Let be a bounded function on and let satisfying
is integrable in iff, it is in and in Provided the existence of those integrals, prove that
First let us prove that if .
Since we can find partitions of respectively such that
Adding the two together gives
Now there must exist parition of the form . Where the result follows since
The proof for follows by assuming the opposite and using the above.
Now to prove the integral one again consider partitions of respectively. Now once again there must exist a partition of of the form , where it follows
Taking the supremum over all partitions of the appropriate intervals gives
But it must also be true that
Taking the infimum over all the partitions of the appropriate intervals gives
Any criticism would be greatly appreciated!