what is the difference between the Riemann integral and the Stieltjes integral definition wise?

Question

I know what the definition of a Riemann integral is from my previous question here but now when I look up definitions of the Stieltjes integral all I find is that it is derived from the Riemann integral and instead of integrating $f(x) dx$ you are integrating $f(x) dg(x)$, I know how to integrate Stieltjes integrals what in terms of defining what it is, how do I state what a Stieltjes integral is without proofs?

Answer 1

From Wiki:

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If I understand right, all you do pretty much is replace $x_{i+1} - x_i$ with $g(x_{i+1}) - g(x_i)$

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Perhaps, I don't know what you mean. Are you looking for intuition? If so, we want to sometimes integrate

$$\int f(x) g'(x) dx$$

If g isn't differentiable (such as a Cantor cumulative distribution function), we have the Stieltjes integral:

$$\int f(x) dg(x)$$

which generalises Riemann integrals of the form $\int f(x) g'(x) dx$ to allow for non-differentiable g