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
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.