Theorems · Theorem · real analysis

BoxIntegral.Integrable.cauchy_map_integralSum_toFilteriUnion

∀ {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) ⊤},
  BoxIntegral.Integrable I l f vol →
    ∀ (π₀ : BoxIntegral.Prepartition I),
      Cauchy (Filter.map (BoxIntegral.integralSum f vol) (BoxIntegral.IntegrationParams.toFilteriUnion I π₀))

If f is integrable on a box I along l, then for any fixed subset s of I that can be represented as a finite union of boxes, the integral sums of f over tagged prepartitions that cover exactly s form a Cauchy “sequence” along l.

Defined in

Mathlib.Analysis.BoxIntegral.Basic

Cited by

1 results in Mathlib

Foundations

Depth 176 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceFintype