  Directionally convex ($dcx$) ordering is a tool for comparison of dependence
structure of random vectors that also takes into account the variability of the
marginal distributions. When extended to random fields it concerns comparison
of all finite dimensional distributions. Viewing locally finite measures as
non-negative fields of measure-values indexed by the bounded Borel subsets of
the space, in this paper we formulate and study the $dcx$ ordering of random
measures on locally compact spaces. We show that the $dcx$ order is preserved
under some of the natural operations considered on random measures and point
processes, such as deterministic displacement of points, independent
superposition and thinning as well as independent, identically distributed
marking. Further operations such as position dependent marking and displacement
of points though do not preserve the $dcx$ order on all point processes, are
shown to preserve the order on Cox point processes. We also examine the impact
of $dcx$ order on the second moment properties, in particular on clustering and
on Palm distributions. Comparisons of Ripley's functions, pair correlation
functions as well as examples seem to indicate that point processes higher in
$dcx$ order cluster more. As the main result, we show that non-negative
integral shot-noise fields with respect to $dcx$ ordered random measures
inherit this ordering from the measures. Numerous applications of this result
are shown, in particular to comparison of various Cox processes and some
performance measures of wireless networks, in both of which shot-noise fields
appear as key ingredients. We also mention a few pertinent open questions.
