Starting with a trajectory space, providing a non-stochastic analogue of a discrete time martingale process, we use the notion of super-replication to introduce definitions for null and full sets and the associated notion of a property holding almost everywhere (a.e.). The latter providing what can be seen as the worst case analogue of sets of measure zero in a stochastic setting. The a.e. notion is used to prove the pointwise convergence, on a full set of the original trajectory space, of the limit of a trajectorial transform sequence. The setting also allows to construct a natural integration operator which we study in detail.

The authors would like to thank Dr. Pedro Catuogno for stimulating conversations on the subject of the paper. S. Ferrando would like to thank Professor König for providing a personal copy of reference [7].