Tagged partitions of $[a,b]$, with a tag $\xi_i$ in each subinterval, and the Riemann sum $S(f,P,\xi) = \sum_i f(\xi_i)\,\Delta_i$

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 $\Delta_i = t_{i+1} - t_i$ for $i < n$ (Partition of $[a,b]$ as a finite strictly increasing list $a = t_0 < t_1 < \dots < t_n = b$, 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

$\xi_i \;\in\; I_i \qquad \text{for every } i < n, \qquad \xi_k := b \ \text{ for } k \ge n ,$

the second clause being the same bookkeeping tail convention that Partition of $[a,b]$ as a finite strictly increasing list $a = t_0 < t_1 < \dots < t_n = b$, 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 $\|P\|$ 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 $\varepsilon > 0$ there is a real $\delta > 0$ such that $|S(f,P,\xi) - I| < \varepsilon$ 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

$S(f,P,\xi) \;:=\; \sum_{i<n} f(\xi_i)\,\Delta_i ,$

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 $L(f,P) = \sum_i m_i \Delta_i$ and $U(f,P) = \sum_i M_i \Delta_i$ are defined. Then for every tagging $\xi$ of $P$,

$L(f,P) \;\le\; S(f,P,\xi) \;\le\; U(f,P) .$

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 $L(f,P) = \sum_i m_i \Delta_i$ and $U(f,P) = \sum_i M_i \Delta_i$), and multiplying by $\Delta_i > 0$ 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 $\varepsilon > 0$ there is a real $\delta > 0$ such that $|S(f,P,\xi) - I| < \varepsilon$ 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 $\varepsilon > 0$ there is a real $\delta > 0$ such that $|S(f,P,\xi) - I| < \varepsilon$ 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 $\varepsilon > 0$ there is a real $\delta > 0$ such that $|S(f,P,\xi) - I| < \varepsilon$ for every tagged partition of mesh below $\delta$.