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 SeveriniâÂÂEgoroff theorem or SeveriniâÂÂEgorov theorem, after Carlo Severini, an Italian mathematician, and Dmitri Egorov, a Russian mathematician and geometer, who published independent proofs respectively in 1910 and 1911.
Egorov's theorem can be used along with compactly supported continuous functions to prove Lusin's theorem for integrable functions.
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. 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. A year later Dmitri Egorov published his independently proved results, and the theorem became widely known under his name: however, it is not uncommon to find references to this theorem as the SeveriniâÂÂEgoroff theorem. The first mathematicians to prove independently the theorem in the nowadays common abstract measure space setting were , and in : 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 long (242 pages) paper . 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 nonâÂÂnegative set functions more general than measures.
Let be a sequence of -valued measurable functions, where is a separable metric space, on some measure space , and suppose there is a measurable subset , with finite -measure, such that converges -almost everywhere on to a limit function . The following result holds: for every , there exists a measurable subset of such that , and converges to uniformly on .
Here, denotes the -measure of . In words, the theorem says that pointwise convergence almost everywhere on implies the apparently much stronger uniform convergence everywhere except on some subset of arbitrarily small measure. This type of convergence is called almost uniform convergence.
Fix . For natural numbers n and k, define the set E<sub>n,k</sub> by the union
These sets get smaller as n increases, meaning that E<sub>n+1,k</sub> is always a subset of E<sub>n,k</sub>, because the first union involves fewer sets. A point x, for which the sequence (f<sub>m</sub>(x)) converges to f(x), cannot be in every E<sub>n,k</sub> for a fixed k, because f<sub>m</sub>(x) has to stay closer to f(x) than 1/k eventually. Hence by the assumption of ü-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 n<sub>k</sub> such that
For x in this set we consider the speed of approach into the 1/k-neighbourhood of f(x) as too slow. Define
as the set of all those points x in A, for which the speed of approach into at least one of these 1/k-neighbourhoods of f(x) is too slow. On the set difference we therefore have uniform convergence. Explicitly, for any , let , then for any , we have on all of .
Appealing to the sigma additivity of ü and using the geometric series, we get
Nikolai Luzin's generalization of the SeveriniâÂÂEgorov theorem is presented here according to .
Under the same hypothesis of the abstract SeveriniâÂÂEgorov theorem suppose that A is the union of a sequence of measurable sets of finite ü-measure, and (f<sub>n</sub>) is a given sequence of M-valued measurable functions on some measure space (X,ã,ü), such that (f<sub>n</sub>) converges ü-almost everywhere on A to a limit function f, then A can be expressed as the union of a sequence of measurable sets H, A<sub>1</sub>, A<sub>2</sub>,... such that ü(H) = 0 and (f<sub>n</sub>) converges to f uniformly on each set A<sub>k</sub>.
It is sufficient to consider the case in which the set A is itself of finite ü-measure: using this hypothesis and the standard SeveriniâÂÂEgorov theorem, it is possible to define by mathematical induction a sequence of sets {A<sub>k</sub>}<sub>k=1,2,...</sub> such that
and such that (f<sub>n</sub>) converges to f uniformly on each set A<sub>k</sub> for each k. Choosing
then obviously ü(H) = 0 and the theorem is proved.
The proof of the Korovkin version follows closely the version on , which however generalizes it to some extent by considering admissible functionals instead of non-negative measures and inequalities and respectively in conditions 1 and 2.
Let denote a separable metric space and a measurable space: consider a measurable set and a class containing and its measurable subsets such that their countable in unions and intersections belong to the same class. Suppose there exists a non-negative measure such that exists and
If is a sequence of -valued measurable functions converging -almost everywhere on to a limit function , then there exists a subset of such that 0 and where the convergence is also uniform.
Consider the indexed family of sets whose index set is the set of natural numbers defined as follows:
Obviously
and
therefore there is a natural number m<sub>0</sub> such that putting A<sub>0,m<sub>0</sub></sub>=A<sub>0</sub> the following relation holds true:
Using A<sub>0</sub> 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 A<sub>1,m<sub>1</sub></sub>=A<sub>1</sub>, where m<sub>1</sub> is a surely existing natural number such that
By iterating the shown construction, another indexed family of set {A<sub>n</sub>} is defined such that it has the following properties:
and finally putting
the thesis is easily proved.