feat: Doob maximal inequality for lower semicontinuous submartingales (separable index) - #498
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Add `BrownianMotion/Auxiliary/SeparableSpace.lean`: a separable linearly ordered topological space is hereditarily separable (`Set.separableSpace`), with the supporting order-convex neighbourhood basis (`exists_isOpen_ordConnected_mem_subset`, `isTopologicalBasis_isOpen_ordConnected`) and the countability of the isolated points of a subspace (`countable_setOf_isolated_subtype`). This packages the mathlib PR leanprover-community/mathlib4#41918 as a project auxiliary file; it is a shared dependency of the variation-process and Doob-maximal-inequality developments. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Prove Doob's maximal inequality for submartingales whose paths are lower semicontinuous, indexed by a separable linearly ordered topological space, weakening the `SecondCountableTopology` assumption of the right-continuous version (PR RemyDegenne#257) to `SeparableSpace`. This covers continuous paths in particular. Uses Mathlib's `measurable_iSup_of_lowerSemicontinuous`: a lower semicontinuous path pulls `Set.Ioi` back to an open set, so a countable dense set of times already computes the running supremum; separability of `Set.Iic n` comes from hereditary separability of separable LOTS. Main results (`section Separable`): * `maximal_ineq_ennreal_of_lowerSemicontinuous` * `maximal_ineq_nonneg_of_lowerSemicontinuous` * `maximal_ineq_of_lowerSemicontinuous` * `maximal_ineq_norm_of_continuous` The topology-free plumbing shared with the right-continuous version is factored into `Submartingale.iSup_ofReal_ne_top_of_measurable` and `maximal_ineq_nonneg_of_ne_top`, and the existing right-continuous proofs are refactored to reuse them. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
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Following up on #257 (Doob's maximal inequality for right-continuous sub-martingales on a second-countable index set), this PR proves the analogous statement for submartingales whose paths are lower semicontinuous, on a separable index set. This covers continuous paths in particular, and mirrors how
VariationProcess.leanhandles the second-countable and separable regimes for the measurability of the variation process in #494.Why a separate version
For general linearly ordered topological spaces,
SeparableSpaceis strictly weaker thanSecondCountableTopology(a separable LOTS can have uncountably many right-isolated points), so neither result implies the other:SecondCountableTopology;SeparableSpacesufficesThe measurability of the running supremum is exactly Mathlib's
measurable_iSup_of_lowerSemicontinuous; separability ofSet.Iic ncomes from hereditary separability of separable LOTS (Set.separableSpace, added in #499).Main results (new
section SeparableinDoobLp.lean)maximal_ineq_ennreal_of_lowerSemicontinuousmaximal_ineq_nonneg_of_lowerSemicontinuous,maximal_ineq_of_lowerSemicontinuousmaximal_ineq_norm_of_continuous(continuous martingale, norm form)Depends on #499
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