On the existence of functions with a particular convergence

Question

Is the following scenario possible? Provide an example or argue why not.

Let $\{f_n\}_{n=1}^{\infty}$ be measurable non-negative functions on $[0,1]$ converging to $f(x)$ pointwise Lebesgue-almsot everywhere on $[0,1]$ and $\displaystyle \lim_{n \to \infty} \int_{0}^{1} f_n =2$ and $\displaystyle \int_{0}^{1} f =1$.

Answer 1

Take the functions from the example here and shift them up by $1$. More precisely, let $f \equiv 1$ and $$ f_n = f+n\cdot1_{\left(0,\frac1n\right]} $$