I am working on the following exercise and could use some help.
Prove that if , then is integrable on and .[Hint: Any lower sum is and some lower sum . Hence, the lower integral ].
Here is what I have tried so far...
Let be a lower step function such that for all . Then, let the lower sum for . Then, we have that for any lower step function. Also, . Thus, for any , . So, .
So for , we see that any lower sum and some lower sum is . Thus,
=lower integral .
We can do something similar for ...For we have that and . Therefore, .
Then conclude that .
I'm not sure if I have said enough for the proof to be complete or if what I have is even good. Please help me out.