Theorems · Theorem · real analysis

BoxIntegral.HasIntegral.of_bRiemann_eq_false_of_forall_isLittleO

∀ {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) ⊤},
  l.bRiemann = false →
    ∀ (B : BoxIntegral.BoxAdditiveMap ι ℝ ↑I),
      (∀ (J : BoxIntegral.Box ι), 0 ≤ B J) →
        ∀ (g : BoxIntegral.BoxAdditiveMap ι F ↑I) (s : Set (ι → ℝ)),
          s.Countable →
            (s.Nonempty → l.bHenstock = true) →
              (∀ (c : NNReal),
                  ∀ x ∈ BoxIntegral.Box.Icc I ∩ s,
                    ∀ ε > 0,
                      ∃ δ > 0,
                        ∀ J ≤ I,
                          BoxIntegral.Box.Icc J ⊆ Metric.closedBall x δ →
                            x ∈ BoxIntegral.Box.Icc J →
                              (l.bDistortion = true → J.distortion ≤ c) → dist ((vol J) (f x)) (g J) ≤ ε) →
                (∀ (c : NNReal),
                    ∀ x ∈ BoxIntegral.Box.Icc I \ s,
                      ∀ ε > 0,
                        ∃ δ > 0,
                          ∀ J ≤ I,
                            BoxIntegral.Box.Icc J ⊆ Metric.closedBall x δ →
                              (l.bHenstock = true → x ∈ BoxIntegral.Box.Icc J) →
                                (l.bDistortion = true → J.distortion ≤ c) → dist ((vol J) (f x)) (g J) ≤ ε * B J) →
                  BoxIntegral.HasIntegral I l f vol (g I)

This is an auxiliary lemma used to prove two statements at once. Use one of the next two lemmas instead.

Defined in

Mathlib.Analysis.BoxIntegral.Basic

Cited by

2 results in Mathlib

Foundations

Depth 166 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceFintype