Given a family $\{\mu \}_{i\in I}$ on a Polish space (complete, separable metric space) $X$. When does there exist a measure $\lambda$ such that $$\mu_i=f_i \lambda$$

where the f_i are densities (Radon-Nikodym) of $mu_i$ wrt to $\lambda$.

EDIT: What is a verifyable condition in the case I is uncountable.