Assume that $(X, \mathcal{R}, \mu)$ and $(X, \mathcal{R}, \nu)$ are premeasures on a ring $\mathcal{R}$ such that there exists $\{A_n\} \subset \mathcal{R}$ where $A_n \uparrow X$ and $\mu(A_n) < \infty$. Similarly, there exists $\{B_n\} \subset \mathcal{R}$ such that $B_n \uparrow X$ and $\nu(B_n) < \infty$.
How to prove that if $E$ is measurable with respect to $(\mu + \nu)^*$ in the sense of Caratheodory, then $E$ is also measurable with respect to both $\mu^*$ and $\nu^*$?
I am trying to prove that if $E$ is $(\mu + \nu)^*$-measurable, then for for any $A \subset X$, $\mu^*(A)=\mu^*(A \cap E) + \mu^*(A \cap E^c)$. I am not heading to any direction using just the definition of measurability. Do you have an idea on how to use the assumptions given above?