyle (\Omega ,\Sigma )}
  
 is a <a href="/facts/Measurable_space/AJ98lhT4">measurable space</a> and 
  
    
      
        μ
      
    
    {\displaystyle \mu }
  
 and 
  
    
      
        ν
      
    
    {\displaystyle \nu }
  
 
are <a href="/facts/Sigma-finite_measure/5PXwoAX7">σ-finite</a> <a href="/facts/Signed_measure/vukVf84A">signed measures</a> on 
  
    
      
        Σ
      
    
    {\displaystyle \Sigma }
  
, then there exist two uniquely determined σ-finite signed measures 
  
    
      
        
          ν
          
            0
          
        
      
    
    {\displaystyle \nu _{0}}
  
 and 
  
    
      
        
          ν
          
            1
          
        
      
    
    {\displaystyle \nu _{1}}
  
 such that:<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>
<ul><li>
  
    
      
        ν
        =
        
          ν
          
            0
          
        
        +
        
          ν
          
            1
          
        
        
      
    
    {\displaystyle \nu =\nu _{0}+\nu _{1}\,}
  
</li>
<li>
  
    
      
        
          ν
          
            0
          
        
        ≪
        μ
      
    
    {\displaystyle \nu _{0}\ll \mu }
  
 (that is, 
  
    
      
        
          ν
          
            0
          
        
      
    
    {\displaystyle \nu _{0}}
  
 is <a href="/facts/Absolute_continuity_(measure_theory)/NPKDt9Qn">absolutely continuous</a> with respect to 
  
    
      
        μ
      
    
    {\displaystyle \mu }
  
)</li>
<li>
  
    
      
        
          ν
          
            1
          
        
        ⊥
        μ
      
    
    {\displaystyle \nu _{1}\perp \mu }
  
 (that is, 
  
    
      
        
          ν
          
            1
          
        
      
    
    {\displaystyle \nu _{1}}
  
 and 
  
    
      
        μ
      
    
    {\displaystyle \mu }
  
 are <a href="/facts/Singular_measure/nEKfH33k">singular</a>).</li></ul>
<h3>Refinement</h3>
<p>Lebesgue's decomposition theorem can be refined in a number of ways.
First, as the Lebesgue-Radon-Nikodym theorem. That is, let

(
        Ω
        ,
        Σ
        )
      
    
    {\displaystyle (\Omega ,\Sigma )}
  
 be a measure space, 
  
    
      
        μ
      
    
    {\displaystyle \mu }
  
 a <a href="/facts/Sigma-finite_measure/5PXwoAX7">σ-finite</a> <a href="/facts/Positive_measure/yCq7zlI4">positive measure</a> on 
  
    
      
        Σ
      
    
    {\displaystyle \Sigma }
  
 and 
  
    
      
        λ
      
    
    {\displaystyle \lambda }
  
 a <a href="/facts/Complex_measure/zLY1CMtz">complex measure</a> on 
  
    
      
        Σ
      
    
    {\displaystyle \Sigma }
  
.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p>
<ul><li>There is a unique pair of complex measures on 
  
    
      
        Σ
      
    
    {\displaystyle \Sigma }
  
 such that 
  
    
      
        λ
        =
        
          λ
          
            a
          
        
        +
        
          λ
          
            s
          
        
        ,
        
        
          λ
          
            a
          
        
        ≪
        μ
        ,
        
        
          λ
          
            s
          
        
        ⊥
        μ
        .
      
    
    {\displaystyle \lambda =\lambda _{a}+\lambda _{s},\quad \lambda _{a}\ll \mu ,\quad \lambda _{s}\perp \mu .}
  
 If 
  
    
      
        λ
      
    
    {\displaystyle \lambda }
  
 is positive and finite, then so are 
  
    
      
        
          λ
          
            a
          
        
      
    
    {\displaystyle \lambda _{a}}
  
 and 
  
    
      
        
          λ
          
            s
          
        
      
    
    {\displaystyle \lambda _{s}}
  
.</li>
<li>There is a unique 
  
    
      
        h
        ∈
        
          L
          
            1
          
        
        (
        μ
        )
      
    
    {\displaystyle h\in L^{1}(\mu )}
  
 such that 
  
    
      
        
          λ
          
            a
          
        
        (
        E
        )
        =
        
          ∫
          
            E
          
        
        h
        d
        μ
        ,
        
        ∀
        E
        ∈
        Σ
        .
      
    
    {\displaystyle \lambda _{a}(E)=\int _{E}hd\mu ,\quad \forall E\in \Sigma .}
  
</li></ul>
<p>The first assertion follows from the Lebesgue decomposition, the second is known as the <a href="/facts/Radon-Nikodym_theorem/uWepNG2B">Radon-Nikodym theorem</a>. That is, the function 
  
    
      
        h
      
    
    {\displaystyle h}
  
 is a Radon-Nikodym derivative that can be expressed as

h
        =
        
          
            
              d
              
                λ
                
                  a
                
              
            
            
              d
              μ
            
          
        
        .
      
