I have a general question regarding Riemann-Stieltjes integrals. By definition, the Riemann-Stieltjes integral I exists for a given function if for any partition of the domain of interest, I lies between the lower and upper bounds given by the partition. In the case of the Riemann Integral, using more subdivisions of the interval leads greater precision of the approximation ie. the lower and upper bounds are closer in value to real real value of the integral.
My question is:
Under what conditions do we have greater accuracy in the Riemann-Stieltjes case, since having more subdivisions don't necessarily mean more precision?
In the case of an integrator which is a step function, having points in partition closer to the jump points of the integrator does seem to give greater precision
However, I fail to explain the intuition behind this observation
\begin{equation} \int_{-1}^{1} (1+x) \; d\alpha(x) \mathrm{~~~with~~~} \alpha(x) \begin{cases} 2 & x > 0 \\ 1 & x \leq 0 \end{cases} \end{equation}
In this example, using the partition $P = \{ -1, -0.1, 0.1, 1 \}$ of the domain $[-1,1]$ yields a lower estimate of 0.9 and upper estimate of 1.1 and also note the jump point of the integrator is at $x = 0$.
Knowing that the real value of the integral is 1 the partition $P' = \{ -1, -10^{-n}, 10^{-n}, 1 \}$ where the points to the left and to right of the jump point are closer to it, we get a precision up to $n$ decimal places ie $|L(P') - I| < 10^{-n}$ and $|U(P') - I| < 10^{-n}$.