Characterization of Compact Subsets of ${\mathcal{A}}^p$ with Respect to Weak Topology

In this brief article we characterize the relatively compact subsets of ${\mathcal{A}}^p$ for the topology $\sigma ({\mathcal{A}}^p,{\mathcal{R}}^q)$ (see below), by the weak compact subsets of $L^p$ . The spaces ${\mathcal{R}}^q$ endowed with the weak topology induced by ${\mathcal{A}}^p$, was recently employed to create the convex risk theory of random processes. The weak compact sets of ${\mathcal{A}}^p$ are important to characterize the so-called Lebesgue property of convex risk measures, to give a complete description of the Makcey topology on ${\mathcal{R}}^q$ and for their use in the optimization theory.