I have been stuck in this question for a while. I am not able to prove the first part. I have thought along the following possible directions:
I tried showing that $\mu^{*}(G^{c})=0$. However, I couldn't prove this.
I tried proving that if $B$ is in the given $\sigma$ field $\mathfrak{F}$ and $B$ contains $G \cap A $ then $\mu^{*}(B \setminus A)=0$.
However, I could't proceed much further while proving these. Could you please suggest me a good hint to solve the first part?
Also, for the second question, I aim to show that the set of $\mu^{*}$ measurable sets contains $\mathfrak{G}$. Now, restriction of $\mu^{*}$ to $\mu^{*}$ measurable sets is a measure. Will that be the correct approach?