Convergence in measure

<h2 id="definitions">Definitions</h2>
Let 
 
 
 
 f
 ,
 
 f
 
 n
 
 
  
 (
 n
 ∈
 
 N
 
 )
 :
 X
 →
 
 R
 
 
 
 {\displaystyle f,f_{n}\ (n\in \mathbb {N} ):X\to \mathbb {R} }
 
 be <a href="/facts/Measurable_function/NgIHnb1b">measurable functions</a> on a <a href="/facts/Measure_space/w9Dx4KY1">measure space</a> 
 
 
 
 (
 X
 ,
 Σ
 ,
 μ
 )
 .
 
 
 {\displaystyle (X,\Sigma ,\mu ).}
 
 The sequence 
 
 
 
 
 f
 
 n
 
 
 
 
 {\displaystyle f_{n}}
 
 is said to converge globally in measure to 
 
 
 
 f
 
 
 {\displaystyle f}
 
 if for every 
 
 
 
 ε
 >
 0
 ,
 
 
 {\displaystyle \varepsilon >0,}

lim
          
            n
            →
            ∞
          
        
        μ
        (
        {
        x
        ∈
        X
        :
        
          |
        
        f
        (
        x
        )
        −
        
          f
          
            n
          
        
        (
        x
        )
        
          |
        
        ≥
        ε
        }
        )
        =
        0
        ,
      
    
    {\displaystyle \lim _{n\to \infty }\mu (\{x\in X:|f(x)-f_{n}(x)|\geq \varepsilon \})=0,}

and to converge locally in measure to 
  
    
      
        f
      
    
    {\displaystyle f}
  
 if for every 
  
    
      
        ε
        >
        0
      
    
    {\displaystyle \varepsilon >0}
  
 and every 
  
    
      
        F
        ∈
        Σ
      
    
    {\displaystyle F\in \Sigma }
  
 with

μ
 (
 F
 )
 <
 ∞
 ,
 
 
 {\displaystyle \mu (F)<\infty ,}

lim
          
            n
            →
            ∞
          
        
        μ
        (
        {
        x
        ∈
        F
        :
        
          |
        
        f
        (
        x
        )
        −
        
          f
          
            n
          
        
        (
        x
        )
        
          |
        
        ≥
        ε
        }
        )
        =
        0.
      
    
    {\displaystyle \lim _{n\to \infty }\mu (\{x\in F:|f(x)-f_{n}(x)|\geq \varepsilon \})=0.}

On a finite measure space, both notions are equivalent. Otherwise, convergence in measure can refer to either global convergence in measure or local convergence in measure, depending on the author.

<h2 id="properties">Properties</h2>
Throughout, f and fn (n 
 
 
 
 ∈
 
 
 {\displaystyle \in }
 
 <a href="/facts/Natural_numbers/u65og8JE">N</a>) are measurable functions X → <a href="/facts/Real_numbers/R02gw5Pb">R</a>.

<ul><li>Global convergence in measure implies local convergence in measure. The converse, however, is false; i.e., local convergence in measure is strictly weaker than global convergence in measure, in general.</li>
<li>If, however, 
 
 
 
