Maximal ergodic theorem

The maximal ergodic theorem is a <a href="/facts/Theorem/f2Pe1FMD">theorem</a> in <a href="/facts/Ergodic_theory/rkqCbIy0">ergodic theory</a>, a discipline within <a href="/facts/Mathematics/pxTouaz4">mathematics</a>.
Suppose that 
 
 
 
 (
 X
 ,
 
 
 B
 
 
 ,
 μ
 )
 
 
 {\displaystyle (X,{\mathcal {B}},\mu )}
 
 is a <a href="/facts/Probability_space/dEcv2VrU">probability space</a>, that 
 
 
 
 T
 :
 X
 →
 X
 
 
 {\displaystyle T:X\to X}
 
 is a (possibly noninvertible) <a href="/facts/Measure-preserving_transformation/acIdlpqt">measure-preserving transformation</a>, and that 
 
 
 
 f
 ∈
 
 L
 
 1
 
 
 (
 μ
 ,
 
 R
 
 )
 
 
 {\displaystyle f\in L^{1}(\mu ,\mathbb {R} )}
 
. Define 
 
 
 
 
 f
 
 ∗
 
 
 
 
 {\displaystyle f^{*}}
 
 by

f
          
            ∗
          
        
        =
        
          sup
          
            N
            ≥
            1
          
        
        
          
            1
            N
          
        
        
          ∑
          
            i
            =
            0
          
          
            N
            −
            1
          
        
        f
        ∘
        
          T
          
            i
          
        
        .
      
    
    {\displaystyle f^{*}=\sup _{N\geq 1}{\frac {1}{N}}\sum _{i=0}^{N-1}f\circ T^{i}.}

Then the maximal ergodic theorem states that

∫
          
            
              f
              
                ∗
              
            
            >
            λ
          
        
        f
        
        d
        μ
        ≥
        λ
        ⋅
        μ
        {
        
          f
          
            ∗
          
        
        >
        λ
        }
      
    
    {\displaystyle \int _{f^{*}>\lambda }f\,d\mu \geq \lambda \cdot \mu \{f^{*}>\lambda \}}

for any λ ∈ R.
This theorem is used to prove the point-wise <a href="/facts/Ergodic_theorem/rkqCbIy0">ergodic theorem</a>.

<ul><li>Keane, Michael; Petersen, Karl (2006), "Easy and nearly simultaneous proofs of the Ergodic Theorem and Maximal Ergodic Theorem", Dynamics & Stochastics, Institute of Mathematical Statistics Lecture Notes - Monograph Series, vol. 48, pp. 248–251, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math/0004070">math/0004070</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1214%2F074921706000000266">10.1214/074921706000000266</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-940600-64-1.</li></ul>

Maximal ergodic theorem open-in-new

Maximal ergodic theorem