Theorems · Theorem · real analysis

BoxIntegral.integrable_iff_cauchy_basis

∀ {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 ↔
    ∀ ε > 0,
      ∃ r,
        (∀ (c : NNReal), l.RCond (r c)) ∧
          ∀ (c₁ c₂ : NNReal) (π₁ π₂ : BoxIntegral.TaggedPrepartition I),
            l.MemBaseSet I c₁ (r c₁) π₁ →
              π₁.IsPartition →
                l.MemBaseSet I c₂ (r c₂) π₂ →
                  π₂.IsPartition → dist (BoxIntegral.integralSum f vol π₁) (BoxIntegral.integralSum f vol π₂) ≤ ε

In a complete space, a function is integrable if and only if its integral sums form a Cauchy net. Here we restate this fact in terms of ∀ ε > 0, ∃ r, ....

Defined in

Mathlib.Analysis.BoxIntegral.Basic

Cited by

1 results in Mathlib

Foundations

Depth 168 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceFintypeCompleteSpace