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

Question

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?

Answer 1

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.