Theorems · Theorem · measure theory

MeasureTheory.integral_image_eq_integral_abs_deriv_smul

∀ {F : Type u_1} [inst : NormedAddCommGroup F] [inst_1 : NormedSpace ℝ F] {s : Set ℝ} {f f' : ℝ → ℝ},
  MeasurableSet s →
    (∀ x ∈ s, HasDerivWithinAt f (f' x) s x) →
      Set.InjOn f s → ∀ (g : ℝ → F), ∫ (x : ℝ) in f '' s, g x = ∫ (x : ℝ) in s, |f' x| • g (f x)

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

Defined in

Mathlib.MeasureTheory.Function.JacobianOneDim

Cited by

5 results in Mathlib

Foundations

Depth 278 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpace