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

✍ Scribed by Daniel Wulbert

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 140 KB
• Volume: 182
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.2000.3717

✦ Synopsis

Let (X, 7, +) be a finite, nonatomic, measure space. Let G=span[g 1 , g 2 , ..., g n ] L 1 , and let the support of G be X, a.e. For f # L , put

. This paper characterizes when the functions, s, in Liapunov's theorem can further be restricted to being the signs of continuous functions. That is, suppose there is a topology on X, and that 7 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 _-field in the general theorem is arbitrary, the theorem becomes Liapunov's Theorem. The theorem above results when the _-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.