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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: http://en.wikipedia.org/wiki/Limit_superior_and_limit_inferior2011-02-13
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    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.)2011-02-13
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    @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.2011-02-13
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    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)2011-02-13
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    @Américo: The LaTeX-code for $\varliminf$ is `\varliminf` and the one for $\varlimsup$ is `\varlimsup`2011-02-13
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    I confirm Hans Lundmark's answer. In Portugal it was common until 30 to 20 years ago. Today it is rare.2011-02-13
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    @Ben: I took the liberty of replacing the `\lim` by the proper symbols and crossing out the no longer relevant passages describing them. Hope you don't mind.2011-02-13
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    Where is this from? If it's from a textbook for a first course in analysis, I'd consider this to be really bad form because $\limsup$ at that point probably is only defined for sequences of real numbers (and maybe sets). And even if it's not, what is "$\limsup_{\lVert C \rVert \to 0} \mathcal{L}(f;C)$" supposed to mean? "$\lim_{\epsilon \to 0+} \sup \{\mathcal{L}(f;C) \in \mathbb{R} \;|\; \lVert C \rVert = \epsilon\}$"?2011-02-13
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    BTW, this notation is the subject of Exercise 18.3 of The TeXbook (http://net.ytu.edu.cn/share/%D7%CA%C1%CF/texbook.pdf), which says "some people prefer a different notation" and asks how to realize it in TeX (the answer being "\def\limsup{\mathop{\overline{\rm lim}}}" and "\def\liminf{\mathop{\underline{\rm lim}}}").2011-02-13

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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.2011-02-13
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    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*.2011-02-13
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    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.2011-10-07
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    An alternative, and very good to read article is located in wikipedia: https://en.wikipedia.org/wiki/Limit_superior_and_limit_inferior2017-12-20
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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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    Should be the other way around, Lim Inf and Lim Sup. (according to how the question was originally asked)2017-12-20
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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).2011-02-13
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    Thanks, it has been corrected now.2011-02-14