Let B be a unital monotone sigma-complete C*-algebra, X a Banach space and (T-n) (n=1, 2,...) a sequence of weakly compact operators from B to X. Our key theorem implies that if limT(n)p exists for each projection p in B, then there exists a weakly compact operator T such that parallel to T ''(n)(b)-T ''(b)parallel to -> 0, for each b in B ''. (An immediate corollary of this is that B is a Grothendieck space.) Let mu be any state of B such that each T-n is strongly absolutely continuous with respect to mu. Then, as a consequence of our key theorem, we obtain the following uniform absolute continuity property. For each epsilon > 0, there exists delta > 0, such that, whenever b is in the unit ball of B and mu(bb*+b*b)<delta, then parallel to T(n)b parallel to <=epsilon for all n. This last result is a far reaching non-commutative generalization of the classical Brooks-Jewett theorem for vector measures.