Essential range

<h2 id="formal-definition">Formal definition</h2>
<p>Let 
  
    
      
        (
        X
        ,
        
          
            A
          
        
        ,
        μ
        )
      
    
    {\displaystyle (X,{\cal {A}},\mu )}
  
 be a <a href="/facts/Measure_space/w9Dx4KY1">measure space</a>, and let 
  
    
      
        (
        Y
        ,
        
          
            T
          
        
        )
      
    
    {\displaystyle (Y,{\cal {T}})}
  
 be a <a href="/facts/Topological_space/v2ut7CbK">topological space</a>. For any 
  
    
      
        (
        
          
            A
          
        
        ,
        σ
        (
        
          
            T
          
        
        )
        )
      
    
    {\displaystyle ({\cal {A}},\sigma ({\cal {T}}))}
  
-<a href="/facts/Measurable_function/NgIHnb1b">measurable function</a> 
  
    
      
        f
        :
        X
        →
        Y
      
    
    {\displaystyle f:X\to Y}
  
, we say the essential range of 
  
    
      
        f
      
    
    {\displaystyle f}
  
 to mean the set
</p>

e
          s
          s
          .
          i
          m
        
        ⁡
        (
        f
        )
        =
        
          {
          
            y
            ∈
            Y
            ∣
            0
            <
            μ
            (
            
              f
              
                −
                1
              
            
            (
            U
            )
            )
            
               for all 
            
            U
            ∈
            
              
                T
              
            
            
               with 
            
            y
            ∈
            U
          
          }
        
        .
      
    
    {\displaystyle \operatorname {ess.im} (f)=\left\{y\in Y\mid 0<\mu (f^{-1}(U)){\text{ for all }}U\in {\cal {T}}{\text{ with }}y\in U\right\}.}
  
<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>: Example 0.A.5 <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>Equivalently, 
  
    
      
        
          e
          s
          s
          .
          i
          m
        
        ⁡
        (
        f
        )
        =
        supp
        ⁡
        (
        
          f
          
            ∗
          
        
        μ
        )
      
    
    {\displaystyle \operatorname {ess.im} (f)=\operatorname {supp} (f_{*}\mu )}
  
, where 
  
    
      
        
          f
          
            ∗
          
        
        μ
      
    
    {\displaystyle f_{*}\mu }
  
 is the <a href="/facts/Pushforward_measure/aFljwqr6">pushforward measure</a> onto 
  
    
      
        σ
        (
        
          
            T
          
        
        )
      
    
    {\displaystyle \sigma ({\cal {T}})}
  
 of 
  
    
      
        μ
      
    
    {\displaystyle \mu }
  
 under 
  
    
      
        f
      
    
    {\displaystyle f}
  
 and 
  
    
      
        supp
        ⁡
        (
        
          f
          
            ∗
          
        
        μ
        )
      
    
    {\displaystyle \operatorname {supp} (f_{*}\mu )}
  
 denotes the <a href="/facts/Support_(measure_theory)/mBM9wEAF">support</a> of 
  
    
      
        
          f
          
            ∗
          
        
        μ
        .
      
    
    {\displaystyle f_{*}\mu .}
  
<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p>
<h3>Essential values</h3>
<p>The phrase "essential value of 
  
    
      
        f
      
    
    {\displaystyle f}
  
" is sometimes used to mean an element of the essential range of 
  
    
      
        f
        .
      
    
    {\displaystyle f.}
  
<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>: Exercise 4.1.6 <a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>: Example 7.1.11 
</p>
<h2 id="special-cases-of-common-interest">Special cases of common interest</h2>
<h3><i>Y</i> = C</h3>
<p>Say 
  
    
      
        (
        Y
        ,
        
          
            T
          
        
        )
      
    
    {\displaystyle (Y,{\cal {T}})}
  
 is 
  
    
      
        
          C
        
      
    
    {\displaystyle \mathbb {C} }
  
 equipped with its usual topology. Then the essential range of <i>f</i> is given by
</p>

e
          s
          s
          .
          i
          m
        
        ⁡
        (
        f
        )
        =
        
          {
          
            z
            ∈
            
              C
            
            ∣
            
              for all
            
             
            ε
            ∈
            
              
                R
              
              
                >
                0
              
            
            :
            0
            <
            μ
            {
            x
            ∈
            X
            :
            
              |
            
            f
            (
            x
            )
            −
            z
            
              |
            
            <
            ε
            }
          
          }
        
        .
      
