Egorov's theorem¶
It is also named Severini–Egoroff theorem or Severini–Egorov theorem, after Carlo Severini, an Italian mathematician, and Dmitri Egorov, a Russian mathematician and geometer, who published independent proofs respectively in 1910 and 1911.
Core Idea¶
Egorov's theorem is treated here as the recurring mathematicslogicstatistics identity summarized by this source-grounded definition: It is also named Severini–Egoroff theorem or Severini–Egorov theorem, after Carlo Severini, an Italian mathematician, and Dmitri Egorov, a Russian mathematician and geometer, who published independent proofs respectively in 1910 and 1911. In measure theory, an area of mathematics, Egorov's theorem establishes a condition for the uniform convergence of a pointwise convergent sequence of measurable functions. It is also named Severini–Egoroff theorem or Severini–Egorov theorem, after Carlo Severini, an Italian mathematician.
Scope of Application¶
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Historical note. The first proof of the theorem was given by Carlo Severini in 1910: he used the result as a tool in his research on series of orthogonal functions.
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Documented setting. Egorov's theorem can be used along with compactly supported continuous functions to prove Lusin's theorem for integrable functions.
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Historical note. Further generalizations were given much later by Pavel Korovkin, in the paper , and by Gabriel Mokobodzki in the paper : in particular Korovkin extended the result to a class of non–negative.
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Discussion of assumptions and a counterexample. To see this, it is simple to construct a counterexample when μ is the Lebesgue measure: consider the sequence of real-valued indicator functions fn(x) = 1{[n,n+1]}(x), \qquad.
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Discussion of assumptions and a counterexample. The separability of the metric space is needed to make sure that for M -valued, measurable functions f and g , the distance d(f(x), g(x)) is again a measurable.
Clarity¶
A clear use of Egorov's theorem names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is It is also named Severini–Egoroff theorem or Severini–Egorov theorem, after Carlo Severini, an Italian mathematician, and Dmitri Egorov, a Russian mathematician and geometer, who published independent proofs respectively in 1910 and 1911.
Manages Complexity¶
Egorov's theorem compresses multiple mathematicslogicstatistics details into a stable diagnostic relation. The source shows both the central mechanism—further generalizations were given much later by Pavel Korovkin, in the paper , and by Gabriel Mokobodzki in the paper : in particular Korovkin extended the result to a class of non–negative set functions more general than measures.—and the practical consequence—hence by the assumption of μ-almost everywhere pointwise convergence on A,.
Abstract Reasoning¶
- Type the carrier. Identify the mathematicslogicstatistics entities to which the claim applies.
- State the relation. Use the source-grounded identity: It is also named Severini–Egoroff theorem or Severini–Egorov theorem, after Carlo Severini, an Italian mathematician, and Dmitri Egorov, a Russian mathematician and geometer, who published independent proofs respectively in 1910 and 1911.
- Check operation and conditions. For natural numbers n and k, define the set E n,k by the union.
- Demand recognition evidence.
Knowledge Transfer¶
Within the home domain. Knowledge about Egorov's theorem transfers literally when a new case preserves the same carrier type, relation, and recognition test. The first proof of the theorem was given by Carlo Severini in 1910: he used the result as a tool in his research on series of orthogonal functions. Egorov's theorem can be used along with compactly supported continuous functions to prove Lusin's.
Neighborhood in Abstraction Space¶
Egorov's theorem sits in a crowded region of the domain-specific corpus (38th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Named Analytic Theorems & Operators (39 abstractions)
Nearest neighbors
- Mean Value Theorem — 0.89
- Julia set — 0.88
- Coinduction — 0.87
- Filling radius — 0.87
- Mehler Kernel — 0.87
Computed from structural-signature embeddings · 2026-10-08