I have been struggling with the following problem for many hours now :
Suppose $R$ is an algebra of sets on $X$ and $\mathcal{A}$ is the $\sigma$-algebra generated by $R$. Let $\mu$ be a measure defined on $\mathcal{A}$. Show that $\mu$ is $\sigma$-finite on $\mathcal{A}$ if an only if it is $\sigma$-finite on $R$.
One of the implications is trivial, since if you have a collection of sets in $R$ they are also in $\mathcal{A}$.
The other one, I have been struggling with.
If $\mu$ was actually constructed from $\mu$ on $R$ using the outer measure construction, than the solution would be easy. The problem is, we know nothing about $\mu$ on $\mathcal{A}$ a priori.
In order to show that $\mu$ on $\mathcal{A}$ is equal to $\mu^*$, the outer measure constructed via $\mu$ on $R$, you would need to use the fact that $\mu$ is $\sigma$-finite on $R$, which is precisely what we are trying to prove...