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# References — Chapter ᚓ(E): The cutting

*Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ref-ch-20, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.*

[Guide for AI readers](index.md) · [The chapter](ch20.md)

*Each entry: one plain sentence saying exactly what it is cited for, its status, then every source with a note saying what that source actually carries. "Own computation" / "Own reading" entries are the author's own work, not outside sources.*

### 20.1 {#r-20.1}

The middle-thirds Cantor set is obtained by repeatedly deleting open middle thirds from [0, 1]; it consists of points with a ternary expansion using only 0 and 2, is uncountable, and has the cardinality of the interval.

*Status: confirmed*

- [Wikipedia, Cantor set](https://en.wikipedia.org/wiki/Cantor_set) — the construction, ternary 0/2 description, and equal cardinality with [0, 1] are stated in the construction and cardinality sections.

### 20.2 {#r-20.2}

The lengths removed in the Cantor construction form the geometric series 1/3+2/9+4/27+…=1, so the Cantor set has Lebesgue measure zero although it is nonempty and uncountable.

*Status: confirmed*

- [Wikipedia, Cantor set](https://en.wikipedia.org/wiki/Cantor_set) — the composition section evaluates the removed geometric progression to 1 and explains why endpoints and uncountably many points remain.
- [Wolfram MathWorld, Measure Zero](https://mathworld.wolfram.com/MeasureZero.html) — defines measure zero by covers with intervals of arbitrarily small total length.

### 20.3 {#r-20.3}

Henry John Stephen Smith described the construction in 1874 and Georg Cantor mentioned it in 1883.

*Status: confirmed*

- [Wikipedia, Cantor set](https://en.wikipedia.org/wiki/Cantor_set) — the historical remarks identify Smith’s 1874 discovery and Cantor’s 1883 mention.

### 20.4 {#r-20.4}

Cantor’s 1874 work established that the reals are uncountable while the real algebraic numbers are countable — the countability of the rationals is older and is not the content of that paper; see 20.22 — and the ternary-to-binary digit map explains why the Cantor set has continuum cardinality.

*Status: confirmed*

- [Wikipedia, Cantor set](https://en.wikipedia.org/wiki/Cantor_set) — the article gives the 0/2 ternary representation, the binary-tree/cardinality argument, and the uncountability conclusion, but no date for Cantor’s 1874 paper (it dates the set to Smith, 1874, and Cantor, 1883) and nothing about algebraic numbers. This claim line said until 17 September 2026 that Cantor’s 1874 work showed “the rationals are countable”; the first theorem of that article is that the real algebraic numbers are countable, which is what 20.22 states and what the chapter’s argument needs.

### 20.5 {#r-20.5}

Lebesgue measure extends length, is countably additive, and gives the Cantor set and the rationals in an interval measure zero; countable sets are null, while the Cantor set is uncountable. Kolmogorov’s 1933 axioms of probability take σ-additivity from Lebesgue’s measure theory.

*Status: confirmed*

- [Wikipedia, Lebesgue measure](https://en.wikipedia.org/wiki/Lebesgue_measure) — defines Lebesgue measure and countable additivity, and states that countable sets such as the rationals have measure zero.
- [Null set, Wikipedia](https://en.wikipedia.org/wiki/Null_set) — read as raw wikitext: “all countable sets are null. In particular, the set ℚ of rational numbers is a null set, despite being dense in ℝ.”
- [Probability axioms, Wikipedia](https://en.wikipedia.org/wiki/Probability_axioms) — read as raw wikitext: the axioms were “introduced by Russian mathematician Andrey Kolmogorov in 1933”; “The event space, F, must be a σ-algebra on Ω”; “The third axiom, σ-additivity, is relatively modern, and originates with Lebesgue’s measure theory.”
- [Cantor set, Wikipedia](https://en.wikipedia.org/wiki/Cantor_set) — “As the above summation argument shows, the Cantor set is uncountable but has Lebesgue measure 0.”

