\[ \newcommand{\C}{\mathbb{C}} \newcommand{\cont}{\operatorname{\mathsf{C}}} \newcommand{\haar}{\mathsf{m}} \newcommand{\B}{\mathscr{B}} \newcommand{\D}{\mathscr{D}} \newcommand{\P}{\mathcal{P}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\sph}{\mathsf{S}^\mathsf{1}} \newcommand{\g}{>} \newcommand{\l}{<} \newcommand{\intd}{\,\mathsf{d}} \newcommand{\Re}{\mathsf{Re}} \newcommand{\area}{\mathop{\mathsf{Area}}} \newcommand{\met}{\mathop{\mathsf{d}}} \newcommand{\emptyset}{\varnothing} \DeclareMathOperator{\borel}{\mathsf{Bor}} \DeclareMathOperator{\baire}{\mathsf{Baire}} \newcommand{\orb}{\mathop{\mathsf{orb}}} \newcommand{\symdiff}{\mathop\triangle} \newcommand{\lp}[1][2]{\operatorname{\mathscr{L}^{\mathsf{#1}}}} \newcommand{\Lp}[1][2]{\operatorname{\mathsf{L}^{\!\mathsf{#1}}}} \]

1. Define $T : (0,1] \to (0,1]$ by \[ T(x) = \dfrac{1}{x} \bmod 1 \] for all $x \in (0,1]$. Write $f(x) = 1/(1+x)$. Verify that \[ \int 1_{(a,b]} \cdot f \intd \lambda = \int 1_{(a,b]} \circ T \cdot f \intd \lambda \] for all $0 \le a \l b \le 1$ where $\lambda$ is Lebesgue measure.
2. Criticise the strategy of the following outline of a proof that irrational rotations are ergodic.
   
   Fix an interval $[a,b) \subset [0,1)$. Suppose that $[a,b)$ is $T$ invariant. Then \[ [a,b) = \bigcup_{n=1}^\infty (T^n)^{-1}([a,b)) = [0,1) \] because every point has dense orbit. Since the σ-algebra generated by the intervals equals the Borel σ-algebra, every non-empty invariant set must equal $[0,1)$.