An f-measure is a finitely
additive nonnegative set function defined on a collection of subsets of [0,∞) which
vanishes on bounded Lebesgue measurable sets. We define statistical convergence and
convergence in density relative to an f-measure and use nonnegative regular
integral summability methods to generate f-measures. We observe that, for a
large class of regular integral summability methods, the notions of strong
integral summability, convergence in density and statistical convergence
(relative to the f-measure generated by the method) coincide for bounded
functions.
The support set of an f-measure is a subset of the Stone-Čech compactification
of [0,∞) that is generated by the measure. We characterize f-measures that generate
nowhere dense support sets and f-measures which have P-sets for support sets. The
support set of a nonnegative regular integral summability method is used to
introduce some summability invariants for bounded strong integral summability. We
show that the support sets of f-measures generated by some summability methods
are compact zero-dimensional F-spaces of weight c without isolated points, but that
they need not be P′-spaces.
