Measure Theory Submartingale monotone predictable Part ae
MeasureTheory.Submartingale.monotone_predictablePart_ae
Plain-language statement
For a submartingale indexed by a countable type, the predictable part is monotone a.e.
Exact Lean statement
lemma MeasureTheory.Submartingale.monotone_predictablePart_ae {ι Ω E : Type*} [LinearOrder ι]
[LocallyFiniteOrderBot ι] [SuccOrder ι] [Countable ι] {mΩ : MeasurableSpace Ω} {P : Measure Ω}
[NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] [PartialOrder E]
[IsOrderedAddMonoid E] {S : ι → Ω → E} {𝓕 : Filtration ι mΩ} (hs : Submartingale S 𝓕 P) :
∀ᵐ ω ∂P, Monotone (_root_.predictablePart S 𝓕 P · ω)Formal artifact
Lean source
lemma MeasureTheory.Submartingale.monotone_predictablePart_ae {ι Ω E : Type*} [LinearOrder ι] [LocallyFiniteOrderBot ι] [SuccOrder ι] [Countable ι] {mΩ : MeasurableSpace Ω} {P : Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] [PartialOrder E] [IsOrderedAddMonoid E] {S : ι → Ω → E} {𝓕 : Filtration ι mΩ} (hs : Submartingale S 𝓕 P) : ∀ᵐ ω ∂P, Monotone (_root_.predictablePart S 𝓕 P · ω) := by have hnonneg : ∀ᵐ ω ∂P, ∀ i : ι, 0 ≤ P[S (succ i) - S i | 𝓕 i] ω := ae_all_iff.2 fun i ↦ hs.condExp_sub_nonneg (le_succ i) filter_upwards [hnonneg] with ω hω a b hab simp only [_root_.predictablePart, Finset.sum_apply] exact Finset.sum_le_sum_of_subset_of_nonneg (Finset.Iio_subset_Iio hab) fun i _ _ ↦ hω i- Project
- Brownian motion
- License
- Apache-2.0
- Commit
- 5077304f5e73
- Source
- BrownianMotion/StochasticIntegral/DoobMeyer.lean:156-165
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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.