I am so confusing about measure and especially Lebesgue measure, I read that "Given an open set S = ∑(a,b) containing disjoint intervals, the Lebesgue measure is defined by
"
So can I represent the set like that ..?
If the image is true, then the Lebesgue measure function output will be the total length of all internal intervals which means that the result is the length of set S.. !!
And in outer measure, I had some conflicts during representing it as graph like I mentioned above. I read that"
If E is a subset of real line (R) with the interval length I=[a,b] (or I=(a,b) ) given by l(I)=b-a the Lebesgue outer measure on E can represented as the following:
And here is the conflict, at the first he mentioned that the I interval is a length of E, and in the outer measure fn, he wrote that the set E is subset or equal to the unions of I intervals !!! so what's it mean? .. as I thought, E has small disjoint intervals and I represent the length of set E, then the I times k means the intervals which are represented within the set E ... !!