Birkhoff average of $x mapsto x+1$ in $mathbb R$ with $L^p$ observable












2












$begingroup$


Let $f in mathcal L^p(mathbb R, lambda)$, where $lambda$ is Lebesgue measure and $p in (1,infty)$. And let $T : mathbb R to mathbb R$ be the map $T(x) = x+1$. I want to show:
$$
frac 1 n sum_{k=0}^{n-1} f circ T^k xrightarrow{L^p} 0 quad textrm{as } n to infty.
$$
This exercise is coming from a probability theory textbook, so I want to avoid advanced tools from ergodic theory. I can show the $n$th Birkhoff average is in $mathcal L^p$, but I'm having trouble proving anything about this sequence of $mathcal L^p$ functions (Cauchy, a.e. convergent, etc.). Any suggestions?






real-analysis probability-theory dynamical-systems ergodic-theory







share|cite|improve this question









$endgroup$

















    2












    $begingroup$


    Let $f in mathcal L^p(mathbb R, lambda)$, where $lambda$ is Lebesgue measure and $p in (1,infty)$. And let $T : mathbb R to mathbb R$ be the map $T(x) = x+1$. I want to show:
    $$
    frac 1 n sum_{k=0}^{n-1} f circ T^k xrightarrow{L^p} 0 quad textrm{as } n to infty.
    $$
    This exercise is coming from a probability theory textbook, so I want to avoid advanced tools from ergodic theory. I can show the $n$th Birkhoff average is in $mathcal L^p$, but I'm having trouble proving anything about this sequence of $mathcal L^p$ functions (Cauchy, a.e. convergent, etc.). Any suggestions?






    real-analysis probability-theory dynamical-systems ergodic-theory







    share|cite|improve this question









    $endgroup$















      2












      2








      2





      $begingroup$


      Let $f in mathcal L^p(mathbb R, lambda)$, where $lambda$ is Lebesgue measure and $p in (1,infty)$. And let $T : mathbb R to mathbb R$ be the map $T(x) = x+1$. I want to show:
      $$
      frac 1 n sum_{k=0}^{n-1} f circ T^k xrightarrow{L^p} 0 quad textrm{as } n to infty.
      $$
      This exercise is coming from a probability theory textbook, so I want to avoid advanced tools from ergodic theory. I can show the $n$th Birkhoff average is in $mathcal L^p$, but I'm having trouble proving anything about this sequence of $mathcal L^p$ functions (Cauchy, a.e. convergent, etc.). Any suggestions?






      real-analysis probability-theory dynamical-systems ergodic-theory







      share|cite|improve this question









      $endgroup$




      Let $f in mathcal L^p(mathbb R, lambda)$, where $lambda$ is Lebesgue measure and $p in (1,infty)$. And let $T : mathbb R to mathbb R$ be the map $T(x) = x+1$. I want to show:
      $$
      frac 1 n sum_{k=0}^{n-1} f circ T^k xrightarrow{L^p} 0 quad textrm{as } n to infty.
      $$
      This exercise is coming from a probability theory textbook, so I want to avoid advanced tools from ergodic theory. I can show the $n$th Birkhoff average is in $mathcal L^p$, but I'm having trouble proving anything about this sequence of $mathcal L^p$ functions (Cauchy, a.e. convergent, etc.). Any suggestions?






      real-analysis probability-theory dynamical-systems ergodic-theory




      real-analysis probability-theory dynamical-systems ergodic-theory






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 27 '18 at 0:10









      D FordD Ford

      593213




      593213






















          1 Answer
          1






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          3












          $begingroup$

          First note the result is obvious if $f$ is compactly supported. Then, since compactly supported functions are dense, we are done if we can get a bound of the form $||frac{1}{N}sum_{n le N} f(cdot+n)||_p le C ||f||_p$ with $C$ independent of $f$ and $N$. But we can get $C=1$ by triangle inequality.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Thanks for the tip! I might try using the simple function argument without using Minkowski's inequality, only because that hasn't appeared yet in the textbook I'm referencing. Fingers crossed.
            $endgroup$
            – D Ford
            Dec 27 '18 at 0:29










          • $begingroup$
            @DFord do you mean proving the result is true for $f$ simple?
            $endgroup$
            – mathworker21
            Dec 27 '18 at 0:31










          • $begingroup$
            @DFord if so, I edited my answer to make life easier for you
            $endgroup$
            – mathworker21
            Dec 27 '18 at 0:32










          • $begingroup$
            Well, the argument via simple functions or compactly supported ones still uses the $L^p$ triangle inequality, a.k.a. Minkowski's inequality. Since this theorem isn't proven in this textbook until the following section, I'd like to avoid it if possible.
            $endgroup$
            – D Ford
            Dec 27 '18 at 4:10












