Discrete stopping times in the lattice of continuous functions

A functional calculus for an order complete vector lattice \(\mathcal{E}\) was developed by Grobler in 2014 using the Daniell integral. We show that if one represents the universal completion of \(\mathcal{E}\) as \(C^\infty(K)\), 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.