Homeomorphic measures in metric spaces

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