### 20.6 {#r-20.6}

The Cantor function is continuous and nondecreasing, constant on removed middle-third intervals, has derivative zero almost everywhere, and takes value 1/2 at 1/2. Absolute continuity is exactly the condition under which f(x) = f(a) + ∫ f′, and the Cantor function is not absolutely continuous.

*Status: confirmed*

- [Wikipedia, Cantor function](https://en.wikipedia.org/wiki/Cantor_function) — the article describes the devil’s staircase, its constancy on removed intervals, derivative zero almost everywhere, and midpoint value.
- [Absolute continuity, Wikipedia](https://en.wikipedia.org/wiki/Absolute_continuity) — read as raw wikitext: for f on [a, b], being absolutely continuous is equivalent to f having a derivative f′ almost everywhere that is Lebesgue integrable with f(x) = f(a) + ∫ₐˣ f′(t) dt, the “fundamental theorem of Lebesgue integral calculus”; the Cantor function is listed as not absolutely continuous.

### 20.7 {#r-20.7}

Hausdorff dimension for a self-similar set with m copies scaled by 1/r is ln(m)/ln(r); the Cantor dimension is ln 2/ln 3, about 0.6309, and it is transcendental by Gelfond–Schneider. Felix Hausdorff introduced the dimension in 1918, and Gelfond and Schneider proved the theorem independently in 1934.

*Status: confirmed*

- [Wikipedia, Hausdorff dimension](https://en.wikipedia.org/wiki/Hausdorff_dimension) — gives the self-similar dimension formula and the Cantor-set example, and dates the introduction of the dimension to Felix Hausdorff in 1918.
- [Wikipedia, Gelfond–Schneider theorem](https://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem) — states that if a is algebraic other than 0 or 1 and b is algebraic irrational, every value of a^b is transcendental; applying 3^x=2 proves ln 2/ln 3 transcendental; it dates the two independent proofs, Gelfond’s and Schneider’s, to 1934.

### 20.8 {#r-20.8}

The Smith–Volterra–Cantor set is nowhere dense and has Lebesgue measure one half when middle quarters, then middle sixteenths and so on, are removed. Sources differ on Henry Smith’s year: 1874 (the publisher’s record of the paper, November 1874 issue, and the article on the Cantor set) or 1875 (the article on this set); the book follows the publisher’s record; corrections welcome.

*Status: confirmed*

- [Wikipedia, Smith–Volterra–Cantor set](https://en.wikipedia.org/wiki/Smith%E2%80%93Volterra%E2%80%93Cantor_set) — gives the construction, nowhere-dense property, and measure 1/2; it removes open middle subintervals of width 1/4 to the n at step n, sums the removed lengths to 1/2, and dates Smith’s discussion to 1875.
- [Zenodo, H. J. S. Smith, On the Integration of Discontinuous Functions](https://zenodo.org/records/1932560) — a free record and scan of the paper, Proc. London Math. Soc. s1-6 (1874) 140–153 (doi:10.1112/plms/s1-6.1.140), with publication date 1 November 1874.

### 20.9 {#r-20.9}

The Koch curve uses four copies scaled by one third, has Hausdorff dimension ln 4/ln 3, infinite length, and its snowflake has area 8/5 of the initial triangle.

*Status: confirmed*

- [Wikipedia, Koch snowflake](https://en.wikipedia.org/wiki/Koch_snowflake) — states the four-at-one-third construction, dimension, infinite perimeter, and limiting area 8/5.

### 20.10 {#r-20.10}

The Sierpiński triangle consists of three copies at half scale and has Hausdorff dimension ln 3/ln 2; Wacław Sierpiński described it in 1915.

*Status: confirmed*

- [Wikipedia, Sierpiński triangle](https://en.wikipedia.org/wiki/Sierpi%C5%84ski_triangle) — gives the three-copy construction, dimension, and historical attribution to Sierpiński; “If one takes Pascal’s triangle with 2^n rows and colors the even numbers white, and the odd numbers black, the result is an approximation to the Sierpiński triangle,” and “The states of an n-disk puzzle, and the allowable moves from one state to another, form an undirected graph, the Hanoi graph, that can be represented geometrically as the intersection graph of the set of triangles remaining after the nth step in the construction of the Sierpiński triangle” (exponent as typeset on the page).

