Theorems · Theorem · real analysis

BoxIntegral.Integrable.tendsto_integralSum_sum_integral

∀ {ι : Type u} {E : Type v} {F : Type w} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E]
  [inst_2 : NormedAddCommGroup F] [inst_3 : NormedSpace ℝ F] {I : BoxIntegral.Box ι} [inst_4 : Fintype ι]
  {l : BoxIntegral.IntegrationParams} {f : (ι → ℝ) → E} {vol : BoxIntegral.BoxAdditiveMap ι (E →L[ℝ] F) ⊤}
  [CompleteSpace F],
  BoxIntegral.Integrable I l f vol →
    ∀ (π₀ : BoxIntegral.Prepartition I),
      Filter.Tendsto (BoxIntegral.integralSum f vol) (BoxIntegral.IntegrationParams.toFilteriUnion I π₀)
        (nhds (∑ J ∈ π₀.boxes, BoxIntegral.integral J l f vol))

Integral sum of f over a tagged prepartition π such that π.Union = π₀.Union tends to the sum of integrals of f over the boxes of π₀.

Defined in

Mathlib.Analysis.BoxIntegral.Basic

Cited by

1 results in Mathlib

Foundations

Depth 180 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceFintypeCompleteSpace