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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 ?
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  • $\begingroup$ 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. $\endgroup$ – GEdgar Jan 7 at 12:05
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$L$ 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).

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

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