Is there a dominated convergence theorem for Riemann-Stieltjes integral which is analogous to the usual Lebesgue dominated convergence theorem?

More precisely, let $(f_n)$ be a sequence of continuous functions in $[0,1]$ satisfying $f_n$ converges pointwise to a continuous function $f$ in $[0,1]$ and there exists a Riemann-Stieltjes integrable (or Lebesgue-Stieltjes integrable) function $g$ satisfying $|f_n|\leq g$ for all $n.$

(1) Is it true that $\int_0^1 f_n(r) d\alpha(r)$ converges to $\int_0^1 f(r)d\alpha(r)$? Here, if necessary, $\alpha$ is non-decreasing in $[0,1]$. Note that when f is a continuous function and $\alpha$ is a non-decreasing function, the Lebesgue–Stieltjes integral is equivalent to the Riemann–Stieltjes integral (see (a)).

(2) If $(1)$ is true for more general $\alpha,$ e.g., bounded variation, are there any advantage for non-decreasing function $\alpha$? If $\alpha$ is non-decreasing in $[0,1]$, $\int_0^1 f_1(r)d\alpha(r) \le \int_0^1 f_2(r)d\alpha(r)$ for any Riemann-Stieltjes integrable functions $f_1,f_2$ satisfying $f_1 \leq f_2$ in $[0,1].$ Are there any reason for giving the monotonicity of $\alpha$?

(3) If (1) is true, is it necessary that $\alpha$ is right continuous in this case?

I would be grateful if you give any comment for my question. Thanks in advance.


1 Answer 1


I found this paper [https://www.jstor.org/stable/2323739?seq=1#metadata_info_tab_contents]. It seems that the paper is good reference for my question.

From the paper, I can use standard dominated convergence theorem for Riemann-Stieltjes integral.


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