Theorems · Theorem · measure theory

MeasureTheory.VectorMeasure.continuousWithinAt_of_dominated

∀ {X : Type u_2} {E : Type u_4} {F : Type u_5} {G : Type u_6} {mX : MeasurableSpace X} [inst : NormedAddCommGroup E]
  [inst_1 : NormedSpace ℝ E] [inst_2 : NormedAddCommGroup F] [inst_3 : NormedSpace ℝ F] [inst_4 : NormedAddCommGroup G]
  [inst_5 : NormedSpace ℝ G] {μ : MeasureTheory.VectorMeasure X F} {B : E →L[ℝ] F →L[ℝ] G} {Y : Type u_8}
  [inst_6 : TopologicalSpace Y] [FirstCountableTopology Y] {F_1 : Y → X → E} {x₀ : Y} {bound : X → ℝ} {s : Set Y},
  (∀ᶠ (x : Y) in nhdsWithin x₀ s, MeasureTheory.AEStronglyMeasurable (F_1 x) μ.variation) →
    (∀ᶠ (x : Y) in nhdsWithin x₀ s, ∀ᵐ (a : X) ∂μ.variation, ‖F_1 x a‖ ≤ bound a) →
      MeasureTheory.Integrable bound μ.variation →
        (∀ᵐ (a : X) ∂μ.variation, ContinuousWithinAt (fun x => F_1 x a) s x₀) →
          ContinuousWithinAt (fun x => ∫ᵛ (a : X), F_1 x a ∂[B; μ]) s x₀

Defined in

Mathlib.MeasureTheory.VectorMeasure.Integral

Cited by

0 results in Mathlib

Foundations

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

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceFirstCountableTopology