Almost everywhere convergence of functions which verify a property on the set $C_b$

Question

Let $T>0$. Let $(h_n)_{n\geq 1}\subseteq L^\infty([0, T])$ and $h\in L^\infty([0, T])$. Suppose that for all continuous and bounded functions $f:[0, T]\rightarrow \mathbb R$ we have $$\lim_{n\rightarrow\infty}\int_0^Tf(t)h_n(t)dt = \int_0^Tf(t)h(t)dt.$$ Do we have that $(h_n)_{n\geq 1}$ converges to $h$, $t$-almost everywhere?

Answer 1

No. There is well known example of a sequence $(h_n)$ such that $0 \leq h_n \leq 1$, $h_n \to h$ in measure but does not converge at any point. For this sequence your hypothesis is satisfied. [ There is a version of DCT with a.e. convergence replaced by convergence in measure].