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FALSE: a pointwise limit of a sequence of Riemann integrable functions on $[a, b]$ is Riemann integrable

Statement

False claim: if $(f_{n})_{n \in N}$ is a sequence of Riemann integrable functions 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 $\int_{a}^{b} f$ , Sequences of reals: bounded, eventually, frequently, tails, subsequences) and $f : [a, b] \to R$ satisfies

$f_{n} (x) ⟶ f (x) for every x \in [a, b]$

(Limits and Cauchy sequences of reals), then $f$ is Riemann integrable on $[a, b]$ .

The witness below is the standard one: an increasing sequence of indicators of finite sets of rationals, each integrable because it has only finitely many discontinuities, whose pointwise limit is the Dirichlet function, which is not integrable at all. Every $f_{n}$ takes values in ${0, 1}$ , so no unboundedness is involved, and the convergence is even monotone.

Facts & Assumptions

Given: The set $E : = Q \cap [0, 1]$ , a surjection $s : N \to E$ , the finite sets $F_{n} : = {s (k) : k < n}$ for $n \in N$ , and the indicators $f_{n} : [0, 1] \to R$ with $f_{n} (x) = 1$ for $x \in F_{n}$ and $f_{n} (x) = 0$ otherwise.

[A1]

The false claim: a pointwise limit of Riemann integrable functions on a closed bounded interval with distinct endpoints is Riemann integrable.

[L1]

$Q$ is countably infinite and every subset of an at most countable set is at most countable, so $E$ is at most countable; $E$ is nonempty, since $0 \in E$ ; and a nonempty at most countable set admits a surjection from $N$ ( $Q$ is countably infinite, Every subset of an at most countable set is at most countable, Finite, countably infinite, countable, uncountable, A nonempty set is at most countable iff it is a surjective image of $N$ ).

[L2]

A bounded function on $[a, b]$ that is continuous at every point other than $r$ listed points is Riemann integrable (A bounded function on $[a, b]$ that is continuous except at finitely many points is Riemann integrable, Lower bound, bounded below, bounded set).

[L3]

The Dirichlet function restricted to $[0, 1]$ , that is $g : [0, 1] \to R$ with $g (x) = 1$ for rational $x$ and $g (x) = 0$ for irrational $x$ , is bounded and not Riemann integrable: its lower Darboux integral is $0$ and its upper Darboux integral is $1$ (FALSE: every bounded function on $[a, b]$ is Riemann integrable, The Dirichlet function $1_{Q}$ , and Thomae's function $t$ with $t (x) = 1 / q$ at a rational $x = p / q$ in lowest terms with $q \geq 1$ and $t (x) = 0$ at every irrational $x$ ).

[L4]

A sequence of reals converges to $x$ when for every rational $ε > 0$ there is $K$ with $∣ x_{k} - x ∣ < ε$ for all $k \geq K$ ; an eventually constant sequence converges to that constant (Limits and Cauchy sequences of reals, Sequences of reals: bounded, eventually, frequently, tails, subsequences, Basic properties of the absolute value).

[L5]

Every nonempty finite set of reals has a minimum (Every nonempty finite set of reals has a maximum and a minimum, Maximum and minimum of a set).

[L6]

Continuity at a point: for every real $ε > 0$ there must be a real $δ > 0$ with $∣ h (y) - h (x) ∣ < ε$ for every $y$ in the domain with $∣ y - x ∣ < δ$ (Continuity of $f : A \to R$ at a point of $A$ and on $A$ : the $ε$ - $δ$ condition, its agreement with $\lim_{x \to c} f (x) = f (c)$ at a limit point, and continuity at an isolated point, The $ε$ -neighbourhood and the punctured $ε$ -neighbourhood of a point of $R$ ).

[L7]

Ordered-field arithmetic: the order is total and transitive, $∣ u - v ∣ > 0$ for $u \neq v$ , and $0 < 1$ (Basic properties of the absolute value, 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), Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length). 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.

