Theorems · Theorem · measure theory

MeasureTheory.integral_image_eq_integral_abs_det_fderiv_smul

∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [FiniteDimensional ℝ E]
  [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace ℝ F] {s : Set E} {f : E → E} {f' : E → E →L[ℝ] E}
  [inst_5 : MeasurableSpace E] [BorelSpace E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure],
  MeasurableSet s →
    (∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) →
      Set.InjOn f s → ∀ (g : E → F), ∫ (x : E) in f '' s, g x ∂μ = ∫ (x : E) in s, |(f' x).det| • g (f x) ∂μ

Change of variable formula for differentiable functions: if a function f is injective and differentiable on a measurable set s, then the Bochner integral of a function g : E → F on f '' s coincides with the integral of |(f' x).det| • g ∘ f on s.

Defined in

Mathlib.MeasureTheory.Function.Jacobian

Cited by

6 results in Mathlib

Foundations

Depth 277 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceFiniteDimensionalNormedAddCommGroupNormedSpaceMeasurableSpaceBorelSpaceMeasureTheory.Measure.IsAddHaarMeasure