(Problem) Let $(\Omega, \mathcal{F}, P)$: a probability space, and $\mathcal{A}$: algebra which satisfies $ \sigma (\mathcal{A})=\mathcal{F}$. And we difine $\tilde{P}$ as
$$ \tilde{P}(A) = \inf \left\{ \sum_{i=1}^{\infty} P(A_i) \mid A \subset \cup _{i=1}^{\infty}A_i , A_i \in \mathcal{A} \right\}. $$
We consider a class of sets $\mathcal{G}$ s.t.
$$ \mathcal{G} = \{A\in\mathcal{F} \mid \tilde{P}(A) = P(A), \ \tilde{P}(A^{\text{c}}) = P(A^{\text{c}}) \}. $$
Prove $\mathcal{G} = \mathcal{F}$.
(Question) I managed to show $\mathcal{A} \subset \mathcal{G}$, so I next tried to show $\mathcal{G}$ is a $\sigma$-algebra, where I am stuck. I don't see why a countable union of elements in $\mathcal{G}$ is also an element in $\mathcal{G}$.
How can I solve this? I looked up in some textbooks and found this might have a relationship to completion of a probability space.
Thank you in advance.