Why are these two lemmas worth including and proving?

Question

The following two lemmas are from Stein and Shakarchi (2005).

1. (p4 Lemma 1.1) If a rectangle is the almost disjoint union of finitely many other rectangles, say $R=\cup_{k=1}^n R_k$, then $|R| = \sum_{k=1}^n |R_k|$. They prove this by extending the edges of $R_k$ to get a grid and then use the condition of almost disjoint. I cannot see why this extra layer of complexity is necessary. After all, the key is to use almost disjoint condition no matter how you cut the rectangle.

2. (p12 Example 4) The exterior measure of a rectangle $R$ is equal to its volume. They went throught a great deal of trouble to show that $m_*(R)\leq|R|$. However, this is rather clear since $R$ clearly covers itself.

I want to know what the rationale behind listing these two as lemmas and prove them is. Or it simply is being pedentic for the purpose of being pedantic? Thank you!

Answer 1

1) they have just given a brief formal argument about what is true. The goal is to prove that the notion of volume is additive for almost disjoint collections of rectangles, and that doesn't just follow from definition or 'obviousness' alone. Until you suggest your argument that removes the step (without just appealing to "it is obvious") I'm not sure what you're thinking.

2) R covering itself means little when the exterior measure is phrased in terms of open coverings by cubes.

Answer 2

The elementary area (volume) of rectangles (boxes) is used to define the measure of pretty arbitrary sets using some limiting process. In the end you have to prove that this process also gives rectangles and boxes the measure we are acquainted with.