File: Mathlib/MeasureTheory/Function/Jacobian.lean

full_name: exists_closed_cover_approximatesLinearOn_of_hasFDerivWithinAt
code: /-- Assume that a function `f` has a derivative at every point of a set `s`. Then one may cover `s`
with countably many closed sets `t n` on which `f` is well approximated by linear maps `A n`. -/
theorem exists_closed_cover_approximatesLinearOn_of_hasFDerivWithinAt [SecondCountableTopology F]
    (f : E → F) (s : Set E) (f' : E → E →L[ℝ] F) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x)
    (r : (E →L[ℝ] F) → ℝ≥0) (rpos : ∀ A, r A ≠ 0) :
    ∃ (t : ℕ → Set E) (A : ℕ → E →L[ℝ] F),
      (∀ n, IsClosed (t n)) ∧
        (s ⊆ ⋃ n, t n) ∧
          (∀ n, ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧
            (s.Nonempty → ∀ n, ∃ y ∈ s, A n = f' y) 
goal: ∃ t A,
  (∀ (n : ℕ), IsClosed (t n)) ∧
    s ⊆ ⋃ n, t n ∧
      (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧ (s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y)

full_name: exists_partition_approximatesLinearOn_of_hasFDerivWithinAt
code: /-- Assume that a function `f` has a derivative at every point of a set `s`. Then one may
partition `s` into countably many disjoint relatively measurable sets (i.e., intersections
of `s` with measurable sets `t n`) on which `f` is well approximated by linear maps `A n`. -/
theorem exists_partition_approximatesLinearOn_of_hasFDerivWithinAt [SecondCountableTopology F]
    (f : E → F) (s : Set E) (f' : E → E →L[ℝ] F) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x)
    (r : (E →L[ℝ] F) → ℝ≥0) (rpos : ∀ A, r A ≠ 0) :
    ∃ (t : ℕ → Set E) (A : ℕ → E →L[ℝ] F),
      Pairwise (Disjoint on t) ∧
        (∀ n, MeasurableSet (t n)) ∧
          (s ⊆ ⋃ n, t n) ∧
            (∀ n, ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧
              (s.Nonempty → ∀ n, ∃ y ∈ s, A n = f' y) 
goal: ∃ t A,
  Pairwise (Disjoint on t) ∧
    (∀ (n : ℕ), MeasurableSet (t n)) ∧
      s ⊆ ⋃ n, t n ∧
        (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧ (s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y)

full_name: MeasureTheory.addHaar_image_le_mul_of_det_lt
code: /-- Let `f` be a function which is sufficiently close (in the Lipschitz sense) to a given linear
map `A`. Then it expands the volume of any set by at most `m` for any `m > det A`. -/
theorem addHaar_image_le_mul_of_det_lt (A : E →L[ℝ] E) {m : ℝ≥0}
    (hm : ENNReal.ofReal |A.det| < m) :
    ∀ᶠ δ in 𝓝[>] (0 : ℝ≥0),
      ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → μ (f '' s) ≤ m * μ s 
goal: ∀ᶠ (δ : ℝ≥0) in 𝓝[>] 0, ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → μ (f '' s) ≤ ↑m * μ s

full_name: MeasureTheory.mul_le_addHaar_image_of_lt_det
code: /-- Let `f` be a function which is sufficiently close (in the Lipschitz sense) to a given linear
map `A`. Then it expands the volume of any set by at least `m` for any `m < det A`. -/
theorem mul_le_addHaar_image_of_lt_det (A : E →L[ℝ] E) {m : ℝ≥0}
    (hm : (m : ℝ≥0∞) < ENNReal.ofReal |A.det|) :
    ∀ᶠ δ in 𝓝[>] (0 : ℝ≥0),
      ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → (m : ℝ≥0∞) * μ s ≤ μ (f '' s) 
goal: ∀ᶠ (δ : ℝ≥0) in 𝓝[>] 0, ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → ↑m * μ s ≤ μ (f '' s)

