Let $(X,S,\mu)$ be a measure space s.t. $\mu(X)=1$.
Let $\mu^{*}$ be defined on $X$ by:
$\forall E\subseteq X:\,\mu^{*}(E):=\text{inf}\left\{\sum_{i=1}^{\infty}\mu(A_{i})\,|\, A_{i}\in S,E\subseteq\cup A_{i}\right\}$
I have a set $E$ s.t. $\mu^{*}(E)=1$, does this mean $\mu^{*}(E^{c})=0$?
I have tried to work with the definition, given $\epsilon>0$ there is $N\in\mathbb{N}$ and $\{A_{i}\}_{i=1}^{N}\subseteq S$ s.t. $\sum_{i=1}^{N}\mu(A_{i})\geq1-\epsilon$ and $E\subseteq\cup_{i=1}^{N}A_{i}$
I want to use the $A_{i}$'s to get some set $B$ s.t $E^{c}\subseteq B$ and $\mu(B)<\epsilon$, but I didn't manage to find such a set.
Can someone please help me understand if this claim is true, and if so how to prove it?