to construct a counterexample when μ is the Lebesgue measure: consider the sequence of real-valued indicator functions defined on the real line. This sequence converges pointwise to the zero function everywhere but does not converge uniformly on for any set B of finite measure: a counterexample in the general -dimensional real vector space can be constructed as shown by . 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 real-valued function of x. Proof Fix . For natural numbers n and k, define the set En,k by the union These sets get smaller as n increases, meaning that En+1,k is always a subset of En,k, because the first union involves fewer sets. A point x, for which the sequence (fm(x)) converges to f(x), cannot be in every En,k for a fixed k, because fm(x) has to stay closer to f(x) than 1/k eventually. 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. 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 μ and using the geometric series, we get Generalizations Luzin's version Nikolai Luzin's generalization of the Severini–Egorov theorem is presented here according to . Statement 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. Proof It is sufficient to consider the case in which the set A is