The biting lemma of J. K. Brooks and R. V. Chacon, which is related to H. P. Rosenthal’s subsequence splitting lemma, is extended to Banach-valued measures. The \(\omega^ 2\)-convergence introduced by J. K. Brooks and R. V. Chacon [Adv. Math. 37, 16-26 (1980; Zbl 0463.28003)] is shown not to be topological. A sequence \((f_ n)\) in \(L^ 1(P)\) converges \(\omega^ 2\) to zero if and only if there exists a strictly positive function \(g\in L^ \infty(P)\) such that \((gf_ n)\) is weakly convergent to zero. A characterization of compact subsets of \(L^ 1(X)\) is given. Finally, the authors present an application of \(\omega^ 2\)- convergence to Schmeidler’s \(n\)-dimensional Fatou lemma.