Let $X=\mathbb{Q}\cap[0,1]$. Let $\mathcal{A}$ be the algebra generated by the collection of all sets of the form $(a,b)\cap X$ where $0. I would like to find out whether there is a finitely additive measure $\mu:\mathcal{A}\rightarrow[0,\infty]$ such that $\mu(X\cap(a,b])=b-a$ for all $a,b\in(0,1)$ with $a.
Remark: Previously I was considering the $\sigma$-algebra generated by this collection of sets and whether there is a measure having those same properties but I soon found out that there isn't (basically because $X$ is countable).