Theorems · Theorem · probability
MeasureTheory.Submartingale.exists_ae_tendsto_of_bdd
∀ {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0}
{f : ℕ → Ω → ℝ} {R : NNReal} [MeasureTheory.IsFiniteMeasure μ],
MeasureTheory.Submartingale f ℱ μ →
(∀ (n : ℕ), MeasureTheory.eLpNorm (f n) 1 μ ≤ ↑R) →
∀ᵐ (ω : Ω) ∂μ, ∃ c, Filter.Tendsto (fun n => f n ω) Filter.atTop (nhds c)An L¹-bounded submartingale converges almost everywhere.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 316 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites30
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 · cited by 9,879
- Top.topproof · cited by 9,680
- nhdsstatement · cited by 5,554
- NNRealstatement and proof · cited by 4,310
- Filter.Tendstostatement · cited by 3,814
- Filter.Eventuallystatement · cited by 3,134
- Filter.atTopstatement and proof · cited by 2,405
- MeasureTheory.aestatement · cited by 2,352
- Filter.univ_mem'proof · cited by 1,672
Cited by2
Results whose statement or proof uses this declaration.
- MeasureTheory.Submartingale.exists_ae_trim_tendsto_of_bddproof · cited by 1
- MeasureTheory.Submartingale.exists_tendsto_of_abs_bddAbove_auxproof · cited by 1