The approximate variation of univariate uniform space valued functions and pointwise selection principles

Lobachevskii Journal of Mathematics. 2022. Vol. 43. No. 3. P. 550–563.

V. V. Chistyakov, S. A. Chistyakova

Given a Hausdorff uniform space X with the countable gage of pseudometrics of the 
uniformity of X, we introduce a concept of the approximate variation of a function f mapping a 
subset T of the reals intoX: this is the infimum of the family of Jordan-type variations of all functions 
g : T → X which differ from f in each uniform pseudometric, generated by a pseudometric from 
the gage, not greater than ε > 0. We prove the following compactness theorem in the topology 
of pointwise convergence: if a pointwise relatively sequentially compact sequence of functions is 
such that the limit superior of its approximate variations is finite for all pseudometrics in the gage 
and all ε > 0, then it contains a subsequence which converges pointwise on the domain T to a 
bounded regulated function (in a generalized sense). We illustrate this result by appropriate sharp 
examples and present a new characterization of uniform space valued regulated functions in terms 
of the approximate variation.

Keywords: selection principle, pointwise convergence, uniform space, принцип выбора, поточечная сходимость, равномерное пространство, regulated function, регулярная функция, approximate variation, аппроксимативная вариация