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Let $(\Omega, \mathcal{A}, \mu)$ be a $\sigma$-finite measure space. I want to prove the following (Exercise 7.1.2 in Klenke's Probability book):

Let $p \in (1, \infty), f \in \mathcal{L}^p(\lambda)$, where $\lambda$ is the Lebesgue measure on $\mathbb{R}$. Let $T: \mathbb{R} \to \mathbb{R}, x \to x + 1$. Show that, \begin{align} \frac{1}{n}\sum_{k = 0}^{n - 1}f\circ T^k \to 0 \hspace{0.2cm} \text{as} \hspace{0.2cm} n \to \infty \hspace{0.2cm} \text{in} \hspace{0.2cm} L^p(\lambda). \end{align}

I am lost on how to prove this. I think I am at least able to prove that the sequence is a Cauchy sequence and thus converges, but even there I am not sure. Here is my attempt:

Let $S_n = \frac{1}{n}\sum_{k = 0}^{n - 1}f\circ T^k$. Then \begin{align} \|S_n - S_{n - 1}\|_p &= \|\frac{1}{n} f \circ T^{n-1} - \frac{1}{n(n-1)}\sum_{k = 0}^{n -2}f \circ T^k \|_p \\ & \leq \frac{1}{n} \|f \circ T^{n - 1}\|_p + \frac{1}{n(n - 1)}\sum_{k = 0}^{n -2} \|f \circ T^k \|_p\\ &= \frac{2M}{n} \to 0, \end{align} where $M = \|f\circ T^k\|_p$. Here I am obviously assuming $\|f\circ T^i\|_p = \|f\circ T^j\|_p$ for all $i, j \in \{0, \ldots, n- 1\}$, which I think is true, but can only provide intuitive (as opposed to formal) justification (Since $f \in \mathcal{L}^p (\lambda)$, then the integral of the function should not change by shifting it around horizontally. I guess the idea would be to invoke the translation invariance of the Lebesgue measure?).

I have two questions then:

(1) Is my proof that the sequence converges in $L^p(\lambda)$ correct? Is my above assumption correct? If yes, what would be the formal justification?

(2) How would one prove that the limit is $0$?

Thanks much in advance!

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2 Answers 2

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Here is an alternative solution.

The operator norm of $S_n$ is bounded above by $1$. Let $g$ be function in $L^p$. Then

$$ \|S_n f\| \le \|S_n g\|+ \|S_n (f-g)\|\le \|S_n g \| + \|f-g\|.$$

Thus we will be done if we can show that for every $\epsilon>0$ we can find $g$ such that (i) $\|f-g\|<\epsilon/2$ and (ii) $\|S_n g\|< \epsilon/2$.

(i) For an integer $M>0$, let $g_M= f {\bf 1}_{[-M,M)}$. By dominated convergence, for $M$ large enough $\|f-g_M\|<\epsilon/2$.

Moreover, $g_M$ is the sum of $2M$ functions $h_j$, $j=-M,\dots,M-1$ where $h_{j} = f {\bf 1}_{[j,j+1)}$. Each of these functions is supported on an interval of length $\le 1$.

(ii) Because of the last fact, for each $j$, the functions $h_j, h_j \circ T,\dots$ have non-overlapping supports, and so (that's the key):

$$\|S_n h_j\|^p = n*n^{-p} \|h_ j\|^p \quad \Rightarrow \quad \|S_n h_j\|= n^{1/p-1} \|h_j\|.$$

Thus,

$$\|S_n g_M \|=\|S_n \sum h_j\| \le\sum \|S_n h_j\| \le n^{1/p-1} \sum_{j=-M}^{M-1} \|h_j\|.$$

Since $p>1$, $n^{1/p-1}\to 0$, and therefore we can choose $n$ large enough so that $\|S_n g_M\|\le \epsilon/2$.

$\Box$

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  • $\begingroup$ Thanks! Maybe I'm misunderstanding here, but why are the $h_j=g \mathbf{1}_{[j, j+1)}$ rather than $h_j=f \mathbf{1}_{[j, j+1)}$? $\endgroup$ Oct 17, 2022 at 16:10
  • $\begingroup$ right. corrected! $\endgroup$
    – Fnacool
    Oct 17, 2022 at 20:07
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With regards to the questions of the OP
(1) your approach is not entirely correct since having $\|S_nf-S_{n-1}f\|_p\xrightarrow{n\rightarrow\infty}0$ does not imply that the sequence $S_nf$ is Cauchy in $L_p$

(2) Looking at the material presented in your textbook, the idea it seems is to show that $\{|S_nf|^p:n\in\mathbb{N}\}$ converges to $0$ in measure and also that is uniform integrable, at least for a collection of functions $f$ that is dense in $L_p$.

