Could you explain the definition of mesh?

Question

For the context, I have seen various definitions, like this:

Or this:

I would like to concentrate to the first one (Definition 19.) and understand this definition. I think mesh there is supremum of some metric stuff. However, I don´t know, what is "diam $U$". Could you answer this, please?

Also, have you seen this definition of mesh, or different definitions?

$\mathrm{diam};U=\sup{d(x,y)\mid x,y\in U}$ measures how far two points in $U$ can be. When $U$ is a nice closed geometrical shape this agrees with the usual notion of diameter from geometry — Alessandro Codenotti, Sep 15 '21 at 10:34
@user10354138 I imagine Tereza's intent was to type $\mathrm{mesh}$ rather than italicize. — Mark S., Sep 15 '21 at 10:52
@MarkS. Still against guideline though. Do not use MathJax for text formatting in title (or anywhere really), full-stop. — user10354138, Sep 15 '21 at 10:56
@user10354138 Sorry, I just wanted to make it look different from the other text I guess. Next time I know the guidelines. :) — Tereza Tizkova, Sep 15 '21 at 11:45

score 4 · Accepted Answer · edited Jul 15 '22 at 04:17

4

The mesh is simply the length of the largest sub-interval.

Example: If we divide the interval $[1,2]$ into sub-intervals $[1,1.5]$, $[1.5,2]$, $[2,3]$, then the mesh is equal to $1$, which is the length of the longest (last in this case) sub-interval.

Note that, by length of $[x,y]$, we mean, $|y-x|$.

edited Jul 15 '22 at 04:17

JonathanZ

12,727

answered Sep 15 '21 at 10:34

F. A. Mala

3,547
5
31

1

Not sure that $[2,3]$ is a subinterval of $[1,2]$ – Didier Nov 29 '23 at 08:34

score 2 · Answer 2 · edited Nov 29 '23 at 08:28

2

$\mathrm{diam} U$ is short for diameter. So the mesh is the smallest number where all the diameters of things in $\mathcal{U}$ are less than it.

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edited Nov 29 '23 at 08:28

John D

140

answered Sep 15 '21 at 10:34

Mark S.

25,893

score 2 · Answer 3 · answered Jul 15 '22 at 03:44

The concept of mesh is usually studied in relation with the partition of a closed interval in Riemann Integration. It is the length of the largest sub-interval included in the partition. If the sub-intervals happen to be of equal size, the mesh will the length of any sub-interval.

Usually, for R-integrability, the mesh is desired to be arbitrarily small.

Could you explain the definition of mesh?

3 Answers3