AWhat changed
In his French note, Lusin characterizes measurable real functions on [0,1] by continuity on a perfect nowhere-dense subset whose measure can be chosen greater than1−epsilon for every positive epsilon. Continuity is relative to that retained set. Using Egoroff's almost-uniform convergence result, the note shows that this C-property is preserved by pointwise or almost-everywhere limiting constructions and states the measurable-function characterization and converse.
Scientific scope
In his French note, Lusin characterizes measurable real functions on [0,1] by continuity on a perfect nowhere-dense subset whose measure can be chosen greater than1−epsilon for every positive epsilon. Continuity is relative to that retained set. Using Egoroff's almost-uniform convergence result, the note shows that this C-property is preserved by pointwise or almost-everywhere limiting constructions and states the measurable-function characterization and converse. The large set depends on epsilon; this does not say the original function is continuous at almost every ambient-domain point, or that one fixed full-measure closed set works for all errors. The source uses a finite real interval and relative continuity, not an unrestricted theorem for arbitrary measure spaces and target topologies. Lusin explicitly says this 1912 note collects results from his 1911 Russian memoir and responds to Borel's 1912 note; it is not claimed as first discovery in 1912. Earlier Borel/Lebesgue formulations and Vitali's 1905 work must retain separate historical credit within this contribution family. The final polynomial-series and a.e.-primitive consequences are announced without full proofs here and are outside the established theorem covered by this record.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. The large set depends on epsilon; this does not say the original function is continuous at almost every ambient-domain point, or that one fixed full-measure closed set works for all errors. The source uses a finite real interval and relative continuity, not an unrestricted theorem for arbitrary measure spaces and target topologies. Lusin explicitly says this 1912 note collects results from his 1911 Russian memoir and responds to Borel's 1912 note; it is not claimed as first discovery in 1912. Earlier Borel/Lebesgue formulations and Vitali's 1905 work must retain separate historical credit within this contribution family. The final polynomial-series and a.e.-primitive consequences are announced without full proofs here and are outside the established theorem covered by this record.
Contributors
- The named authors in the research record · SCIENCE: Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.
Provisional WRS assessment, edition 2026-10-23.3. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.
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