Defining the length of an interval in the Riemann sums

Question

In the Riemann integration theory, the partition of the studied interval $[a,b]$ is $P(x,t)$. But how do we define the length of the subdivided intervals $[x_i , x_{i+1}]$ ? Do we say it is the integral of the characteristic function of this interval like this: $$\int\chi_{[x_i,x_{i+1}]}(x)dx$$

In this case what integral are we using? The Riemann one? Using Riemann integrals to explain Riemann integrals seems to me a bit surprising...

Answer 1

The length of an interval $[a,b]$ is $b-a$. No integrals required.