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I am reading Baby Rudin and am wondering why is function y in Riemann steiltjes integral $\int xdy$ required to be monotonic? Can anyone please explain?

I have also heard that y needs to be of bounded variation. Is that right?

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A well-known existence theorem for R.S.-integrals shows that on a compact interval $[a,b] \subset \mathbb{R}$, the R.S.-integral $\int_a^b f(x) dg(x)$ exists when $f$ is continuous on $[a,b]$ and $g$ is of bounded variation on $[a,b]$. Perhaps this is what you're referring to? It is well-known that a function $g$ is of bounded variation on $[a,b]$ if and only if there are monotone functions $h$ and $k$ on $[a,b]$ such that $g(x) = h(x)-k(x)$ for every $x \in [a,b]$. However, $g$ itself is not required to be monotone.

That said, you might consider what properties you want from the integral? For example, if for every $s < t$ in $[a,b]$, you might want $\int_s^t dg$ to be positive (i.e., the measure induced by $dg$ is a positive measure), in this case, since $\int_s^t dg = g(t) - g(s)$, the function $g$ would be required to be monotone. (Note that every monotone function on a compact interval is of bounded variation.)

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