Theorems · Theorem · real analysis
LipschitzWith.integral_lineDeriv_mul_eq
∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [inst_2 : MeasurableSpace E] [BorelSpace E]
{C D : NNReal} {f g : E → ℝ} {μ : MeasureTheory.Measure E} [FiniteDimensional ℝ E] [μ.IsAddHaarMeasure],
LipschitzWith C f →
LipschitzWith D g →
HasCompactSupport g → ∀ (v : E), ∫ (x : E), lineDeriv ℝ f x v * g x ∂μ = ∫ (x : E), lineDeriv ℝ g x (-v) * f x ∂μIntegration by parts formula for the line derivative of Lipschitz functions, assuming one of them is compactly supported.
- Defined in
- Mathlib.Analysis.Calculus.Rademacher
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 287 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites37
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- NormedAddCommGroupstatement and proof · cited by 15,752
- MeasurableSpacestatement and proof · cited by 13,106
- NormedSpacestatement and proof · cited by 12,499
- MeasureTheory.Measurestatement and proof · cited by 10,939
- nhdsproof · cited by 5,554
- NNRealstatement and proof · cited by 4,310
- Filter.Tendstoproof · cited by 3,814
- mul_commproof · cited by 2,262
- nhdsWithinproof · cited by 1,912
- FiniteDimensionalstatement and proof · cited by 1,854
- MeasureTheory.integralstatement and proof · cited by 1,779
Cited by1
Results whose statement or proof uses this declaration.
- LipschitzWith.ae_lineDeriv_sum_eqproof · cited by 1