Continuous transforms of a Lebesgue measure. (Q1974371)

The following statement is proved: Theorem 1. Let \(K\) be a compact metric space which is the continuous image of \([0,1]\) and \(\mu\) be a Borel probability measure on \(K\) whose topological support coincides with \(K\). Then there exists a continuous surjective \(g:[0,1]\to K\) such that \(\lambda\circ g^{-1}=\mu,\) where \(\lambda\) is the Lebesgue measure on \([0,1].\)