    
    {\displaystyle h={\frac {d\lambda _{a}}{d\mu }}.}

</p><p>An alternative refinement is that of the decomposition of a regular <a href="/facts/Borel_measure/8cYR8Ysw">Borel measure</a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a><a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a><a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>

ν
        =
        
          ν
          
            a
            c
          
        
        +
        
          ν
          
            s
            c
          
        
        +
        
          ν
          
            p
            p
          
        
        ,
      
    
    {\displaystyle \nu =\nu _{ac}+\nu _{sc}+\nu _{pp},}

where
</p>
<ul><li>
  
    
      
        
          ν
          
            a
            c
          
        
        ≪
        μ
      
    
    {\displaystyle \nu _{ac}\ll \mu }
  
 is the absolutely continuous part</li>
<li>
  
    
      
        
          ν
          
            s
            c
          
        
        ⊥
        μ
      
    
    {\displaystyle \nu _{sc}\perp \mu }
  
 is the singular continuous part</li>
<li>
  
    
      
        
          ν
          
            p
            p
          
        
      
    
    {\displaystyle \nu _{pp}}
  
 is the pure point part (a <a href="/facts/Discrete_measure/5FUpllMK">discrete measure</a>).</li></ul>
<p>The absolutely continuous measures are classified by the <a href="/facts/Radon%E2%80%93Nikodym_theorem/uWepNG2B">Radon–Nikodym theorem</a>, and discrete measures are easily understood. Hence (singular continuous measures aside), Lebesgue decomposition gives a very explicit description of measures.  The <a href="/facts/Cantor_measure/I8DgNUFo">Cantor measure</a> (the <a href="/facts/Probability_measure/8BjhgoPR">probability measure</a> on the <a href="/facts/Real_line/6eq42xX5">real line</a> whose <a href="/facts/Cumulative_distribution_function/WaKU8tp4">cumulative distribution function</a> is the <a href="/facts/Cantor_function/xTyypgnm">Cantor function</a>) is an example of a singular continuous measure.
</p>
<h2 id="related-concepts">Related concepts</h2>
<h3>Lévy–Itō decomposition</h3>
<p class="note">Main article: <a href="/facts/L%C3%A9vy%E2%80%93It%C5%8D_decomposition/m9A3YQW8">Lévy–Itō decomposition</a></p>
<p>The analogous decomposition for a <a href="/facts/Stochastic_processes/Ng1n1dnB">stochastic processes</a> is the <a href="/facts/L%C3%A9vy%E2%80%93It%C5%8D_decomposition/m9A3YQW8">Lévy–Itō decomposition</a>: given a <a href="/facts/L%C3%A9vy_process/m9A3YQW8">Lévy process</a> <i>X,</i> it can be decomposed as a sum of three independent <a href="/facts/L%C3%A9vy_process/m9A3YQW8">Lévy processes</a> 
  
    
      
        X
        =
        
          X
          
            (
            1
            )
          
        
        +
        
          X
          
            (
            2
            )
          
        
        +
        
          X
          
            (
            3
            )
          
        
      
    
    {\displaystyle X=X^{(1)}+X^{(2)}+X^{(3)}}
  
 where:
</p>
<ul><li>
  
    
      
        
          X
          
            (
            1
            )
          
        
      
    
    {\displaystyle X^{(1)}}
  
 is a <a href="/facts/Brownian_motion/Th0g0wkr">Brownian motion</a> with drift, corresponding to the absolutely continuous part;</li>
<li>
  
    
      
        
          X
          
            (
            2
            )
          
        
      
    
    {\displaystyle X^{(2)}}
  
 is a <a href="/facts/Compound_Poisson_process/DrfL22v3">compound Poisson process</a>, corresponding to the pure point part;</li>
<li>
  
    
      
        
          X
          
            (
            3
            )
          
        
      
