Theorems · Theorem · probability
MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part
∀ {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {f : ℕ → Ω → ℝ}
{ℱ : MeasureTheory.Filtration ℕ m0} [MeasureTheory.IsFiniteMeasure μ] (a b : ℝ),
MeasureTheory.Submartingale f ℱ μ →
ENNReal.ofReal (b - a) * ∫⁻ (ω : Ω), MeasureTheory.upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μA variant of Doob's upcrossing estimate obtained by taking the supremum on both sides.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 313 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
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Cites46
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
- ENNRealstatement and proof · cited by 9,879
- LE.le.transproof · cited by 3,151
- iSupstatement and proof · cited by 2,415
- LT.lt.leproof · cited by 2,189
- MeasureTheory.integralproof · cited by 1,779
- Filter.univ_mem'proof · cited by 1,672
- MulZeroClass.zero_mulproof · cited by 1,625
- MeasureTheory.Integrableproof · cited by 1,367
- MeasureTheory.lintegralstatement and proof · cited by 1,152
Cited by1
Results whose statement or proof uses this declaration.
- MeasureTheory.Submartingale.upcrossings_ae_lt_top'proof · cited by 1