What is recorded here?
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.
Source-linked catalogue note. A detailed explanatory guide has not yet been prepared for this entry.
Scientific scope
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.
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People and systems
N. Lusin
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.
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English catalogue narrative
Lusin: measurable functions are continuous on arbitrarily large closed subsets
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
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.
Reading status
This is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.
People and systems
N. Lusin
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.