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How can we see Riemann-Stieltjes integral geometrically?

Also how defining a function $\alpha$ monotonically increasing on $[a, b]$ confirms it to be bounded? It may be silly thing to ask but I am not getting it atleast right now. Please help me.

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  • $\begingroup$ $\alpha(a) \leq \alpha(x) \leq \alpha(b)$ for all $x \in [a,b]$. $\endgroup$ – Bungo Sep 6 at 4:13
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If a monotonically increasing function on a closed interval $[a,b]$ would be unbounded, then it must be necessarily (also) in $b$, right? But what would $\alpha(b)$ need to be? Is that possible? (Just very heuristic but maybe you can get this into something rigorous.)

For the first question, it might bet helpful (at least it was for me) to calculate the following Riemann-Stiltjes integral to get a more intuitiv idea about this integral. Take $f: [-1,1] \to \mathbb{R}$ to be a continuous function and take $$ g : [0,1] \to \mathbb{R}, \qquad x \mapsto \begin{cases} 0 & x < 0\\ 1 & x \geq 0 \end{cases} . $$ So what is $\int_{-1}^1 f(x) dg(x)$?

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