Around Taylor's theorem on the convergence of sequences of functions (Q2067286)

Let \((f_n)_n\) be a sequence of measurable real-valued functions defined on a Lebesgue measurable set \(E\) and \(\lim_{n\to+\infty}f_n=f\), the well-known theorem due to Taylor states that there exists a decreasing sequence \((\delta_n)_n\) of positive reals converging to zero and a subset \(A\) of \(E\) such that \(E\backslash A\) is a null set and \[\lim_{n\to+\infty}\frac{|f_n(x)-f(x)|}{\delta_n}=0 \quad\text{ for all}\;\; x\in A. \] Let \(J(A,{f_n})\) denote the set of all such sequences \((\delta_n)_n\). In the paper under review, the authors study some properties of sets of all such sequences \((\delta_n)_n\) which appear in Taylor's theorem. More precisely, they proved that Theorem 1. Suppose that \(E\) is a Lebesgue measurable subset of \(n\)-dimensional Euclidean space and \(\left\{f_{n}\right\}_{n \in \mathbb{N}}\) is a sequence of measurable functions such that \(f_{n}(x) \rightarrow f(x)\) as \(n \rightarrow+\infty\) for all \(x \in E\). Then, there exsists a subset \(\lambda(E \backslash A)=0\) such that \(J\left(A,\left\{f_{n}\right\}\right) \neq \emptyset\). Theorem 2. Let \(\left\{f_{n}\right\}_{n \in \mathbb{N}}\) be a sequence of functions uniformly convergent to a function \(f\) on a set \(E\). Then, \(J\left(E,\left\{f_{n}\right\}\right) \neq \emptyset\).