    
    {\displaystyle X^{(3)}}
  
 is a <a href="/facts/Square_integrable/IASblcJJ">square integrable</a> pure jump <a href="/facts/Martingale_(probability_theory)/wk9qdXFn">martingale</a> that almost surely has a countable number of jumps on a finite interval, corresponding to the singular continuous part.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Decomposition_of_spectrum_(functional_analysis)/6jQ7KdsE">Decomposition of spectrum</a></li>
<li><a href="/facts/Hahn_decomposition_theorem/FW5oGvJe">Hahn decomposition theorem</a> and the corresponding Jordan decomposition theorem</li>
<li><a href="/facts/Spectrum_(functional_analysis)/cA1evTo6">Spectrum (functional analysis) § Classification of points in the spectrum</a></li>
<li><a href="/facts/Spectral_measure/nFN3JjlB">Spectral measure</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Paul_Halmos/mR11lcap">Halmos, Paul R.</a> (1974) [1950], <a href="https://archive.org/details/measuretheory00halm"><i>Measure Theory</i></a>, <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">Graduate Texts in Mathematics</a>, vol. 18, New York, Heidelberg, Berlin: Springer-Verlag, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-90088-9, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0033869">0033869</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0283.28001">0283.28001</a></li>
<li><a href="/facts/Edwin_Hewitt/vY1ff529">Hewitt, Edwin</a>; Stromberg, Karl (1965), <a href="https://archive.org/details/realabstractanal00hewi_0"><i>Real and Abstract Analysis. A Modern Treatment of the Theory of Functions of a Real Variable</i></a>, Graduate Texts in Mathematics, vol. 25, Berlin, Heidelberg, New York: Springer-Verlag, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-90138-1, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0188387">0188387</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0137.03202">0137.03202</a></li>
<li>Reed, Michael; Simon, Barry (1981-01-11), <i>I: Functional Analysis</i>, San Diego, Calif.: Academic Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-12-585050-6</li>
<li><a href="/facts/Walter_Rudin/9mxRAECY">Rudin, Walter</a> (1974), <a href="https://archive.org/details/realcomplexanaly00rudi_0"><i>Real and Complex Analysis</i></a>, McGraw-Hill Series in Higher Mathematics (2nd ed.), New York, Düsseldorf, Johannesburg: McGraw-Hill Book Comp., <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-07-054233-3, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0344043">0344043</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0278.26001">0278.26001</a></li>
<li><a href="/facts/Barry_Simon/BImmcE2I">Simon, Barry</a> (2005), <i>Orthogonal polynomials on the unit circle. Part 1. Classical theory</i>, American Mathematical Society Colloquium Publications, vol. 54, Providence, R.I.: <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-3446-6, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2105088">2105088</a></li>
<li>Swartz, Charles (1994), <i>Measure, Integration and Function Spaces</i>, WORLD SCIENTIFIC, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1142%2F2223">10.1142/2223</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-981-02-1610-8</li></ul>
<p><i>This article incorporates material from Lebesgue decomposition theorem on <a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a>, which is licensed under the Creative Commons Attribution/Share-Alike License.</i>
</p>

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

<ol>
<li id="fn:1"><p>Hewitt & Stromberg 1965, Chapter V, § 19, (19.42) Lebesgue Decomposition Theorem. - Hewitt, Edwin; Stromberg, Karl (1965), Real and Abstract Analysis. A Modern Treatment of the Theory of Functions of a Real Variable, Graduate Texts in Mathematics, vol. 25, Berlin, Heidelberg, New York: Springer-Verlag, ISBN 978-0-387-90138-1, MR 0188387, Zbl 0137.03202 <a href="https://archive.org/details/realabstractanal00hewi_0" target="_blank">https://archive.org/details/realabstractanal00hewi_0</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Halmos 1974, Section 32, Theorem C. - Halmos, Paul R. (1974) [1950], Measure Theory, Graduate Texts in Mathematics, vol. 18, New York, Heidelberg, Berlin: Springer-Verlag, ISBN 978-0-387-90088-9, MR 0033869, Zbl 0283.28001 <a href="https://archive.org/details/measuretheory00halm" target="_blank">https://archive.org/details/measuretheory00halm</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Swartz 1994, p. 141. - Swartz, Charles (1994), Measure, Integration and Function Spaces, WORLD SCIENTIFIC, doi:10.1142/2223, ISBN 978-981-02-1610-8 <a href="https://doi.org/10.1142%2F2223" target="_blank">https://doi.org/10.1142%2F2223</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Rudin 1974, Section 6.9, The Theorem of Lebesgue-Radon-Nikodym. - Rudin, Walter (1974), Real and Complex Analysis, McGraw-Hill Series in Higher Mathematics (2nd ed.), New York, Düsseldorf, Johannesburg: McGraw-Hill Book Comp., ISBN 0-07-054233-3, MR 0344043, Zbl 0278.26001 <a href="https://archive.org/details/realcomplexanaly00rudi_0" target="_blank">https://archive.org/details/realcomplexanaly00rudi_0</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Hewitt & Stromberg 1965, Chapter V, § 19, (19.61) Theorem. - Hewitt, Edwin; Stromberg, Karl (1965), Real and Abstract Analysis. A Modern Treatment of the Theory of Functions of a Real Variable, Graduate Texts in Mathematics, vol. 25, Berlin, Heidelberg, New York: Springer-Verlag, ISBN 978-0-387-90138-1, MR 0188387, Zbl 0137.03202 <a href="https://archive.org/details/realabstractanal00hewi_0" target="_blank">https://archive.org/details/realabstractanal00hewi_0</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Reed & Simon 1981, pp. 22–25. - Reed, Michael; Simon, Barry (1981-01-11), I: Functional Analysis, San Diego, Calif.: Academic Press, ISBN 978-0-12-585050-6 <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Simon 2005, p. 43. - Simon, Barry (2005), Orthogonal polynomials on the unit circle. Part 1. Classical theory, American Mathematical Society Colloquium Publications, vol. 54, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3446-6, MR 2105088 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2105088" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=2105088</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
</ol>

Lebesgue's decomposition theorem open-in-new

Lebesgue's decomposition theorem