Let $A\subset\mathbb{}\mathbb{R}^N$ and $u$ a measurable function on $A$ bounded by above in every compact set contained in $A$. Suppose $x\in A$ and define:
$\liminf_{y\rightarrow x}u(y)=\lim_{r\rightarrow 0}\inf\{u(y):\ y\in B(x,r)\cap (A\setminus\{x\})\}$
Define:
$ess\liminf_{y\rightarrow x}u(y)=\lim_{r\rightarrow 0}ess\inf_{B(x,r)}u$
Is true that:
$\liminf_{y\rightarrow x}u(y)\leq ess\liminf_{y\rightarrow x}u(y)$
UPD: I think the answer is more easy than o thougth, because by definition $u(y)\geq \inf u$ almost everywhere, hence i\m gonna change the question. I just want to know a example where the inequality is strict.