    
    {\displaystyle \operatorname {ess.im} (f)=\left\{z\in \mathbb {C} \mid {\text{for all}}\ \varepsilon \in \mathbb {R} _{>0}:0<\mu \{x\in X:|f(x)-z|<\varepsilon \}\right\}.}
  
<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>: Definition 4.36 <a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a><a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a>: cf. Exercise 6.11 
<p>In other words: The essential range of a complex-valued function is the set of all complex numbers <i>z</i> such that the inverse image of each ε-neighbourhood of <i>z</i> under <i>f</i> has positive measure.
</p>
<h3>(<i>Y</i>,<i>T</i>) is discrete</h3>
<p>Say 
  
    
      
        (
        Y
        ,
        
          
            T
          
        
        )
      
    
    {\displaystyle (Y,{\cal {T}})}
  
 is <a href="/facts/Discrete_space/hpvXA23u">discrete</a>, i.e., 
  
    
      
        
          
            T
          
        
        =
        
          
            P
          
        
        (
        Y
        )
      
    
    {\displaystyle {\cal {T}}={\cal {P}}(Y)}
  
 is the <a href="/facts/Power_set/FVuf8PDD">power set</a> of 
  
    
      
        Y
        ,
      
    
    {\displaystyle Y,}
  
 i.e., the <a href="/facts/Discrete_space/hpvXA23u">discrete topology</a> on 
  
    
      
        Y
        .
      
    
    {\displaystyle Y.}
  
 Then the essential range of <i>f</i> is the set of values <i>y</i> in <i>Y</i> with strictly positive 
  
    
      
        
          f
          
            ∗
          
        
        μ
      
    
    {\displaystyle f_{*}\mu }
  
-measure:
</p>

e
          s
          s
          .
          i
          m
        
        ⁡
        (
        f
        )
        =
        {
        y
        ∈
        Y
        :
        0
        <
        μ
        (
        
          f
          
            pre
          
        
        {
        y
        }
        )
        }
        =
        {
        y
        ∈
        Y
        :
        0
        <
        (
        
          f
          
            ∗
          
        
        μ
        )
        {
        y
        }
        }
        .
      
    
    {\displaystyle \operatorname {ess.im} (f)=\{y\in Y:0<\mu (f^{\text{pre}}\{y\})\}=\{y\in Y:0<(f_{*}\mu )\{y\}\}.}
  
<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a>: Example 1.1.29 <a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a><a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a>
<h2 id="properties">Properties</h2>
<ul><li>The essential range of a measurable function, being the <a href="/facts/Support_(measure_theory)/mBM9wEAF">support of a measure</a>, is always closed.</li>
<li>The essential range ess.im(f) of a measurable function is always a subset of 
  
    
      
        
          
            
              im
              ⁡
              (
              f
              )
            
            ¯
          
        
      
    
    {\displaystyle {\overline {\operatorname {im} (f)}}}
  
.</li>
<li>The essential image cannot be used to distinguish functions that are almost everywhere equal: If 
  
    
      
        f
        =
        g
      
    
    {\displaystyle f=g}
  
 holds 
  
    
      
        μ
      
    
    {\displaystyle \mu }
  
-<a href="/facts/Almost_everywhere/1quRFtJE">almost everywhere</a>, then 
  
    
      
        
          e
          s
          s
          .
          i
          m
        
        ⁡
        (
        f
        )
        =
        
          e
          s
          s
          .
          i
          m
        
        ⁡
        (
        g
        )
      
    
    {\displaystyle \operatorname {ess.im} (f)=\operatorname {ess.im} (g)}
  
.</li>
<li>These two facts characterise the essential image: It is the biggest set contained in the closures of 
  
    
      
        im
        ⁡
        (
        g
        )
      
    
    {\displaystyle \operatorname {im} (g)}
  
 for all g that are a.e. equal to f:</li></ul>

e
          s
          s
          .
          i
          m
        
        ⁡
        (
        f
        )
        =
        
          ⋂
          
            f
            =
            g
            
            
              a.e.
            