full_name: ApproximatesLinearOn.norm_fderiv_sub_le
code: /-- If a differentiable function `f` is approximated by a linear map `A` on a set `s`, up to `δ`,
then at almost every `x` in `s` one has `‖f' x - A‖ ≤ δ`. -/
theorem _root_.ApproximatesLinearOn.norm_fderiv_sub_le {A : E →L[ℝ] E} {δ : ℝ≥0}
    (hf : ApproximatesLinearOn f A s δ) (hs : MeasurableSet s) (f' : E → E →L[ℝ] E)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) : ∀ᵐ x ∂μ.restrict s, ‖f' x - A‖₊ ≤ δ 
goal: ∀ᵐ (x : E) ∂μ.restrict s, ‖f' x - A‖₊ ≤ δ

full_name: MeasureTheory.addHaar_image_eq_zero_of_differentiableOn_of_addHaar_eq_zero
code: /-- A differentiable function maps sets of measure zero to sets of measure zero. -/
theorem addHaar_image_eq_zero_of_differentiableOn_of_addHaar_eq_zero (hf : DifferentiableOn ℝ f s)
    (hs : μ s = 0) : μ (f '' s) = 0 
goal: μ (f '' s) = 0

full_name: MeasureTheory.addHaar_image_eq_zero_of_det_fderivWithin_eq_zero_aux
code: /-- A version of **Sard's lemma** in fixed dimension: given a differentiable function from `E`
to `E` and a set where the differential is not invertible, then the image of this set has
zero measure. Here, we give an auxiliary statement towards this result. -/
theorem addHaar_image_eq_zero_of_det_fderivWithin_eq_zero_aux
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (R : ℝ) (hs : s ⊆ closedBall 0 R) (ε : ℝ≥0)
    (εpos : 0 < ε) (h'f' : ∀ x ∈ s, (f' x).det = 0) : μ (f '' s) ≤ ε * μ (closedBall 0 R) 
goal: μ (f '' s) ≤ ↑ε * μ (Metric.closedBall 0 R)

full_name: MeasureTheory.addHaar_image_eq_zero_of_det_fderivWithin_eq_zero
code: /-- A version of Sard lemma in fixed dimension: given a differentiable function from `E` to `E` and
a set where the differential is not invertible, then the image of this set has zero measure. -/
theorem addHaar_image_eq_zero_of_det_fderivWithin_eq_zero
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (h'f' : ∀ x ∈ s, (f' x).det = 0) :
    μ (f '' s) = 0 
goal: μ (f '' s) = 0

full_name: MeasureTheory.aemeasurable_fderivWithin
code: /-- The derivative of a function on a measurable set is almost everywhere measurable on this set
with respect to Lebesgue measure. Note that, in general, it is not genuinely measurable there,
as `f'` is not unique (but only on a set of measure `0`, as the argument shows). -/
theorem aemeasurable_fderivWithin (hs : MeasurableSet s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) : AEMeasurable f' (μ.restrict s) 
goal: AEMeasurable f' (μ.restrict s)

full_name: MeasureTheory.aemeasurable_ofReal_abs_det_fderivWithin
code: theorem aemeasurable_ofReal_abs_det_fderivWithin (hs : MeasurableSet s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) :
    AEMeasurable (fun x => ENNReal.ofReal |(f' x).det|) (μ.restrict s) 
goal: AEMeasurable (fun x => ENNReal.ofReal |(f' x).det|) (μ.restrict s)

full_name: MeasureTheory.aemeasurable_toNNReal_abs_det_fderivWithin
code: theorem aemeasurable_toNNReal_abs_det_fderivWithin (hs : MeasurableSet s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) :
    AEMeasurable (fun x => |(f' x).det|.toNNReal) (μ.restrict s) 
goal: AEMeasurable (fun x => |(f' x).det|.toNNReal) (μ.restrict s)

full_name: MeasureTheory.measurable_image_of_fderivWithin
code: /-- If a function is differentiable and injective on a measurable set,
then the image is measurable. -/
theorem measurable_image_of_fderivWithin (hs : MeasurableSet s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) : MeasurableSet (f '' s) 
goal: MeasurableSet (f '' s)

