exists_ae_tendsto_of_bdd — Mathlib

English

If a<b and f frequently lies below a and above b, then upcrossings a b f ω = ∞.

Русский

Если f часто принимает значения ниже a и выше b, то upcrossings(a,b,f,ω) = ∞.

LaTeX

$$$\\text{If } a

Lean4

/-- An L¹-bounded submartingale converges almost everywhere. -/ theorem exists_ae_tendsto_of_bdd [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) (hbdd : ∀ n, eLpNorm (f n) 1 μ ≤ R) : ∀ᵐ ω ∂μ, ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) := by filter_upwards [hf.upcrossings_ae_lt_top hbdd, ae_bdd_liminf_atTop_of_eLpNorm_bdd one_ne_zero (fun n => (hf.stronglyMeasurable n).measurable.mono (ℱ.le n) le_rfl) hbdd] with ω h₁ h₂ exact tendsto_of_uncrossing_lt_top h₂ h₁

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