  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.
