I have seen this in textbooks, and indeed found this Stack thread about how to create the bar in LaTEX, but don't know what it means:


  • 4
    $\begingroup$ In textbooks, they probably introduce/explain the notation...? It's likely to be an upper (Darboux) integral (and with a bar below for the lower integral); see here. $\endgroup$
    – StackTD
    May 10 '17 at 14:23
  • 3
    $\begingroup$ Could they be Darboux integrals? $\endgroup$
    – lioness99a
    May 10 '17 at 14:24
  • $\begingroup$ Ah thank you, turns out it was Riemann integrals which is the same thing :) $\endgroup$
    – user296950
    May 10 '17 at 14:27
  • $\begingroup$ As a side note, sometimes $\overline{\lim} a_n$ and $\underline{\lim} a_n$ are used to denote limit superior and limit inferior respectively. $\endgroup$
    – Alex Vong
    May 10 '17 at 16:27

These are denoted as the upper and lower Riemann integrals respectively.

More so, we can construct the following:

Let $P=\left\{x_0,x_1,...,x_n\right\}$ be a partition on $[a,b]$ for $n\in \mathbb{N}$.

Notating the lower and upper sums of some function $f$ with respect to its partition $P$, as $L(P,f)$ and $U(P,f)$, we can define the following:

$$\underline {\int_a^b}f=sup\left\{L(P,f):\forall \ partitions \ P \ on \ [a,b]\right\}$$

$$\overline {\int_a^b}f=inf\left\{U(P,f):\forall \ partitions \ P \ on \ [a,b]\right\}$$

Furthermore, we note that for a reimann integral to exist on some bounded interval $[a,b]$,

$$\underline {\int_a^b}f=\int_a^bf=\overline {\int_a^b}f$$


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