Theorems · Theorem · measure theory

MeasureTheory.lintegral_abs_det_fderiv_eq_addHaar_image

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

Change of variable formula for differentiable functions, set version: if a function f is injective and differentiable on a measurable set s, then the measure of f '' s is given by the integral of |(f' x).det| on s. Note that the measurability of f '' s is given by measurable_image_of_fderivWithin.

Defined in

Mathlib.MeasureTheory.Function.Jacobian

Cited by

1 results in Mathlib

Foundations

Depth 273 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceFiniteDimensionalMeasurableSpaceBorelSpaceMeasureTheory.Measure.IsAddHaarMeasure