# dominated convergence theorem for Riemann-Stieltjes integral

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.