Let $\scr{A}$ be an algebra of subsets. Let $(\Omega, \sigma(\mathscr{A}), P)$ be a probability space. Then for each $B \in \sigma(\scr{A})$ and $\epsilon > 0$, there exists $A \in \scr{A}$ such that $$P(A \Delta B) := P((A \setminus B) \cup (B \setminus A)) < \epsilon.$$
There is a suggestion to consider a collection of such all sets $B$ satisfying the condition. I guess I should look at $$G = \{B \subseteq \Omega :\ \mbox{there exists} \ A \in \mathscr{A} \ \mbox{such that} \ P^*(A \Delta B) < \epsilon \ \forall \epsilon > 0\}$$ (I am not sure here that I should use the measure $P$, or its corresponding outer measure $P^*$ instead).
I find that the hint does not much helpful for me, can anyone suggest more about this problem ?
