This Problem Sheet is based on Lectures 21 and 22.

• Problem 1 asks you to prove that circle rotations are never weakly mixing with respect to Lebesgue measure.
• Problem 2 asks you to improve Corollary 21.7 under the additional assumption that the probability space has a countable basis.
• Problem 3 continues with the assumption that the probability space has a countable basis, and asks you give (yet another) characterisation of weakly mixing.
• Problem 4 asks you to prove an $L^p$ version of the Birkhoff Ergodic Theorem from Lecture 20.