Theorems · Theorem · measure theory

MeasureTheory.hasFiniteIntegral_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},
  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))) → MeasureTheory.HasFiniteIntegral f μ

Defined in

Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral

Cited by

1 results in Mathlib

Foundations

Depth 180 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroup