The definition of the outer measure of a set $A\subseteq\mathbb{R}$ is as follows:
$$ |A| = \inf \left\{ \Sigma_{k=1}^{\infty}\ \mathscr{l}(I_k): I_1, I_2,\dots\text{ are open intervals such that }A\subseteq\bigcup_{k=1}^{\infty} I_k \right\}. $$
My quick observation is that if two open intervals intersect, we can join them together to create another open interval. So starting from $I_1$, apply this process for every $I_k$ and we get a countable list of disjoint open intervals $O_1, O_2,\dots$ such that $A\subseteq\bigcup_{k=1}^{\infty} O_k$ and $\Sigma_{k=1}^{\infty}\ \mathscr{l}(I_k)\ge\Sigma_{k=1}^{\infty}\ \mathscr{l}(O_k)$. Since we're taking the $\inf$ of the length sum, we can replace $I_k$ by $O_k$.
So I wonder why the definition uses "open intervals" instead of "disjoint open intervals", or if using "disjoint open interval", we will get a different set function other than the outer measure?
