Theorems · Theorem · real analysis

BoxIntegral.HasIntegral.of_mul

∀ {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) ⊤} {y : F}
  (a : ℝ),
  (∀ (ε : ℝ),
      0 < ε →
        ∃ r,
          (∀ (c : NNReal), l.RCond (r c)) ∧
            ∀ (c : NNReal) (π : BoxIntegral.TaggedPrepartition I),
              l.MemBaseSet I c (r c) π → π.IsPartition → dist (BoxIntegral.integralSum f vol π) y ≤ a * ε) →
    BoxIntegral.HasIntegral I l f vol y

Quite often it is more natural to prove an estimate of the form a * ε, not ε in the RHS of BoxIntegral.hasIntegral_iff, so we provide this auxiliary lemma.

Defined in

Mathlib.Analysis.BoxIntegral.Basic

Cited by

2 results in Mathlib

Foundations

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

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceFintype