What does limit notation with an underline or an overline mean?
I've never seen this notation before, and I'm having trouble finding a reference through search. Could someone explain what these notations mean for me?
In context, the statement they're in is the following: a bounded $f$ is Riemann integrable iff $$\varliminf_{||C||\to 0}\mathcal{L}(f; C)\ge\varlimsup_{||C||\to 0}\mathcal{U}(f;C)$$ where $C$ is a non-overlapping, finite, exact cover of a rectangular region $J$ in $\mathbb{R}^N$, $||C||$ denotes mesh size, and $\mathcal{L}, \mathcal{U}$ represent the lower and upper Riemann sums, respectively.
$\endgroup$ 93 Answers
$\begingroup$It's definitely liminf and limsup. Maybe this notation is more common in Europe than in America? For example, the German Wikipedia page mentions it as an alternative. A well-known book that uses this notation is Hörmander's The Analysis of Linear Partial Differential Operators.
$\endgroup$ 4 $\begingroup$I agree : it is definitely Lim Sup, Lim Inf.; I have seen it used many times. If you do not see the top or bottom, you still have a Lim, but ---No Sup For You!
$\endgroup$ 1 $\begingroup$I, however, have seen this notation, called upper and lower limits defined as follows.
Given a sequence of numbers, ${a_n}$, we may consider the lower bound: inf{$a_n$|n>m} as $b_m$. And then take the upper bound of it, $sup_m$$b_m$, called as the greatest lower bound denoted by the limit notation with an underline.
If, nevertheless, you have been acquainted with this notation and found that it's not what you want, then it must be because of my limited sight; if it is the case, please inform me, thanks.
