If a Borel measure is positive on every nonempty set, can it be finite? Let $\mu$ be a positive measure on $\mathscr{B}_R$, which is strictly positive on any non empty set. Can it be finite?
I thought about finding a very fine partition of $R$, and claim that the measure is infinity if you sum up all the measure on each subset. But I think in order to do that, we need uncountably many positive terms to sum, but additivity of measure only holds for countable sum. 
I'm very stuck. Could you give me some suggestions or comments?
Thank you. 

For every $x$ you have $a_x=\mu(\{x\})>0$. Since there are uncountably many $x$ this implies that $\mu$ is infinite.
Except you're worried that we only have countable additivity. Fine: Say $E_n$ is the set of $x$ with $a_x>1/n$. There exists $n$ so that $E_n$ is uncountable. In particular $E_n$ contains an infinite countable subset $F$. And so countable additivity shows that $$\mu(F)\ge\sum_{j=1}^\infty 1/n=\infty.$$
