Week 10 Coursework Test

Fix a measure-preserving transformation $T$ of a probability space $(X, B, μ)$ . Suppose $f \in L^{2} (X, B, μ)$ 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 $x \in X$ .
For each $r, s \in Z$ define $χ_{r, s}$ on $[0, 1)^{2}$ by $χ_{r, s} (x, y) = e^{2 π i (r x + s y)}$ for all $x, y \in [0, 1)^{2}$ . Take for granted that ${χ_{r, s} : r, s \in Z}$ is an orthonormal basis of $L^{2, C} (X, B, μ)$ where $X = [0, 1)^{2}$ and $B$ is the Borel σ-algebra and $μ$ is the measure defined by assigning rectangles their Eucliden areas. Prove that $T (x, y) = (2 x + y mod 1, x + y mod 1)$ is ergodic.