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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

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Your (3) is wrong. It should say $E \in \overline{M}$, not $E \in M$. – Robert Israel May 6 '12 at 18:12

Let's start with (1). The only non-obvious part is to prove $\overline{M}$ is closed under countable unions. So if $A_n$ is a sequence of members of $\overline{M}$, there will be corresponding $B_n$ and $C_n$ with $B_n \subset A_n \subset C_n$ and $\mu(C_n \sim B_n) = 0$. Now what can you say about $\bigcup_n A_n$, $\bigcup_n B_n$ and $\bigcup_n C_n$?

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