### 20.11 {#r-20.11}

The Menger sponge keeps twenty of twenty-seven subcubes at each step, has Hausdorff dimension ln 20/ln 3, zero volume, infinite surface area, and is a universal curve. The limit is neither a solid nor a surface: it has topological dimension 1, so it is a curve.

*Status: confirmed*

- [Wikipedia, Menger sponge](https://en.wikipedia.org/wiki/Menger_sponge) — states the 20-of-27 construction, dimension, zero volume, infinite surface area, and universal-curve property.

### 20.12 {#r-20.12}

Banach and Tarski’s 1924 theorem, using the axiom of choice, decomposes a three-dimensional ball into finitely many pieces, some of them non-measurable, that can be reassembled into two congruent balls; Robinson proved in 1947 that five pieces suffice and fewer than five do not, one of the five being a single point.

*Status: confirmed*

- [Wikipedia, Banach–Tarski paradox](https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox) — states the 1924 publication, axiom-of-choice dependence, non-measurable pieces, and Robinson’s sharp five-piece result: doubling the ball can be done with five pieces, and fewer than five will not suffice.
- [R. M. Robinson, On the decomposition of spheres, Fundamenta Mathematicae 34 (1947), 246–260](https://eudml.org/doc/213130) — the bibliographic record dates the five-piece minimum to 1947.
- [Encyclopedia of Mathematics, Banach–Tarski paradox](https://encyclopediaofmath.org/wiki/Banach%E2%80%93Tarski_paradox) — identifies Banach and Tarski’s 1924 paper and the equidecomposability theorem in three dimensions.
- [MathWorld, Banach–Tarski Paradox](https://mathworld.wolfram.com/Banach-TarskiParadox.html) — states that five pieces are minimal and four suffice when the single point at the center is omitted, carrying the claim that one of the five pieces is a singleton.

### 20.13 {#r-20.13}

Robert May’s 1976 Nature paper made the logistic map f(x)=rx(1−x) a standard example of simple rules with complicated dynamics.

*Status: confirmed*

- [Wikipedia, Logistic map](https://en.wikipedia.org/wiki/Logistic_map) — identifies May’s 1976 Nature paper and discusses the logistic map as a simple nonlinear recurrence with complex behavior; “In the general case with r between 1 and 2, the population will quickly approach the value (r − 1)/r, independent of the initial population,” and “With r between 2 and 3, the population will also eventually approach the same value” (fraction as typeset on the page); in the demographic example, “xₙ is a number between zero and one, which represents the ratio of existing population to the maximum possible population.”

### 20.14 {#r-20.14}

For the logistic map, r=2 has attracting fixed point 1/2; for 1<r<3 there is one attracting fixed point, at r=3 period doubling begins, and chaos begins near r=3.5699.

*Status: confirmed*

- [Wikipedia, Logistic map](https://en.wikipedia.org/wiki/Logistic_map) — gives the fixed point 1−1/r, stability range through r=3, period-doubling cascade, and accumulation parameter about 3.5699456.

### 20.15 {#r-20.15}

Feigenbaum’s delta is approximately 4.6692 for the period-doubling parameter spacings, while the magnitude of alpha is approximately 2.5029 for spatial scaling of the cycles; neither constant has been proved irrational, let alone transcendental.

*Status: confirmed*

- [Wikipedia, Feigenbaum constants](https://en.wikipedia.org/wiki/Feigenbaum_constants) — gives δ≈4.6692016 and α≈2.5029079 and describes their parameter- and coordinate-scaling roles; it states that both are believed transcendental but that there is no known proof that either is even irrational. δ “is the limiting ratio of each bifurcation interval to the next between every period doubling,” and α “is the ratio between the width of a tine and the width of one of its two subtines (except the tine closest to the fold).”

### 20.16 {#r-20.16}

Lebesgue integration is built from measurable level sets; the indicator of the rationals has Lebesgue integral zero although it is not Riemann integrable, and convergence theorems are central to analysis and probability.

