Theorems · Theorem · measure theory

MeasureTheory.hasSum_integral_of_summable_integral_norm

∀ {α : Type u_1} {E : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {m : MeasurableSpace α}
  {μ : MeasureTheory.Measure α} {ι : Type u_4} [Countable ι] {F : ι → α → E},
  (∀ (i : ι), MeasureTheory.Integrable (F i) μ) →
    (Summable fun i => ∫ (a : α), ‖F i a‖ ∂μ) → HasSum (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), ∑' (i : ι), F i a ∂μ)

Defined in

Mathlib.MeasureTheory.Integral.DominatedConvergence

Cited by

2 results in Mathlib

Foundations

Depth 255 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceCountable