Exam Practice Questions

1. 1. Fix a set $X$ and a set $A\subset X$. Prove that $$\{\varnothing ,A,X\setminus A,X\}$$ is a σ-algebra on $X$.
   2. Show by example that the union of two σ-algebras need not be a σ-algebra.
2. 1. Prove that $f:\mathbb{R}\to \mathbb{R}$ defined by $$f(x)={\displaystyle \frac{1}{1+x^2}}$$ belongs to ${\mathsf{L}}^{\mathsf{1}}(\mathbb{R},\mathcal{B},\lambda )$ where $\mathcal{B}$ is the Borel σ-algebra on $\mathbb{R}$ and $\lambda$ is Lebesgue measure.
   2. With $$f_n(x)=\{\begin{array}{ll}{\displaystyle \frac{\sin (\frac{x}{n})}{\frac{x}{n}(x^2+1)}}& x\ne 0\\ 1& x=0\end{array}$$ for all $n\in \mathbb{N}$ and all $x\in \mathbb{R}$ evaluate $$\underset{n\to \mathrm{\infty }}{lim}\int f_n\mathsf{d}\lambda$$ using the dominated convergence theorem.
3. Fix a measure-preserving transformation $T$ on a probability space $(X,\mathcal{B},\mu )$.
   1. What is the definition of ergodicity for $T$?
   2. Write down a characterization of ergodicity using the Koopman operator of $T$.
   3. Suppose $T$ is ergodic. Prove that if $T^2$ is not ergodic then $T$ has an eigenfunction with eigenvalue $-1$.
4. Fix a measure-preserving transformation $T$ on a probability space $(X,\mathcal{B},\mu )$.
   1. State the pointwise ergodic theorem.
   2. Let $T$ be the full shift on $\{0,1{\}}^{\mathbb{N}}$ and let $\mu$ be the fair-coin measure. Use $f:\{0,1{\}}^{\mathbb{N}}\to [0,\mathrm{\infty })$ defined by $$f(x)=\{\begin{array}{ll}1& x(1)=1\\ 1& x(1)=0\text{ and }x(2)=1\\ 0& \text{otherwise}\end{array}$$ for all $x$ in $\{0,1{\}}^{\mathbb{N}}$ to deduce a statistical statement about coin tosses from the pointwise ergodic theorem.