Given two or more invariant measures on a structure, there are various ways to combine them to form another invariant measure on the structure. For example, given two invariant measures on a structure, one may construct another invariant measure by taking a mixture, the distribution of the following probabilistic process: first flip a weighted coin to determine which measure to use, and then independently sample from that measure. This provides a general method for finding new invariant measures, but since it is always available to us, we might search for ergodic measures, i.e., ones that are not decomposable as a nontrivial mixture.
Question 1: What Ergodic invariant measures exist on the Rado graph? What about the Henson graph?
Question 2: Can you devise ways of using invariant measures on one structure to produce invariant measures on another? For example, the measures described on the generic bipartite graph can be thought of as elaborations of the measures on the Rado graph. Can such methods for creating invariant measures tell you anything nonobvious about the algebraic or definable closures of the second structure?