I was reading van der Vaart's text on asymptotic statistics and found the following theorem.
The text only provides a proof for the direction where we assume asymptotic uniform integrability and want to prove the convergence of means. For the other direction, it refers to van der Vaart's book on Weak Convergence and Empirical Processes (page 69), but I was unable to find the proof of it there as well.
Can anyone help complete the specified direction of the proof? I tried to complete it myself, but I didn't get very far.
EDIT: in case it helps, the definition in the textbook of asymptotic uniform integrability is as follows