Sorry if my English isn't good I'm from Croatia.
Offer an example of function bounded g and a non-decreasing function $\beta$ defined in $[a,b]$ such that $\mid g\mid \in R(\beta)$ but for which $\int_a^b g d\beta$ doesn't exist.
2026-03-25 14:18:32.1774448312
Bounded function and integration
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Hint:
If $g$ and $\beta $ are discontinuous at the same point, the Riemann-Stieltjes integral does not exist. See this answer.
Take
$$g(x) = \begin{cases}-1, & 0 \leqslant x < 1/2\\ 1, & 1/2 \leqslant x \leqslant 1 \end{cases}$$
and find a suitable $\beta$ to finish.