1. I am only currently finishing up Riemann-Stieltjes integration...I am still a novice. I would not even be posting if this were not for "fun"
2.I assume that a knowledge of the defintion of the Riemann integral is knownw
3. I assume you mean integrable in respect to x
Suppose that then
Proof: We must merely prove that given we may find a partion of such that .
Now for each partition of choose (note this can be done because x is continuous).
The last step is true for sufficiently large n.
Now I will show that if is monotone then is bounded:
Without loss of generality assume that is montonically increasing. Then we can see that
And because Riemann integration is only applicable to bounded functions we can see that if is montonically increasing that is bounded. The proof for montonically decreasing functions is almost identical.
Similarly we must just show that for any there exists a corresponding partition of such that .
Let be a continuous function on the compact interval Prove that is integrable on
So for every choose such that . Now since continuity and uniform continuity are equivalent on compact spaces we may choose a such that . Now choose a partition of such that , so then
is justified by