*Status: confirmed*

- [Wikipedia, Lebesgue integral](https://en.wikipedia.org/wiki/Lebesgue_integral) — discusses level-set construction, the Dirichlet function, and monotone/dominated convergence.

### 20.17 {#r-20.17}

A sigma-algebra is closed under complements and countable unions; Borel and Lebesgue sigma-algebras provide standard measurable domains, and Kolmogorov formulates probability on such a domain. The Lebesgue-measurable sets are the completion of the Borel σ-algebra under Borel measure; Kolmogorov’s 1933 axioms require the events to form a σ-algebra.

*Status: confirmed*

- [Wikipedia, Sigma-algebra](https://en.wikipedia.org/wiki/%CE%A3-algebra) — defines sigma-algebras and describes Borel/Lebesgue examples and their use in probability.
- [Lebesgue measure, Wikipedia (completion)](https://en.wikipedia.org/wiki/Lebesgue_measure) — read as raw wikitext: “The Lebesgue-measurable sets are obtained by completing this Borel measure.”
- [Probability axioms, Wikipedia](https://en.wikipedia.org/wiki/Probability_axioms) — read as raw wikitext: the axioms were “introduced by Russian mathematician Andrey Kolmogorov in 1933”; “The event space, F, must be a σ-algebra on Ω”; “The third axiom, σ-additivity, is relatively modern, and originates with Lebesgue’s measure theory.”

### 20.18 {#r-20.18}

Vitali’s 1905 set is obtained by choosing one representative from each rational-translation class and is non-measurable; box-counting dimension uses the asymptotic growth of covering boxes and can differ from Hausdorff dimension on the rationals. The Hausdorff dimension is often hard to compute; for compact self-similar fractals the two dimensions coincide. Sets such as Vitali’s, for which no length exists, are why measures are defined on a σ-algebra; Banach and Tarski name Vitali’s 1905 construction among the precursors of their paradox.

*Status: confirmed*

- [Wikipedia, Vitali set](https://en.wikipedia.org/wiki/Vitali_set) — gives Vitali’s construction, non-measurability, and dependence on choice.
- [Wikipedia, Box counting](https://en.wikipedia.org/wiki/Box_counting) — the empirical box-counting procedure, its scan types and edge effects. It carries neither the N(ε) dimension formula nor any Hausdorff comparison; both are on the entry below.
- [Wikipedia, Minkowski–Bouligand dimension](https://en.wikipedia.org/wiki/Minkowski%E2%80%93Bouligand_dimension) — defines the box-counting dimension by the asymptotic growth of N(ε), names it for Hermann Minkowski and Georges Bouligand, and gives the rationals box dimension 1 against Hausdorff dimension 0.
- [Scott Sutherland, Fractal dimension (Stony Brook University, MAT 331 notes)](https://www.math.stonybrook.edu/~scott/Book331/Fractal_Dimension.html) — “One difficulty with the Hausdorff dimension is that it is often very hard to compute. The Hausdorff and box dimensions coincide for compact, self-similar fractals.”
- [σ-algebra, Wikipedia](https://en.wikipedia.org/wiki/%CE%A3-algebra) — read as raw wikitext: “there exist sets for which no size exists, for example, the Vitali sets. For this reason, one considers instead a smaller collection of privileged subsets of X.”
- [Banach–Tarski paradox, Wikipedia (precursors)](https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox) — read as raw wikitext: “Banach and Tarski explicitly acknowledge Giuseppe Vitali’s 1905 construction of the set bearing his name, Hausdorff’s paradox (1914), and a 1923 paper by Banach as the precursors to their work.”

### 20.19 {#r-20.19}

The derivative of the Cantor function at x=1/2 is exactly zero because the function is constant on the entire open interval (1/3, 2/3), where its value is 1/2.

*Status: confirmed*

- [Wikipedia, Cantor function](https://en.wikipedia.org/wiki/Cantor_function) — states that the Cantor function is constant at 1/2 on [1/3, 2/3], so the derivative at the interior point 1/2 is zero.

