$A_t$ is a BV right continuous function on $[0,a]$.
$f(t)$ is a borel measurable function on $[0,a]$ which satisfy: $$\int_0^a|f(s)|\,|dA_s|<\infty$$
I want to show $$g(t):=\int_0^tf(s)\,dA_s$$ is right continuous on $[0,a]$.
$$|g(t+h)-g(t)|=|\int_t^{t+h}f(s)\,dA_s|\le\int_0^a1_{(t,t+h]}(s)|f(s)|\,|dA_s|$$
then by DCT,we get the conclusion.
My question is we can also get the LEFT continuity of $g$ by the same procedure,but in general this conclusion is false .Moreover,I didn't use the right continuity of $A$.I think the right continuity of $A$ is only used to construct the L-S measure,right?