full_name: MeasureTheory.measurableEmbedding_of_fderivWithin
code: /-- If a function is differentiable and injective on a measurable set `s`, then its restriction
to `s` is a measurable embedding. -/
theorem measurableEmbedding_of_fderivWithin (hs : MeasurableSet s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) :
    MeasurableEmbedding (s.restrict f) 
goal: MeasurableEmbedding (s.restrict f)

full_name: MeasureTheory.addHaar_image_le_lintegral_abs_det_fderiv_aux1
code: theorem addHaar_image_le_lintegral_abs_det_fderiv_aux1 (hs : MeasurableSet s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) {ε : ℝ≥0} (εpos : 0 < ε) :
    μ (f '' s) ≤ (∫⁻ x in s, ENNReal.ofReal |(f' x).det| ∂μ) + 2 * ε * μ s 
goal: μ (f '' s) ≤ ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ + 2 * ↑ε * μ s

full_name: MeasureTheory.addHaar_image_le_lintegral_abs_det_fderiv_aux2
code: theorem addHaar_image_le_lintegral_abs_det_fderiv_aux2 (hs : MeasurableSet s) (h's : μ s ≠ ∞)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) :
    μ (f '' s) ≤ ∫⁻ x in s, ENNReal.ofReal |(f' x).det| ∂μ 
goal: μ (f '' s) ≤ ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ

full_name: MeasureTheory.addHaar_image_le_lintegral_abs_det_fderiv
code: theorem addHaar_image_le_lintegral_abs_det_fderiv (hs : MeasurableSet s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) :
    μ (f '' s) ≤ ∫⁻ x in s, ENNReal.ofReal |(f' x).det| ∂μ 
goal: μ (f '' s) ≤ ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ

full_name: MeasureTheory.lintegral_abs_det_fderiv_le_addHaar_image_aux1
code: theorem lintegral_abs_det_fderiv_le_addHaar_image_aux1 (hs : MeasurableSet s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) {ε : ℝ≥0} (εpos : 0 < ε) :
    (∫⁻ x in s, ENNReal.ofReal |(f' x).det| ∂μ) ≤ μ (f '' s) + 2 * ε * μ s 
goal: ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ ≤ μ (f '' s) + 2 * ↑ε * μ s

full_name: MeasureTheory.lintegral_abs_det_fderiv_le_addHaar_image_aux2
code: theorem lintegral_abs_det_fderiv_le_addHaar_image_aux2 (hs : MeasurableSet s) (h's : μ s ≠ ∞)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) :
    (∫⁻ x in s, ENNReal.ofReal |(f' x).det| ∂μ) ≤ μ (f '' s) 
goal: ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ ≤ μ (f '' s)

full_name: MeasureTheory.lintegral_abs_det_fderiv_le_addHaar_image
code: theorem lintegral_abs_det_fderiv_le_addHaar_image (hs : MeasurableSet s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) :
    (∫⁻ x in s, ENNReal.ofReal |(f' x).det| ∂μ) ≤ μ (f '' s) 
goal: ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ ≤ μ (f '' s)

full_name: MeasureTheory.lintegral_abs_det_fderiv_eq_addHaar_image
code: /-- 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`. -/
theorem lintegral_abs_det_fderiv_eq_addHaar_image (hs : MeasurableSet s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) :
    (∫⁻ x in s, ENNReal.ofReal |(f' x).det| ∂μ) = μ (f '' s) 
goal: ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ = μ (f '' s)

