I am trying to figure out how to prove this problem.
Since $A \subseteq (A\cap Y)\cup (A\cap Y^c)$, by subbadditivit, $\mu^*(A)\leq \mu^*(A\cap B)+\mu^*(A\cap B^c)$. I can't figure out the other direction. I keep messing around with $\mu^*(A)=\mu^*(A\cap Y)+\mu^*(A\cap Y^c)$, but I keep hitting dead ends. Any suggestions? Thanks