Relations between \(\mathcal L^{p}\)- and pointwise convergence of families of functions indexed by the unit interval (Q1952423)

The main result of this paper is to establish a Lusin-type theorem and present counterexamples which are \(\mathcal L^{p}\) continuous but have various degrees of pointwise discontinuity. It is an interesting discussion, beginning with a theorem and counterexamples of the pathologies one may encounter when comparing pointwise properties to those that hold with respect to measure. Interestingly, when considering (indexed) families of functions, the topology of the index set of a sequence can interact with the regularity properties of the family. Finally, by a mild strengthening of hypotheses (continuity in one parameter) it is shown that the arbitrarily small set which contains the discontinuities is a simple ``slice'', rather than a possibly much more complicated set.