Invariant measures on products and on the space of linear orders

Let M be an ℵ 0 -categorical structure and assume that has no algebraicity weak elimination of imaginaries. Generalizing classical theorems de Finetti Ryll-Nardzewski, we show any ergodic, Aut(M)-invariant measure on [0,1] is a product measure. We also investigate the action Aut(M) compact space LO(M) linear orders M. If moreover Aut(M)↷M transitive, prove Aut(M)↷LO(M) either fixed point or uniquely ergodic.