I was thinking if it is possible to come up with a $\sigma$-finite measure on $\mathbb R$ which is positive on any uncountable set. I think that I have a proof that there is no such measure - but I am not sure if it is formal enough.
The proof: let $\mu$ be such measure, then for any $[a,b]$ such that $a0$. Now, if there is only finitely many $x$ such that $\mu(f_x)>1/n$ for all $n\in\mathbb N$ then we obtain that there are only countably many $x$. As a result, for some $n\in\mathbb N$ there are infinitely many $x$ such that $\mu(f_x)>1/n$ and hence $\mu([a,b])=\infty$.