full_name: MeasureTheory.map_withDensity_abs_det_fderiv_eq_addHaar
code: /-- Change of variable formula for differentiable functions, set version: if a function `f` is
injective and differentiable on a measurable set `s`, then the pushforward of the measure with
density `|(f' x).det|` on `s` is the Lebesgue measure on the image set. This version requires
that `f` is measurable, as otherwise `Measure.map f` is zero per our definitions.
For a version without measurability assumption but dealing with the restricted
function `s.restrict f`, see `restrict_map_withDensity_abs_det_fderiv_eq_addHaar`.
-/
theorem map_withDensity_abs_det_fderiv_eq_addHaar (hs : MeasurableSet s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) (h'f : Measurable f) :
    Measure.map f ((μ.restrict s).withDensity fun x => ENNReal.ofReal |(f' x).det|) =
      μ.restrict (f '' s) 
goal: MeasureTheory.Measure.map f ((μ.restrict s).withDensity fun x => ENNReal.ofReal |(f' x).det|) = μ.restrict (f '' s)

full_name: MeasureTheory.restrict_map_withDensity_abs_det_fderiv_eq_addHaar
code: /-- Change of variable formula for differentiable functions, set version: if a function `f` is
injective and differentiable on a measurable set `s`, then the pushforward of the measure with
density `|(f' x).det|` on `s` is the Lebesgue measure on the image set. This version is expressed
in terms of the restricted function `s.restrict f`.
For a version for the original function, but with a measurability assumption,
see `map_withDensity_abs_det_fderiv_eq_addHaar`.
-/
theorem restrict_map_withDensity_abs_det_fderiv_eq_addHaar (hs : MeasurableSet s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) :
    Measure.map (s.restrict f) (comap (↑) (μ.withDensity fun x => ENNReal.ofReal |(f' x).det|)) =
      μ.restrict (f '' s) 
goal: MeasureTheory.Measure.map (s.restrict f)
    (MeasureTheory.Measure.comap Subtype.val (μ.withDensity fun x => ENNReal.ofReal |(f' x).det|)) =
  μ.restrict (f '' s)

full_name: MeasureTheory.lintegral_image_eq_lintegral_abs_det_fderiv_mul
code: theorem lintegral_image_eq_lintegral_abs_det_fderiv_mul (hs : MeasurableSet s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) (g : E → ℝ≥0∞) :
    ∫⁻ x in f '' s, g x ∂μ = ∫⁻ x in s, ENNReal.ofReal |(f' x).det| * g (f x) ∂μ 
goal: ∫⁻ (x : E) in f '' s, g x ∂μ = ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| * g (f x) ∂μ

full_name: MeasureTheory.integrableOn_image_iff_integrableOn_abs_det_fderiv_smul
code: /-- Integrability in the change of variable formula for differentiable functions: if a
function `f` is injective and differentiable on a measurable set `s`, then a function
`g : E → F` is integrable on `f '' s` if and only if `|(f' x).det| • g ∘ f` is
integrable on `s`. -/
theorem integrableOn_image_iff_integrableOn_abs_det_fderiv_smul (hs : MeasurableSet s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) (g : E → F) :
    IntegrableOn g (f '' s) μ ↔ IntegrableOn (fun x => |(f' x).det| • g (f x)) s μ 
goal: MeasureTheory.IntegrableOn g (f '' s) μ ↔ MeasureTheory.IntegrableOn (fun x => |(f' x).det| • g (f x)) s μ

full_name: MeasureTheory.integral_image_eq_integral_abs_det_fderiv_smul
code: /-- 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`. -/
theorem integral_image_eq_integral_abs_det_fderiv_smul (hs : MeasurableSet s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) (g : E → F) :
    ∫ x in f '' s, g x ∂μ = ∫ x in s, |(f' x).det| • g (f x) ∂μ 
goal: ∫ (x : E) in f '' s, g x ∂μ = ∫ (x : E) in s, |(f' x).det| • g (f x) ∂μ

full_name: MeasureTheory.det_one_smulRight
code: theorem det_one_smulRight {𝕜 : Type*} [NormedField 𝕜] (v : 𝕜) :
    ((1 : 𝕜 →L[𝕜] 𝕜).smulRight v).det = v 
goal: (ContinuousLinearMap.smulRight 1 v).det = v

