For $f$ continuous on $[a,b]$ There is no difference between $\int_{[a,b]} f dx$ and $\int_{[a,b)} f dx$ with respect to the usual Lebesgue measure, since a single point is a set of measure zero.
Here is an intuitive way to look at this: The area under a single point is zero, since the rectangle will have no width. Because of this, it does not matter if we add that point in or not, so both integrals are the same.
Remark: As pointed out by Arturo in the above comments, we have to be careful when dealing with the improper Riemann integral. (Which is defined to be a limit of Riemann integrals.)