Consider Durrett’s proof of the Skorokhod embedding (theorem 8.1.1 in the fifth edition).

1. Show by example that the stopping time $T_{U,V}$ constructed in this proof is not in general the same as the one we constructed in class to prove the same theorem, even though $B_T \stackrel{d}{=} X$ in both cases.

2. Rewrite Durrett’s proof in your own words, by doing something like the following: (a) understand the proof and write down notes of how it works, and then (b) looking at your notes but not the book, write out the proof, making sure to explain the various steps with complete sentences.