Theorems · Theorem · probability

MeasureTheory.Submartingale.setIntegral_le

∀ {Ω : Type u_1} {E : Type u_2} {ι : Type u_3} [inst : Preorder ι] {m0 : MeasurableSpace Ω}
  {μ : MeasureTheory.Measure Ω} [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace ℝ E]
  {ℱ : MeasureTheory.Filtration ι m0} [CompleteSpace E] [inst_4 : PartialOrder E] [IsOrderedAddMonoid E]
  [IsOrderedModule ℝ E] [ClosedIciTopology E] [MeasureTheory.SigmaFiniteFiltration μ ℱ] {f : ι → Ω → E},
  MeasureTheory.Submartingale f ℱ 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

The converse of this lemma is MeasureTheory.submartingale_of_setIntegral_le.

Defined in

Mathlib.Probability.Martingale.Basic

Cited by

3 results in Mathlib

Foundations

Depth 300 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

PreorderNormedAddCommGroupNormedSpaceCompleteSpacePartialOrderIsOrderedAddMonoidIsOrderedModuleClosedIciTopologyMeasureTheory.SigmaFiniteFiltration