Upper/Lower Integral
and ; is Riemann integrable iff they agree, and the common value is .
Definition 4.3 (Upper/Lower Integral)
Let be bounded. We say
to be the lower integral and upper integral of on respectively.
We say is Riemann integrable on if , and in this case we set
Ordering of the two integrals
Always for any two partitions, so . Hence is Riemann integrable iff .
With the convention built into the setup of the sums, the definition extends to oriented intervals.
Related
Stated in
- Definition 4.3 (Upper/Lower Integral)§4.1 Basics