A theorem about Borel measures

[Aerostd]

Homework Statement

There is a theorem in Folland's book (Real Analysis) in which he gives three statements that are equivalent. In the proof he decides to take the first statement and prove the other two, but in the end he derives a statement that looks different than what is given in the theorem. This means that the statement he derived is the same as the statement in the proof, but I am not able to go from one to the other.

Theorem: Let $$E \in R$$(reals). Then the following are equivalent.

(a) $$E$$ is measurable.
(b) $$E = V$$ $$- N_{1}$$ where $$V$$ is a $$G_{\delta}$$ set and $$\mu(N_{1})=0$$.

There is a (c) but let us ignore that for now. The measure is the Lebesgue measure and the domain I think is the extended Borel sigma algebra, so both $$V$$ and $$N_{1}$$ are measurable.

In the book, the author assumes (a) and then shows that:

(d) there exists a $$G_{\delta}$$ set $$V \subset E$$ such that $$\mu(V-E)=0$$.

I am trying to either go from this statement to (b) or from (b) to (d), but I don't know how.

2. The attempt at a solutionI don't know how to go from (d) to (b) because I don't know how one can conclude that sets are equal based on their measures. So I tried to go from (b) to (d). I have to show that $$E \in V$$ and $$\mu(E-V)$$ is zero.

If $$E = V-N_{1}$$, then

$$\mu(E) = \mu(V) - \mu(V \cap N_{1})$$,

$$\mu(E) = \mu(V)$$.

I don't know what to do next.

 

[mighty2000]

I thought about using the fact that E is measurable and V is a G_{\delta} set, but I don't know how to connect those to conclude that E is also a G_{\delta} set. I also tried using the fact that \mu(N_{1})=0, but I couldn't see how that could help either. Can someone please give me some hints or guidance on how to proceed?

First of all, it's important to note that in the proof of the theorem, the author is not trying to show that (b) is equivalent to (d). Instead, they are showing that (b) implies (d) and (d) implies (b), which is enough to establish the equivalence of (a), (b), and (d).

To go from (b) to (d), you can use the fact that V is a G_{\delta} set, which means it can be written as a countable intersection of open sets. Let's say V = \bigcap_{n=1}^{\infty} U_n. Then, by definition, V \subset U_n for all n. Since E = V - N_{1}, it follows that E \subset U_n - N_{1} for all n. But U_n - N_{1} is an open set, since N_{1} has measure zero and thus does not affect the openness of U_n. Therefore, E is contained in a countable union of open sets, and hence it is a G_{\delta} set.

To go from (d) to (b), you can use the fact that V \subset E. Since V is a G_{\delta} set, it can be written as a countable intersection of open sets, say V = \bigcap_{n=1}^{\infty} U_n. Then, by definition, V \subset U_n for all n, and hence E \subset U_n for all n. This implies that E is contained in a countable union of open sets, and therefore it is a G_{\delta} set. Additionally, since \mu(V-E)=0, it follows that \mu(V \cap E) = \mu(V) - \mu(V-E) = \mu(V) = \mu(E), which shows that E and V have the same measure, as required in (b).