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Mesh — Definition, Formula & Examples

Mesh of a Partition
Norm of a Partition

The width of the largest sub-interval in a partition.

Example: Consider the partition {0,0.2,0.9,1.1,1.6,2}\{0, 0.2, 0.9, 1.1, 1.6, 2\} of the interval [0,2].[0,2]\text{.}
The sub-intervals are [0,0.2],[0,0.2]\text{,} [0.2,0.9],[0.2,0.9]\text{,} [0.9,1.1],[0.9,1.1]\text{,} [1.1,1.6],[1.1,1.6]\text{,} and [1.6,2].[1.6,2]\text{.}
The widest of these is [0.2,0.9].[0.2,0.9]\text{.} It has width =0.7.= 0.7\text{.}
Thus the mesh of the partition is 0.7.

See also

Riemann sum

Key Formula

∥P∥=max⁡1≤i≤nΔxi=max⁡1≤i≤n(xi−xi−1)\|P\| = \max_{1 \le i \le n} \Delta x_i = \max_{1 \le i \le n} (x_i - x_{i-1})
Where:
  • ∥P∥\|P\| = The mesh (norm) of partition P
  • Δxi\Delta x_i = The width of the i-th subinterval
  • xi−1,xix_{i-1}, x_i = The left and right endpoints of the i-th subinterval
  • nn = The total number of subintervals in the partition

Worked Example

Problem: Find the mesh of the partition P = {0, 1, 3, 4, 6} on the interval [0, 6].
Step 1: Identify each subinterval and compute its width.
Δx1=1−0=1,Δx2=3−1=2,Δx3=4−3=1,Δx4=6−4=2\Delta x_1 = 1 - 0 = 1,\quad \Delta x_2 = 3 - 1 = 2,\quad \Delta x_3 = 4 - 3 = 1,\quad \Delta x_4 = 6 - 4 = 2
Step 2: The mesh is the maximum of all subinterval widths.
∥P∥=max⁡(1, 2, 1, 2)=2\|P\| = \max(1,\, 2,\, 1,\, 2) = 2
Answer: The mesh of the partition is 2.

Another Example

Problem: Find the mesh of the regular (equally spaced) partition of [0, 10] into 5 subintervals.
Step 1: For a regular partition, every subinterval has the same width.
Δx=10−05=2\Delta x = \frac{10 - 0}{5} = 2
Step 2: Since all widths are equal, the maximum width is simply that common value.
∥P∥=2\|P\| = 2
Answer: The mesh is 2. For any regular partition of [a, b] into n equal parts, the mesh is always (b − a)/n.

Frequently Asked Questions

What is the difference between mesh and norm of a partition?
They are two names for the same thing. 'Mesh' and 'norm of a partition' both refer to the width of the largest subinterval. The notation ‖P‖ is standard for both terms.
Why does the mesh need to approach zero for a Riemann integral to exist?
As the mesh approaches zero, every subinterval becomes arbitrarily narrow, which forces the Riemann sum to approximate the true area under the curve with increasing precision. The formal definition of the Riemann integral requires that the limit of Riemann sums exists as ‖P‖ → 0, regardless of how the partition points and sample points are chosen.

Mesh (norm) of a partition vs. Number of subintervals (n)

The mesh measures the width of the widest subinterval, while n counts how many subintervals there are. Increasing n does not always decrease the mesh — if the partition is irregular, one subinterval could remain wide even as n grows. For a regular partition, however, the mesh equals (b − a)/n, so increasing n automatically shrinks the mesh. The rigorous definition of the Riemann integral uses ‖P‖ → 0 rather than n → ∞ because it handles irregular partitions correctly.

Why It Matters

The mesh is central to the formal definition of the Riemann integral. A function is Riemann integrable on [a,b][a,b] precisely when the limit of its Riemann sums exists as the mesh approaches zero. This criterion is more general than simply letting n→∞n \to \infty, because it guarantees convergence for every possible way of partitioning the interval, not just equally spaced ones.

Common Mistakes

Mistake: Confusing the mesh with the average subinterval width.
Correction: The mesh is the maximum subinterval width, not the average. For example, a partition with widths 1, 1, 1, 5 has mesh 5, even though the average width is only 2.
Mistake: Assuming that adding more partition points always reduces the mesh.
Correction: Adding a point only reduces the mesh if it splits the widest subinterval. Inserting a new point into a narrow subinterval leaves the mesh unchanged.

Related Terms