  In this paper, almost Dunford-Pettis operators with ranges in $c_0$ are used
to identify totally bounded sets in the absolute weak topology. That is, a
bounded subset $A$ of a Banach lattice $E$ is $|\sigma|(E,E^\prime)$-totally
bounded if and only if $T(A)\subset c_0$ is relatively compact for every almost
Dunford-Pettis operator $T:E\to{c_0}$. As an application, we show that for two
Banach lattices $E$ and $F$ every positive operator from $E$ to $F$ dominated
by a PL-compact operator is PL-compact if and only if either the norm of
$E^{\,\prime}$ is order continuous or every order interval in $F$ is
$|\sigma|(F,F^\prime)$-totally bounded.
