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I need some help with the proof of the following.

Let $(X,\mathcal{A},\mu)$ be a measure space with $\mu$ being $\sigma$-finite, $\mu^*$ be the outer measure given by the formula $\mu^*(E)=\inf\{\sum_{n}\mu(A_n): E\subset\bigcup A_n, (A_n)\subset\mathcal{A}\}$ and $\mathcal{M}$ the $\sigma$-algebra of the $\mu^*$-measurable sets. Also let $\mathcal{A}_{\mu}=\{A\subset X: \exists E,F\in\mathcal{A},$ with $E\subset A\subset F$ and $ \mu(F\setminus E)=0\}$ and $\overline{\mu}:\mathcal{A}_\mu\to [0,+\infty]$ given by $\overline{\mu}(E)=\sup\{\mu(B): B\in\mathcal{A}, B\subset E\}.$

Then $\mathcal{A}_\mu=\mathcal{M}$ and $\overline{\mu}=\mu^*\vert_{\mathcal{M}}$.

Okay, so it is known that the measure space $(X,\mathcal{A}_\mu, \overline\mu)$ is the completion of $(X,\mathcal{A},\mu)$ and by the facts that $\mu^*\vert_\mathcal{A}=\mu$ and $\mathcal{A}\subset\mathcal{M}$, we have that $\mathcal{A}_\mu\subset\mathcal{M}$ and that $\mu^*\vert_{\mathcal{A}_\mu}=\overline\mu$. So in order to prove the statement above, it would be enough to show that $\mathcal{M}\subset\mathcal{A}_\mu$. I'm trying to prove that if $A\in\mathcal{M}$ and $\mu^*(A)<\infty$ then $A\in\mathcal{A}_{\mu}$ but I'm stuck. My progress is the following:

For each $n\in\mathbb{N}$ there exists a sequence $(A_k^{(n)})\subset\mathcal{A}$ such that $\sum_{k}\mu(A_k^{(n)})<\mu^*(A)+\frac{1}{n}$. Let $A_n=\bigcup_{k}A_k^{(n)}\in\mathcal{A}$. We have that $\mu(A_n)\leq\sum_{k}\mu(A_k^{(n)})<\mu^*(A)+\frac{1}{n}$. Finally let $F=\bigcap_{n} A_n$. Then $\mu(F)\leq\mu(A_n)<\mu^*(A)+\frac{1}{n}$ for all $n$. Taking limits we have that $\mu(F)\leq\mu^*(A)$, but $A\subset F$, so $\mu^*(A)\leq\mu^*(F)=\mu(F)$ so we found the first desirable set. But what about the other? I'm stuck here and I can't seem to be able to use the sigma-finiteness. Any help?

EDIT: Maybe considering the fact that the sigma-algebra $\mathcal{A}_1=\{A\cup E: A\in\mathcal{A}, E\subset F,$ where $F\in\mathcal{A}, \mu(F)=0\}$ is equal to $\mathcal{A}_\mu$ helps. It is easy to show this equality.

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Denote by $(X_n)_{n \in \mathbb{N}} \subseteq \mathcal{A}$ a sequence of increasing sets, $X_n \uparrow X$, such that $\mu(X_n)< \infty$ for all $n \in \mathbb{N}$. You have already shown that

$$B \in \mathcal{M} \implies \exists F \in \mathcal{A}, F \supseteq B: \mu(F) = \mu^*(B). \tag{1}$$

Roughly speaking, the idea is to use $(1)$ for $B:=A^c \in \mathcal{M}$ and then define $E:=F^c$; if $\mu$ is a finite measure, you can easily verify that the so-defined set has all desired properties. For $\sigma$-finite measures the reasoning is somewhat more complicated.


Fix $A \in \mathcal{M}$. Applying $(1)$ for $B:=A^c \cap X_n \in \mathcal{M}$ we find that there exists a set $H_n \in \mathcal{A}$, $H_n \supseteq A^c \cap X_n$, such that

$$\mu(H_n) = \mu^*(A^c \cap X_n)<\infty.$$

Note that $E_n := X_n \backslash H_n \in \mathcal{A}$ satisfies $E_n \subseteq A \cap X_n$ and

$$\mu(E_n) = \mu(X_n)-\mu(H_n) = \mu^*(X_n)-\mu^*(A^c \cap X_n) = \mu^*(X_n \backslash A^c) = \mu^*(X_n \cap A)$$

where we have used in the penultimate step that $A^c \in \mathcal{M}$. Combining this with the fact that $X_n \cap A^c \in \mathcal{M}$, we find that

$$\mu^*((X_n \cap A) \backslash E_n)=0. \tag{2}$$

Now set $E:= \bigcup_{n \geq 1} E_n \in \mathcal{A}$, then $E \subseteq A$ and, by the monotonicity and subadditivity of $\mu^*$,

$$\begin{align*} \mu^*(A \backslash E) &= \mu^* \left( \left( \bigcup_{n \geq 1} A \cap X_n \right) \backslash \left( \bigcup_{k \geq 1} E_k \right) \right) \\ &\leq \mu^* \left( \bigcup_{n \geq 1} (A \cap X_n) \backslash E_n \right) \\ &\leq \sum_{n \geq 1} \mu^*((A \cap X_n) \backslash E_n) \\ &\stackrel{(2)}{=} 0.\end{align*}$$

Thus,

$$\mu^*(A) = \mu^*(E) = \mu(E).$$

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