Amarts on Riesz spaces

The concepts of conditional expectations, martingales and stopping times were extended to the Riesz space context by Kuo, Labuschagne and Watson (Discrete time stochastic processes on Riesz spaces, Indag. Math., 15 (2004), 435–451). Here we extend the definition of an asymptotic martingale (amart) to the Riesz spaces context, and prove that Riesz space amarts can be decomposed into the sum of a martingale and an adapted sequence convergent to zero. Consequently an amart convergence theorem is deduced.