Theorems · Theorem · probability

MeasureTheory.tendsto_sum_indicator_atTop_iff

∀ {Ω : Type u_2} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0}
  {f : ℕ → Ω → ℝ} {R : NNReal} [MeasureTheory.IsFiniteMeasure μ],
  (∀ᵐ (ω : Ω) ∂μ, ∀ (n : ℕ), f n ω ≤ f (n + 1) ω) →
    MeasureTheory.StronglyAdapted ℱ f →
      (∀ (n : ℕ), MeasureTheory.Integrable (f n) μ) →
        (∀ᵐ (ω : Ω) ∂μ, ∀ (n : ℕ), |f (n + 1) ω - f n ω| ≤ ↑R) →
          ∀ᵐ (ω : Ω) ∂μ,
            Filter.Tendsto (fun n => f n ω) Filter.atTop Filter.atTop ↔
              Filter.Tendsto (fun n => MeasureTheory.predictablePart f ℱ μ n ω) Filter.atTop Filter.atTop

An a.e. monotone strongly adapted process f with uniformly bounded differences converges to +∞ if and only if its predictable part also converges to +∞.

Defined in

Mathlib.Probability.Martingale.BorelCantelli

Cited by

1 results in Mathlib

Foundations

Depth 322 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasureTheory.IsFiniteMeasure