The Range of Vector Measures over Topological Sets and a Unified Liapunov Theorem

Let (X, Σ, μ) be a finite, nonatomic, measure space. Let G=span{g1, g2, …, gn}⊆L1, and let the support of G be X, a.e. For f∈L∞, put M(f)=(∫X fg1 dμ, ∫X fg2 dμ, …, ∫X fgn dμ). Then Q={M(f): f∈B(L∞)} is a compact convex set. Liapunovʹs classsical theorem is that also Q={M(s): s∈ext B(L∞)}. This paper characterizes when the functions, s, in Liapunovʹs theorem can further be restricted to being the signs of continuous functions. That is, suppose there is a topology on X, and that Σ is the Baire sets. Let S be the collection of supports of continuous non-negative functions. Theorem.The following are equivalent: (i)Q={M(s): s∈ext B(L∞), and s=sgn f for some f∈C(X)}, (ii)for every g∈G, {x: g(x)>0} and {x: g(x)⩽0} are in S, a.e. The theorem is proved in a general setting. If the σ-field in the general theorem is arbitrary, the theorem becomes Liapunovʹs Theorem. The theorem above results when the σ-field is the Baire sets. A setting with the Borel sets produces Q as the range of M over extreme functions that are both lower semi-continuous a.e. and upper semi-continuous a.e.