AWhat changed
A normal number has the expected long-run frequencies of digit blocks; absolute normality requires this in every integer base. Borel had already shown that almost every number has this property. Sierpiński supplies an elementary exceptional-interval argument and specifies one such number using the infimum of a carefully defined complement. Lebesgue's companion article explains a complementary selection method using nested decimal intervals and a controllable bound on the lengths excluded. Their papers bridge a bare existence statement and the specification of an individual example. They are separate works in one historical contribution family.
Scientific scope
A normal number has the expected long-run frequencies of digit blocks; absolute normality requires this in every integer base. Borel had already shown that almost every number has this property. Sierpiński supplies an elementary exceptional-interval argument and specifies one such number using the infimum of a carefully defined complement. Lebesgue's companion article explains a complementary selection method using nested decimal intervals and a controllable bound on the lengths excluded. Their papers bridge a bare existence statement and the specification of an individual example. They are separate works in one historical contribution family. Historical effective determination is not automatically an efficient or formally certified computable-digit algorithm. Lebesgue's operational selection needs numerical cover information and control of its remaining length; it is not an algorithm for every abstract measurable set. His normal-tail calculation is a sketch with an explicit error-control qualification. No normality claim for pi, e or the square root of two follows. Borel retains credit for the earlier full-measure theorem.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. Historical effective determination is not automatically an efficient or formally certified computable-digit algorithm. Lebesgue's operational selection needs numerical cover information and control of its remaining length; it is not an algorithm for every abstract measurable set. His normal-tail calculation is a sketch with an explicit error-control qualification. No normality claim for pi, e or the square root of two follows. Borel retains credit for the earlier full-measure theorem.
Contributors
- The named authors in the research record · SCIENCE: Original-paper scientific authors for this exact mathematical contribution. Independent papers grouped in one historical family are not a joint byline. Earlier theorems, supporting proof contributions and later generalizations remain separate. Individual effort shares, author nationality and unresolved cross-record identities are not inferred.
- Démonstration élémentaire du théorème de M. Borel sur les nombres absolument normaux et détermination effective d'un tel nombre, BSMF45 pp125–132
- Sur certaines démonstrations d'existence, BSMF45 pp132–144
Provisional WRS assessment, edition 2026-10-23.7. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.
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