### 20.20 {#r-20.20}

Georg Cantor introduced the Cantor function in 1884, mentioning Scheeffer’s point that it is a counterexample to an extension of the fundamental theorem of calculus claimed by Harnack; Scheeffer (1884), Lebesgue (1904) and Vitali (1905) discussed and popularized it.

*Status: confirmed*

- [Wikipedia, Cantor function](https://en.wikipedia.org/wiki/Cantor_function) — the history section credits Cantor with introducing the function in 1884 and with reporting Scheeffer’s counterexample to Harnack’s claimed extension of the fundamental theorem of calculus, and names Scheeffer (1884), Lebesgue (1904) and Vitali (1905) as those who discussed and popularized it.

### 20.21 {#r-20.21}

In 1877 Cantor wrote to Dedekind that he had found a one-to-one correspondence between the points of the unit interval and the points of a square, and of any n-dimensional space, and exclaimed: je le vois, mais je ne le crois pas. The letter carrying the exclamation is dated 29 June 1877. The two encyclopaedia records give the year, the result and the French wording but not the day; the day is Gouvêa’s, in the standard scholarly account of the remark, and that is the dating this book uses.

*Status: confirmed*

- [Wikipedia, Georg Cantor](https://en.wikipedia.org/wiki/Georg_Cantor) — in an 1877 letter to Dedekind, Cantor proved that for any positive integer n there is a one-to-one correspondence between the points of the unit line segment and all the points of an n-dimensional space, and wrote to Dedekind: je le vois, mais je ne le crois pas.
- [MacTutor, Georg Cantor](https://mathshistory.st-andrews.ac.uk/Biographies/Cantor/) — dates the correspondence with Dedekind on the one-to-one correspondence between the unit interval and p-dimensional space to 1877, and gives the exclamation in translation.
- [Akihiro Kanamori, Set Theory from Cantor to Cohen (Elsevier sample chapter, PDF)](https://booksite.elsevier.com/samplechapters/9780444516213/sample.pdf) — “In a letter to Dedekind of 29 June 1877 Cantor … wrote, in French in the text, ‘I see it, but I don’t believe it.’” (The reading that the remark voiced doubt about the proof, not surprise, is Gouvêa’s: Was Cantor Surprised?, Amer. Math. Monthly 118 (2011) 198–209, doi:10.4169/amer.math.monthly.118.03.198, paywalled).

### 20.22 {#r-20.22}

Cantor’s article of 1874 proved that the set of real algebraic numbers is countable while the set of real numbers is not, which is the result that infinite sets come in different sizes. Only countably many reals are definable, since there are only countably many formulas.

*Status: confirmed*

- [Wikipedia, Cantor’s first set theory article](https://en.wikipedia.org/wiki/Cantor%27s_first_set_theory_article) — states that the article was published in 1874, that its first theorem is that the set of real algebraic numbers is countable, and that its second is the discovery that the set of all real numbers is uncountably rather than countably infinite.
- [Definable real number, Wikipedia](https://en.wikipedia.org/wiki/Definable_real_number) — read as raw wikitext: “This follows from the fact that there are only countably many formulas, and so only countably many elements of M can be definable over M.”

Editorial note: The chapter’s 20.4 carries the ternary-to-binary cardinality argument; its source, the Cantor set article, does not date Cantor’s uncountability proof, so the 1874 date is carried here instead.

### 20.23 {#r-20.23}

A property holds almost everywhere when the set on which it fails has measure zero; the failing set may be uncountable, as the Cantor set is.

*Status: confirmed*

- [Wikipedia, Almost everywhere](https://en.wikipedia.org/wiki/Almost_everywhere) — defines a property holding almost everywhere as holding for all elements except a subset of measure zero, formally a measurable set N with measure 0, and gives monotone functions being differentiable almost everywhere as an example.

