Using the representation
T
f
(
y
)
=
∫
f
d
v
y
Tf(y) = \smallint f\;d{v_y}
, where
(
v
y
)
({v_y})
is a random measure, we characterize some interesting bands in the lattice of all order-continuous operators on a space of measurable functions. For example, an operator
T
T
is (lattice-)orthogonal to all integral operators (i.e. all
v
y
{v_y}
are singular) or belongs to the band generated by all Riesz homomorphisms (i.e. all
v
y
{v_y}
are atomic) if and only if
T
T
satisfies certain properties which are modeled after the Riesz homomorphism property [31] and continuity with respect to convergence in measure. On the other hand, for operators orthogonal to all Riesz homomorphisms (i.e. all
v
y
{v_y}
are diffuse) we give characterizations analogous to the characterizations of Dunford and Pettis, and Buhvalov for integral operators. The latter results are related to Enflo operators, to a result of J. Bourgain on Dunford-Pettis operators and martingale representations of operators.