 μ
 (
 X
 )
 <
 ∞
 
 
 {\displaystyle \mu (X)<\infty }
 
 or, more generally, if f and all the fn vanish outside some set of finite measure, then the distinction between local and global convergence in measure disappears.</li>
<li>If μ is <a href="/facts/Sigma-finite/5PXwoAX7">σ-finite</a> and (fn) converges (locally or globally) to f in measure, there is a subsequence converging to f <a href="/facts/Almost_everywhere/1quRFtJE">almost everywhere</a>. The assumption of σ-finiteness is not necessary in the case of global convergence in measure.</li>
<li>If μ is σ-finite, (fn) converges to f locally in measure <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> every subsequence has in turn a subsequence that converges to f almost everywhere.</li>
<li>In particular, if (fn) converges to f almost everywhere, then (fn) converges to f locally in measure. The converse is false.</li>
<li><a href="/facts/Fatou%27s_lemma/LwERbS3L">Fatou's lemma</a> and the <a href="/facts/Monotone_convergence_theorem/mYFH9yGY">monotone convergence theorem</a> hold if almost everywhere convergence is replaced by (local or global) convergence in measure.</li>
<li>If μ is σ-finite, Lebesgue's <a href="/facts/Dominated_convergence_theorem/EDCoCBJY">dominated convergence theorem</a> also holds if almost everywhere convergence is replaced by (local or global) convergence in measure.</li>
<li>If X = [a,b] ⊆ R and μ is <a href="/facts/Lebesgue_measure/8us9cGoW">Lebesgue measure</a>, there are sequences (gn) of step functions and (hn) of continuous functions converging globally in measure to f.</li>
<li>If f and fn (n ∈ N) are in <a href="/facts/Lp_space/gbad0Rv1">Lp(μ)</a> for some p > 0 and (fn) converges to f in the p-norm, then (fn) converges to f globally in measure. The converse is false.</li>
<li>If fn converges to f in measure and gn converges to g in measure then fn + gn converges to f + g in measure. Additionally, if the measure space is finite, fngn also converges to fg.</li></ul>
<h2 id="counterexamples">Counterexamples</h2>
Let 
 
 
 
 X
 =
 
 R
 
 
 
 {\displaystyle X=\mathbb {R} }
 
 μ be Lebesgue measure, and f the constant function with value zero.

<ul><li>The sequence 
 
 
 
 
 f
 
 n
 
 
 =
 
 χ
 
 [
 n
 ,
 ∞
 )
 
 
 
 
 {\displaystyle f_{n}=\chi _{[n,\infty )}}
 
 converges to f locally in measure, but does not converge to f globally in measure.</li>
<li>The sequence 
 
 
 
 
 f
 
 n
 
 
 =
 
 χ
 
 
 [
 
 
 
 j
 
 2
 
 k
 
 
 
 
 ,
 
 
 
 j
 +
 1
 
 
 2
 
 k
 
 
 
 
 
 ]
 
 
 
 
 
 {\displaystyle f_{n}=\chi _{\left[{\frac {j}{2^{k}}},{\frac {j+1}{2^{k}}}\right]}}
 
 where 
 
 
 
 k
 =
 ⌊
 
 log
 
 2
 
 
 ⁡
 n
 ⌋
 
 
 {\displaystyle k=\lfloor \log _{2}n\rfloor }
 
 and 
 
 
 
 j
 =
 n
 −
 
 2
 
 k
 
 
 
 
 {\displaystyle j=n-2^{k}}
 
 (The first five terms of which are 
 
 
 
 
 χ
 
 
 [
 
 0
 ,
 1
 
 ]
 
 
 
 ,
 
 
 χ
 
 
 [
 
 0
 ,
 
 
 1
 2
 
 
 
 ]
 
 
 
 ,
 
 
 χ
 
 
 [
 
 
 
 1
 2
 
 
 ,
 1
 
 ]
 
 
 
 ,
 
 
 χ
 
 
 [
 
 0
 ,
 
 
 1
 4
 
 
 
 ]
 
 
 
 ,
 
 
 χ
 
 
 [
 
 
 
 1
 4
 
 
 ,
 
 
 1
 2
 
 
 
 ]
 
 
 
 
 
 {\displaystyle \chi _{\left[0,1\right]},\;\chi _{\left[0,{\frac {1}{2}}\right]},\;\chi _{\left[{\frac {1}{2}},1\right]},\;\chi _{\left[0,{\frac {1}{4}}\right]},\;\chi _{\left[{\frac {1}{4}},{\frac {1}{2}}\right]}}
 
) converges to 0 globally in measure; but for no x does fn(x) converge to zero. Hence (fn) fails to converge to f almost everywhere.</li></ul>
<ul><li>The sequence 
 
 
 
 
 f
 
 n
 
 
 =
 n
 
 χ
 
 
 [
 
 0
 ,
 
 
 1
 n
 
 
 
