Relations Between 𝓛<sup><em>p</em></sup>- and Pointwise Convergence of Families of Functions Indexed by the Unit Interval

Abstract

We construct a variety of mappings from the unit interval \(\mathcal{I}\) into \(\mathcal{L}^p([0,1]),1\leq p<\infty,\) to generalize classical examples of \(\mathcal{L}^p\)-converging sequences of functions with simultaneous pointwise divergence. By establishing relations between the regularity of the functions in the image of the mappings and the topology of \(\mathcal{I}\), we obtain examples which are \(\mathcal{L}^p\)-continuous but exhibit discontinuity in a pointwise sense to different degrees. We conclude by proving a Lusin-type theorem, namely that if almost every function in the image is continuous, then we can remove a set of arbitrarily small measure from the index set \(\mathcal{I}\) and establish pointwise continuity in the remainder.