\[
\newcommand{\C}{\mathbb{C}}
\newcommand{\haar}{\mathsf{m}}
\newcommand{\B}{\mathscr{B}}
\newcommand{\D}{\mathscr{D}}
\newcommand{\P}{\mathcal{P}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\g}{>}
\newcommand{\l}{<}
\newcommand{\intd}{\,\mathsf{d}}
\newcommand{\Re}{\mathsf{Re}}
\newcommand{\area}{\mathop{\mathsf{Area}}}
\newcommand{\met}{\mathop{\mathsf{d}}}
\newcommand{\emptyset}{\varnothing}
\DeclareMathOperator{\borel}{\mathsf{Bor}}
\DeclareMathOperator{\baire}{\mathsf{Baire}}
\newcommand{\symdiff}{\mathop\triangle}
\]
Exam Practice Questions
-
- Fix a set $X$ and a set $A \subset X$. Prove that
\[
\{ \emptyset, A, X \setminus A, X \}
\]
is a σ-algebra on $X$.
- Show by example that the union of two σ-algebras need not be a σ-algebra.
-
- Prove that $f : \R \to \R$ defined by
\[
f(x) = \dfrac{1}{1+x^2}
\]
belongs to $\mathsf{L}^{\!\mathsf{1}}(\R,\B,\lambda)$ where $\B$ is the Borel σ-algebra on $\R$ and $\lambda$ is Lebesgue measure.
- With
\[
f_n(x) = \begin{cases} \dfrac{\sin(\tfrac{x}{n})}{\tfrac{x}{n} (x^2+1)} & x \ne 0 \\ 1 & x = 0 \end{cases}
\]
for all $n \in \N$ and all $x \in \R$ evaluate
\[
\lim_{n \to \infty} \int f_n \intd \lambda
\]
using the dominated convergence theorem.
-
Fix a measure-preserving transformation $T$ on a probability space $(X,\B,\mu)$.
- What is the definition of ergodicity for $T$?
- Write down a characterization of ergodicity using the Koopman operator of $T$.
- Suppose $T$ is ergodic. Prove that if $T^2$ is not ergodic then $T$ has an eigenfunction with eigenvalue $-1$.
-
Fix a measure-preserving transformation $T$ on a probability space $(X,\B,\mu)$.
- State the pointwise ergodic theorem.
- Let $T$ be the full shift on $\{0,1\}^\N$ and let $\mu$ be the fair-coin measure. Use $f : \{0,1\}^\N \to [0,\infty)$ defined by
\[
f(x) = \begin{cases} 1 & x(1) = 1 \\ 1 & x(1) = 0 \textup{ and } x(2) = 1 \\ 0 & \textup{otherwise} \end{cases}
\]
for all $x$ in $\{0,1\}^\N$ to deduce a statistical statement about coin tosses from the pointwise ergodic theorem.