Let $f: \mathbb{R}\to\mathbb{R}$ and the essential supremum of $f$ $$e=\inf\{\alpha\ge 0 \mid |f|\le\alpha\text{ almost everywhere}\}$$
I can't see why $$\lambda^*\left(\{x\in\mathbb{R} \mid |f(x)|\ge e-\epsilon\}\right)>0$$ for $a0$, where $\lambda$ is the Lebesgue measure.
Could someone help me ?
Fixed after Didier Piau's comment.