Theorems · Theorem · probability

MeasureTheory.submartingale_of_setIntegral_le

∀ {Ω : Type u_1} {ι : Type u_3} [inst : Preorder ι] {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω}
  {ℱ : MeasureTheory.Filtration ι m0} [MeasureTheory.SigmaFiniteFiltration μ ℱ] {f : ι → Ω → ℝ},
  MeasureTheory.StronglyAdapted ℱ f →
    (∀ (i : ι), MeasureTheory.Integrable (f i) μ) →
      (∀ (i j : ι), i ≤ j → ∀ (s : Set Ω), MeasurableSet s → ∫ (ω : Ω) in s, f i ω ∂μ ≤ ∫ (ω : Ω) in s, f j ω ∂μ) →
        MeasureTheory.Submartingale f ℱ μ

Defined in

Mathlib.Probability.Martingale.Basic

Cited by

2 results in Mathlib

Foundations

Depth 298 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

PreorderMeasureTheory.SigmaFiniteFiltration