In Riemann-Stieltjes integral,
$ \int_{0}^{\infty} f(x) w(x)dx$
why the weight function should be nontrivial when the interval is $[0,\infty)$?
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That is just a Riemann integral with integrand $f(x)w(x)$. A Riemann-Stieltjes integral has the form $$\int_0^\infty f(x),dw(x)$$ If $w$ is continuously differentiable, then the Riemann-Stieltjes integral can be converted to a Riemann integral by $$\int_0^\infty f(x),dw(x) = \int_0^\infty f(x)w'(x),dx$$ But otherwise they are distinct from each other. In the actual Riemann-Stieltjes integral, if $w$ is constant ($0$ or not), then $dw(x) \equiv 0$, and thus the whole integral is $0$, regardless of $f$. – Paul Sinclair Jul 30 '21 at 13:23
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Thank you so much. – soso Jul 30 '21 at 14:07