Theorems · Theorem · functional analysis

ContinuousLinearMap.integrable_of_bilin_of_bdd_left

∀ {α : Type u_1} {𝕜 : Type u_2} {E : Type u_3} {F : Type u_4} {G : Type u_5} {m : MeasurableSpace α}
  {μ : MeasureTheory.Measure α} [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedAddCommGroup F] [inst_3 : NormedAddCommGroup G] [inst_4 : NormedSpace 𝕜 E] [inst_5 : NormedSpace 𝕜 F]
  [inst_6 : NormedSpace 𝕜 G] (B : E →L[𝕜] F →L[𝕜] G) {f : α → E} {g : α → F} (C : ℝ),
  MeasureTheory.AEStronglyMeasurable f μ →
    (∀ᵐ (a : α) ∂μ, ‖f a‖ ≤ C) → MeasureTheory.Integrable g μ → MeasureTheory.Integrable (fun x => (B (f x)) (g x)) μ

Defined in

Mathlib.MeasureTheory.Function.Holder

Cited by

3 results in Mathlib

Foundations

Depth 216 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedAddCommGroupNormedAddCommGroupNormedSpaceNormedSpaceNormedSpace