Originally Posted by

**ThePerfectHacker** I am not really sure what you mean by "addition laws". It happens to be really easy to prove this by using the definition of the Riemann integral. For any subinterval $\displaystyle [x_{j-1},x_j]$ (in the partition) there exists a rational and an irrational point. If for any partition we choose only the irrational points then the Riemann sum has value $\displaystyle \Sigma_{j=1}^n (1)(x_{j-1}-x_j) = b-a$. If for any partition we choose only the rational points then the Riemann sum has value $\displaystyle \Sigma_{j=1}^n (0)(x_{j-1} - x_j) = 0$. The problem is that $\displaystyle b-a$ and $\displaystyle 0$ cannot be made arbitrary close to one another with a fine enough parition (since it is independent of $\displaystyle ||P||$).