Hi to everyone, The ingredients of my problem are the following: I have a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, a set (continuum cardinality) $\mathcal{Q}$ of probability measures on $(\Omega, \mathcal{F})$ and a filtration $(\mathcal{F_n} )_{n \in \mathbb{N}}$

Assumption 1 $\forall P \in \mathcal{Q}$ it holds $P \sim \mathbb{P}$ on $\mathcal{F_n} \ \ \forall n \in \mathbb{N}$

Thanks to this assumption and Radon-Nikodym theorem we can define the likelihood process $$\left( \frac{dP}{d\mathbb{P}}_{|\mathcal{F_n}} \right)_{n \in \mathbb{N}} $$ each of these of course is in $L^1 (\Omega, \mathcal{F_n}, \mathbb{P}) \subset L^1 (\Omega, \mathcal{F}, \mathbb{P})$

Assumption 2 The set $$D_n := \left( \frac{dP}{d\mathbb{P}}_{|\mathcal{F_n}} \middle| \ \ P \in \mathcal{Q} \right) $$ is weakly compact in $L^1 (\Omega, \mathcal{F}, \mathbb{P})$ for all $n \in \mathbb{N}$

This last assumption is my problem. Exactly what mean weakly compactness in $L^1$? How can it be characterized? There is a relation with a sort of compactness of the set $\mathcal{Q}$?

Thank You very much

definitionof weak compactness is part of general functional analysis, namely Banach space theory. However, if you just want a characterization in probabilist's language, then I think UI as Tomek mentions is what you are after. – Yemon Choi Jun 22 '13 at 9:22