          
        
        
          
            
              im
              ⁡
              (
              g
              )
            
            ¯
          
        
      
    
    {\displaystyle \operatorname {ess.im} (f)=\bigcap _{f=g\,{\text{a.e.}}}{\overline {\operatorname {im} (g)}}}
  
.
<ul><li>The essential range satisfies 
  
    
      
        ∀
        A
        ⊆
        X
        :
        f
        (
        A
        )
        ∩
        
          e
          s
          s
          .
          i
          m
        
        ⁡
        (
        f
        )
        =
        ∅
        
        ⟹
        
        μ
        (
        A
        )
        =
        0
      
    
    {\displaystyle \forall A\subseteq X:f(A)\cap \operatorname {ess.im} (f)=\emptyset \implies \mu (A)=0}
  
.</li>
<li>This fact characterises the essential image: It is the <i>smallest</i> closed subset of 
  
    
      
        
          C
        
      
    
    {\displaystyle \mathbb {C} }
  
 with this property.</li>
<li>The <a href="/facts/Essential_supremum/pN5r06dj">essential supremum</a> of a real valued function equals the supremum of its essential image and the essential infimum equals the infimum of its essential range. Consequently, a function is essentially bounded if and only if its essential range is bounded.</li>
<li>The essential range of an essentially bounded function f is equal to the <a href="/facts/Spectrum_(functional_analysis)/cA1evTo6">spectrum</a> 
  
    
      
        σ
        (
        f
        )
      
    
    {\displaystyle \sigma (f)}
  
 where f is considered as an element of the <a href="/facts/C*-algebra/B3tnGHIo">C*-algebra</a> 
  
    
      
        
          L
          
            ∞
          
        
        (
        μ
        )
      
    
    {\displaystyle L^{\infty }(\mu )}
  
.</li></ul>
<h2 id="examples">Examples</h2>
<ul><li>If 
  
    
      
        μ
      
    
    {\displaystyle \mu }
  
 is the zero measure, then the essential image of all measurable functions is empty.</li>
<li>This also illustrates that even though the essential range of a function is a subset of the closure of the range of that function, equality of the two sets need not hold.</li>
<li>If 
  
    
      
        X
        ⊆
        
          
            R
          
          
            n
          
        
      
    
    {\displaystyle X\subseteq \mathbb {R} ^{n}}
  
 is open, 
  
    
      
        f
        :
        X
        →
        
          C
        
      
    
    {\displaystyle f:X\to \mathbb {C} }
  
 continuous and 
  
    
      
        μ
      
    
    {\displaystyle \mu }
  
 the <a href="/facts/Lebesgue_measure/8us9cGoW">Lebesgue measure</a>, then 
  
    
      
        
          e
          s
          s
          .
          i
          m
        
        ⁡
        (
        f
        )
        =
        
          
            
              im
              ⁡
              (
              f
              )
            
            ¯
          
        
      
    
    {\displaystyle \operatorname {ess.im} (f)={\overline {\operatorname {im} (f)}}}
  
 holds. This holds more generally for all <a href="/facts/Borel_measure/8cYR8Ysw">Borel measures</a> that assign non-zero measure to every non-empty open set.</li></ul>
<h2 id="extension">Extension</h2>
<p>The notion of essential range can be extended to the case of 
  
    
      
        f
        :
        X
        →
        Y
      
    
    {\displaystyle f:X\to Y}
  
, where 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 is a <a href="/facts/Separable_space/2bmU73bk">separable</a> <a href="/facts/Metric_space/gnBBRXtc">metric space</a>.
If 
  
    
      
        X
      
    
    {\displaystyle X}
  
 and 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 are <a href="/facts/Differentiable_manifold/aSnbXKcV">differentiable manifolds</a> of the same dimension, if 
  
    
      
        f
        ∈
      
    
    {\displaystyle f\in }
  
 <a href="/facts/Bounded_mean_oscillation/mPQQHDIN">VMO</a>
  
    
      
        (
        X
        ,
        Y
        )
      
    
    {\displaystyle (X,Y)}
  
 and if 
  
    
      
        
          e
          s
          s
          .
          i
          m
        
        ⁡
        (
        f
        )
        ≠
        Y
      
    
    {\displaystyle \operatorname {ess.im} (f)\neq Y}
  
, then 
  
    
      
        deg
        ⁡
        f
        =
        0
      
    
    {\displaystyle \deg f=0}
  
.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a>
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Essential_supremum_and_essential_infimum/pN5r06dj">Essential supremum and essential infimum</a></li>
<li><a href="/facts/Measure_(mathematics)/yCq7zlI4">measure</a></li>
<li><a href="/facts/Lp_space/gbad0Rv1">Lp space</a></li></ul>