### 20.24 {#r-20.24}

The chapter’s dated attributions are as stated. Helge von Koch’s curve is from a 1904 paper whose title is On a Continuous Curve Without Tangents, Constructible from Elementary Geometry, so the date and the absence of tangents come from the same place. Karl Menger first described his sponge in 1926, in his work on topological dimension. Vito Volterra introduced his nowhere-dense set of positive measure in 1881. Henri Lebesgue’s theory of integration was published in his 1902 dissertation Integrale, longueur, aire, supervised by Emile Borel. Borel’s Lecons sur la theorie des fonctions, the course in which his measure appears, is of 1898.

*Status: confirmed*

- [Koch snowflake, Wikipedia](https://en.wikipedia.org/wiki/Koch_snowflake) — read as raw HTML: the curve “appeared in a 1904 paper titled ‘On a Continuous Curve Without Tangents, Constructible from Elementary Geometry’ by the Swedish mathematician Helge von Koch,” and the page adds that “drawing a tangent line to any point is impossible” and that the closed snowflake itself appears neither in the 1904 article nor in the extended 1906 memoir, and may be due to Edward Kasner.
- [Menger sponge, Wikipedia](https://en.wikipedia.org/wiki/Menger_sponge) — read as raw HTML: “It was first described by Karl Menger in 1926, in his studies of the concept of topological dimension,” and “Menger showed, in the 1926 construction, that the sponge is a universal curve”; the reference given is Menger, Karl (1926), “Allgemeine Räume und Cartesische Räume. I.,” Communications to the Amsterdam Academy of Sciences.
- [Smith–Volterra–Cantor set, Wikipedia](https://en.wikipedia.org/wiki/Smith%E2%80%93Volterra%E2%80%93Cantor_set) — read as raw HTML: “In an 1875 paper, Smith discussed a nowhere-dense set of positive measure on the real line, and Volterra introduced a similar example in 1881. The Cantor set as we know it today followed in 1883.” Note the Smith year: this page says 1875 where r-20.8 gives 1874 after a documented tie-break to the London Mathematical Society issue of November 1874; r-20.8’s ruling stands and only Volterra’s 1881 is taken from here.
- [Henri Lebesgue, Wikipedia](https://en.wikipedia.org/wiki/Henri_Lebesgue) — read as raw HTML: the infobox gives the thesis as Intégrale, longueur, aire, thesis year 1902, doctoral advisor Émile Borel, and the text says “His theory was published originally in his dissertation Intégrale, longueur, aire (‘Integral, length, area’) at the University of Nancy during 1902.”
- [Émile Borel, Wikipedia](https://en.wikipedia.org/wiki/%C3%89mile_Borel) — read as raw HTML: Borel is described as “known for his founding work in the areas of measure theory and probability,” and his list of works dates “A course on the theory of functions” — Leçons sur la théorie des fonctions, the course in which the Borel sets and the Borel measure appear — to 1898. The page dates the book and does not separately date the σ-algebra, so the card’s “Borel 1898” is the book’s year.
- [G. H. Greco, S. Mazzucchi, E. M. Pagani, Peano on derivative of measures, arXiv:1002.4098](https://arxiv.org/abs/1002.4098) — read from the PDF: additivity of set functions, “expressed in terms of partitions of sets, was introduced for the first time in Borel [2, (1898), pp. 46-50], and, more clearly, in Lebesgue [26, (1902), p. 6]”; [2] is Borel’s Leçons sur la théorie des fonctions.

Editorial note: Added 17 September 2026 for the B6 pass. Six dates on five cards stood on entries that carry none of them: #koch’s 1904 and its no-tangent claim on r-20.9 (four copies, dimension, infinite length, area 8/5), #menger’s 1926 on r-20.11, #fatcantor’s Volterra 1881 on r-20.8 (which carries Smith 1874 and nothing else), and the 1902 Lebesgue thesis on r-20.5, whose claim line has no date, in three places at once (#lebesgue, #lebesgueint, #lamQ). r-20.7 was the model for the fix: it states Hausdorff 1918 and Gelfond-Schneider 1934 in the claim line itself, which is why #dimH’s two dates were already clean. Kolmogorov’s 1933 axioms, the sixth date, are carried by r-13.2 and the #sigmaalg card now cites it across chapters, which this book does routinely.

