Upper/Lower Riemann Sums
For bounded on and a partition , weights interval infima and weights interval suprema of .
Definition 4.1 (Upper/Lower Riemann Sums)
Let be bounded. Given a partition, with , we set
where is the lower Riemann sum and is the upper Riemann sum of associated with .
Bounds for bounded functions
If is bounded, then for some , and every lower and upper sum obeys
In particular , so both
are non-empty bounded sets.
Related
Stated in
- Definition 4.1 (Upper/Lower Riemann Sums)§4.1 Basics