Abstract
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.
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Acknowledgements
I would like to thank my advisor, Prof. Vladimir Troitsky, for his patient guidance, from the construction of the general idea and the argument to the details of writing this paper.
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Polavarapu, A.R. Discrete stopping times in the lattice of continuous functions. Positivity 28, 25 (2024). https://doi.org/10.1007/s11117-024-01044-5
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DOI: https://doi.org/10.1007/s11117-024-01044-5