Theorems · Theorem · measure theory

MeasureTheory.tendsto_setToFun_filter_of_dominated_convergence

∀ {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) {ι : Type u_7} {l : Filter ι}
  [l.IsCountablyGenerated] {fs : ι → α → E} {f : α → E} (bound : α → ℝ),
  (∀ᶠ (n : ι) in l, MeasureTheory.AEStronglyMeasurable (fs n) μ) →
    (∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a) →
      MeasureTheory.Integrable bound μ →
        (∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun n => fs n a) l (nhds (f a))) →
          Filter.Tendsto (fun n => MeasureTheory.setToFun μ T hT (fs n)) l (nhds (MeasureTheory.setToFun μ T hT f))

Lebesgue dominated convergence theorem for filters with a countable basis

Defined in

Mathlib.MeasureTheory.Integral.SetToL1

Cited by

6 results in Mathlib

Foundations

Depth 243 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceFilter.IsCountablyGenerated