Discrete stopping times in the lattice of continuous functions

A functional calculus for an order complete vector lattice \({\mathcal {E}}\) was developed by Grobler (Indag Math (NS) 25(2):275–295, 2014) using the Daniell integral. We show that if one represents the universal completion of \({\mathcal {E}}\) as \(C^\infty (K)\), where K is an extremally disconnected compact Hausdorff topological space, then the Daniell functional calculus for continuous functions is exactly the pointwise composition of functions in \(C^\infty (K)\). This representation allows an easy deduction of the various properties of the functional calculus. Afterwards, we study discrete stopping times and stopped processes in \(C^\infty (K)\). We obtain a representation that is analogous to what is expected in probability theory.