Let $(X,\Sigma,\mu)$ be a signed measure space.
If $\mu^*$ denotes the exterior measure of $\mu$ and $\Lambda_{\mu^*}$ denotes the set of $\mu^*$ measurables sets, i'm trying to find a relation between the sets $\Lambda_{{\mu^+}^*}$ and $\Lambda_{{\mu^-}^*}$, where $\mu = \mu^+ - \mu^{-}$ and $|\mu|$ is the total variation.
I think that $$\Lambda_{{\mu^+}^*} \cap \Lambda_{{\mu^-}^*} = \Lambda_{{|\mu|}^*}$$ Is it true?
Thanks