I wanted to add a quick follow up question to one I asked earlier here.
To summarize, there I let $\mathcal{R}$ be the set ring of subsets of $\mathbb{Q}$ consisting of finite unions of left-open, right closed intervals on the rational line.
With George Lowther's help, I know there exists a finitely additive $\mu\colon\mathcal{R}\to[0,\infty]$ where $\mu=\nu\circ f^{-1}$ is a function such that $\mu((a,b])=b-a$, where $\nu$ is the restriction of the Lebesgue measure on $\mathbb{R}$ to finite unions of half-open real intervals $(a,b]$, and $f$ is the function mapping finite unions of half-open real intervals to their intersection with $\mathbb{Q}$. That is, $f(A)=A\cap\mathbb{Q}$ where $A$ is a finite union of left-open, right-closed intervals on the real line.
One additional thing I'm curious about, with the $\mu$ constructed previously, is it possible to extend $\mu$ to a measure on the $\sigma$-algebra $\mathcal{T}$ generated by $\mathcal{R}$, that is, the intersection of all $\sigma$-algebras containing $\mathcal{R}$?
My suspicion is that countable additivity does not come along with $\mu$, so no such extension is possible, but I'm not 100% sure.
Thanks.