Suppose that $f_1\ge f_2\ge \ldots f\ge 0$ are bounded Borel measurable functions from $[0,1]$ to $\mathbb R$ such that $\lim_n f_n(x)=f(x)$ for every $x\in [0,1]$. Given a Borel measurable function $g\colon [0,1]\to\mathbb R$, define $$ \text{ess.sup} g:=\inf\{ r\colon \lambda(g> r)=0\}, $$ where $\lambda$ is the Lebesgue measure.
Is it true that $\lim_n \text{ess.sup} f_n=\text{ess.sup} f$?
It is not true.
Consider $f_n(x) = \mathbf{1}_{[0,\frac 1 n]}$.
It converges pointwise to $f(x) = \delta_{0}(x)$ (the function whose value is $1$ at $0$ and $0$ otherwise) but its essential supremum is $0$.