Theorems · Theorem · probability

MeasureTheory.smul_le_stoppedValue_hittingBtwn

∀ {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {𝒢 : MeasureTheory.Filtration ℕ m0}
  {f : ℕ → Ω → ℝ} [MeasureTheory.IsFiniteMeasure μ],
  MeasureTheory.Submartingale f 𝒢 μ →
    ∀ {ε : NNReal} (n : ℕ),
      ε • μ {ω | ↑ε ≤ (Finset.range (n + 1)).sup' ⋯ fun k => f k ω} ≤
        ENNReal.ofReal
          (∫ (ω : Ω) in {ω | ↑ε ≤ (Finset.range (n + 1)).sup' ⋯ fun k => f k ω},
            MeasureTheory.stoppedValue f (fun ω => ↑(MeasureTheory.hittingBtwn f {y | ↑ε ≤ y} 0 n ω)) ω ∂μ)

Defined in

Mathlib.Probability.Martingale.OptionalStopping

Cited by

1 results in Mathlib

Foundations

Depth 297 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasureTheory.IsFiniteMeasure