Let ($\Omega,F$) be sample and event spaces. My instructor said the proof has some holes so I’m hoping to get a second look.
First, $\Omega\in G$ is trivial. A probability measure needs to include an entire sample space.
Second, let $A,B\in G, A \subset B$. $P(B-A)=P(B)-P(A)$ which is in $[0,1]$. Thus, $B-A \in G$. My idea is if two sets are in $G$, their difference are in $G$ too.
Lastly, $A_i \uparrow A$ and $A_i \in G$. Using property of measures, $P(A_i) \uparrow P(A)$ too. $P(A) \in [0,1]$ so $A\in G$. However, I was told that I need to use property of measures twice here.