Theorems · Theorem · real analysis

BoxIntegral.Integrable.dist_integralSum_sum_integral_le_of_memBaseSet_of_iUnion_eq

∀ {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}
  {π : BoxIntegral.TaggedPrepartition I} [inst_4 : Fintype ι] {l : BoxIntegral.IntegrationParams} {f : (ι → ℝ) → E}
  {vol : BoxIntegral.BoxAdditiveMap ι (E →L[ℝ] F) ⊤} {c : NNReal} {ε : ℝ} [CompleteSpace F]
  (h : BoxIntegral.Integrable I l f vol),
  0 < ε →
    l.MemBaseSet I c (h.convergenceR ε c) π →
      ∀ {π₀ : BoxIntegral.Prepartition I},
        π.iUnion = π₀.iUnion →
          dist (BoxIntegral.integralSum f vol π) (∑ J ∈ π₀.boxes, BoxIntegral.integral J l f vol) ≤ ε

Henstock-Sacks inequality. Let r : ℝⁿ → (0, ∞) be a function such that for any tagged partition of I subordinate to r, the integral sum of f over this partition differs from the integral of f by at most ε. Then for any tagged prepartition π subordinate to r, the integral sum of f over this prepartition differs from the integral of f over the part of I covered by π by at most ε. The actual statement - uses BoxIntegral.Integrable.convergenceR instead of a predicate assumption on r; - uses BoxIntegral.IntegrationParams.MemBaseSet instead of “subordinate to r” to account for additional requirements like being a Henstock partition or having a bounded distortion; - takes an extra argument π₀ : prepartition I and an assumption π.Union = π₀.Union instead of using π.to_prepartition.

Defined in

Mathlib.Analysis.BoxIntegral.Basic

Cited by

2 results in Mathlib

Foundations

Depth 179 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceFintypeCompleteSpace