Theorems · Theorem · measure theory

MeasureTheory.VectorMeasure.Integrable.tendsto_setIntegral_nhds_zero

∀ {X : Type u_2} {E : Type u_3} {F : Type u_4} {G : Type u_5} {mX : MeasurableSpace X} [inst : NormedAddCommGroup E]
  [inst_1 : NormedAddCommGroup F] [inst_2 : NormedAddCommGroup G] {μ : MeasureTheory.VectorMeasure X F} {f : X → E}
  [inst_3 : NormedSpace ℝ E] [inst_4 : NormedSpace ℝ F] [inst_5 : NormedSpace ℝ G] {B : E →L[ℝ] F →L[ℝ] G}
  {ι : Type u_7},
  μ.Integrable f →
    ∀ {l : Filter ι} {s : ι → Set X},
      Filter.Tendsto (⇑μ.variation ∘ s) l (nhds 0) → Filter.Tendsto (fun i => ∫ᵛ (x : X) in s i, f x ∂[B; μ]) l (nhds 0)

If f is integrable, then ∫ᵛ x in s, f x ∂[B; μ] is absolutely continuous in s: it tends to zero as μ.variation s tends to zero.

Defined in

Mathlib.MeasureTheory.VectorMeasure.SetIntegral

Cited by

0 results in Mathlib

Foundations

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

Assumes

NormedAddCommGroupNormedAddCommGroupNormedAddCommGroupNormedSpaceNormedSpaceNormedSpace