Given a continuous map s ↦ μs, from a compact metric space into the space of nonatomic measures on T, we show the existence of a family (Aαs)α∈[0,1], increasing in α and continuous in s, such that μs(Aαs)=αμs(T)(α∈[0,1]).
A continuous version of Liapunov's convexity theorem
COLOMBO, GIOVANNI;
1988
Abstract
Given a continuous map s ↦ μs, from a compact metric space into the space of nonatomic measures on T, we show the existence of a family (Aαs)α∈[0,1], increasing in α and continuous in s, such that μs(Aαs)=αμs(T)(α∈[0,1]).
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