full_name: MeasureTheory.integrableOn_image_iff_integrableOn_abs_deriv_smul
code: /-- Integrability in the 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 a
function `g : ℝ → F` is integrable on `f '' s` if and only if `|(f' x)| • g ∘ f` is integrable on
`s`. -/
theorem integrableOn_image_iff_integrableOn_abs_deriv_smul {s : Set ℝ} {f : ℝ → ℝ} {f' : ℝ → ℝ}
    (hs : MeasurableSet s) (hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (hf : InjOn f s)
    (g : ℝ → F) : IntegrableOn g (f '' s) ↔ IntegrableOn (fun x => |f' x| • g (f x)) s 
goal: MeasureTheory.IntegrableOn g (f '' s) MeasureTheory.volume ↔
  MeasureTheory.IntegrableOn (fun x => |f' x| • g (f x)) s MeasureTheory.volume

full_name: MeasureTheory.integral_image_eq_integral_abs_deriv_smul
code: /-- 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`. -/
theorem integral_image_eq_integral_abs_deriv_smul {s : Set ℝ} {f : ℝ → ℝ} {f' : ℝ → ℝ}
    (hs : MeasurableSet s) (hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x)
    (hf : InjOn f s) (g : ℝ → F) : ∫ x in f '' s, g x = ∫ x in s, |f' x| • g (f x) 
goal: ∫ (x : ℝ) in f '' s, g x = ∫ (x : ℝ) in s, |f' x| • g (f x)

full_name: MeasureTheory.integral_target_eq_integral_abs_det_fderiv_smul
code: theorem integral_target_eq_integral_abs_det_fderiv_smul {f : PartialHomeomorph E E}
    (hf' : ∀ x ∈ f.source, HasFDerivAt f (f' x) x) (g : E → F) :
    ∫ x in f.target, g x ∂μ = ∫ x in f.source, |(f' x).det| • g (f x) ∂μ 
goal: ∫ (x : E) in f.target, g x ∂μ = ∫ (x : E) in f.source, |(f' x).det| • g (↑f x) ∂μ

full_name: MeasurableEmbedding.withDensity_ofReal_comap_apply_eq_integral_abs_det_fderiv_mul
code: lemma _root_.MeasurableEmbedding.withDensity_ofReal_comap_apply_eq_integral_abs_det_fderiv_mul
    (hs : MeasurableSet s) (hf : MeasurableEmbedding f)
    {g : E → ℝ} (hg : ∀ᵐ x ∂μ, x ∈ f '' s → 0 ≤ g x) (hg_int : IntegrableOn g (f '' s) μ)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) :
    (μ.withDensity (fun x ↦ ENNReal.ofReal (g x))).comap f s
      = ENNReal.ofReal (∫ x in s, |(f' x).det| * g (f x) ∂μ) 
goal: (MeasureTheory.Measure.comap f (μ.withDensity fun x => ENNReal.ofReal (g x))) s =
  ENNReal.ofReal (∫ (x : E) in s, |(f' x).det| * g (f x) ∂μ)

full_name: MeasurableEquiv.withDensity_ofReal_map_symm_apply_eq_integral_abs_det_fderiv_mul
code: lemma _root_.MeasurableEquiv.withDensity_ofReal_map_symm_apply_eq_integral_abs_det_fderiv_mul
    (hs : MeasurableSet s) (f : E ≃ᵐ E)
    {g : E → ℝ} (hg : ∀ᵐ x ∂μ, x ∈ f '' s → 0 ≤ g x) (hg_int : IntegrableOn g (f '' s) μ)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) :
    (μ.withDensity (fun x ↦ ENNReal.ofReal (g x))).map f.symm s
      = ENNReal.ofReal (∫ x in s, |(f' x).det| * g (f x) ∂μ) 
goal: (MeasureTheory.Measure.map (⇑f.symm) (μ.withDensity fun x => ENNReal.ofReal (g x))) s =
  ENNReal.ofReal (∫ (x : E) in s, |(f' x).det| * g (f x) ∂μ)

