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, and Dmitri Egorov, a Russian mathematician and geometer, who published independent proofs respectively in 1910 and 1911.
The first proof of the theorem was given by Carlo Severini in 1910:[1][2] he used the result as a tool in his research on series of orthogonal functions.
His work remained apparently unnoticed outside Italy, probably due to the fact that it is written in Italian, appeared in a scientific journal with limited diffusion and was considered only as a means to obtain other theorems.
The first mathematicians to prove independently the theorem in the nowadays common abstract measure space setting were Frigyes Riesz (1922, 1928), and in Wacław Sierpiński (1928):[4] an earlier generalization is due to Nikolai Luzin, who succeeded in slightly relaxing the requirement of finiteness of measure of the domain of convergence of the pointwise converging functions in the ample[further explanation needed] paper (Luzin 1916).
Let (fn) be a sequence of M-valued measurable functions, where M is a separable metric space, on some measure space (X,Σ,μ), and suppose there is a measurable subset A ⊆ X, with finite μ-measure, such that (fn) converges μ-almost everywhere on A to a limit function f. The following result holds: for every ε > 0, there exists a measurable subset B of A such that μ(B) < ε, and (fn) converges to f uniformly on A \ B.
Hence by the assumption of μ-almost everywhere pointwise convergence on A, for every k. Since A is of finite measure, we have continuity from above; hence there exists, for each k, some natural number nk such that For x in this set we consider the speed of approach into the 1/k-neighbourhood of f(x) as too slow.
Appealing to the sigma additivity of μ and using the geometric series, we get Nikolai Luzin's generalization of the Severini–Egorov theorem is presented here according to Saks (1937, p. 19).
Under the same hypothesis of the abstract Severini–Egorov theorem suppose that A is the union of a sequence of measurable sets of finite μ-measure, and (fn) is a given sequence of M-valued measurable functions on some measure space (X,Σ,μ), such that (fn) converges μ-almost everywhere on A to a limit function f, then A can be expressed as the union of a sequence of measurable sets H, A1, A2,... such that μ(H) = 0 and (fn) converges to f uniformly on each set Ak.
It is sufficient to consider the case in which the set A is itself of finite μ-measure: using this hypothesis and the standard Severini–Egorov theorem, it is possible to define by mathematical induction a sequence of sets {Ak}k=1,2,... such that and such that (fn) converges to f uniformly on each set Ak for each k. Choosing then obviously μ(H) = 0 and the theorem is proved.
183–184), which however generalizes it to some extent by considering admissible functionals instead of non-negative measures and inequalities
containing A and its measurable subsets such that their countable in unions and intersections belong to the same class.
Suppose there exists a non-negative measure μ such that μ(A) exists and If (fn) is a sequence of M-valued measurable functions converging μ-almost everywhere on
defined as follows: Obviously and therefore there is a natural number m0 such that putting A0,m0=A0 the following relation holds true: Using A0 it is possible to define the following indexed family satisfying the following two relationships, analogous to the previously found ones, i.e. and This fact enable us to define the set A1,m1=A1, where m1 is a surely existing natural number such that By iterating the shown construction, another indexed family of set {An} is defined such that it has the following properties: and finally putting the thesis is easily proved.