Working in the cantor space $2^\omega$. Giving two measurable spaces $(2^\omega, T, \mu)$ and $(2^\omega, T, \nu)$ an inverse-measure-preserving function $f:2^\omega \rightarrow 2^\omega$ is such that $\mu(A)=\nu(f^{-1}(A))$ for any borel set $A$.

Giving a class of measure $\mathcal{M}$, I wonder if one can find some conditions on $\mathcal{M}$ such that there exists a inverse-measure-preserving function $F$ between any measures on $\mathcal{M}$ and the Lebesgue measure :

i.e. there exists $F$ such that for any $\mu \in \mathcal{M}$ we have $\lambda(A)=\mu(F^{-1}(A))$

Does anyone know if there is some results in that direction, or any references I could look at ?

Thanks in advance.