Derivation of Borel measures w.r.t a “bigger” measure?

Question

Given two Borel measures $\mu_1$ and $\mu_2$ on $\mathbb R$, is there always a Borel measure $\mu$ on $\mathbb R$ such that

$$ d\mu_1=w_1 d\mu,\qquad d\mu_2=w_2 d\mu, $$ for some functions $w_1$ and $w_2$ on $\mathbb R$ ?

Answer 1

Yes, if both measures are $\sigma$-finite. Then for such measure to exist it should be s.t. $\mu_i$ are absolute continuous w.r.t. $\mu. $ Just take $$ \mu = \mu_1+\mu_2. $$

If at least one of the measures is not $\sigma$-finite then as @GEdgar mentioned, there is a counterexample. Take $\mu_1$ to be Lebesgue measure and $\mu_2$ to be counting measure. Assume that measure $\mu$ exists, then

1. by considering $1=\mu_2(\{a\})$ prove that $\mu(\{a\})>0$ for any singleton set $\{a\}$;

2. by considering $1=\mu_1([0,1])$ prove that $w_1(x)>0$ in only countably many $x\in[0,1]$ (say, on the set $K$);

3. prove that $\int\limits_{[0,1]\setminus K} w_1(x)\mu(dx)= 0$ while $\mu_1([0,1]\setminus K) = 1$.