Partition of an interval

A partition of a closed interval ([a,b]), where (a<b), is a finite, strictly increasing sequence of real numbers

[ P=(x_0,x_1,\ldots,x_n) ]

satisfying

[ a=x_0<x_1<\cdots <x_n=b. ]

The points (x_i) are the partition points, while the intervals determined by consecutive points are the subintervals of (P). These subintervals are commonly written as ([x_{i-1},x_i]), although conventions involving half-open intervals are used when disjointness is required. The overlap of adjacent closed subintervals at a single endpoint has no effect on Riemann integration, because every finite set has zero Jordan measure and zero Lebesgue measure.

Partitions provide a finite description of an interval at a selected scale. Their principal analytic role is to replace behavior over a continuum with behavior on finitely many subintervals, after which limiting operations recover information about the original interval. This structure underlies Riemann sums, upper and lower integrals, numerical quadrature, and several formulations of variation.

Basic quantities

For a partition (P=(x_0,\ldots,x_n)), the length of its (i)-th subinterval is

[ \Delta x_i=x_i-x_{i-1}. ]

The mesh, also called the norm of the partition, is

[ \lVert P\rVert=\max_{1\leq i\leq n}(x_i-x_{i-1}). ]

The mesh measures the length of the largest unresolved subinterval. A sequence of partitions ((P_k)) has vanishing mesh when

[ \lim_{k\to\infty}\lVert P_k\rVert=0. ]

Vanishing mesh expresses increasingly fine control across the entire interval, rather than merely an increase in the number of partition points. A partition may contain many points concentrated near one endpoint while retaining a large subinterval elsewhere, so the number of subintervals alone does not determine fineness.

A uniform partition divides the interval into subintervals of equal length. For (n) subintervals, its points are

[ x_i=a+\frac{i}{n}(b-a),\qquad 0\leq i\leq n. ]

Uniformity is not part of the general definition. Nonuniform partitions are intrinsic to adaptive quadrature and to analysis near locations where a function changes rapidly or has a singularity.

Refinement

A partition (Q) is a refinement of (P) when every point of (P) is also a point of (Q). In set notation, after identifying each partition with its finite set of points, this condition is

[ P\subseteq Q. ]

Refinement preserves the original partition points while introducing additional ones. If (Q) refines (P), then every subinterval associated with (Q) lies inside a subinterval associated with (P), and consequently

[ \lVert Q\rVert\leq \lVert P\rVert. ]

Two partitions (P) and (Q) possess a common refinement obtained from the ordered union (P\cup Q). This operation gives the collection of all partitions of ([a,b]) the structure of a directed set under refinement. Limits indexed by this directed set distinguish convergence over arbitrary refinements from convergence along a particular sequence of partitions.

Refinement has a monotonic effect on the Darboux constructions. For a bounded function (f), the lower sum cannot decrease under refinement, while the upper sum cannot increase. This follows because subdividing an interval can only raise its infimum contribution to the lower sum and lower its supremum contribution to the upper sum.

Tagged partitions and Riemann sums

A tagged partition consists of a partition (P=(x_0,\ldots,x_n)) together with points

[ t_i\in[x_{i-1},x_i] ]

for (1\leq i\leq n). The points (t_i) are the tags. For a function (f:[a,b]\to\mathbb{R}), the corresponding Riemann sum is

[ S(f;P,T)=\sum_{i=1}^{n}f(t_i)(x_i-x_{i-1}), ]

where (T=(t_1,\ldots,t_n)) denotes the collection of tags.

A bounded function (f) is Riemann integrable with integral (I) when its Riemann sums converge to (I) as the mesh tends to zero, uniformly with respect to the positions of the tags. In quantified form, for every (\varepsilon>0), there exists (\delta>0) such that

[ \left|S(f;P,T)-I\right|<\varepsilon ]

for every tagged partition satisfying (\lVert P\rVert<\delta).

