I'm studying measure theory, but I'm hard to understand some notation below.
*M is a $\sigma$-algebra.
Notation $\bar M$ : $A \in \bar M \Longleftrightarrow$ there exist $B,C \in M$ such that
$B \subset A \subset C$ and $\mu (C \sim B) = 0$
and here, $\mu (B)=\mu (C)$
so let, $\bar{\mu}(a)=\mu(b)=\mu(C)$
In above Notation, I wanna prove the things below. how can I do that?
1. $\bar M$ is a $\sigma$-algebra
2. $\bar\mu$ is well defined and is measure.
3. $E \subset A \in \bar M$ and $\bar \mu (A) =0$, then $E \in M$
It's easy to know in intution but I don't know how can I prove it.. I wanna your help, T.T