Is Stable has Integrable Sup
ProbabilityTheory.isStable_hasIntegrableSup
Plain-language statement
The class of processes with integrable supremum is stable.
Exact Lean statement
lemma isStable_hasIntegrableSup [SecondCountableTopology ι] :
IsStable 𝓕 (HasIntegrableSup (E := E) · P)Formal artifact
Lean source
lemma isStable_hasIntegrableSup [SecondCountableTopology ι] : IsStable 𝓕 (HasIntegrableSup (E := E) · P) := by refine fun X hX τ hτ ↦ ⟨isStable_hasStronglyMeasurableSupProcess X hX.1 τ hτ, ?_⟩ refine fun t ↦ ⟨(isStable_hasStronglyMeasurableSupProcess X hX.1 τ hτ).comp_measurable (measurable_const.prodMk measurable_id) |>.aestronglyMeasurable, ?_⟩ have h_bound := (hX.2 t).hasFiniteIntegral simp_rw [hasFiniteIntegral_def, enorm_eq_self] at h_bound ⊢ refine lt_of_le_of_lt (lintegral_mono fun ω ↦ ?_) h_bound apply iSup₂_le intro s hs simp only [stoppedProcess, Set.indicator_apply, Set.mem_ofPred_eq] split_ifs with h_bot · refine le_iSup₂_of_le (min ↑s (τ ω)).untopA ?_ le_rfl · rw [WithTop.untopA_le_iff] · exact le_trans (min_le_left _ _) (WithTop.coe_le_coe.mpr hs) · exact ne_top_of_le_ne_top WithTop.coe_ne_top (min_le_left _ _) · simp- Project
- Brownian motion
- License
- Apache-2.0
- Commit
- 5077304f5e73
- Source
- BrownianMotion/StochasticIntegral/ClassD.lean:479-495
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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.