Theorems · Theorem · real analysis
BoxIntegral.integrable_of_continuousOn
∀ {ι : Type u} {E : Type v} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [inst_2 : Fintype ι]
(l : BoxIntegral.IntegrationParams) [CompleteSpace E] {I : BoxIntegral.Box ι} {f : (ι → ℝ) → E},
ContinuousOn f (BoxIntegral.Box.Icc I) →
∀ (μ : MeasureTheory.Measure (ι → ℝ)) [inst_4 : MeasureTheory.IsLocallyFiniteMeasure μ],
BoxIntegral.Integrable I l f μ.toBoxAdditive.toSMulA continuous function is box-integrable with respect to any locally finite measure. This is true for any volume with bounded variation.
- Defined in
- Mathlib.Analysis.BoxIntegral.Basic
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 205 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites38
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Setstatement · cited by 53,352
- Realstatement and proof · cited by 25,697
- NormedAddCommGroupstatement and proof · cited by 15,752
- NormedSpacestatement and proof · cited by 12,499
- MeasureTheory.Measurestatement and proof · cited by 10,939
- ENNRealproof · cited by 9,879
- Top.topstatement · cited by 9,680
- Fintypestatement and proof · cited by 7,736
- Set.imageproof · cited by 5,609
- Norm.normproof · cited by 5,413
- WithTopstatement · cited by 3,754
Cited by3
Results whose statement or proof uses this declaration.
- BoxIntegral.hasIntegral_GP_pderivproof · cited by 1
- BoxIntegral.norm_volume_sub_integral_face_upper_sub_lower_smul_leproof · cited by 1
- MeasureTheory.ContinuousOn.hasBoxIntegralproof · cited by 0