Define functional $\psi(x_n):=\limsup_n x_n$ on a sequence of positive numbers $x_1,x_2...$. We can check that $\psi$ is convex.
Then if $f_n \to f$ pointwise, by Jensen's inequality, $$\limsup_n \int |f_n-f|\le \int \limsup_n|f_n-f|=0$$
This proof is obviously wrong because I used no dominance conditions for $f_n$. But where did make a mistake? Is it because $\limsup_n|f_n-f|\neq 0$?
You're trying to say that if $\phi$ is a convex function $\mathbb R^{\mathbb N} \to \mathbb R$ then $$ \phi\left(\left\{\int g_n(x) \mu(\text{d}x)\right\}_n\right) \leq \int \phi(\{g_n(x)\}_n) \mu(\text{d}x) .$$ for a measure $\mu$ on $X$ and any sequence of functions $g_n:X \to \mathbb R$. This isn't the statement of Jensen's inequality and more importantly isn't true.