# Prove using the definition of Lebesgue outer measure

If $B = X \cup Y$, and $X$ and $Y$ are disjoint compact intervals, then

$$m^*(B) = m^*(X) + m^*(Y)$$

(This is not always true for disjoint subsets of $\mathbb{R}$, but it is for intervals)

I understand conceptually what is happening I think, but I can't express it mathematically.

I'm not very proficient at writing proofs, at least not to what is expected of me. I really want to learn, I'm just having to play catch-up this semester. I want to study the answer, not just have it solved for me. Any help is appreciated.

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two disjoint compact intervals are "separated" by some distance, so you can assume that none of your open balls intersect both $X$ and $Y$. – Deven Ware Jan 21 '13 at 19:00
Does $m^*$ denote the Lebesgue outer-measure? And is this a homework problem? – T. Eskin Jan 21 '13 at 19:12
@Thomas, yes it is, I just found out about the hw tag – user59144 Jan 21 '13 at 20:15
This can be generalized to an arbitrary number of pairwise disjoint measurable sets. A hint: show that the LHS is both greater than or equal to and less than or equal to the right hand side. – cats Jan 21 '13 at 20:28
@lyj Okay, that's what I was thinking, but I'm not really sure how to start with that method. My proof skills are currently very poor... – user59144 Jan 21 '13 at 20:56