  In this paper we look for the existence of large linear and algebraic
structures of sequences of measurable functions with different modes of
convergence. Concretely, the algebraic size of the family of sequences that are
convergent in measure but not a.e.~pointwise, uniformly but not pointwise
convergent, and uniformly convergent but not in $L^1$-norm, are analyzed. These
findings extend and complement a number of earlier results by several authors.
