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

Let I⊂P(N) be an ideal. We say that a sequence (yn)n∈N of real numbers is I-convergent to y∈R if for every neighborhood U of y the set of n's satisfying yn∉U is in I. Basing upon this notion we define pointwise I-convergence and I-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 I-convergence implies the I-convergence in measure. We also present examples of very regular ideals such that this implication does not hold.