Theorems · Theorem · probability

MeasureTheory.submartingale_of_setIntegral_le_succ

∀ {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {𝒢 : MeasureTheory.Filtration ℕ m0}
  [MeasureTheory.IsFiniteMeasure μ] {f : ℕ → Ω → ℝ},
  MeasureTheory.StronglyAdapted 𝒢 f →
    (∀ (i : ℕ), MeasureTheory.Integrable (f i) μ) →
      (∀ (i : ℕ) (s : Set Ω), MeasurableSet s → ∫ (ω : Ω) in s, f i ω ∂μ ≤ ∫ (ω : Ω) in s, f (i + 1) ω ∂μ) →
        MeasureTheory.Submartingale f 𝒢 μ

Defined in: Mathlib.Probability.Martingale.Basic
Cited by: 1 results in Mathlib
Foundations: Depth 299 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: MeasureTheory.IsFiniteMeasure

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Cites17

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Setstatement and proof · cited by 53,352
Realstatement and proof · cited by 25,697
MeasurableSpacestatement and proof · cited by 13,106
MeasureTheory.Measurestatement and proof · cited by 10,939
LE.le.transproof · cited by 3,151
MeasurableSetstatement and proof · cited by 3,075
le_reflproof · cited by 2,061
MeasureTheory.integralstatement and proof · cited by 1,779
MeasureTheory.Measure.restrictstatement and proof · cited by 1,646
MeasureTheory.Integrablestatement and proof · cited by 1,367
MeasureTheory.IsFiniteMeasurestatement and proof · cited by 1,078
MeasureTheory.Filtrationstatement and proof · cited by 425

Cited by1

Results whose statement or proof uses this declaration.

MeasureTheory.supermartingale_of_setIntegral_succ_leproof · cited by 0