Selection Principle for Pointwise Bounded Sequences of Functions

For a number $\varepsilon>0$ and a real function $f$ on an interval $[a,b]$, denote by $N(\varepsilon,f,[a,b])$ the least upper bound of the set of indices $n$ for which there is a family of disjoint intervals $[a_i,b_i]$, $i=1,\dots,n$, on $[a,b]$ such that $|f(a_i)-f(b_i)|>\varepsilon$ for any $i=1,\dots,n$ ($\sup\varnothing=0$). The following theorem is proved: \emph{if $\{f_j\}$ is a pointwise bounded sequence of real functions on the interval $[a,b]$ such that $n(\varepsilon)\equiv\limsup_{j\to\infty}N(\varepsilon,f_j,[a,b])<\infty$ for any $\varepsilon>0$, then the sequence $\{f_j\}$ contains a subsequence which converges, everywhere on $[a,b]$, to some function $f$ such that $N(\varepsilon,f,[a,b])\le n(\varepsilon)$ for any $\varepsilon>0$}. It is proved that the main condition in this theorem related to the upper limit is necessary for any uniformly convergent sequence $\{f_j\}$ and is “almost” necessary for any everywhere convergent sequence of measurable functions, and many pointwise selection principles generalizing Helly's classical theorem are consequences of our theorem. Examples are presented which illustrate the sharpness of the theorem.