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
 A: 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.
A: If it's any help, convergence in measure is a metrizable criterion.  For the case of a finite measure space, see Exercise 2.32 in Folland, "Real Analysis", 2nd. ed., or for a general measure space see Exercise 3.22 in my lecture notes here.
A: 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. 
