Theorems · Theorem · probability
MeasureTheory.Martingale.bddAbove_range_iff_bddBelow_range
∀ {Ω : Type u_2} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0}
{f : ℕ → Ω → ℝ} {R : NNReal} [MeasureTheory.IsFiniteMeasure μ],
MeasureTheory.Martingale f ℱ μ →
(∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R) →
∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => f n ω) ↔ BddBelow (Set.range fun n => f n ω)- Cited by
- 2 results in Mathlib
- Foundations
- Depth 320 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites31
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
- MeasurableSpacestatement and proof · cited by 13,106
- MeasureTheory.Measurestatement and proof · cited by 10,939
- Filterproof · cited by 8,121
- nhdsproof · cited by 5,554
- Set.rangestatement and proof · cited by 4,705
- NNRealstatement and proof · cited by 4,310
- Filter.Tendstoproof · cited by 3,814
- Filter.Eventuallystatement and proof · cited by 3,134
- Filter.atTopproof · cited by 2,405
- MeasureTheory.aestatement and proof · cited by 2,352
- absstatement and proof · cited by 1,814
Cited by2
Results whose statement or proof uses this declaration.
- MeasureTheory.Martingale.ae_not_tendsto_atTop_atBotproof · cited by 1
- MeasureTheory.Martingale.ae_not_tendsto_atTop_atTopproof · cited by 1