In Chapter 2 of Axler's Measure Theory book, found here: http://measure.axler.net/MIRA.pdf, he defines "length of an open set", "outer measure" (denoted $|\cdot |$), and proves some basic theorems about them, mainly that
- In $\mathbb R$, if $A\subset B$, then $|A|\leq |B|$; and
- In $\mathbb R$ for a sequence of sets $A_1, A_2, \ldots$ the following holds: $$\left|\bigcup\limits_{k=1}^\infty A_k\right|\leq\sum\limits_{k=1}^\infty |A_k|$$
He then goes into a huge spiel about the Heine-Borel Theorem and more, for the sake of proving that $|[a,b]| = b-a$. Now my question is this: why couldn't he prove that using the above two theorems like so? $$(a,b)\subset [a,b] \implies b-a=|(b,a)|\leq |[a,b]|$$ and $$[a,b]\subset (a-\epsilon, b+\epsilon) \implies |[a,b]|\leq (b-a+2\epsilon) \to (b-a)$$ (I'm not sure where the second theorem comes in; he seems to have used it to prove $|[a,b]|\leq b-a$ though). So why did Axler need to introduce the Heine-Borel Theorem when it seems the theorems he already proved suffice? Where did I go wrong?
The key here is that Axler has not yet shown that the length of an open interval and the outer measure of an open interval are the same. As said in the comments, this works if you already know $|(a, b) |=b-a$, but we don't know that. All we know so far is that $l(a, b)=b-a$. This is a very different statement from the infimum definition required for the actual outer measure.