Integral representations for a class of operators (Q2517694)

The authors give very simple proofs of the main results of [\textit{L.\,Meziani}, Proc.\ Am.\ Math.\ Soc.\ 130, No.\,7, 2067--2077 (2002; Zbl 1011.28009), J.~Math.\ Anal.\ Appl.\ 340, No.\,2, 817--824 (2008; Zbl 1152.46031)], without the need of Pettis integration. Let \(X\) be a locally compact and \(E\) a quasi-complete locally convex space. Let \(C_b(X,E)\) be the space of all \(E\)-valued bounded functions on \(X\) with strict topologies \(\beta_t, \beta_\infty', \beta_\tau'\). With this notation, the authors prove that a linear continuous mapping \(T:C_b(X,E)\to E\) arises from a scalar measure \(\mu\in(C_b(X),\beta_z)'\) (\(z=t,\infty,\tau\)) if and only if \(g(T(f))=0\) whenever \(g\circ f=0\) for any \(f\in C_b(X,E)\), \(g\in E'\).