full_name: MeasurableEmbedding.withDensity_ofReal_comap_apply_eq_integral_abs_deriv_mul
code: lemma _root_.MeasurableEmbedding.withDensity_ofReal_comap_apply_eq_integral_abs_deriv_mul
    {f : ℝ → ℝ} (hf : MeasurableEmbedding f) {s : Set ℝ} (hs : MeasurableSet s)
    {g : ℝ → ℝ} (hg : ∀ᵐ x, x ∈ f '' s → 0 ≤ g x) (hg_int : IntegrableOn g (f '' s))
    {f' : ℝ → ℝ} (hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) :
    (volume.withDensity (fun x ↦ ENNReal.ofReal (g x))).comap f s
      = ENNReal.ofReal (∫ x in s, |f' x| * g (f x)) 
goal: (MeasureTheory.Measure.comap f (MeasureTheory.volume.withDensity fun x => ENNReal.ofReal (g x))) s =
  ENNReal.ofReal (∫ (x : ℝ) in s, |f' x| * g (f x))

full_name: MeasurableEquiv.withDensity_ofReal_map_symm_apply_eq_integral_abs_deriv_mul
code: lemma _root_.MeasurableEquiv.withDensity_ofReal_map_symm_apply_eq_integral_abs_deriv_mul
    (f : ℝ ≃ᵐ ℝ) {s : Set ℝ} (hs : MeasurableSet s)
    {g : ℝ → ℝ} (hg : ∀ᵐ x, x ∈ f '' s → 0 ≤ g x) (hg_int : IntegrableOn g (f '' s))
    {f' : ℝ → ℝ} (hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) :
    (volume.withDensity (fun x ↦ ENNReal.ofReal (g x))).map f.symm s
      = ENNReal.ofReal (∫ x in s, |f' x| * g (f x)) 
goal: (MeasureTheory.Measure.map (⇑f.symm) (MeasureTheory.volume.withDensity fun x => ENNReal.ofReal (g x))) s =
  ENNReal.ofReal (∫ (x : ℝ) in s, |f' x| * g (f x))

full_name: MeasurableEmbedding.withDensity_ofReal_comap_apply_eq_integral_abs_deriv_mul'
code: lemma _root_.MeasurableEmbedding.withDensity_ofReal_comap_apply_eq_integral_abs_deriv_mul'
    {f : ℝ → ℝ} (hf : MeasurableEmbedding f) {s : Set ℝ} (hs : MeasurableSet s)
    {f' : ℝ → ℝ} (hf' : ∀ x, HasDerivAt f (f' x) x)
    {g : ℝ → ℝ} (hg : 0 ≤ᵐ[volume] g) (hg_int : Integrable g) :
    (volume.withDensity (fun x ↦ ENNReal.ofReal (g x))).comap f s
      = ENNReal.ofReal (∫ x in s, |f' x| * g (f x)) 
goal: (MeasureTheory.Measure.comap f (MeasureTheory.volume.withDensity fun x => ENNReal.ofReal (g x))) s =
  ENNReal.ofReal (∫ (x : ℝ) in s, |f' x| * g (f x))

full_name: MeasurableEquiv.withDensity_ofReal_map_symm_apply_eq_integral_abs_deriv_mul'
code: lemma _root_.MeasurableEquiv.withDensity_ofReal_map_symm_apply_eq_integral_abs_deriv_mul'
    (f : ℝ ≃ᵐ ℝ) {s : Set ℝ} (hs : MeasurableSet s)
    {f' : ℝ → ℝ} (hf' : ∀ x, HasDerivAt f (f' x) x)
    {g : ℝ → ℝ} (hg : 0 ≤ᵐ[volume] g) (hg_int : Integrable g) :
    (volume.withDensity (fun x ↦ ENNReal.ofReal (g x))).map f.symm s
      = ENNReal.ofReal (∫ x in s, |f' x| * g (f x)) 
goal: (MeasureTheory.Measure.map (⇑f.symm) (MeasureTheory.volume.withDensity fun x => ENNReal.ofReal (g x))) s =
  ENNReal.ofReal (∫ (x : ℝ) in s, |f' x| * g (f x))