Let $a<b$. Let $(f_n)_n\subseteq C^1([a, b])$ and $g\in \operatorname{BV}([a, b])$. By Stanislaw Hartman and Jan Mikusinski, "The Theory of Lebesgue Measure and Integration" (p. 165), we can integrate by parts:
$$\int_a^b f_n'(t)g(t)dt = f_n(b)g(b) - f_n(a)g(a) - \int_a^b f_n(t)dg(t).$$
Suppose that:
- For all $n$, $0 \leq f_n\leq 1$, $f_n(a)=0$ and $f_n(b)=1$.
- For all $t\in [a, b[$, $f_n(t)\rightarrow 0$.
- $g$ is continuous on $b$.
Can we pass to the limit in the integration by parts formula (using dominated convergence for example) to get
$$\lim_{n\rightarrow \infty}\int_a^b f_n'(t)g(t)dt = g(b) \quad ?$$