  The notion of uniform equicontinuity in measure at zero for sequences of
additive maps from a normed space into the space of measurable operators
associated with a semifinite von Neumann algebra is discussed. It is shown that
uniform equicontinuity in measure at zero on a dense subset implies the uniform
equicontinuity on the entire space, which is then applied to derive some
non-commutative ergodic theorems
