Theorems · Theorem · real analysis

MeasureTheory.AEContinuous.hasBoxIntegral

∀ {ι : Type u} {E : Type v} [inst : Fintype ι] [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace ℝ E]
  [CompleteSpace E] {f : (ι → ℝ) → E} (μ : MeasureTheory.Measure (ι → ℝ))
  [inst_4 : MeasureTheory.IsLocallyFiniteMeasure μ] {I : BoxIntegral.Box ι},
  (∃ C, ∀ x ∈ BoxIntegral.Box.Icc I, ‖f x‖ ≤ C) →
    (∀ᵐ (x : ι → ℝ) ∂μ, ContinuousAt f x) →
      ∀ (l : BoxIntegral.IntegrationParams),
        BoxIntegral.HasIntegral I l f μ.toBoxAdditive.toSMul (∫ (x : ι → ℝ) in ↑I, f x ∂μ)

If f : ℝⁿ → E is a.e. continuous and bounded on a rectangular box I, then it is Box integrable on I w.r.t. a locally finite measure μ with the same integral.

Defined in

Mathlib.Analysis.BoxIntegral.Integrability

Cited by

1 results in Mathlib

Foundations

Depth 258 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FintypeNormedAddCommGroupNormedSpaceCompleteSpaceMeasureTheory.IsLocallyFiniteMeasure