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\]
Week 9 Worksheet
- Suppose $T$ is measure-preserving and ergodic on $(X,\B,\mu)$. Fix $B \in \B$ and deduce from
\[
\mu(B \triangle T^{-1} B) = 0
\]
that $\mu(B) \in \{0,1\}$.
- Fix a measure space $(X,\B,\mu)$ and $f \in \mathscr{L}^1(X,\B,\mu)$. Prove that
\[
\int f \intd \mu = \int 1_B \cdot f \intd \mu
\]
whenever $B \in \B$ with $\mu(X \setminus B) = 0$.
- Define $f : [0,1) \to \R$ by
\[
f(x) = \begin{cases} 1 & x \in \Q \\ 0 & x \not\in \Q \end{cases}
\]
for all $x \in [0,1)$.
- Fix $\alpha$ irrational. What does the pointwise ergodic theorem tell you about
\[
\lim_{N \to \infty} \dfrac{1}{N} \sum_{n=0}^{N-1} f(n\alpha + x \bmod 1)
\]
for points $x \in [0,1)$.
- Prove that
\[
\lim_{N \to \infty} \dfrac{1}{N} \sum_{n=0}^{N-1} f(n\alpha + x \bmod 1)
\]
for all $x \in [0,1)$ without using the pointwise ergodic theorem.
- Does the uniform distribution theorem imply the same conclusion?
- Apply the pointwise ergodic theorem to say something about the limit
\[
\lim_{N \to \infty} \dfrac{|\{ 1 \le n \le N : x(n) + x(n+1) \ge 2 x(n+2) \}|}{N}
\]
for points $x \in \{0,1\}^\N$.
- Let $X$ be a compact metric space and let $\mu$ be a probability measure on the Borel σ-algebra $\B$ of $X$. Suppose that $T : X \to X$ is measure-preserving and ergodic.
Prove that there is a set $\Omega \subset X$ with
\[
\forall x \in \Omega \; \forall f \in \mathsf{C}(X) \; \lim_{N \to \infty} \dfrac{1}{N} \sum_{n=0}^{N-1} f(T^n x) = \int f \intd \mu
\]
by an application of the Stone-Weierstrass theorem.