Let u ∈ L 1 ( μ ), T A measure preserving map s.t.: ∫ ( u ) d μ = ∫ u ∘ T d μ. A n , ϵ = { | u ( T n ( x ) ) | n 2 &gt; ϵ }. T n = T ∘ T ∘ T … n times.Prove ∑ n ≥ 1 μ ( A n , ϵ ) is finite.My attempt:By Markov inequality we get: μ ( A n , ϵ ) ≤ 1 ϵ ∫ | u ( T n ( x ) ) | n 2 d μ, since T is measure preserving we get: | u ( T n ( x ) ) | ∈ L 1 ( μ ) . Summing over n gives: ∑ n ≥ 1 μ ( A n , ϵ ) ≤ ∑ n ≥ 1 ϵ − 1 n − 2 ∫ | u ( T n ( x ) ) | d μ .This is the part where I am stuck. I know that all the integrals are finite but how do I ensure that summing over all the integrals stays finite? Can I make the integral independent of n? Or can I just state by assumption: ∫ | u ( T n ( x ) ) | d μ = ∫ | u | d μ?

Question

Let   u  ∈            L        1    (  μ  ),   T A measure preserving map s.t.:   ∫  (  u  )  d  μ  =  ∫  u  ∘  T  d  μ.       A          n      ,      ϵ        =  {                    |            u      (              T        n            (      x      )      )              |                    n      2        &amp;gt;  ϵ  }.       T    n    =  T  ∘  T  ∘  T  … n times.Prove       ∑          n      ≥      1        μ  (      A          n      ,      ϵ        ) is finite.My attempt:By Markov inequality we get:   μ  (      A          n      ,      ϵ        )  ≤      1    ϵ    ∫                    |            u      (              T        n            (      x      )      )              |                    n      2        d  μ, since   T is measure preserving we get:       |    u  (      T    n    (  x  )  )      |    ∈            L      1        (    μ    )  . Summing over n gives:      ∑          n      ≥      1        μ  (      A          n      ,      ϵ      )        ≤      ∑          n      ≥      1            ϵ          −      1            n          −      2        ∫      |    u  (      T    n    (  x  )  )      |    d  μ  .This is the part where I am stuck. I know that all the integrals are finite but how do I ensure that summing over all the integrals stays finite? Can I make the integral independent of n? Or can I just state by assumption:   ∫      |    u  (      T    n    (  x  )  )      |    d  μ  =  ∫      |    u      |    d  μ?

Daniel Valdez · Accepted Answer

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