          • $begingroup$
            @DFord the textbook hasn't proven that $||cdot||_p$ is a norm?? also, you should be precise with language. you should say "the argument showing that the result for simple functions implies the result for all $L^p$ functions" rather than just saying "the argument via simple functions"
            $endgroup$
            – mathworker21
            Dec 27 '18 at 4:23











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          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          3












          $begingroup$

          First note the result is obvious if $f$ is compactly supported. Then, since compactly supported functions are dense, we are done if we can get a bound of the form $||frac{1}{N}sum_{n le N} f(cdot+n)||_p le C ||f||_p$ with $C$ independent of $f$ and $N$. But we can get $C=1$ by triangle inequality.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Thanks for the tip! I might try using the simple function argument without using Minkowski's inequality, only because that hasn't appeared yet in the textbook I'm referencing. Fingers crossed.
            $endgroup$
            – D Ford
            Dec 27 '18 at 0:29










          • $begingroup$
            @DFord do you mean proving the result is true for $f$ simple?
            $endgroup$
            – mathworker21
            Dec 27 '18 at 0:31










          • $begingroup$
            @DFord if so, I edited my answer to make life easier for you
            $endgroup$
            – mathworker21
            Dec 27 '18 at 0:32










          • $begingroup$
            Well, the argument via simple functions or compactly supported ones still uses the $L^p$ triangle inequality, a.k.a. Minkowski's inequality. Since this theorem isn't proven in this textbook until the following section, I'd like to avoid it if possible.
            $endgroup$
            – D Ford
            Dec 27 '18 at 4:10












          • $begingroup$
            @DFord the textbook hasn't proven that $||cdot||_p$ is a norm?? also, you should be precise with language. you should say "the argument showing that the result for simple functions implies the result for all $L^p$ functions" rather than just saying "the argument via simple functions"
            $endgroup$
            – mathworker21
            Dec 27 '18 at 4:23
















          3












          $begingroup$

          First note the result is obvious if $f$ is compactly supported. Then, since compactly supported functions are dense, we are done if we can get a bound of the form $||frac{1}{N}sum_{n le N} f(cdot+n)||_p le C ||f||_p$ with $C$ independent of $f$ and $N$. But we can get $C=1$ by triangle inequality.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Thanks for the tip! I might try using the simple function argument without using Minkowski's inequality, only because that hasn't appeared yet in the textbook I'm referencing. Fingers crossed.
            $endgroup$
            – D Ford
            Dec 27 '18 at 0:29










          • $begingroup$
            @DFord do you mean proving the result is true for $f$ simple?
            $endgroup$
            – mathworker21
            Dec 27 '18 at 0:31










          • $begingroup$
            @DFord if so, I edited my answer to make life easier for you
            $endgroup$
            – mathworker21
            Dec 27 '18 at 0:32










          • $begingroup$
            Well, the argument via simple functions or compactly supported ones still uses the $L^p$ triangle inequality, a.k.a. Minkowski's inequality. Since this theorem isn't proven in this textbook until the following section, I'd like to avoid it if possible.
            $endgroup$
            – D Ford
            Dec 27 '18 at 4:10












          • $begingroup$
            @DFord the textbook hasn't proven that $||cdot||_p$ is a norm?? also, you should be precise with language. you should say "the argument showing that the result for simple functions implies the result for all $L^p$ functions" rather than just saying "the argument via simple functions"
            $endgroup$
            – mathworker21
            Dec 27 '18 at 4:23














          3












          3








          3





          $begingroup$

          First note the result is obvious if $f$ is compactly supported. Then, since compactly supported functions are dense, we are done if we can get a bound of the form $||frac{1}{N}sum_{n le N} f(cdot+n)||_p le C ||f||_p$ with $C$ independent of $f$ and $N$. But we can get $C=1$ by triangle inequality.






          share|cite|improve this answer











          $endgroup$



          First note the result is obvious if $f$ is compactly supported. Then, since compactly supported functions are dense, we are done if we can get a bound of the form $||frac{1}{N}sum_{n le N} f(cdot+n)||_p le C ||f||_p$ with $C$ independent of $f$ and $N$. But we can get $C=1$ by triangle inequality.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Dec 27 '18 at 0:32

























          answered Dec 27 '18 at 0:18









          mathworker21mathworker21

          8,9421928




          8,9421928












          • $begingroup$
            Thanks for the tip! I might try using the simple function argument without using Minkowski's inequality, only because that hasn't appeared yet in the textbook I'm referencing. Fingers crossed.
            $endgroup$
            – D Ford
            Dec 27 '18 at 0:29










