Theorems · Theorem · measure theory

MeasureTheory.tendsto_lintegral_norm_of_dominated_convergence

∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : NormedAddCommGroup β]
  {F : ℕ → α → β} {f : α → β} {bound : α → ℝ},
  (∀ (n : ℕ), MeasureTheory.AEStronglyMeasurable (F n) μ) →
    MeasureTheory.HasFiniteIntegral bound μ →
      (∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a) →
        (∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun n => F n a) Filter.atTop (nhds (f a))) →
          Filter.Tendsto (fun n => ∫⁻ (a : α), ENNReal.ofReal ‖F n a - f a‖ ∂μ) Filter.atTop (nhds 0)

Defined in

Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral

Cited by

2 results in Mathlib

Foundations

Depth 213 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroup