This Problem Sheet is based on Lectures 23 and 24.

• Problem 1 asks you to show that the natural inclusion $X \hookrightarrow \mathcal{M}(X)$ that sends a point $x \in X$ to the corresponding Dirac measure $\delta_x$ is a topological embedding.
• Problem 2 asks you to prove that a family of commuting dynamical systems always has a common invariant measure.
• Problems 3 and 4 are both about the periodic orbit measures $\wp_{x,p}$ associated to a periodic point $x$ of (not necessarily minimal) period $p$.