the Borel set is the $\sigma$-ring generated by the open sets. One possible Borel measure on the real line is defined, for a closed interval, as:
But, from my understanding, intervals of the type $[a,b)$, or $(a,b)$ are also part of the Borel set, as it is also generated by the compact sets.
How do you compute the measure of those half-open and open intervals ? Is it $b-a$ too ?