Abstract:
This study undertakes to determine the existence or nonexistence
of an implication in either direction between any two out of
nine different modes of convergence, with the use of any subset of a
set of ten auxiliary hypotheses. The functions are real finite-valued
measurable functions defined on an arbitrary abstract measure space.
A collection of 25 counterexamples is used to establish the invalidity
of the unproved implications. When the desired convergence cannot
be concluded, the existence of a convergent subsequence is investigated.
An appendix investigates the possible non-uniqueness of the
limit function when the definitions of two of the modes are slightly
relaxed.
