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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.

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What is $f_i$ supposed to be? –  Aaron Tikuisis Aug 20 '12 at 12:49
    
the densities/radon nikodym derivative –  warsaga Aug 20 '12 at 12:58

1 Answer 1

A countable family of sigma-finite measures, yes.

Can we drop a condition?

Drop sigma-finite: consider two measures on $\mathbb R$: Lebesgue measure and counting measure.

Drop countable: on $\mathbb R$ consider the family of measures, one of them is Lebesgue measure, and the rest are the unit point masses at all the points of $\mathbb R$.

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I think that the OP has something about the dependence $i\mapsto \mu_i$ in mind. I could imagine positive answers for an interval $I$ and continuous dependence if the topology considered on all (finite) measures on $X$ is strong enough. –  Jochen Wengenroth Aug 20 '12 at 14:17

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