Here I outlined the ideas in more detail. The tricky part is proving uniform integrability of $\{|S_nf|^p:n\in\mathbb{N}\}$. I suggest an approach at the end. Finally, I should mentioned that applying tools from Ergodic Theorem (not covered in the textbook until much later and then only in the context of probability measures) give a short solution to the problem, in particular the Dunford-Hopf-Schwartz) ergodic theorem which states:

Theorem (Dunford--Hopf--Schwartz). Let $(S,\mathscr{B},\mu)$ a $\sigma$-finite measure space. Suppose $T$ is a positive $\mathcal{L}_1(\mu)$-$\mathcal{L}_\infty(\mu)$ contraction. Define $A_n=\frac1n\sum^{n-1}_{k=1}T^k$. If $1\leq p<\infty$, then

  1. For any $f\in\mathcal{L}_p(\mu)$, $Af:=\lim_nA_nf$ exists $\mu$-a.s., and $Af\in \mathcal{L}_p(\mu)$.
  2. The map $A:f\mapsto Af$ is a positive $\mathcal{L}_1(\mu)$--$\mathcal{L}_{\infty-}(\mu)$ contraction and $AT=TA=A$.
  3. If $1<p<\infty$, $\|A_nf-Af\|_p\xrightarrow{n\rightarrow\infty}0$.

If $\mu(S)<\infty$, then:

  1. $A$ is an $\mathcal{L}_\infty(\mu)$ contraction.
  2. $\|A_nf-Af\|_1\xrightarrow{n\rightarrow\infty}0$ for all $f\in\mathcal{L}_1(\mu)$

In the case of the OP, the operator is $Tf(x)=f(x+1)$. Clearly $Tf\geq0$ when $f\geq0$, and $\|Tf\|_p\leq\|f\|_p$ (in fact equal) for any $1\leq p\leq \infty$ (This means that $T$ is an positive $L_1-L_\infty$ contraction).


Fix $1<p<\infty$, and as before, $S_nf(x):=\frac1n\sum^{n-1}_{k=0}f(x+k)$ for any function $f\in L_p$.

Recall that the space $\mathcal{C}_{00}(\mathbb{R})$ of continuous functions of compact support is dense in $L_p(\lambda)$. For $g\in C_{00}(\mathbb{R})$ $S_ng(x)=\frac1n\sum^{n-1}_{k=1}g(x+k)\xrightarrow{n\rightarrow\infty}0$ and so $S_ng$ also converges in measure to $0$.

Showing that $\|S_ng\|_p\xrightarrow{n\rightarrow\infty}0$ for all $g\in C_{00}(\mathbb{R})$ would imply the result for all $f\in L_p$. Indeed, given $f\in L_p$ and $\varepsilon>0$ choose $g\in C_{00}(\mathbb{R})$ such that $\|f-g\|_p<\varepsilon$. Then $$\|S_nf\|_p\leq \|S_n(f-g)\|_p+\|S_ng\|_p<\varepsilon+\|S_ng\|_p$$ whence we obtain $$\limsup_n\|S_nf\|_p\leq\varepsilon, \qquad\varepsilon>0$$ Thus, $\|S_nf\|_p\xrightarrow{n\rightarrow\infty}0$.


Edit:

After more thinking I realized that we can do without having to check for uniform integrability and instead directly prove that $\|S_ng\|_p\xrightarrow{n\rightarrow\infty}0$ for all $g\in\mathcal{C}_{00}(\mathbb{R})$. The support of $g$ is contained in some intercal of the form $I:=(k,m]$ for $k,m\in\mathbb{Z}$. Splitting $I$ in $m-k$ of length $1$ we write $g=\sum^{m-k-1}_{\ell=0}g\mathbb{1}_{(k+\ell,k+\ell+1]}$. Let $\phi_{\ell}:=g\mathbb{1}_{(k+\ell,k+\ell+1]}$. Since the operator $S_n$ is linear, we have that $S_ng=\sum^{m-k-1}_{\ell=0}S_n(\phi_\ell)$

The advantage of breaking $g$ in pieces of $\phi_\ell$ of support of length one is that successive integer translations yield functions whose supports whose interiors are pairwise disjoint, for $$\operatorname{supp}(\phi_\ell\circ T^j)=[k+\ell-j,k+\ell+1-j].$$ Then $$\|S_ng\|_p\leq\sum^{m-k-1}_{\ell}\|S_n\phi_\ell\|_p$$ and by translation invariance, for each $0\leq \ell < m-k$ \begin{align} \|S_n\phi_\ell\|^p_p&=\frac{1}{n^p}\int\Big|\sum^{n-1}_{j=0}\phi_\ell\circ T^k\Big|^p=\frac{1}{n^p}\int\sum^{n-1}_{j=0}|\phi_\ell\circ T^k|^p=\frac{1}{n^{p-1}}\|\phi_\ell\|^p_p \end{align} Putting things together, we obtain that $$\|S_ng\|_p\leq n^{\tfrac1p -1}(m-k)\|g\|_p\xrightarrow{n\rightarrow\infty}0$$

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