Let \([a,b]\) be a closed real interval, \(f:[a,b]\mapsto\mathbb R\) be a bounded real function, and \(a=x_0 < x_1 < \ldots < x_{n-1} < x_n=b\) be a partition of the interval \([a,b]\). Further, let \(\xi\in[x_{k-1},x_k]\) (i.e. \(\xi\) is any point in the interval \([x_{k-1},x_k]\), \(1\le k\le n\)). We call
\[\sum_{k=1}^n f(\xi_k)(x_k-x_{k-1})\]
a Riemann sum of \(f\) with respect to the partition \(a=x_0 < x_1 < \ldots < x_{n-1} < x_n=b\).
Please note that we can have different Riemann sums of \(f\) with respect to the same partition, since the choice of the supporting points \(\xi_k\) is arbitrary in each interval \([x_{k-1},x_k]\).
The mesh \(\mu\) of the partition \(a=x_0 < x_1 < \ldots < x_{n-1} < x_n=b\) is defined as the maximum length of a single partition interval
\[\mu:=\max_{1\le k\le n}(x_k-x_{k-1}).\]
Propositions: 1