Around Taylor’s Theorem on the Convergence of Sequences of Functions

Abstract

Abstract Egoroff’s classical theorem shows that from a pointwise convergence we can get a uniform convergence outside the set of an arbitrary small measure. Taylor’s theorem shows the possibility of controlling the convergence of the sequences of functions on the set of the full measure. Namely, for every sequence of real-valued measurable factions |fn } n∈ℕ pointwise converging to a function f on a measurable set E, there exist a decreasing sequence |δn } n∈ℕ of positive reals converging to 0 and a set A ⊆ E such that E \ A is a nullset and lim n → + ∞ | f n ( x ) − f ( x ) | δ n = 0   for   all   x ∈ A .   Let   J ( A ,   { f n } ) {\lim _{n \to + \infty }}\frac{{|{f_n}(x) - f(x)|}}{{{\delta _n}}} = 0\,{\rm{for}}\,{\rm{all}}\,x \in A.\,{\rm{Let}}\,J(A,\,\{ {f_n}\} ) denote the set of all such sequences |δn } n∈ℕ. The main results of the paper concern basic properties of sets of all such sequences for a given set A and a given sequence of functions. A relationship between pointwise convergence, uniform convergence and the Taylor’s type of convergence is considered.