# A strange Weakly Compactness in $L^1 ( \Omega, \mathcal{F}, \mathbb{P})$

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

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en.wikipedia.org/wiki/Uniform_integrability and the Dunford-Pettis theorem is something you might be interested in. Alas, I am not sure whether I should elaborate on this. Could you please specify your question? What kind of characterisation you are looking for? – Tomek Kania Jun 21 '13 at 23:50
I'm not sure at all. What does mean weakly compactness in $(L^1 , \sigma(L^1,L^\infty))$? We are in the field of coherent risk measures, indeed this assumptions ensures about continuity from below for a coherent risk measure (Follmer-Schied, Stochastic Finance, Corollary 4.35). My setting is more general, think about conditional risk measures. I have never attended a course in functional an. hence I'm not sure about the concept and the characterizazion of weakly compactness in $L^1$ – Jerry Jun 22 '13 at 9:12
The definition of 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
@jerry weak compactness is a pretty standard notion in functional analysis dealing with locally convex spaces. For $L_1$ there are some nice characterisations. I'm not sure that the theory can be covered here. I'd recommend that you have a look at this book: commonsenseatheism.com/wp-content/uploads/2013/03/… – Rabee Tourky Jun 22 '13 at 10:22
I've read the Follmer-Schied paper. They assume that there is an underlying probability distribution $\mu$ and that each $P$ in $Q$ is represented by means of a density function. Then they assume that $D_n$ is convex and weakly compact. For that paper this is only used for the property that if $f$ is $\mu$-essentially bounded function and $P_n$ is a sequence of distributions in $Q$, then for some subsequence $\int f dP_n$ converges. – Rabee Tourky Jun 22 '13 at 10:50