 ]
 
 
 
 
 
 {\displaystyle f_{n}=n\chi _{\left[0,{\frac {1}{n}}\right]}}
 
 converges to f almost everywhere and globally in measure, but not in the p-norm for any 
 
 
 
 p
 ≥
 1
 
 
 {\displaystyle p\geq 1}
 
.</li></ul>
<h2 id="topology">Topology</h2>
There is a <a href="/facts/Topological_space/v2ut7CbK">topology</a>, called the topology of (local) convergence in measure, on the collection of measurable functions from X such that local convergence in measure corresponds to convergence on that topology.
This topology is defined by the family of <a href="/facts/Pseudometric_space/uy1o2lhk">pseudometrics</a>

{
 
 ρ
 
 F
 
 
 :
 F
 ∈
 Σ
 ,
  
 μ
 (
 F
 )
 <
 ∞
 }
 ,
 
 
 {\displaystyle \{\rho _{F}:F\in \Sigma ,\ \mu (F)<\infty \},}

where

ρ
          
            F
          
        
        (
        f
        ,
        g
        )
        =
        
          ∫
          
            F
          
        
        min
        {
        
          |
        
        f
        −
        g
        
          |
        
        ,
        1
        }
        
        d
        μ
        .
      
    
    {\displaystyle \rho _{F}(f,g)=\int _{F}\min\{|f-g|,1\}\,d\mu .}

In general, one may restrict oneself to some subfamily of sets F (instead of all possible subsets of finite measure). It suffices that for each 
 
 
 
 G
 ⊂
 X
 
 
 {\displaystyle G\subset X}
 
 of finite measure and 
 
 
 
 ε
 >
 0
 
 
 {\displaystyle \varepsilon >0}
 
 there exists F in the family such that 
 
 
 
 μ
 (
 G
 ∖
 F
 )
 <
 ε
 .
 
 
 {\displaystyle \mu (G\setminus F)<\varepsilon .}
 
 When 
 
 
 
 μ
 (
 X
 )
 <
 ∞
 
 
 {\displaystyle \mu (X)<\infty }
 
, we may consider only one metric 
 
 
 
 
 ρ
 
 X
 
 
 
 
 {\displaystyle \rho _{X}}
 
, so the topology of convergence in finite measure is metrizable. If 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
 is an arbitrary measure finite or not, then

d
        (
        f
        ,
        g
        )
        :=
        
          inf
          
            δ
            >
            0
          
        
        μ
        (
        {
        
          |
        
        f
        −
        g
        
          |
        
        ≥
        δ
        }
        )
        +
        δ
      
    
    {\displaystyle d(f,g):=\inf \limits _{\delta >0}\mu (\{|f-g|\geq \delta \})+\delta }

still defines a metric that generates the global convergence in measure.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>
Because this topology is generated by a family of pseudometrics, it is <a href="/facts/Uniform_space/GKumHlHo">uniformizable</a>.
Working with uniform structures instead of topologies allows us to formulate <a href="/facts/Uniform_properties/vSPsMykx">uniform properties</a> such as
<a href="/facts/Cauchy_sequences/7Q5c8TuF">Cauchyness</a>.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Convergence_space/FJcNDeog">Convergence space</a></li></ul>

<ul><li>D.H. Fremlin, 2000. <a href="https://web.archive.org/web/20101101220236/http://www.essex.ac.uk/maths/people/fremlin/mt.htm">Measure Theory</a>. Torres Fremlin.</li>
<li>H.L. Royden, 1988. Real Analysis. Prentice Hall.</li>
<li><a href="/facts/Gerald_Folland/xw8PjEnc">G. B. Folland</a> 1999, Section 2.4. Real Analysis. John Wiley & Sons.</li></ul>

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

<ol>
<li id="fn:1">Vladimir I. Bogachev, Measure Theory Vol. I, Springer Science & Business Media, 2007 <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
</ol>

Convergence in measure open-in-new

Convergence in measure