Theorems · Theorem · measure theory

MeasureTheory.continuousWithinAt_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} {x₀ : X} {bound : α → ℝ} {s : Set X},
  (∀ᶠ (x : X) in nhdsWithin x₀ s, MeasureTheory.AEStronglyMeasurable (fs x) μ) →
    (∀ᶠ (x : X) in nhdsWithin x₀ s, ∀ᵐ (a : α) ∂μ, ‖fs x a‖ ≤ bound a) →
      MeasureTheory.Integrable bound μ →
        (∀ᵐ (a : α) ∂μ, ContinuousWithinAt (fun x => fs x a) s x₀) →
          ContinuousWithinAt (fun x => MeasureTheory.setToFun μ T hT (fs x)) s x₀

Defined in

Mathlib.MeasureTheory.Integral.SetToL1

Cited by

3 results in Mathlib

Foundations

Depth 244 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceFirstCountableTopology