# 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.

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I've never seen this notation, but my guess in context is it means liminf and limsup: en.wikipedia.org/wiki/Limit_superior_and_limit_inferior –  Qiaochu Yuan Feb 13 '11 at 13:44
I thought so too, but if that's the case the theorem has liminf of L and limsup of U, which doesn't make sense. (It's possible that's a typo, but I don't want to assume that without looking into alternatives.) –  Ben Lerner Feb 13 '11 at 13:52
@Ben: It seems right to me; if liminf of L is less than limsup of U, then there's a gap which makes the function nonintegrable. –  Hans Lundmark Feb 13 '11 at 14:03
The best I achieved was to write $\underset{||C||\to 0}{\underline{\text{lim} }}\mathcal{L}(f; C)$ by using \underset{||C||\to 0}{\underline{\text{lim} }}\mathcal{L}(f; C) and similarly to $\underset{||C||\to 0}{\overline{\text{lim} }}\mathcal{U}(f; C)$ \underset{||C||\to 0}{\overline{\text{lim} }}\mathcal{U}(f; C) –  Américo Tavares Feb 13 '11 at 14:26
@Américo: The LaTeX-code for $\varliminf$ is \varliminf and the one for $\varlimsup$ is \varlimsup –  t.b. Feb 13 '11 at 14:27

## 3 Answers

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.

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+1 for region distinction. In France, it's a common notation. –  Sam Feb 13 '11 at 14:19
I've seen it used many times in lectures and seminars (in Europe). However, I think it's rarely printed because it looks rather ugly. Another example of a book using this notation is Royden's classic Real Analysis. –  t.b. Feb 13 '11 at 14:20
I've seen it before, and my experience is limited to the U.S. However, I did take a real analysis course out of Royden, so that's probably where I saw it. –  Mike Spivey Oct 7 '11 at 2:58

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!

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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.

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It should be $b_m$ (in the defining expression, $n$ is bound and $m$ is free). –  joriki Feb 13 '11 at 15:16
Thanks, it has been corrected now. –  awllower Feb 14 '11 at 2:38