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Riemann's criterion: a bounded $f$ on $[a,b]$ is Darboux integrable if and only if for every real $\varepsilon > 0$ there is a partition $P$ with $U(f,P) - L(f,P) < \varepsilon$

Statement

Let $a < b$ be reals and let $f : [a,b] \to \mathbb{R}$ be bounded (Lower bound, bounded below, bounded set). Then $f$ is Darboux integrable on $[a,b]$ (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 $\int_a^b f$ ) if and only if

$\text{for every real } \varepsilon > 0 \text{ there is a partition } P \text{ of } [a,b] \text{ with } U(f,P) - L(f,P) < \varepsilon$

(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 $U(f,P) = \sum_i M_i \Delta_i$ , 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).

This is the criterion every later integrability proof on this page uses. It replaces a statement about a supremum and an infimum over all partitions, which cannot be checked directly, by the exhibition of a single partition for each $\varepsilon$ . The criterion says nothing about the value of the integral; that is located separately, by If $m \le f \le M$ on $[a,b]$ then $m(b-a) \le L(f,P) \le \underline{\int_a^b} f \le \overline{\int_a^b} f \le U(f,P) \le M(b-a)$ for every partition $P$ ; in particular every constant function is integrable, with $\int_a^b c = c(b-a)$ , between $L(f,P)$ and $U(f,P)$ for the same $P$ .

Facts & Assumptions

Given: Reals $a < b$ and a bounded $f : [a,b] \to \mathbb{R}$ .

[A1]

The criterion: for every real $\varepsilon > 0$ there is a partition $P$ of $[a,b]$ with $U(f,P) - L(f,P) < \varepsilon$ .

[L1]

$L(f,P) \le \underline{\int_a^b} f \le \overline{\int_a^b} f \le U(f,P)$ for every partition $P$ ; $\underline{\int_a^b} f = \sup \mathcal{L}$ and $\overline{\int_a^b} f = \inf \mathcal{U}$ over the nonempty sets $\mathcal{L}$ of lower sums and $\mathcal{U}$ of upper sums; $f$ is integrable exactly when the two are equal (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 $\int_a^b f$ , 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 $U(f,P) = \sum_i M_i \Delta_i$ ).

[L2]

$\varepsilon$ -characterisation of the supremum: if $u = \sup S$ with $S$ nonempty then for every real $\varepsilon > 0$ there is $s \in S$ with $s > u - \varepsilon$ (Epsilon characterisation of the supremum). Dually, if $\ell = \inf S$ then for every real $\varepsilon > 0$ there is $s \in S$ with $s < \ell + \varepsilon$ (Epsilon characterisation of the infimum, Greatest lower bound (infimum)).

[L3]

If $P'$ refines $P$ then $L(f,P) \le L(f,P')$ and $U(f,P') \le U(f,P)$ ; the common refinement $P_1 \vee P_2$ refines both $P_1$ and $P_2$ (Refining a partition raises the lower Darboux sum and lowers the upper one, and every lower sum is at most every upper sum: $L(f,P) \le L(f,P') \le U(f,P') \le U(f,P)$ when $P'$ refines $P$ , and $L(f,P) \le U(f,Q)$ for arbitrary partitions $P$ and $Q$ ; moreover the two changes are at most $2M(n' - n)\|P\|$ , 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).

[L4]

Ordered-field arithmetic: adding a constant to both sides preserves an inequality, the order is total and transitive, and $t \cdot 2^{-1} > 0$ for $t > 0$ (Order is preserved by adding a constant and by adding inequalities, Sign rules for products and monotonicity of multiplication, Ordered field, Complete ordered field (least-upper-bound property)). These order-arithmetic facts are stated by their sources for the strict order only; the nonstrict forms used below follow by adjoining the equality case, in which the two sides coincide.

Proof

technique · direct

1.1

Write $D := \overline{\int_a^b} f - \underline{\int_a^b} f$ , a real number with $D \ge 0$ by [L1]; $f$ is integrable exactly when $D = 0$ .

