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I'm studying Rudin-PMA and the definition here is different from that of Wikipedia.

Rudin PMA page 122

Wikipedia

Rudin says the condition that "integrator should be monotonically increasing and integrand should be bounded on $[a,b]$" is necessary to define Stieltjes Integral.

But wikipedia defined Stieltjes integral, not assuming above condition.

I'm confused now.. Which definition should i use?

EDIT:

Let $g:[a,b]\to \mathbb{R}$ and $f:[a,b]\to \mathbb{R}$ be functions.

Suppose $\forall \epsilon>0, \exists \delta>0$ such that $\operatorname{mesh}(P)<\delta \implies |S(P,f,g)-A|<\epsilon$.

Then is $f$ bounded and $\inf U(P,f,g) = \sup L(P,f,g)=A$? (where $U(P,f,g)=\sum_{i=1}^n M_i |g(x_i) - g(x_{i-1})|$).

2 Answers 2