How are the essential upper and lower limits defined?

Question

What means \begin{equation} \operatorname*{ess\,lim\,inf}_{x\to x^*} F(x) \end{equation} and \begin{equation} \operatorname*{ess\,lim\,sup}_{x\to x^*} F(x)? \end{equation} Sorry I also do not know in what conditions this notation make sense for some $F$.

Answer 1

Recall that upper limit can be expressed in terms of suprema over shrinking punctured neighborhoods of $x^*$: $$ \limsup_{x\to x^*} F(x) = \lim_{r\to 0} \sup_{0<|x-x^*|<r} F(x) $$ Replacing supremum with essential supremum, we get the concept of essential upper limit: $$ \operatorname*{ess\,lim\,sup}_{x\to x^*} F(x) = \lim_{r\to 0} \operatorname*{ess\,sup}_{0<|x-x^*|<r} F(x) $$ Similarly, $$ \operatorname*{ess\,lim\,inf}_{x\to x^*} F(x) = \lim_{r\to 0} \operatorname*{ess\,inf}_{0<|x-x^*|<r} F(x) $$

The notation makes sense for every real-valued function; the value of the limits may be $\pm \infty$, but they are always defined. Above I assumed that the domain of definition is a Euclidean space. The same definition works for metric spaces. And even for general topological spaces, since the limit $r\to 0$ is really just the infimum or supremum over all neighborhoods $N$ of $x^*$.

$$ \operatorname*{ess\,lim\,sup}_{x\to x^*} F(x) = \inf_{N} \operatorname*{ess\,sup}_{x\in N\setminus \{x^*\}} F(x) $$ $$ \operatorname*{ess\,lim\,inf}_{x\to x^*} F(x) = \sup_{N} \operatorname*{ess\,inf}_{x\in N\setminus \{x^*\}} F(x) $$