Week 8 Coursework Test

Define $T : (0, 1] \to (0, 1]$ by $T (x) = \frac{1}{x} mod 1$ for all $x \in (0, 1]$ . Write $f (x) = 1 / (1 + x)$ . Verify that $\int 1_{(a, b]} \cdot f d λ = \int 1_{(a, b]} \circ T \cdot f d λ$ for all $0 \leq a < b \leq 1$ where $λ$ is Lebesgue measure.
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 $[a, b) = ⋃_{n = 1}^{\infty} (T^{n})^{- 1} ([a, b)) = [0, 1)$ 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 $[0, 1)$ .