Theorems · Theorem · measure theory

MeasureTheory.tendsto_setToFun_approxOn_of_measurable

∀ {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} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) [inst_4 : MeasurableSpace E]
  [inst_5 : BorelSpace E] {f : α → E} {s : Set E} [inst_6 : TopologicalSpace.SeparableSpace ↑s],
  MeasureTheory.Integrable f μ →
    ∀ (hfm : Measurable f),
      (∀ᵐ (x : α) ∂μ, f x ∈ closure s) →
        ∀ {y₀ : E} (h₀ : y₀ ∈ s),
          MeasureTheory.Integrable (fun x => y₀) μ →
            Filter.Tendsto (fun n => MeasureTheory.setToFun μ T hT ⇑(MeasureTheory.SimpleFunc.approxOn f hfm s y₀ h₀ n))
              Filter.atTop (nhds (MeasureTheory.setToFun μ T hT f))

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

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceMeasurableSpaceBorelSpaceTopologicalSpace.SeparableSpace