Random Shifts Which Preserve Measure

Given a flow <!-- MATH ${\theta _g},g \in G$ --> a group, over a probability space <!-- MATH $(\Omega ,\mathfrak{F},P)$ --> and a -valued random variable , we exhibit the Lebesgue decomposition of the measure <!-- MATH $P \circ \theta _Z^{ - 1}$ --> relative to , and give necessary and sufficient conditions for equality <!-- MATH $(P \circ \theta _Z^{ - 1} = P)$ --> , absolute continuity <!-- MATH $(P \circ \theta _Z^{ - 1} \ll P)$ --> , and singularity <!-- MATH $(P \circ \theta _Z^{ - 1} \bot P)$ --> in terms of the Haar measure. The proof rests on the theory of ``Palm measures'' as developed by Mecke and the authors. Specializing the group , we retrieve some known results for the integers and real line, and compute the Radon-Nikodým derivatives in various cases.