### 20.25 {#r-20.25}

Absolute continuity of a real function on an interval is the condition that for every ε > 0 there is a δ > 0 such that every finite collection of pairwise disjoint subintervals of total length under δ has Σ|f(bᵢ) − f(aᵢ)| < ε. A function on [a, b] is absolutely continuous exactly when it is the indefinite Lebesgue integral of an L¹ function (a condition stronger than uniform continuity), and that equivalence is the fundamental theorem of Lebesgue integral calculus, due to Lebesgue. The Cantor function is continuous and of bounded variation but not absolutely continuous, which is why it is differentiable almost everywhere with derivative zero and still climbs from 0 to 1. The Banach–Zaretsky theorem characterizes absolute continuity as continuity together with bounded variation and the Luzin N property.

*Status: confirmed*

- [Absolute continuity, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Absolute_continuity) — read as raw HTML: gives the ε–δ definition over finite collections of pairwise disjoint intervals, states that the fundamental theorem of calculus holds for absolutely continuous functions and is yet another characterization of them, and gives the Cantor ternary function as the notable example that is differentiable almost everywhere and still fails it.
- [Absolute continuity, Wikipedia](https://en.wikipedia.org/wiki/Absolute_continuity) — read as raw HTML: names the equivalence of the ε–δ condition with f(x) = f(a) + ∫ₐˣ g the “fundamental theorem of Lebesgue integral calculus, due to Lebesgue,” lists the Cantor function as uniformly continuous, of bounded variation and not absolutely continuous, and states the Banach–Zareckiĭ characterization; its opening calls absolute continuity “a regularity property of functions that is stronger than uniform continuity.”
- [Absolutely Continuous, MathWorld](https://mathworld.wolfram.com/AbsolutelyContinuous.html) — the companion notion for measures, and the Radon–Nikodym theorem that turns it into an integral.
- [Cantor function, Wikipedia](https://en.wikipedia.org/wiki/Cantor_function) — read as raw wikitext: the function is “non-decreasing,” and though it has zero derivative almost everywhere it “goes from 0 to 1 as x goes from 0 to 1, and takes on every value in between.”

### 20.26 {#r-20.26}

A subset is dense when its closure is the whole space, equivalently when the interior of its complement is empty; the rationals are dense in the reals. A subset is nowhere dense when the closure has empty interior, equivalently when the complement of its closure is dense. The Baire category theorem says that in a complete metric space the complement of a countable union of nowhere dense sets is never empty. The rationals are dense and null; the Cantor set is nowhere dense and null.

*Status: confirmed*

- [Nowhere-dense set, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Nowhere-dense_set) — read as raw HTML: gives the equivalent definitions — “The closure of A has empty interior” and “The complement of the closure of A is dense” — and states the Baire category theorem in the form used here.
- [Dense set, Wikipedia](https://en.wikipedia.org/wiki/Dense_set) — read as raw HTML: cl_X A = X and int_X(X ∖ A) = ∅ as equivalent definitions, the rationals as the standard dense subset of the reals, and nowhere dense as “the interior of its closure is empty.”
- [Null set, Wikipedia](https://en.wikipedia.org/wiki/Null_set) — read as raw wikitext: “all countable sets are null. In particular, the set ℚ of rational numbers is a null set, despite being dense in ℝ.”
- [Cantor set, Wikipedia](https://en.wikipedia.org/wiki/Cantor_set) — read as raw wikitext: “Cantor mentioned this ternary construction only in passing, as an example of a perfect set that is nowhere dense”; “the Cantor set is uncountable but has Lebesgue measure 0.”

### 20.27 {#r-20.27}

Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Volterra’s nowhere-dense set of positive measure, dated 1881.

*Status: confirmed*

- [Smith–Volterra–Cantor set, Wikipedia](https://en.wikipedia.org/w/index.php?title=Smith%E2%80%93Volterra%E2%80%93Cantor_set) — read raw: “In an 1875 paper, Smith discussed a nowhere-dense set of positive measure on the real line, and Volterra introduced a similar example in 1881..”

Editorial note: Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.