          • $begingroup$
            @DFord do you mean proving the result is true for $f$ simple?
            $endgroup$
            – mathworker21
            Dec 27 '18 at 0:31










          • $begingroup$
            @DFord if so, I edited my answer to make life easier for you
            $endgroup$
            – mathworker21
            Dec 27 '18 at 0:32










          • $begingroup$
            Well, the argument via simple functions or compactly supported ones still uses the $L^p$ triangle inequality, a.k.a. Minkowski's inequality. Since this theorem isn't proven in this textbook until the following section, I'd like to avoid it if possible.
            $endgroup$
            – D Ford
            Dec 27 '18 at 4:10












          • $begingroup$
            @DFord the textbook hasn't proven that $||cdot||_p$ is a norm?? also, you should be precise with language. you should say "the argument showing that the result for simple functions implies the result for all $L^p$ functions" rather than just saying "the argument via simple functions"
            $endgroup$
            – mathworker21
            Dec 27 '18 at 4:23


















          • $begingroup$
            Thanks for the tip! I might try using the simple function argument without using Minkowski's inequality, only because that hasn't appeared yet in the textbook I'm referencing. Fingers crossed.
            $endgroup$
            – D Ford
            Dec 27 '18 at 0:29










          • $begingroup$
            @DFord do you mean proving the result is true for $f$ simple?
            $endgroup$
            – mathworker21
            Dec 27 '18 at 0:31










          • $begingroup$
            @DFord if so, I edited my answer to make life easier for you
            $endgroup$
            – mathworker21
            Dec 27 '18 at 0:32










          • $begingroup$
            Well, the argument via simple functions or compactly supported ones still uses the $L^p$ triangle inequality, a.k.a. Minkowski's inequality. Since this theorem isn't proven in this textbook until the following section, I'd like to avoid it if possible.
            $endgroup$
            – D Ford
            Dec 27 '18 at 4:10












          • $begingroup$
            @DFord the textbook hasn't proven that $||cdot||_p$ is a norm?? also, you should be precise with language. you should say "the argument showing that the result for simple functions implies the result for all $L^p$ functions" rather than just saying "the argument via simple functions"
            $endgroup$
            – mathworker21
            Dec 27 '18 at 4:23
















          $begingroup$
          Thanks for the tip! I might try using the simple function argument without using Minkowski's inequality, only because that hasn't appeared yet in the textbook I'm referencing. Fingers crossed.
          $endgroup$
          – D Ford
          Dec 27 '18 at 0:29




          $begingroup$
          Thanks for the tip! I might try using the simple function argument without using Minkowski's inequality, only because that hasn't appeared yet in the textbook I'm referencing. Fingers crossed.
          $endgroup$
          – D Ford
          Dec 27 '18 at 0:29












          $begingroup$
          @DFord do you mean proving the result is true for $f$ simple?
          $endgroup$
          – mathworker21
          Dec 27 '18 at 0:31




          $begingroup$
          @DFord do you mean proving the result is true for $f$ simple?
          $endgroup$
          – mathworker21
          Dec 27 '18 at 0:31












          $begingroup$
          @DFord if so, I edited my answer to make life easier for you
          $endgroup$
          – mathworker21
          Dec 27 '18 at 0:32




          $begingroup$
          @DFord if so, I edited my answer to make life easier for you
          $endgroup$
          – mathworker21
          Dec 27 '18 at 0:32












          $begingroup$
          Well, the argument via simple functions or compactly supported ones still uses the $L^p$ triangle inequality, a.k.a. Minkowski's inequality. Since this theorem isn't proven in this textbook until the following section, I'd like to avoid it if possible.
          $endgroup$
          – D Ford
          Dec 27 '18 at 4:10






          $begingroup$
          Well, the argument via simple functions or compactly supported ones still uses the $L^p$ triangle inequality, a.k.a. Minkowski's inequality. Since this theorem isn't proven in this textbook until the following section, I'd like to avoid it if possible.
          $endgroup$
          – D Ford
          Dec 27 '18 at 4:10














          $begingroup$
          @DFord the textbook hasn't proven that $||cdot||_p$ is a norm?? also, you should be precise with language. you should say "the argument showing that the result for simple functions implies the result for all $L^p$ functions" rather than just saying "the argument via simple functions"
          $endgroup$
          – mathworker21
          Dec 27 '18 at 4:23




          $begingroup$
          @DFord the textbook hasn't proven that $||cdot||_p$ is a norm?? also, you should be precise with language. you should say "the argument showing that the result for simple functions implies the result for all $L^p$ functions" rather than just saying "the argument via simple functions"
          $endgroup$
          – mathworker21
          Dec 27 '18 at 4:23


















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