In my textbook, we define the the existence of an integral to be satisfied if and only if the supremum of the lower sum (minimum sum) is equal to the infimum of the upper sum (maximum sum). Moreover, the unique number $L \leq \int f \leq U$ is defined to be the integral of $f$. Clearly a Riemann sum is such that $L \leq R \leq U$ --- so why do we say that $R$ is only approximately equal to $\int f$ ? Isn't it equal?
Edit: Let me clarify that I understand why Riemann sums should only be considered approximately equal to the integral. My question rather is semantic. To me it seems justified using the definitions of the text (given above) to prove what it is not --- equal.
Edit for Context:
The author seems to be saying that 'Riemann sum' of the $g'(x_i)$ yielded by the MVT is equal to the integral