Tagged partitions of $[a,b]$, with a tag ${\xi }_i$ in each subinterval, and the Riemann sum $S(f,P,\xi )={\sum }_{i}f({\xi }_i){\mathrm{\Delta }}_i$

Let $a<b$ be reals and let $P=(n,t)$ be a partition of $[a,b]$, with subintervals $I_i=[t_i,t_{i+1}]$ and lengths ${\mathrm{\Delta }}_i=t_{i+1}-t_i$ for $i<n$ (Partition of $[a,b]$ as a finite strictly increasing list $a=t_0<t_1<\cdots <t_n=b$, its subintervals and their lengths, its mesh, refinement, and the common refinement of two partitions).

A tagging of $P$ is a sequence $\xi :\mathbb{N}\to \mathbb{R}$ (Sequences of reals: bounded, eventually, frequently, tails, subsequences) with

${\xi }_i\in I_i\text{for every }i<n,{\xi }_k:=b\text{ for }k\ge n,$

the second clause being the same bookkeeping tail convention that Partition of $[a,b]$ as a finite strictly increasing list $a=t_0<t_1<\cdots <t_n=b$, its subintervals and their lengths, its mesh, refinement, and the common refinement of two partitions uses, so that $\xi$ is a genuine sequence and no index above $n$ is ever read. The pair $(P,\xi )$ is a tagged partition of $[a,b]$, and ${\xi }_i$ is the tag of the $i$-th subinterval. The mesh of a tagged partition is the mesh $\mathrm{\parallel }P\mathrm{\parallel }$ of its underlying partition.

Taggings exist, and no choice is involved in producing one. Setting ${\xi }_i:=t_i$ for $i<n$ and ${\xi }_k:=b$ for $k\ge n$ defines a tagging, since $t_i\in [t_i,t_{i+1}]=I_i$ (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length). So every partition carries at least one tagging, exhibited by a formula. What is a selection is choosing a tag in each subinterval subject to a condition, as The Darboux and Riemann definitions agree: a bounded $f$ on $[a,b]$ is Darboux integrable with integral $I$ if and only if for every real $\epsilon >0$ there is a real $\delta >0$ such that $\mathrm{\mid }S(f,P,\xi )-I\mathrm{\mid }<\epsilon$ for every tagged partition of mesh below $\delta$ does; there the family of choices is finite and the selection is a theorem of ZF.

For $f:[a,b]\to \mathbb{R}$ and a tagged partition $(P,\xi )$ the Riemann sum of $f$ is

$S(f,P,\xi ):={\sum }_{i<n}f({\xi }_i){\mathrm{\Delta }}_i,$

the finite sum of Finite sums and finite products, by recursion, indexed by $i\in \mathbb{N}$ with $i<n$. It is a real number, being a finite sum of reals, and it is defined for every $f$, bounded or not: no supremum or infimum of $f$ occurs in it.

A Riemann sum lies between the Darboux sums of the same partition

Suppose in addition that $f$ is bounded (Lower bound, bounded below, bounded set), so that the Darboux sums of For bounded $f$ on $[a,b]$ and a partition $P$: the infimum $m_i$ and supremum $M_i$ of $f$ on the $i$-th subinterval, and the lower and upper Darboux sums $L(f,P)={\sum }_i{m}_i{\mathrm{\Delta }}_i$ and $U(f,P)={\sum }_i{M}_i{\mathrm{\Delta }}_i$ are defined. Then for every tagging $\xi$ of $P$,

$L(f,P)\le S(f,P,\xi )\le U(f,P)\mathrm{.}$

Indeed ${\xi }_i\in I_i$ gives $m_i\le f({\xi }_i)\le M_i$ (For bounded $f$ on $[a,b]$ and a partition $P$: the infimum $m_i$ and supremum $M_i$ of $f$ on the $i$-th subinterval, and the lower and upper Darboux sums $L(f,P)={\sum }_i{m}_i{\mathrm{\Delta }}_i$ and $U(f,P)={\sum }_i{M}_i{\mathrm{\Delta }}_i$), and multiplying by ${\mathrm{\Delta }}_i>0$ and summing over $i<n$ preserves the two inequalities, by monotonicity of finite sums, clause 4 of Laws of finite sums and finite products, and the order axioms (Ordered field, Complete ordered field (least-upper-bound property)).

This one line is the whole of the easy half of The Darboux and Riemann definitions agree: a bounded $f$ on $[a,b]$ is Darboux integrable with integral $I$ if and only if for every real $\epsilon >0$ there is a real $\delta >0$ such that $\mathrm{\mid }S(f,P,\xi )-I\mathrm{\mid }<\epsilon$ for every tagged partition of mesh below $\delta$: whatever the tags, a Riemann sum is trapped between the two Darboux sums, so control of $U(f,P)-L(f,P)$ is control of every Riemann sum over $P$ at once.

• The tags are unconstrained beyond membership. In particular a tag may be an endpoint, two adjacent subintervals may share their tag at the common endpoint, and the tags need not be increasing. The three standard specialisations — left endpoints ${\xi }_i=t_i$, right endpoints ${\xi }_i=t_{i+1}$, midpoints ${\xi }_i=(t_i+t_{i+1})\cdot 2^{-1}$ — are all taggings, and each is given by a formula in $P$.

• Convergence of Riemann sums is a mesh condition, not a sequence condition. The Darboux and Riemann definitions agree: a bounded $f$ on $[a,b]$ is Darboux integrable with integral $I$ if and only if for every real $\epsilon >0$ there is a real $\delta >0$ such that $\mathrm{\mid }S(f,P,\xi )-I\mathrm{\mid }<\epsilon$ for every tagged partition of mesh below $\delta$ quantifies over all tagged partitions of mesh below $\delta$. Weakening that to a single sequence of tagged partitions whose meshes tend to $0$ gives a strictly weaker condition, and the weakening is not harmless: the companion page of this pair exhibits a non-integrable function whose Riemann sums are constant along such a sequence.

• Why Riemann sums and Darboux sums are both kept. The Darboux sums are canonical functions of $f$ and $P$ and make suprema and infima available (The lower and upper Darboux integrals of a bounded $f$ on $[a,b]$ as ${\sup }_{P}L(f,P)$ and ${\inf }_{P}U(f,P)$, Darboux integrability as their equality, and the notation ${\int }_a^{b}f$); the Riemann sums are defined without any completeness of $\mathbb{R}$ and are what a numerical approximation actually computes. The theorem that the two routes give the same integral is The Darboux and Riemann definitions agree: a bounded $f$ on $[a,b]$ is Darboux integrable with integral $I$ if and only if for every real $\epsilon >0$ there is a real $\delta >0$ such that $\mathrm{\mid }S(f,P,\xi )-I\mathrm{\mid }<\epsilon$ for every tagged partition of mesh below $\delta$.