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

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