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[Merged by Bors] - feat(probability/upcrossing): Doob's upcrossing estimate #14933
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…rover-community/mathlib into JasonKYi/disc_stoch_int
sgouezel
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Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr>
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…rty of the conditional expectation (#15274) We prove this result: ```lean lemma condexp_strongly_measurable_mul {f g : α → ℝ} (hf : strongly_measurable[m] f) (hfg : integrable (f * g) μ) (hg : integrable g μ) : μ[f * g | m] =ᵐ[μ] f * μ[g | m] := ``` This could be extended beyond multiplication, to any bounded bilinear map, but we leave this to a future PR. For now we only prove the real multiplication case, which is needed for #14909 and #14933.
sgouezel
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src/probability/upcrossing.lean
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| variables {α ι : Type*} {m0 : measurable_space α} {μ : measure α} |
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Is there a reason you call the space α and not Ω, and your measure μ and not ℙ (and the points in the space x and not ω)? I know it doesn't make any difference, but it can help your probability-inclined reader.
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I've changed α to Ω and x to ω. I've not switched μ to ℙ since this file had opened probability.notation and it seems the notation for ℙ being the volume doesn't allow me to declare it as a measure (I can't use measure_space here since I want to specify the measurable_space).
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This PR proves Doob's upcrossing estimate which is central for the martingale convergence theorems. Co-authored-by: RemyDegenne <Remydegenne@gmail.com>
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This PR proves Doob's upcrossing estimate which is central for the martingale convergence theorems.
mul_indicator_finset_bUnion#14919