Pointwise convergence becomes uniform after a small exceptional set is removed

What changed

Imagine that each point of an interval settles toward its own final value, but different points settle at different speeds. Egoroff shows that, for measurable functions converging almost everywhere on a finite interval, one can remove a set with arbitrarily small positive measure and obtain uniform convergence on what remains. Uniform means that one sufficiently late stage works for every retained point. Severini had developed the small-exceptional-set mechanism in a 1910 study of convergent orthogonal-function expansions. Egoroff's 1911 note gives the general measurable-sequence statement. The two original papers form one historical contribution family, with separate author roles.

Scientific scope

Imagine that each point of an interval settles toward its own final value, but different points settle at different speeds. Egoroff shows that, for measurable functions converging almost everywhere on a finite interval, one can remove a set with arbitrarily small positive measure and obtain uniform convergence on what remains. Uniform means that one sufficiently late stage works for every retained point. Severini had developed the small-exceptional-set mechanism in a 1910 study of convergent orthogonal-function expansions. Egoroff's 1911 note gives the general measurable-sequence statement. The two original papers form one historical contribution family, with separate author roles. The removed set can depend on the allowed measure loss. The conclusion does not generally follow by deleting only one null set, does not give a universal convergence rate and does not hold on arbitrary infinite-measure domains. Severini's paper has weighted square-integrable orthogonal-expansion assumptions; it does not prove that all Fourier series converge. Earlier work by Lebesgue, Fischer and others retains its own credit.

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 removed set can depend on the allowed measure loss. The conclusion does not generally follow by deleting only one null set, does not give a universal convergence rate and does not hold on arbitrary infinite-measure domains. Severini's paper has weighted square-integrable orthogonal-expansion assumptions; it does not prove that all Fourier series converge. Earlier work by Lebesgue, Fischer and others retains its own credit.

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.

• Sopra gli sviluppi in serie di funzioni ortogonali (1910MemoriaXI,1–7;black-and-white full-volume container)
• Sur les suites de fonctions mesurables (CR152,244–246;30January1911 issue container)

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