Is a measure with non-negative value on open sets is non-negative measure?

Question

If a measure has non-negative measure value on open sets, then is it true that measure is non-negative?

Answer 1

No: let $\nu$ be counting measure on $\mathbb{Q}\subset\mathbb{R}$, and define $$ \mu=\nu-\delta_{\sqrt{2}}$$

Then $\mu(O)\geq 0$ for any open subset $O$ of $\mathbb{R}$, but $\mu(\{\sqrt{2}\})=-1<0$.