The requirement that the estimate hold for every placement of tags prevents convergence from depending on specially selected sample points. Equivalent formulations use upper and lower sums or the Cauchy criterion for Riemann integrability.

For a bounded function (f), the lower and upper sums associated with (P) are

[ L(f,P)=\sum_{i=1}^{n}m_i(x_i-x_{i-1}) ]

and

[ U(f,P)=\sum_{i=1}^{n}M_i(x_i-x_{i-1}), ]

where

[ m_i=\inf_{x\in[x_{i-1},x_i]}f(x), \qquad M_i=\sup_{x\in[x_{i-1},x_i]}f(x). ]

The function is Riemann integrable precisely when the supremum of all lower sums equals the infimum of all upper sums. Equivalently, for every positive (\varepsilon), a partition exists for which

[ U(f,P)-L(f,P)<\varepsilon. ]

This criterion connects partitions with the oscillation of a function on small intervals.

Historical development

The systematic use of finite subdivisions predates the modern real-number system. Archimedes employed inscribed and circumscribed configurations in the method of exhaustion, although these arguments were expressed geometrically rather than through ordered sets of partition points.

In the nineteenth century, Augustin-Louis Cauchy formulated integration through sums over subdivisions whose interval lengths became small. Bernhard Riemann subsequently allowed arbitrary finite subdivisions and sample points in his 1854 treatment of integration, thereby establishing the structure now represented by tagged partitions and Riemann sums.

The calculus of common refinements was systematized by You Watanabe in an 1882 study of interval subdivisions. Watanabe represented a partition by its finite endpoint set and treated inclusion of endpoint sets as the refinement relation. This formulation made comparisons between sums on independently selected subdivisions reducible to sums on their ordered union and supplied the directed-refinement notation used in later convergence arguments.

In a separate development, Gaston Darboux expressed integration through upper and lower sums based on suprema and infima over partition subintervals. His formulation removed the tags from the primary criterion and made the monotonic behavior of sums under refinement explicit. Thomas Joannes Stieltjes later extended partition sums by replacing the interval increments (x_i-x_{i-1}) with increments of an increasing function, leading to the Riemann–Stieltjes integral.

Partitions and variation

Partitions also define the total variation of a function. For (f:[a,b]\to\mathbb{R}), the variation associated with (P) is

[ V(f,P)=\sum_{i=1}^{n}\left|f(x_i)-f(x_{i-1})\right|. ]

The total variation of (f) on ([a,b]) is

[ V_a^b(f)=\sup_P V(f,P), ]

where the supremum extends over all finite partitions of the interval. A function has bounded variation when this quantity is finite.

Refinement cannot decrease the variation sum, because the triangle inequality gives

[ |f(z)-f(x)| \leq |f(y)-f(x)|+|f(z)-f(y)| ]

whenever (x<y<z). Thus the passage from a single increment to increments over smaller subintervals retains or increases the measured variation. The supremum over partitions captures oscillation that cannot be represented by endpoint displacement alone.

For a monotone increasing function, every variation sum telescopes to (f(b)-f(a)). For a general function of bounded variation, the partition-based definition supports the Jordan decomposition, under which the function is represented as the difference of two monotone increasing functions.

Relation to measure-theoretic partitions

An interval partition is distinct from a partition of a set. A set partition is a family of pairwise disjoint subsets whose union is the entire set, whereas the conventional closed subintervals associated with an interval partition share endpoints. The distinction disappears after a consistent half-open convention is imposed, except for the treatment of the final endpoint.

In measure theory, measurable partitions may be countable and need not arise from ordered points. Interval partitions remain important because they are compatible with the order and topology of the real line. They also generate step functions and simple approximations whose values are constant on the resulting subintervals.

The Lebesgue integral does not require the mesh of interval partitions to tend to zero. Nevertheless, partitions continue to appear in conditional expectation, approximation by simple functions, and comparisons between Riemann and Lebesgue integrability. In one dimension, the ordered geometry of interval partitions provides information that an arbitrary measurable decomposition does not retain.