Theorems · Theorem · measure theory

MeasureTheory.tendsto_integral_norm_approxOn_sub

∀ {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_7} [inst : NormedAddCommGroup E]
  [inst_1 : MeasurableSpace E] [inst_2 : BorelSpace E] {f : α → E} (fmeas : Measurable f),
  MeasureTheory.Integrable f μ →
    ∀ [inst_3 : TopologicalSpace.SeparableSpace ↑(Set.range f ∪ {0})],
      Filter.Tendsto
        (fun n => ∫ (x : α), ‖(MeasureTheory.SimpleFunc.approxOn f fmeas (Set.range f ∪ {0}) 0 ⋯ n) x - f x‖ ∂μ)
        Filter.atTop (nhds 0)

Defined in

Mathlib.MeasureTheory.Integral.Bochner.Basic

Cited by

1 results in Mathlib

Foundations

Depth 255 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupMeasurableSpaceBorelSpaceTopologicalSpace.SeparableSpace