The two theorems in the title give conditions on Banach lattices E and F under which a positive operator from E into F, dominated by another positive operator with some property, must also have that property. The Dodds-Fremlin theorem says that this is true for compactness provided both E′ and F have order continuous norms, whilst the Kalton–Saab theorem establishes such a result for Dunford–Pettis operators provided F has an order continuous norm. These results were originally provided, in their full generality, in [3] and [5], respectively, whilst very readable proofs may be found in chapter 5 of [2] or §3·7 of [6].

A fundamental theorem of Khinchin says that every limit of an infinitesimal triangular system of probability measures on R is infinitely divisible. This was generalized to all divisible locally compact second countable abelian groups by Parthasarathy et al. (cf. [PRV]). Recently, Ruzsa eliminated the second countability condition and also proved the theorem for all Banach spaces (cf. [R2]). A similar theorem was also proved by Gangolli for certain symmetric spaces (cf. [G]). A result of Carnal shows that infinite divisibility of limits holds for commutative infinitesimal triangular systems on compact groups (cf. [C]). The same was recently proved by Neuenschwander for simply connected step-2 nilpotent Lie groups, provided the system is symmetric or supported on a discrete subgroup (cf. [N]).