Theorems · Theorem · measure theory

MeasureTheory.tendsto_setToFun_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) {fs : ℕ → α → E} {f : α → E}
  (bound : α → ℝ),
  (∀ (n : ℕ), MeasureTheory.AEStronglyMeasurable (fs n) μ) →
    MeasureTheory.Integrable bound μ →
      (∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a) →
        (∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun n => fs n a) Filter.atTop (nhds (f a))) →
          Filter.Tendsto (fun n => MeasureTheory.setToFun μ T hT (fs n)) Filter.atTop
            (nhds (MeasureTheory.setToFun μ T hT f))

Lebesgue dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies the convergence of their image by setToFun. We could weaken the condition bound_integrable to require HasFiniteIntegral bound μ instead (i.e. not requiring that bound is measurable), but in all applications proving integrability is easier.

Defined in

Mathlib.MeasureTheory.Integral.SetToL1

Cited by

5 results in Mathlib

Foundations

Depth 242 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace