# Definition of “essential limit sup”?

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 ?
• Yes, French "lim sup ess" is the same as English "ess lim sup". In French, an adjective is usually placed after the noun it modifies. – GEdgar Jan 7 at 12:05

## 1 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).

• Thanks ! So it is the same as $\inf_{\delta>0} (\text{ess sup} \{f(x) \quad | \quad |x-x_0|<\delta\})$ ? – perturbation Jan 7 at 14:16
• @perturbation Yes, that is correct. – Kavi Rama Murthy Jan 7 at 23:15