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. Korovkin's version 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. Statement Let (M,d) denote a separable metric space and (X,Σ) a measurable space: consider a measurable set A and a class 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 with for all n if with . If (fn) is a sequence of M-valued measurable functions converging μ-almost everywhere on to a limit function f, then there exists a subset A′ of A such that 0 < μ(A) − μ(A′) < ε and where the convergence is also uniform. Proof 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 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: for all for each there exists km such that for all then for all and finally putting the thesis is easily proved. Notes References Historical references , available at Gallica. . . . Published by the Accademia Gioenia in Catania. . , available from the Biblioteca Digitale Italiana di Matematica. The obituary of Carlo Severini. . A short note in which Leonida Tonelli credits Severini for the first proof of Severini–Egorov theorem. Scientific references . A definitive monograph on integration and measure theory: the treatment of the limiting behavior of the integral of various kind of sequences of measure-related structures (measurable functions, measurable sets, measures and their