A generalization of the Helly theorem for functions with values in a uniform space (Q1956616)

Let \(T\) be a subset of the real line and \(Y\) a Hausdorff uniform space. In terms of generalized \(p\)-variation a sufficient condition for the existence of a pointwise convergent subsequence for a relatively sequentially compact sequence \(f_n: T\to Y\) is proved. This condition is also necessary for the uniform convergence of \(f_n\). A selection principle for the a.e. convergence is deduced. The results may be useful e.g. for proving the existence of selections of bounded \(p\)-variation for multifunctions, in study of Niemytski superposition operator, stochastic processes, harmonic analysis etc.