One word, six integrals – and Lebesgue does not contain improper Riemann.
This paper takes a census of the word integral and shows in numbers that the advantage of the Lebesgue integral is not that more functions can be integrated. NO NEW MATHEMATICAL THEOREM IS CLAIMED: Riemann’s criterion, Thomae’s function, the fat Cantor set, the improper integrability of the sine cardinal, the derivative of x squared times the sine of one over x squared, the Henstock-Kurzweil integral and the convergence theorems are all standard. Nor does this paper judge which integral is right: what can be written is what each of them demands.
First, what decides Riemann integrability is not the density of the discontinuities but their measure. The Dirichlet function, one on the rationals and zero elsewhere, has a rational and an irrational in every subinterval, so every partition gives upper sum one and lower sum zero and it is not integrable. Yet Thomae’s function, one over q at p over q in lowest terms and zero at the irrationals, is discontinuous at every rational and is integrable. Only finitely many points exceed a given epsilon, so they can be covered by thin intervals: there are thirty-four points with denominator at most ten, three thousand and forty-six with denominator at most a hundred, and three hundred and four thousand one hundred and ninety-four with denominator at most a thousand, and the resulting bounds on the upper integral are 0.2, 0.02 and 0.002. The bound can be made as small as one likes, so Thomae’s function is Riemann integrable with integral zero.
The opposite example is needed too. Build a fat Cantor set by removing from the unit interval the open middle of length four to the minus n at step n. After twelve steps there are four thousand and ninety-six pieces, the remaining measure is 0.500122070313, and the longest piece is 0.0001221. The measure tends to one half and the longest piece to zero, so the set is nowhere dense and still has measure one half. Its indicator is discontinuous exactly on the set, so upper minus lower stays at one half for every partition and it is not Riemann integrable. Thomae is densely discontinuous and integrable; the fat Cantor set is nowhere dense and is not. So density is not what decides: the measure of the discontinuities is, which is Lebesgue’s criterion. Everywhere and how much are different questions.
Second, and this is the core. Lebesgue does not contain improper Riemann. Integrating the sine cardinal over the half-line, interval by interval, gives 1.53902907958 by ten pi, 1.56761329243 by a hundred pi and 1.57047801697 by a thousand pi, settling on pi over two, which is 1.57079632679. Over the same ranges the absolute value integrates to 1.45572998476, 2.92148797826 and 4.38735804234, and keeps climbing. Its per-decade increments are 1.465757993 and 1.465870064, agreeing to five digits with the predicted two over pi times the logarithm of ten, which is 1.465871198. The signed integral settles; the absolute one adds a fixed amount per decade, for ever. So the sine cardinal is improperly Riemann integrable and is not Lebesgue integrable. The separator is absolute convergence: since Lebesgue’s definition demands that the integral of the absolute value be finite, a convergence living on cancellation of signs is outside its scope from the start. The sentence that Lebesgue extends Riemann is true only with the improper case removed.
Third, neither of them contains the gauge integral. The function x squared times the sine of one over x squared, with value zero at the origin, is differentiable everywhere including at the origin, so its derivative is the derivative of some function and one would like the fundamental theorem to hold for it. The signed side is the fundamental theorem itself: the difference between the value at one and the value at epsilon is 0.84150154624679 at epsilon a hundredth, 0.84147097549151 at a ten-thousandth, and 0.8414709848079, the sine of one, from a hundred-millionth onwards. The absolute side is not. The function reaches plus or minus x squared where the sine equals plus or minus one, that is at one over the square root of pi over two plus k pi, so the variation is at least the sum of those differences: 2.10984075491 by k equal to ten, 5.01132130405 by a thousand and 7.94274870532 by a hundred thousand, against two over pi times the logarithm of k, which is 1.46587, 4.39761 and 7.32936. The variation diverges, so the integral of the absolute value of the derivative is infinite and the derivative is not Lebesgue integrable. The gauge integral of Henstock and Kurzweil integrates it and returns the difference of the endpoint values. The separator is the fundamental theorem of calculus: only this integral recovers every derivative from its function without exception, and it does so by loosening exactly one thing, namely that the fineness of the cut may vary from point to point.
Fourth, then what was bought? Enumerate the rationals in the unit interval and let the n-th function be the indicator of the first n of them. At one, ten and a thousand each is Riemann integrable, being supported on finitely many points, and each integral is zero. Only the limit steps outside the class, and that limit is the Dirichlet function. Every term lies inside Riemann’s class and only the limit leaves it. Lebesgue’s class does not leak, and what was bought is not the size of the class but the fact that it is closed under limits: monotone convergence, dominated convergence and Fubini are only other names for that closure.
Even so, the hypotheses are not decoration. Take the function equal to n on the interval from zero to one over n. Its integral is one for every n, while its value at a fixed point, say a thousandth, is ten at n equal to ten, a thousand at n equal to a thousand, and zero at n equal to a million. It tends to zero at every point and every integral is one. It is neither monotone nor dominated by an integrable function, so neither theorem applies and the conclusion fails. This is no defect of the integral: a theorem says only what it says, and the hypotheses were asked for by name, the same shape as the misreading of the equals sign in Paper 16.
A census. Riemann cuts the domain, the measure of the discontinuities is what moves it, and it is not closed under limits. Lebesgue cuts the range, absolute integrability is what moves it, and it is closed. The improper Riemann integral is defined as a limit, cancellation of signs is what moves it, and it is not inside Lebesgue. The gauge integral lets the cut vary with the point, the fundamental theorem is what moves it, and it contains the three above. The integral of a differential form lives on an oriented chain, orientation is what moves it, and it is not a measure at all, as Paper 92 wrote. The Itô integral answers to the roughness of a path and is a further distinct root, treated already in Papers 132 and 148, so only its address is given here. These six are not a staircase of strength: Lebesgue is wider than Riemann and yet excludes the improper case, and the gauge integral contains three of them without cutting the range at all. Which one to use is settled by what one demands.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
⚠️ Scope ── what is not claimed
a census of established standard facts (Layer 2). What is not shown: any new mathematical theorem, none is claimed; which integral is correct, no judgment is made. What is added: separating density from measure with two examples set side by side, measuring the per-decade increment of the absolute sine cardinal and matching it against two over pi times the logarithm of ten, putting the convergent signed integral and the divergent absolute one into one table, and writing, for each of the six, what must be moved for it to move. What remains: the sentence that the Lebesgue integral is stronger than the Riemann integral looks along one direction of strength only. What increased was not the supply of functions but the supply of theorems, and the price paid for the sentence that the limit and the integral may be exchanged was absolute convergence.
Reference
Yamagishi, Y. (2026). What Lebesgue Bought Was Not a Bigger Class but Closure under Limits [Paper 168]. Zenodo.
DOI: 10.5281/zenodo.22093320
ORCID: 0009-0006-8587-3450