Refutation

technique · direct

1.1

By [L1] fix a surjection $s : N \to E$ and define $F_{n}$ and $f_{n}$ as in the Given. Each $f_{n}$ takes only the values $0$ and $1$ , so it is bounded.

givenL1chooseconstruct

2.1

Each $f_{n}$ is continuous at every point of $[0, 1]$ outside the $n$ listed points $s (0), \dots, s (n - 1)$ . Let $x \in [0, 1]$ with $x \neq s (k)$ for all $k < n$ , so $f_{n} (x) = 0$ . If $n = 0$ put $δ : = 1$ ; otherwise put $δ : = \min {∣ x - s (k) ∣ : k < n}$ , which exists by [L5] and is positive by [L7]. Every $y \in [0, 1]$ with $∣ y - x ∣ < δ$ then differs from each $s (k)$ with $k < n$ , so $f_{n} (y) = 0$ and $∣ f_{n} (y) - f_{n} (x) ∣ = 0 < ε$ for every $ε > 0$ .

step 1.1L5L6L7

2.2

$(f_{n})$ converges pointwise to $g$ on $[0, 1]$ . Let $x \in [0, 1]$ . If $x$ is rational then $x \in E$ , so $x = s (k)$ for some $k \in N$ by surjectivity, and then $x \in F_{n}$ and $f_{n} (x) = 1$ for every $n > k$ ; the sequence is eventually constant with value $1 = g (x)$ , so it converges to $g (x)$ by [L4]. If $x$ is irrational then $x \notin E$ and hence $x \notin F_{n}$ for any $n$ , so $f_{n} (x) = 0 = g (x)$ for every $n$ and again the sequence converges to $g (x)$ .

step 1.1L1L3L4

3.1

By [L2], applied with the $n$ listed points $s (0), \dots, s (n - 1)$ , each $f_{n}$ is Riemann integrable on $[0, 1]$ .

step 1.1step 2.1L2

4.1

So $(f_{n})$ is a sequence of Riemann integrable functions on $[0, 1]$ , an interval with $0 < 1$ , converging pointwise to $g$ , and $g$ is not Riemann integrable by [L3]. Hence [A1] fails at this sequence and the claim is false.

step 3.1step 2.2A1L3∎

Remarks

Monotone convergence does not help either. The sequence above is nondecreasing in $n$ at every point, since $F_{n} \subseteq F_{n + 1}$ , and uniformly bounded by $1$ . So no monotonicity or boundedness hypothesis on the sequence rescues the claim; what rescues it is uniform convergence, or a wider notion of integral, and neither is developed here.
What actually degrades in the limit. Each $f_{n}$ is discontinuous at $n$ points at most, so its discontinuity set is null and Lebesgue's criterion for Riemann integrability: a bounded $f$ on $[a, b]$ is Riemann integrable if and only if its set of discontinuities has measure zero passes it. The union of those finite sets is $E = Q \cap [0, 1]$ , which is still null; but the discontinuity set of the limit is all of $[0, 1]$ (The Dirichlet function is continuous at no point of $R$ , and Thomae's function is continuous at every irrational and at no rational, so its set of continuity points is exactly the set of irrationals and its oscillation at $c$ equals $t (c)$ ), which is not null (A sequence of intervals covering $[a, b]$ has total length at least $b - a$ , so no interval of positive length has measure zero). Discontinuity sets do not pass to pointwise limits, and that is the whole failure.
The refutation incurs no choice of its own; what it costs is inherited. A nonempty set is at most countable iff it is a surjective image of $N$ produces a single surjection $s$ , and every $f_{n}$ is defined from it by a formula; the $δ$ of step 2.1 is a minimum of a finite set, not a selection. The one countable choice behind this item is the one inside Heine-Cantor in $R$ : a continuous real function on a compact subset of $R$ is uniformly continuous, proved $R$ -natively from sequential compactness, reached through A bounded function on $[a, b]$ that is continuous except at finitely many points is Riemann integrable at step 3.1, and that is how 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 records it.

Depends on

Used by

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Sources

Riemann integral (Wikipedia) (standard reference, not scraped)
Dirichlet function (Wikipedia) (standard reference, not scraped)
E. Schechter, Gauge Integral (standard reference, not scraped)

FALSE: a pointwise limit of a sequence of Riemann integrable functions on [a,b][a,b] is Riemann integrable