solving existence problems in measure theory

Question

i'm here to ask for some tips/books,writes or anything that helps to solve a specifically type of problem:to prove that exists something in measure theory.
Even more specifically, i am stuck in chapter 9 of bartle's book (measure theory and integration) because there are a lot of these problems and altough i had seem(and understand) some solutions, i just cant formulate one by myself.
The exercise 9.G ,for example, say:
If A is a Lebesgue measurable subset of R and $\epsilon > 0$, show that there exists an open set $G_{\epsilon} \supseteq A$ such that $l^{*}(A) < l^{*}(G_{\epsilon}) < l^{*}(A) + \epsilon$
I just write the hypotesis and what they implie, but everytime i try to go any further i never get anything closer to a concrete case.
The thing is that this happens to almost every question of this kind(show existence), the other ones i dont have this much trouble. If someone can help i would really apreciate, its really frustating. The other questions at least after some more study i am able to solve in general, but not this ones.