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

TL;DR

This paper characterizes relatively compact subsets of the space ^p in the weak topology, linking them to weakly compact subsets of L^p, with implications for risk theory and optimization.

Contribution

It provides a characterization of relatively compact sets in ^p under a specific weak topology, connecting to weak compactness in L^p and applications in risk measures.

Findings

Relatively compact subsets of ^p are characterized via weak compactness in L^p.

The results aid in understanding the Lebesgue property of convex risk measures.

Applications include the Mackey topology and optimization theory.

Abstract

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.