Compositions of conditional expectations, Amemiya-Andô conjecture and paradoxes of thermodynamics

We investigate properties of compositions of conditional expectations on a non-atomic probability space (Ω,F,μ). Let 1≤p<∞ and X,Y∈Lp(Ω,F,μ). If for any convex f:R→R we have Ef(X)≥Ef(Y), then for each ε>0 there exist conditional expectations P1,P2,P3 and E1,…,En∈{P1,P2,P3} such that ‖Y−En…E1X‖p<ε. This theorem seems particularly surprising if one simplifies the assumption by taking X and Y to have the same distribution since, intuitively, it seems to contradict the second law of thermodynamics. We use this theorem to build the following counterexample for the Amemiya-Andô conjecture: There exist f∈L1(R)∩L2(R), conditional expectations P1,…,P5 on (R,Borel(R),λ) and E1,E2,…∈{P1,…,P5} such that the sequence of iterations (En…E2E1f) diverges in L2(R).