I am reading the proof of Theorem 2.4 in the book "Probability in Banach spaces" by Ledoux and Talagrand. There's a lot of confusion so I really need help here.
And the proof is given below.
I don't understand why "By difference, $(X_i)$ is weakly relative compact", this is just too quick and I could fathom the reasoning behind.
The Fubini part highlighted in the pic seems questionable, how do we prove this rigorously? Perhaps I'm confused about the notation. They used $\mathbb P_\epsilon$ (resp. $\mathbb P_X$) to denote the condition expectation w.r.t $X = (X_i)$ (resp. $\epsilon = (\epsilon_i)$). I tried to get an estimate, $\forall \delta$, $\exists \gamma$, s.t $\mathbb P(X_i \in f^{-1}[-\gamma, \gamma]) > 1- \delta $ but the set $ f^{-1}[-\gamma, \gamma] \subset B$ may not be compact...
The implication from $f(X_i) \rightarrow 0$ a.s. to $(X_i)$ is a tight sequence also seems questionable and I cannot see why.
What contradiction do we have at the end?
Thank you!