1.2

The criterion is sufficient. Assume [A1] and let a real $\varepsilon > 0$ be given. Fix a partition $P$ with $U(f,P) - L(f,P) < \varepsilon$ . By [L1], $\underline{\int_a^b} f \ge L(f,P)$ and $\overline{\int_a^b} f \le U(f,P)$ , so $D \le U(f,P) - L(f,P) < \varepsilon$ .

A1L1L4choose

1.3

The criterion is necessary; this half of the proof is steps 1.3, 2.2 and 3.1, and its symbols are its own. Assume $f$ is integrable and write $I$ for the common value $\underline{\int_a^b} f = \overline{\int_a^b} f$ . Let a real $\eta > 0$ be given; then $\eta \cdot 2^{-1} > 0$ by [L4].

L1L4

2.1

So $0 \le D < \varepsilon$ for every real $\varepsilon > 0$ . If $D > 0$ , taking $\varepsilon := D$ gives $D < D$ , which is false; hence $D = 0$ and $f$ is integrable by step 1.1.

step 1.1step 1.2L4

2.2

By [L2] applied to $\mathcal{L}$ , whose supremum is $I$ , there is a partition $P_1$ with $L(f,P_1) > I - \eta \cdot 2^{-1}$ ; by [L2] applied to $\mathcal{U}$ , whose infimum is $I$ , there is a partition $P_2$ with $U(f,P_2) < I + \eta \cdot 2^{-1}$ .

step 1.3L1L2choose

3.1

Put $P_{\ast} := P_1 \vee P_2$ , which refines both by [L3]. Then $L(f,P_{\ast}) \ge L(f,P_1) > I - \eta \cdot 2^{-1}$ and $U(f,P_{\ast}) \le U(f,P_2) < I + \eta \cdot 2^{-1}$ , so $U(f,P_{\ast}) - L(f,P_{\ast}) < \eta$ by [L4]. Since $\eta > 0$ was arbitrary, the criterion holds.

step 2.2L3L4

4.1

Steps 1.2 and 2.1 give the implication from the criterion to integrability, and steps 1.3, 2.2 and 3.1 give the converse; the two halves are independent and use no symbol in common, and together they are the stated equivalence.

step 2.1step 3.1∎

Remarks

The strict inequality is not essential. Requiring $U(f,P) - L(f,P) \le \varepsilon$ for every $\varepsilon > 0$ defines the same condition, since a partition working for $\varepsilon \cdot 2^{-1}$ works strictly for $\varepsilon$ . Statements below use whichever form is convenient and no consequence turns on the difference.
The common refinement is where the two partitions are reconciled. Step 2.2 produces one partition good for the lower sum and another good for the upper sum, and there is no reason for them to be the same. Refining a partition raises the lower Darboux sum and lowers the upper one, and every lower sum is at most every upper sum: $L(f,P) \le L(f,P') \le U(f,P') \le U(f,P)$ when $P'$ refines $P$ , and $L(f,P) \le U(f,Q)$ for arbitrary partitions $P$ and $Q$ ; moreover the two changes are at most $2M(n' - n)\|P\|$ is exactly what allows a single partition to inherit both properties, and it is the only place in this proof where anything beyond the definitions is used.
No choice principle is used. Steps 1.2, 2.2 and 3.1 instantiate finitely many existential statements, which is ordinary first-order reasoning. See What this page costs in choice: Riemann's criterion, the Darboux-Riemann equivalence and integrability of a monotone function are theorems of ZF; integrability of a continuous function inherits the single use of countable choice inside Heine-Cantor; and only the forward half of the Lebesgue criterion spends countable choice, once, at the countable union of null sets.

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Sources

Riemann integral (Wikipedia) (standard reference, not scraped)
Darboux integral (Wikipedia) (standard reference, not scraped)
W. Rudin, Principles of Mathematical Analysis, 3rd ed., Ch. 6 (standard reference, not scraped)
J. Lebl, Basic Analysis I, The Riemann Integral (standard reference, not scraped)
J. Hunter, Chapter 11: The Riemann Integral (standard reference, not scraped)

Riemann's criterion: a bounded fff on [a,b][a,b][a,b] is Darboux integrable if and only if for every real ε>0\varepsilon > 0ε>0 there is a partition PPP with U(f,P)−L(f,P)<εU(f,P) - L(f,P) < \varepsilonU(f,P)−L(f,P)<ε