Pointwise -convergence and -convergence in measure of sequences of functions

Let be an ideal. We say that a sequence of real numbers is -convergent to if for every neighborhood U of y the set of n's satisfying y_nU is in . Basing upon this notion we define pointwise -convergence and -convergence in measure of sequences of measurable functions defined on a measure space with finite measure. We discuss the relationship between these two convergences. In particular we show that for a wide class of ideals including Erdős–Ulam ideals and summable ideals the pointwise -convergence implies the -convergence in measure. We also present examples of very regular ideals such that this implication does not hold.