Theorems · Theorem · real analysis

BoxIntegral.integrable_iff_cauchy

∀ {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 ↔
    Cauchy (Filter.map (BoxIntegral.integralSum f vol) (BoxIntegral.IntegrationParams.toFilteriUnion I ⊤))

Defined in

Mathlib.Analysis.BoxIntegral.Basic

Cited by

1 results in Mathlib

Foundations

Depth 167 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceFintypeCompleteSpace