Weak precompactness in the space of Pettis integrable functions (Q810824)

We give a necessary and sufficient condition for weak precompactness in the space of Pettis integrable functions with relatively compact range of their indefinite integrals. The main result is formulated in the spirit of previous works by \textit{J.K. Brooks} and \textit{N. Dinculeanu} [Contemp. Math. 2, 161-187 (1980; Zbl 0553.46027) and Adv. Math. 45, 255-258 (1982; Zbl 0532.46024)], where such a theorem was obtained for strongly measurable and Pettis integrable functions. Also we show that, in a sense, our result is the best possible.