Countable not cadlag Modif ae eq
ProbabilityTheory.countable_not_cadlagModif_ae_eq
Plain-language statement
The set of points where the cadlag modification of a real quasimartingale along a countable dense set T disagrees with X is countable.
Exact Lean statement
lemma countable_not_cadlagModif_ae_eq [SecondCountableTopology ι] [IsFiniteMeasure μ]
(hX : IsRealQuasimartingale 𝓕 X μ) :
{t | ¬ cadlagModif X t =ᵐ[μ] X t}.CountableFormal artifact
Lean source
lemma countable_not_cadlagModif_ae_eq [SecondCountableTopology ι] [IsFiniteMeasure μ] (hX : IsRealQuasimartingale 𝓕 X μ) : {t | ¬ cadlagModif X t =ᵐ[μ] X t}.Countable := by refine (countable_not_rightLimWithin_ae_eq hX countable_denseCountable dense_denseCountable).mono fun t ht ↦ ?_ simp only [Set.mem_ofPred_eq] at ht ⊢ refine fun h_contra ↦ ht ?_ filter_upwards [cadlagModif_ae_eq_rightContModif hX, rightContModif_ae_eq_of_rightLimWithin_ae_eq hX h_contra] with ω hω hωc grind- Project
- Brownian motion
- License
- Apache-2.0
- Commit
- 5077304f5e73
- Source
- BrownianMotion/StochasticIntegral/Quasimartingale/CadlagModification.lean:1097-1106
Reuse this declaration
Bring the exact result into your workflow
The import identifies the source module. Your project still needs the pinned package dependency shown on this page.
What this badge means
This completion status comes from the project or community source. It has not yet been represented here as an independent rebuild and axiom audit.
Continue in this project
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.