reference for "compactness" coming from topology of convergence in measure I have found this sentence in a paper of F. Delbaen and W. Schachermayer with the title: A compactness principle for bounded sequences of martingales with applications. (can be found here)

On page 2, I quote: "If one passes to the case of non-reflexive Banach spaces there is—in general—no
analogue to theorem 1.2 pertaining to any bounded sequence $(x_n )_{n\ge 1} $ , the main
obstacle being that the unit ball fails to be weakly compact. But sometimes there
are Hausdorff topologies on the unit ball of a (non-reflexive) Banach space which
have some kind of compactness properties. A noteworthy example is the Banach
space $ L^1 (Ω, F, P) $ and the topology of convergence in measure."

So I'm looking for a good reference for topology of convergence in measure and this property of "compactness" for $ L^1 $ in probability spaces.

Thx

math
So that this question has an answer: t.b.'s comment suggests that the quotes passage relates to the paper's Theorem 1.3, which states:


  Theorem. Given a bounded sequence $(f_n)_{n \ge 1} \in L^1(\Omega, \mathcal{F}, \mathbb{P})$ then there are convex combinations
  $$g_n \in \operatorname{conv}(f_n, f_{n+1}, \dots)$$
  such that $(g_n)_{n \ge 1}$ converges in measure to some $g_0 \in L^1(\Omega, \mathcal{F}, \mathbb{P})$.


This is indeed "some kind of compactness property" as it guarantees convergence after passing to convex combinations.
Under the right assumptions about the measure space, any sequence of measurable functions that is Cauchy in measure actually converges in measure to some target function. Here is a set of lecture notes by C. Heil that has more information. The full webpage for that course is here.