I was reading a proof ("my question is also about the steps in the proof itself which I will type out here also") regarding a necessary and sufficient condition for measurability. In particular the theorem says:
A subset A $\subset$ $\mathbb{R}^{n}$ is Lebesgue measurable if and only if for every $\epsilon > 0$, there is an open set $G \supset A$ such that $\mu ^{*}(G \setminus A) < \epsilon$.
Proof:
First we assume that A is measurable and show that it satisfies the condition given in the theorem. Suppose that $\mu(A) < \infty$ and let $\epsilon >0$. From (2.12) there is an open set $G \supset A$ such that $\mu(G) < \mu^{*}(A) + \epsilon$. Then since A is measurable,
$\mu^{*}(G\setminus A) = \mu^{*}(G)- \mu^{*}(G\cap A) = \mu(G)-\mu^{*}(A) < \epsilon$
My question is specifically regarding the second last step of the expression which is this:
$\mu(G)-\mu^{*}(A)$
I am not sure how the $\mu^{*}(G)$ becomes $\mu(G)$ because I was reading that if G is a rectangle in $R^n$, then yes we have $\mu^{*}(G)$ would equal $\mu(G)$. But in this case, the conditions for the proof only states that G is an open set that contains A (it did not say G is a rectangle, all it says is G is an open set in the statement of the theorem). So I am not sure how the author can use $\mu^{*}(G)$ = $\mu(G)$ in the proof.
Could someone kindly give me some hints as to what the author has used or which part am I missing? The symbol $\mu^{*} (G)$ means the outer-measure of the set $G$. The symbol without the star (i.e. $\mu(G)$) means measure (i.e. it means the usual sense of volume for a box, or the usual sense of area for a 2D-rectangle, or the length for a 1D line). For example, $\mu(G)$ = 27, if G is a rectangle of sides 3 and 9, and in this case $\mu^{*}(G)= \mu(G)$ because we have G being a rectangle. But the theorem only says G is an open set, so I am not sure how the author proves the theorem in particular the steps in the second last statement.
Thank you for your help
$\mu(G) = \mu^*(G)$ by definition of $\mu$. See Definition 2.10 in your notes. This only makes sense where $\mu$ is defined, which is on measurable sets. All Borel sets, including open sets, are measurable (theorem 2.21).
Possible confusion arises, I speculate, because "$\mu$" is used for the volume of "rectangles" before defining $\mu$ as Lebesgue measure. So the context of following Definition 2.10 is important. The stuff about proving that $\mu^*(R)=\mu(R)$ for rectangles ensures consistency of the notation later, but it is overloaded/abused and takes getting used to. Get used to knowing that $\mu = \mu^*$ wherever the former is defined.