Continous measure-preserving maps onto Peano spaces

Abstract

In this paper we obtain measure-theoretic versions of some topological existence theorems relating to continuous maps, in particular the Hahn-Mazurkiewitz Theorem. Let X be a Peano Space and λ a Borel measure on X with λ(X) = 1. There is a continuous measure-preserving surjection from the unit interval (with Lebesque measure) to X if and only if the support of Λ is X. 1. Introduction. Peano's construction in 1890 of a continuous surjection from I = [0,1] to 5 = I x I led twenty-five years later to the Hahn-Mazurkiewitz theorem, which characterizes those topological spaces which are the continuous image of the unit interval.1 Since the original paper, there have been many elegant constructions of space-filling curves, including one by Hubert.2 Perhaps most striking about Hubert's construction is its remarkable symmetry: one sees easily that for each of the intervals A,, = [ί/4', (ί + 1)/4;L the image of Au is a square of area 4~'; and the images of distinct intervals Au and Aktj intersect in a set of (planar Lebesgue) measure zero. It is easy, in fact, to verify that Hubert's space-filling curve is measure-preserving. This suggests the possibility that, under suitable restrictions, a Peano space X which is also a measure space might be the image of a continuous measure-preserving map from the unit interval. Clearly a necessary condition for the existence of such a map is that open subsets of X have positive measure. The aim of this paper is to show that this condition is sufficient as well. We will prove THEOREM 1. Let I = [0,1] and μ be Lebesgue measure on I. Let Xbe a Peano space and λ a Borel measure on X with λ (X) = 1. Then a (necessary and) sufficient condition that there be a continuous measurepreserving surjection f: (I,μ)-*(X,λ) is that X be the support of λ3. 1. The Hahn-Mazurkiewitz theorem states: