In the article of Lions https://www.jstor.org/stable/2045002?seq=1 the following notion is used:
$$L=\underset{y \to x_0}{\text{lim sup ess }} f(y),$$
for a measurable function $f:\Omega \rightarrow \mathbb{R}$ but I have no idea what it means.
- What is the precise definition of this $L$ with quantifiers ?
- How to pronounce it, is it "essential supremum limit" ?
- Where to find it in classical textbooks ?
- I have also seen "$\text{ess lim sup }$" in another article, is it the same object ?
Best Answer
$L$ is defined by the following properties:
a) For every $\epsilon >0$ ther exist $\delta >0$ such that $|y-x_0| <\delta$ implies $f(x) \leq L+\epsilon$ almost everywhere.
b) No smaller number has property a).
'ess lim sup' is same as 'lim sup ess'.
You pronounce it as 'essential lim sup' (or eseential limit superior).