<ul><li><a href="/facts/Walter_Rudin/9mxRAECY">Walter Rudin</a> (1974). <a href="https://archive.org/details/realcomplexanaly00rudi_0"><i>Real and Complex Analysis</i></a> (2nd ed.). <a href="/facts/McGraw-Hill/oG8VpKJT">McGraw-Hill</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-07-054234-1.</li></ul>

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

<ol>
<li id="fn:1"><p>Zimmer, Robert J. (1990). Essential Results of Functional Analysis. University of Chicago Press. p. 2. ISBN 0-226-98337-4. <a href="0-226-98337-4" target="_blank">0-226-98337-4</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Kuksin, Sergei; Shirikyan, Armen (2012). Mathematics of Two-Dimensional Turbulence. Cambridge University Press. p. 292. ISBN 978-1-107-02282-9. <a href="978-1-107-02282-9" target="_blank">978-1-107-02282-9</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Kon, Mark A. (1985). Probability Distributions in Quantum Statistical Mechanics. Springer. pp. 74, 84. ISBN 3-540-15690-9. <a href="3-540-15690-9" target="_blank">3-540-15690-9</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Driver, Bruce (May 7, 2012). Analysis Tools with Examples (PDF). p. 327. Cf. Exercise 30.5.1. <a href="https://mathweb.ucsd.edu/~bdriver/240C-S2018/Lecture_Notes/2012%20Notes/240Lecture_Notes_Ver8.pdf" target="_blank">https://mathweb.ucsd.edu/~bdriver/240C-S2018/Lecture_Notes/2012%20Notes/240Lecture_Notes_Ver8.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Segal, Irving E.; Kunze, Ray A. (1978). Integrals and Operators (2nd revised and enlarged ed.). Springer. p. 106. ISBN 0-387-08323-5. <a href="0-387-08323-5" target="_blank">0-387-08323-5</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Bogachev, Vladimir I.; Smolyanov, Oleg G. (2020). Real and Functional Analysis. Moscow Lectures. Springer. p. 283. ISBN 978-3-030-38219-3. ISSN 2522-0314. <a href="978-3-030-38219-3" target="_blank">978-3-030-38219-3</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Weaver, Nik (2013). Measure Theory and Functional Analysis. World Scientific. p. 142. ISBN 978-981-4508-56-8. <a href="978-981-4508-56-8" target="_blank">978-981-4508-56-8</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>Bhatia, Rajendra (2009). Notes on Functional Analysis. Hindustan Book Agency. p. 149. ISBN 978-81-85931-89-0. <a href="978-81-85931-89-0" target="_blank">978-81-85931-89-0</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>Folland, Gerald B. (1999). Real Analysis: Modern Techniques and Their Applications. Wiley. p. 187. ISBN 0-471-31716-0. <a href="0-471-31716-0" target="_blank">0-471-31716-0</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
<li id="fn:10"><p>Cf. Tao, Terence (2012). Topics in Random Matrix Theory. American Mathematical Society. p. 29. ISBN 978-0-8218-7430-1. <a href="978-0-8218-7430-1" target="_blank">978-0-8218-7430-1</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li>
<li id="fn:11"><p>Cf. Freedman, David (1971). Markov Chains. Holden-Day. p. 1. <a href="/wiki/David_A._Freedman" target="_blank">/wiki/David_A._Freedman</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li>
<li id="fn:12"><p>Cf. Chung, Kai Lai (1967). Markov Chains with Stationary Transition Probabilities. Springer. p. 135. <a href="/wiki/Chung_Kai-lai" target="_blank">/wiki/Chung_Kai-lai</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li>
<li id="fn:13"><p>Brezis, Haïm; Nirenberg, Louis (September 1995). "Degree theory and BMO. Part I: Compact manifolds without boundaries". Selecta Mathematica. 1 (2): 197–263. doi:10.1007/BF01671566. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li>
</ol>

Essential range open-in-new

Essential range