Homeomorphic measures in metric spaces

Abstract

For any nonatomic, normalized Borel measure μ \mu in a complete separable metric space X X there exists a homeomorphism h : N → X h:\mathfrak {N} \to X such that μ = λ h − 1 \mu = \lambda {h^{ - 1}} on the domain of μ \mu , where N \mathfrak {N} is the set of irrational numbers in ( 0 , 1 ) (0,1) and λ \lambda denotes Lebesgue-Borel measure in N \mathfrak {N} . A Borel measure in N \mathfrak {N} is topologically equivalent to λ \lambda if and only if it is nonatomic, normalized, and positive for relatively open subsets.