An announcement of the results presented here formed a portion of a talk given at the 2003 Positivity Conference dedicated to Y. A. Abramovich and A. C. Zaanen. In this note we shall present a few aspects of equicontinuity, uniform absolute continuity and weak compactness in measure space (the commutative setting) and von Neumann algebras (the non-commutative setting). Recall that a family K of finitely-additive Banach-valued set functions defined on a ring of sets is equicontinuous if limm(Ri) = 0, uniformly for m ∈ K, whenever (Ri) is a disjoint sequence of sets from the ring. The C∗-algebra notion of equicontinuity deals with K as a subset of the dual space of a C∗-algebra A, and (Ri) is an orthogonal sequence of self-adjoint elements in the unit ball of A. The relationship between equicontinuity and uniform absolute continuity is the author’s main theorem in [4], which states that if each element in K is absolutely continuous with respect to a finitely additive measure, then if K (not assumed to be bounded) is equicontinuous, the absolute continuity is uniform. The non-commutative counterpart has been established by the author, K. Saito and J. D. M. Wright in [10], under the assumption of boundedness for K. The unbounded case has not yet been resolved. The techniques in proving this result are completely different from the commutative arguments. A Biting Lemma for vector measures is given in Section 2. The scalar version [5,6] is the following. If (fn) is an L1(μ)-bounded sequence, then there exists a subsequence (fni ) and a sequence of ‘bites’ Bk ↘ B, μ(B) = 0, such that (fni ) is weakly convergent outside Bk for each k. Thus, bounded L 1