For and , is integrable iff its restrictions and are both integrable, and in that case

Write , , and .

Suppose is integrable and let . There is a partition of with ; WLOG for some (otherwise add to , which can only make the gap smaller). Then with and partitions of and , and

Hence

so both restrictions are integrable.

Conversely, given partitions of and whose gaps are each , the same equalities give with , so is integrable on . Finally,

and since is arbitrary, forces equality.