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

1. Fix a measure-preserving transformation $T$ of a probability space $(X,\B,\mu)$. Suppose $f \in \Lp[2](X,\B,\mu)$ is fixed by the Koopman operator. Prove there is a function $g : X \to \R$ in the equivalence class defined by $f$ such that \[ g(T(x)) = g(x) \] for all points $x \in X$.
2. For each $r,s \in \Z$ define $\chi_{r,s}$ on $[0,1)^2$ by \[ \chi_{r,s}(x,y) = e^{2 \pi i (rx + sy)} \] for all $x,y \in [0,1)^2$. Take for granted that\[ \{ \chi_{r,s} : r,s \in \Z \} \] is an orthonormal basis of $\Lp[2,\C](X,\B,\mu)$ where $X = [0,1)^2$ and $\B$ is the Borel σ-algebra and $\mu$ is the measure defined by assigning rectangles their Eucliden areas. Prove that \[ T(x,y) = (2x+y \bmod 1, x+y \bmod 1) \] is ergodic.