Disjoint Sets Seemingly Not Disjoint in Description of Lebesgue Measure I am currently working through Knapp's Basic Real Analysis.  I am currently working in Chapter 5 on "Lebesgue Measure and Abstract Measure Theory."  The book states the following, giving Lebesgue Measure as an example of a measure. 
"$\bf{Lebesgue \ Measure}$ $m$ on the ring $R$ of elementary sets of $\mathbb{R}$.  If $E$ is a finite disjoint union of bounded intervals, we let $m(E)$ be the sum of the lengths of the intervals...Consider the case that $J=I_{1} \cup...\cup I_{r}$ disjointly with $I_{k}$ extending from $a_{k}$ to $b_{k}$, with or without endpoints.  Then we can arrange the intervals in order so that $b_{k}=a_{k+1}$ for $k=1,...,r-1$."
I believe that I am misunderstanding something.  If the union of the intervals is disjoint, how can it be that $b_{k}=a_{k+1}$?  Then, wouldn't $I_{k} \cap I_{k+1}\neq \emptyset$?  This seems counterintuitive, and any help would be much appreciated.

There are several ways we can have disjoint intervals with coinciding end-points. For example, we could have an interval $(a,b)$ or $(a,b]$ next to an interval $(b,c)$. If points are admitted as intervals, $[a,a]=\{a\}$, then we could have situations such as $(a,b)$ next to $\{b\}$ next to $(b,c)$, etc. 
