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 Erd˝os–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. © 2007 Elsevier Inc. All rights reserved