Doob's upcrossing inequality for Countable and NNReal - #361
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…nset.orderBotOfBotMem helper Co-authored-by: Copilot <copilot@github.com>
Co-authored-by: Copilot <copilot@github.com>
…ENat theorems Co-authored-by: Copilot <copilot@github.com>
Co-authored-by: Copilot <copilot@github.com>
…proof outline Co-authored-by: Copilot <copilot@github.com>
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First on Countable index sets - via finite ones and thanks to the existing code for Nat - and then in continuous case, the Doob's upcrossing inequality has been formalized and proved, short of measurability in continuous case, which requires the debut theorem. With this latter restriction, the number of upcrossings is a.s. finite, which is important for Martingale Convergence Theorem.
As the recursive-hitting-time definition of upcrossing number is unsuitable for NNRat, we use an equivalent one for the sake of proving the Countable case: the one found in the Kallenberg's "Foundations of Modern Probability" (2021), Theorem 9.18.