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?

Answer 1

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, $|y-x|$.

Answer 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.

Note: If you can't read the box above, remember to follow https://math.meta.stackexchange.com/a/35899/491852

Answer 3

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.