\[
\newcommand{\C}{\mathbb{C}}
\newcommand{\haar}{\mathsf{m}}
\newcommand{\B}{\mathscr{B}}
\newcommand{\P}{\mathcal{P}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\g}{>}
\newcommand{\l}{<}
\newcommand{\intd}{\,\mathsf{d}}
\newcommand{\Re}{\mathsf{Re}}
\newcommand{\area}{\mathop{\mathsf{Area}}}
\newcommand{\met}{\mathop{\mathsf{d}}}
\newcommand{\emptyset}{\varnothing}
\DeclareMathOperator{\borel}{\mathsf{Bor}}
\newcommand{\symdiff}{\mathop\triangle}
\DeclareMathOperator{\leb}{\mathsf{Leb}}
\DeclareMathOperator{\cont}{\mathsf{C}}
\DeclareMathOperator{\lpell}{\mathsf{L}}
\newcommand{\lp}[1]{\lpell^{\!\mathsf{#1}}}
\]
Week 7 Worksheet
-
- Let $(T(x))(n) = x(n+1)$ be the shift map on $X = \{0,1\}^\N$. Describe a point $x \in \{0,1\}^\N$ with dense orbit.
- Let $(T(x))(n) = x(n+1)$ be the shift map on $X = \{0,1,2\}^\N$. Describe a point $x \in \{0,1,2\}^\N$ with dense orbit.
- Let $(T(x))(n) = x(n+1)$ be the shift map on $X = \{0,1\}^\N$. Describe a point $x \in \{0,1\}^\N$ with
- $\displaystyle\lim_{N \to \infty} \dfrac{1}{N} \sum_{n=1}^N x(n) = \dfrac{1}{2}$
- $\displaystyle\lim_{N \to \infty} \dfrac{1}{N} \sum_{n=1}^N x(n) = \dfrac{1}{4}$
- $\displaystyle\lim_{N \to \infty} \dfrac{1}{N} \sum_{n=1}^N x(n) = \dfrac{1}{3}$
- $\displaystyle\lim_{N \to \infty} \dfrac{1}{N} \sum_{n=1}^N x(n) = \dfrac{1}{\sqrt{2}}$
- $\displaystyle\lim_{N \to \infty} \dfrac{1}{N} \sum_{n=1}^N x(n)$ not existing.
- Let $\mu$ be the fair coin measure on $\{0,1\}^\N$. Put
\[
Y = \bigcap_{n \in \N} \{x \in \{0,1\}^\N : x(n) = 1 \Rightarrow x(n+1) = 0 \}
\]
- Describe what it means for $x \in \{0,1\}^\N$ to belong to $Y$.
- Calculate $\mu(Y)$.
- Does
\[
\lim_{N \to \infty} \dfrac{1}{N} \sum_{n=1}^N y(n) = \dfrac{1}{2}
\]
for some point $y \in Y$?
- Does
\[
\lim_{N \to \infty} \dfrac{1}{N} \sum_{n=1}^N y(n) y(n+1) = \dfrac{1}{4}
\]
for some point $y \in Y$?
- Describe a measure $\nu$ on $\{0,1\}^\N$ with $\nu(Y) = 1$.
-
Define $f : \{0,1\}^\N \to [0,1]$ by
\[
f(x) = \sum_{n=1}^\infty \dfrac{x(n)}{2^n}
\]
for all $x \in \{0,1\}^\N$.
- Check that $f$ is measurable.
- Let $\mu$ be the fair coin measure on $\{0,1\}^\N$. Define $\nu$ on $\borel([0,1])$ by $\nu(B) = \mu(f^{-1}(B))$. Check that $\nu$ is a measure.
- What does the strong law of large numbers tell you about
\[
\lim_{N \to \infty} \dfrac{1}{N} \sum_{n=0}^{N-1} 1_{[0,\tfrac{1}{2}]}(2^n x \bmod 1)
\]
for $\nu$ almost-every $x \in [0,1]$?