Set-valued measures generated by set-valued increasing processes (Q2757113)

Let \(X\) be a reflexive Banach space with separable dual and \(P(X)\) the space of all weakly compact convex sets. A set-valued function \(S: [0,\infty)\to P(X)\) is said to be increasing if \(S(s)\subset S(t)\) whenever \(s< t\). In this paper, it is shown that if \(S\) is increasing and absolutely continuous (in some sense), then \(S\) induces a set-valued Borel measure \(M\) on \([0,\infty)\) and \(M[0,t]= S(t)\). A similar result is also proved for \(S:\Omega\times [0,\infty)\to P(X)\), where \(\Omega\) is a measure space. However, the proof is more involved.