Conditional expectations and weak and strong compactness in spaces of Bochner integrable functions

In [6, theorem IV.8.18], relatively norm compact sets K in Lp([mu]) are characterized by means of strong convergence of conditional expectations, E[pi]f --> f in Lp([mu]), uniformly for f [set membership, variant] K, where (E[pi]) is the family of conditional expectations corresponding to the net of all finite measurable partitions. In this paper we extend the above result in several ways: we consider nets of not necessarily finite partitions; we consider spaces of vector valued pth power Bochner integrable functions (and spaces M([Sigma], E) of vector valued measures with finite variation); we characterize relatively strong compact sets K in by means of uniform strong convergence E[pi]f --> f, as well as relatively weak compact sets Kby means of uniform weak convergence E[pi]f --> f. Previously, in [4], uniform strong convergence (together with some other conditions) was proved to be sufficient (but not necessary) for relative weak compactness.