in other words, i am not sure how i can make valid of your statement of "then take the finite subcover $O_{\lambda}$ of K" as that was never given, for that family of sets $O_{\lambda}$

I think the idea of Lebesgue measure is inner regular is really really close to what I am actually wondering about. The difference is that, the idea that Lebesgue measure is inner regular, are having the intervals, or the compact sets, lots of freedom of construction, possibly i believe it actually covers the set A in a sense of vitali, whereas, in my question, the intervals don't have that freedom, the only thing that they have is their union is an open cover of A

You are right, the last paragraph is not making sense

i am not sure if i understand what you mean by considering a specific example of A. Thanks for an editing of better formatting btw.

yep it's perfectly correct, thx!

I think it might be helpful to mention vitali's lemma as a good comparison. Vitali's lemma, as far as i understand, from royden's real analysis, states the conditions roughtly the same, except that, the sets Oλ covers A in a sense of vitali, and on top of that, the finite subcollection of Oλ are disjoint. Which are two very very interesting differences