Let $(f_n)_{n\in \mathbb{N}}$ be a sequence of nonnegative measurable functions in $L_1[0,1]$. Assume that $$f_n \to f, \text{ a.e.}$$ and $$\int f_n h \to \int g h,\, \forall h \in C[0,1].$$

Question: Do we have $$f = g, \text{ a.e.?}$$

Remark: Of course, it is a standard result that if we allow $h$ to be any bounded measurable function, then the almost sure limit coincides with the weak limit.