The MathWorld page on Stieltjes integrals here contains the following:
"...consider the Riemann sum $$\sum _{i=0}^{n-1} f\left(\xi _i\right) \left(g\left(x_{i+1}\right)-g\left(x_i\right)\right)$$ with $\xi _i \in[x_i,x_{i+1}]$."
I am not 100% sure of the meaning of $\xi _i \in[x_i,x_{i+1}]$ in this context. Does it mean that $\xi _i$ is contained within the closed interval $[x_i,x_{i+1}]$, or does it mean that $\xi _i$ is equal to either $x_i$ or $x_{i+1}$?
In this context, the square brackets would ordinarily denote an interval. But then one would write $\xi _i = [x_i,x_{i+1}]$, no? And, surely, if it means either/or one would use curly brackets for the list $x_i,x_{i+1}$? So, the expression as written on MathWorld is half one thing, half another...
Given that I'm trying to learn about Stieltjes integrals from first principles, I'd like to be sure.
This notation $\xi_i \in [x_i,x_{i+1}]$ means it belongs to that closed interval, i.e. $\xi_i$ is in that interval. So the notation you are reading is fine, your understanding of it seems to be off. One would not write $\xi_i = [x_i,x_{i+1}]$ since this does not make sense, a number cannot be equal to an interval, it usually belongs to an interval which is what they wrote.