A common example of a semiring of sets is the family of half open interals $(a,b]\subseteq\mathbb{R}$. Also, the premeasure $\rho((a,b])=b-a$ is well known to extend to a measure on a $\sigma$-algebra.
With a little tinkering, I believe the "mirror images" across the origin of these intervals also form a semiring. That is, the sets of form $[-b,-a)\cup(a,b]$ for $0 are also a semiring. Say $I_{a,b}=[-b,-a)\cup(a,b]$. If I put nearly the same premeasure $\rho(I_{a,b})=b-a$ on this semiring, then I think that $\rho$ can be extended to a measure on a $\sigma$-algebra by taking the measure which sends a set $A$ to $\mu(A)/2$ for the usual Lebesgue measure $\mu$ on $\mathbb{R}$. (I hope this is correct?)
Is there a way to tell if a closed interval $[a,b]$ is $\rho^*$ measurable? I'm interested in seeing maybe an example first to figure this out. Take an interval $[1,2]$ for example. I know that $[1,2]$ is $\rho^*$ measurable if for any $I\subseteq\mathbb{R}$, then $\rho^*(I)=\rho^*(I\cap[1,2])+\rho^*(I\setminus[1,2]).$
My feeling is that $[1,2]$ is $\rho^*$ measurable just by testing it with a few subsets $I$ of the real line. Is there a way to prove or disprove whether this is true?
Thank you.