Theorems · Theorem · measure theory

MeasureTheory.continuousOn_setToFun_of_dominated

∀ {α : Type u_1} {E : Type u_2} {F : Type u_3} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E]
  [inst_2 : NormedAddCommGroup F] [inst_3 : NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α}
  {T : Set α → E →L[ℝ] F} {C : ℝ} {X : Type u_7} [inst_4 : TopologicalSpace X] [FirstCountableTopology X]
  (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) {fs : X → α → E} {bound : α → ℝ} {s : Set X},
  (∀ x ∈ s, MeasureTheory.AEStronglyMeasurable (fs x) μ) →
    (∀ x ∈ s, ∀ᵐ (a : α) ∂μ, ‖fs x a‖ ≤ bound a) →
      MeasureTheory.Integrable bound μ →
        (∀ᵐ (a : α) ∂μ, ContinuousOn (fun x => fs x a) s) →
          ContinuousOn (fun x => MeasureTheory.setToFun μ T hT (fs x)) s

Defined in

Mathlib.MeasureTheory.Integral.SetToL1

Cited by

2 results in Mathlib

Foundations

Depth 245 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceFirstCountableTopology