Theorems · Theorem · probability
MeasureTheory.Supermartingale.le_zero_of_predictable
∀ {Ω : Type u_1} {E : Type u_2} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : NormedAddCommGroup E]
[inst_1 : NormedSpace ℝ E] {𝒢 : MeasureTheory.Filtration ℕ m0} [inst_2 : Preorder E]
[MeasureTheory.SigmaFiniteFiltration μ 𝒢] {f : ℕ → Ω → E},
MeasureTheory.Supermartingale f 𝒢 μ → (MeasureTheory.StronglyAdapted 𝒢 fun n => f (n + 1)) → ∀ (n : ℕ), f n ≤ᵐ[μ] f 0A predictable supermartingale is a.e. less than or equal to its initial state.
- Defined in
- Mathlib.Probability.Martingale.Basic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 297 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites21
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- NormedAddCommGroupstatement and proof · cited by 15,752
- MeasurableSpacestatement and proof · cited by 13,106
- NormedSpacestatement and proof · cited by 12,499
- MeasureTheory.Measurestatement and proof · cited by 10,939
- Preorderstatement and proof · cited by 7,952
- MeasureTheory.aestatement and proof · cited by 2,352
- MeasureTheory.Filtrationstatement and proof · cited by 425
- Filter.EventuallyEq.symmproof · cited by 408
- Filter.EventuallyLEstatement and proof · cited by 383
- MeasureTheory.StronglyAdaptedstatement and proof · cited by 81
- Filter.Germ.ofFunproof · cited by 73
Cited by1
Results whose statement or proof uses this declaration.
- MeasureTheory.Supermartingale.le_zero_of_predictable'proof · cited by 0