In real analysis, the mesh (also known as the norm) of a partition is a crucial measure that describes the "fineness" or "coarseness" of how an interval has been divided. Specifically, it is defined as the maximum length of any subinterval within that partition.
Understanding Partitions
Before delving deeper into the mesh, it's essential to understand what a partition is. When working with an interval, say $[a, b]$, in real analysis, a partition divides this interval into a finite number of smaller, non-overlapping subintervals.
Consider an interval $[a, b]$. A partition $P$ of this interval is a finite sequence of points $x_0, x_1, x_2, \ldots, x_n$ such that:
- $a = x_0 < x_1 < x_2 < \ldots < x_n = b$
- These points define $n$ subintervals: $[x_0, x_1], [x_1, x2], \ldots, [x{n-1}, x_n]$.
Each of these subintervals has a specific length. For instance, the length of the $j$-th subinterval is $xj - x{j-1}$.
Calculating the Mesh (Norm)
The mesh or norm of a partition $P$, often denoted as $|P|$, quantifies the largest "gap" or segment in the partition. It is calculated by finding the length of each subinterval and then identifying the maximum among them.
Mathematically, for a partition $P = {x_0, x_1, \ldots, x_n}$ of the interval $[a, b]$, the mesh is given by:
$$|P| = \max{xj - x{j-1} \text{ for } j = 1, 2, \ldots, n}$$
This means that the mesh is the length of the longest subinterval created by the partition points.
Significance in Real Analysis
The concept of the mesh is fundamental, particularly in the study of Riemann integration. For a function to be Riemann integrable, the limit of Riemann sums must exist as the mesh of the partitions approaches zero.
Here's why the mesh is important:
- Convergence of Riemann Sums: In the definition of the definite integral, we take the limit of Riemann sums as the number of subintervals goes to infinity and the mesh approaches zero. This ensures that the chosen sample points in each subinterval accurately represent the function's behavior across the entire interval, regardless of how the partition points are spaced.
- Accuracy of Approximation: A smaller mesh implies finer subintervals, leading to a more accurate approximation of the area under a curve. As $|P| \to 0$, the approximation becomes exact.
- Generalization: It allows for flexibility in partition choice. Instead of strictly requiring subintervals of equal length, the mesh condition ensures that even partitions with varying subinterval lengths will yield the correct integral value as long as the largest subinterval shrinks to zero.
Example of Mesh Calculation
Let's consider an example to illustrate how to calculate the mesh of a partition.
Problem: Find the mesh of the partition $P = {0, 0.5, 0.7, 1.2, 1.5}$ on the interval $[0, 1.5]$.
Solution:
-
Identify the subintervals and their lengths:
- Subinterval 1: $[0, 0.5]$, Length = $0.5 - 0 = 0.5$
- Subinterval 2: $[0.5, 0.7]$, Length = $0.7 - 0.5 = 0.2$
- Subinterval 3: $[0.7, 1.2]$, Length = $1.2 - 0.7 = 0.5$
- Subinterval 4: $[1.2, 1.5]$, Length = $1.5 - 1.2 = 0.3$
-
Find the maximum length:
- The lengths are $0.5, 0.2, 0.5, 0.3$.
- The maximum length among these is $0.5$.
Therefore, the mesh of the partition $P$ is $|P| = 0.5$.
