Theorems · Theorem · measure theory

MeasureTheory.integrableOn_image_iff_integrableOn_deriv_smul_of_monotoneOn

∀ {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) →
      MonotoneOn f s →
        ∀ (g : ℝ → F),
          MeasureTheory.IntegrableOn g (f '' s) MeasureTheory.volume ↔
            MeasureTheory.IntegrableOn (fun x => f' x • g (f x)) s MeasureTheory.volume

Integrability in the change of variable formula for differentiable functions: if a real function f is monotone and differentiable on a measurable set s, then a function g : ℝ → F is integrable on f '' s if and only if f' x • g ∘ f is integrable on s .

Defined in

Mathlib.MeasureTheory.Function.JacobianOneDim

Cited by

3 results in Mathlib

Foundations

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

Assumes

NormedAddCommGroupNormedSpace