Measure Theory Submartingale integrable On const tau Mesh lt top
MeasureTheory.Submartingale.integrableOn_const_tauMesh_lt_top
Plain-language statement
The constant c is integrable on the event where τₙ(c) hits before the top element.
Exact Lean statement
lemma MeasureTheory.Submartingale.integrableOn_const_tauMesh_lt_top {ι Ω : Type*}
[TopologicalSpace ι] [SecondCountableTopology ι] [LinearOrder ι] [OrderBot ι] [OrderTop ι]
{mΩ : MeasurableSpace Ω} {P : Measure Ω} {S : ι → Ω → ℝ} {𝓕 : Filtration ι mΩ}
(hs : Submartingale S 𝓕 P) (n : ℕ) {c : ℝ} (hc : 0 ≤ c) :
IntegrableOn (fun _ : Ω => c) {ω | tauMesh S 𝓕 P n c ω < (⊤ : mesh ι n)} PFormal artifact
Lean source
lemma MeasureTheory.Submartingale.integrableOn_const_tauMesh_lt_top {ι Ω : Type*} [TopologicalSpace ι] [SecondCountableTopology ι] [LinearOrder ι] [OrderBot ι] [OrderTop ι] {mΩ : MeasurableSpace Ω} {P : Measure Ω} {S : ι → Ω → ℝ} {𝓕 : Filtration ι mΩ} (hs : Submartingale S 𝓕 P) (n : ℕ) {c : ℝ} (hc : 0 ≤ c) : IntegrableOn (fun _ : Ω => c) {ω | tauMesh S 𝓕 P n c ω < (⊤ : mesh ι n)} P := by by_cases! hc0 : c = 0 · simp [hc0] · refine integrableOn_const (LT.lt.ne ?_) rw [measure_congr (hs.tauMesh_lt_top_eq_lt_predictableSeqTop n hc)] exact (integrable_predictableSeqTop S 𝓕 P n).measure_gt_lt_top (lt_of_le_of_ne hc hc0.symm)- Project
- Brownian motion
- License
- Apache-2.0
- Commit
- 5077304f5e73
- Source
- BrownianMotion/StochasticIntegral/DoobMeyer.lean:426-435
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Related declarations
Continuous Within At Iio indicator Ioc
continuousWithinAt_Iio_indicator_Ioc
Plain-language statement
The indicator of a half-open interval Ioc a b with constant value c is left-continuous: when approached from the left it is eventually constant, so it is continuous within Iio t at t for every t.
Source project: Brownian motion
Person-level attribution pending.
Dense comap val nhds Within Iio ne Bot
Dense.comap_val_nhdsWithin_Iio_neBot
Plain-language statement
This is the dual of Dense.comap_val_nhdsWithin_Ioi_neBot.
Source project: Brownian motion
Person-level attribution pending.
Dense comap val nhds Within Ioi ne Bot
Dense.comap_val_nhdsWithin_Ioi_neBot
Project documentation
This is an auxillary lemma used to prove Dense.monotone_of_isRightContinuous. It is saying that if D is a dense set and a, b are two points such that a < b, then the comap of 𝓝[Set.Ioi a] a under the inclusion D → α is nontrivial. Note that a < b is necessary as this is clearly not true if a is a top element.
Source project: Brownian motion
Person-level attribution pending.