question about sequence of measures

Question

Let {$μ_n$} be a sequence of probability measures on a measure space X.
Then there exist a probability measure ν on X and a sequence {$f_n$} in $L^1(ν)$ such that
$μ_n(A)$=$\int_A f_n dν$ for all $n$ and measurable set $A$.

I think it can be proved by finding ν such that $μ_n$<< ν for all $n$ and apply Radon-Nikodym theorem, but I cannot find it.

Can anyone help?

Answer 1

Let $\nu (E)=\sum_n \frac 1 {2^{n}} \mu_n(E)$. Countable additivity follows by the fact that we can interchange two infinite sums when the terms are non-neagtive. It is obvious that $\mu_n <<\nu$ for each $n$.