A note on regular martingales in Riesz spaces

We consider regular martingales (i.e., martingales which can be written as differences of two positive martingales) in the measure-free setting of Riesz spaces. The aim of this note is to show that the space of such martingales is a Riesz space. We derive an analogue of a result of Troitsky's on regular norm bounded martingales in Banach lattices.