Theorems · Theorem · real analysis

BoxIntegral.Integrable.dist_integralSum_integral_le_of_memBaseSet

∀ {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} (h : BoxIntegral.Integrable I l f vol),
  0 < ε →
    l.MemBaseSet I c (h.convergenceR ε c) π →
      π.IsPartition → dist (BoxIntegral.integralSum f vol π) (BoxIntegral.integral I l f vol) ≤ ε

Defined in

Mathlib.Analysis.BoxIntegral.Basic

Cited by

2 results in Mathlib

Foundations

Depth 173 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceFintype