Week 9 Worksheet

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1. Suppose $T$ is measure-preserving and ergodic on $(X,\mathcal{B},\mu )$. Fix $B\in \mathcal{B}$ and deduce from $$\mu (B\mathrm{△}{T}^{-1}B)=0$$ that $\mu (B)\in \{0,1\}$.
2. Fix a measure space $(X,\mathcal{B},\mu )$ and $f\in {\mathcal{L}}^1(X,\mathcal{B},\mu )$. Prove that $$\int f\mathsf{d}\mu =\int 1_B\cdot f\mathsf{d}\mu$$ whenever $B\in \mathcal{B}$ with $\mu (X\setminus B)=0$.
3. Define $f:[0,1)\to \mathbb{R}$ by $$f(x)=\{\begin{array}{ll}1& x\in \mathbb{Q}\\ 0& x\notin \mathbb{Q}\end{array}$$ for all $x\in [0,1)$.
   1. Fix $\alpha$ irrational. What does the pointwise ergodic theorem tell you about $$\underset{N\to \mathrm{\infty }}{lim}{\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}f(n\alpha +xmod1)$$ for points $x\in [0,1)$.
   2. Prove that $$\underset{N\to \mathrm{\infty }}{lim}{\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}f(n\alpha +xmod1)$$ for all $x\in [0,1)$ without using the pointwise ergodic theorem.
   3. Does the uniform distribution theorem imply the same conclusion?
4. Apply the pointwise ergodic theorem to say something about the limit $$\underset{N\to \mathrm{\infty }}{lim}{\displaystyle \frac{|\{1\le n\le N:x(n)+x(n+1)\ge 2x(n+2)\}|}{N}}$$ for points $x\in \{0,1{\}}^{\mathbb{N}}$.
5. Let $X$ be a compact metric space and let $\mu$ be a probability measure on the Borel σ-algebra $\mathcal{B}$ of $X$. Suppose that $T:X\to X$ is measure-preserving and ergodic. Prove that there is a set $\mathrm{\Omega }\subset X$ with $$\mathrm{\forall }x\in \mathrm{\Omega }\mathrm{\forall }f\in \mathsf{C}(X)\underset{N\to \mathrm{\infty }}{lim}{\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}f(T^{n}x)=\int f\mathsf{d}\mu$$ by an application of the Stone-Weierstrass theorem.