Finding the Riemann stieltjes integral using partitions

Question

Please take a look at the following example :

$α(x)=x$ when $0≤x≤1$

$α(x)=x+2$ when $1< x≤2$

Evaluate $\int_{0}^{2}xdα(x)$

I solved this using the integration by parts formula and got the answer as 4. But what I am curious to know is how to solve this using partitions. Say I take a partition as follows;

P={$0,1/n,2/n,...,1,(n+1)/n,(n+2)/n,...,2$}.

I want to know the best way to proceed forward. Is it by using Riemann sums and take the limit or use upper sums and lower sums? If I choose to go with Riemann sums can i take the choice point as $i/n$ where $i=1,...,2n$ and calculate a limit? or should it be a arbitrary point within $[i-1/n,i/n]$? Do we always have to use arbitrary point or depending on the functions can we make that choice? I hope my question is not confusing. Thanks in advance !

Answer 1

Is it by using Riemann sums and take the limit or use upper sums and lower sums?

Upper and lower sums are preferable: fewer choices to make.

If I choose to go with Riemann sums can i take the choice point as $i/n$ where $i=1,...,2n$ and calculate a limit? or should it be a arbitrary point within $[i−1/n,i/n]$?

Depends. Do you already know (from some theorem) that the integral exists? If yes, you can take a convenient point such as $i/n$, because the limit of Riemann sums is already known to be independent of your choice. If not, you must consider arbitrary points.

Do we always have to use arbitrary point or depending on the functions can we make that choice?

Usually, neither: there are better ways to evaluate integrals than looking at partitions. Also the answer depends on whether we know if the integral exists: see above.