  We introduce and study invariant (weighted) transport-kernels balancing
stationary random measures on a locally compact Abelian group. The first main
result is an associated fundamental invariance property of Palm measures,
derived from a generalization of Neveu's exchange formula. The second main
result is a simple sufficient and necessary criterion for the existence of
balancing invariant transport-kernels. We then introduce (in a nonstationary
setting) the concept of mass-stationarity with respect to a random measure,
formalizing the intuitive idea that the origin is a typical location in the
mass. The third main result of the paper is that a measure is a Palm measure if
and only if it is mass-stationary.
