l3lab/ntp-mathlib huggingface dataset:
/-- Radon-Nikodym derivative of the scalar multiple of a measure.
See also `rnDeriv_smul_left_of_ne_top'`, which requires sigma-finite `ν` and `μ`. -/
theorem rnDeriv_smul_left_of_ne_top (ν μ : Measure α) [IsFiniteMeasure ν]
    [ν.HaveLebesgueDecomposition μ] {r : ℝ≥0∞} (hr : r ≠ ∞) :
    (r • ν).rnDeriv μ =ᵐ[μ] r • ν.rnDeriv μ := by
  have h : (r.toNNReal • ν).rnDeriv μ =ᵐ[μ] r.toNNReal • ν.rnDeriv μ :=
    rnDeriv_smul_left ν μ r.toNNReal
  

Proof state:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : HaveLebesgueDecomposition ν μ
r : ℝ≥0∞
hr : r ≠ ⊤
h : rnDeriv (ENNReal.toNNReal r • ν) μ =ᶠ[ae μ] ENNReal.toNNReal r • rnDeriv ν μ
⊢ rnDeriv (r • ν) μ =ᶠ[ae μ] r • rnDeriv ν μ

Next tactic:
simpa [ENNReal.smul_def, ENNReal.coe_toNNReal hr] using h
/-- Radon-Nikodym derivative with respect to the scalar multiple of a measure.
See also `rnDeriv_smul_right'`, which requires sigma-finite `ν` and `μ`. -/
theorem rnDeriv_smul_right (ν μ : Measure α) [IsFiniteMeasure ν]
    [ν.HaveLebesgueDecomposition μ] {r : ℝ≥0} (hr : r ≠ 0) :
    ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ := by
  

Proof state:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : HaveLebesgueDecomposition ν μ
r : ℝ≥0
hr : r ≠ 0
⊢ rnDeriv ν (r • μ) =ᶠ[ae μ] r⁻¹ • rnDeriv ν μ

Next tactic:
refine (absolutelyContinuous_smul $ ENNReal.coe_ne_zero.2 hr).ae_le
    (?_ : ν.rnDeriv (r • μ) =ᵐ[r • μ] r⁻¹ • ν.rnDeriv μ)
/-- Radon-Nikodym derivative with respect to the scalar multiple of a measure.
See also `rnDeriv_smul_right'`, which requires sigma-finite `ν` and `μ`. -/
theorem rnDeriv_smul_right (ν μ : Measure α) [IsFiniteMeasure ν]
    [ν.HaveLebesgueDecomposition μ] {r : ℝ≥0} (hr : r ≠ 0) :
    ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ := by
  refine (absolutelyContinuous_smul $ ENNReal.coe_ne_zero.2 hr).ae_le
    (?_ : ν.rnDeriv (r • μ) =ᵐ[r • μ] r⁻¹ • ν.rnDeriv μ)
  

Proof state:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : HaveLebesgueDecomposition ν μ
r : ℝ≥0
hr : r ≠ 0
⊢ rnDeriv ν (r • μ) =ᶠ[ae (r • μ)] r⁻¹ • rnDeriv ν μ

Next tactic:
rw [← withDensity_eq_iff]
/-- Radon-Nikodym derivative with respect to the scalar multiple of a measure.
See also `rnDeriv_smul_right'`, which requires sigma-finite `ν` and `μ`. -/
theorem rnDeriv_smul_right (ν μ : Measure α) [IsFiniteMeasure ν]
    [ν.HaveLebesgueDecomposition μ] {r : ℝ≥0} (hr : r ≠ 0) :
    ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ := by
  refine (absolutelyContinuous_smul $ ENNReal.coe_ne_zero.2 hr).ae_le
    (?_ : ν.rnDeriv (r • μ) =ᵐ[r • μ] r⁻¹ • ν.rnDeriv μ)
  rw [← withDensity_eq_iff]
  

Proof state:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : HaveLebesgueDecomposition ν μ
r : ℝ≥0
hr : r ≠ 0
⊢ withDensity (r • μ) (rnDeriv ν (r • μ)) = withDensity (r • μ) (r⁻¹ • rnDeriv ν μ)
case hf
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : HaveLebesgueDecomposition ν μ
r : ℝ≥0
hr : r ≠ 0
⊢ AEMeasurable (rnDeriv ν (r • μ))
case hg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : HaveLebesgueDecomposition ν μ
r : ℝ≥0
hr : r ≠ 0
⊢ AEMeasurable (r⁻¹ • rnDeriv ν μ)
case hfi
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : HaveLebesgueDecomposition ν μ
r : ℝ≥0
hr : r ≠ 0
⊢ ∫⁻ (x : α), rnDeriv ν (r • μ) x ∂r • μ ≠ ⊤

Next tactic:
rotate_left
/-- Radon-Nikodym derivative with respect to the scalar multiple of a measure.
See also `rnDeriv_smul_right'`, which requires sigma-finite `ν` and `μ`. -/
theorem rnDeriv_smul_right (ν μ : Measure α) [IsFiniteMeasure ν]
    [ν.HaveLebesgueDecomposition μ] {r : ℝ≥0} (hr : r ≠ 0) :
    ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ := by
  refine (absolutelyContinuous_smul $ ENNReal.coe_ne_zero.2 hr).ae_le
    (?_ : ν.rnDeriv (r • μ) =ᵐ[r • μ] r⁻¹ • ν.rnDeriv μ)
  rw [← withDensity_eq_iff]
  rotate_left
  · 

Proof state:
case hf
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : HaveLebesgueDecomposition ν μ
r : ℝ≥0
hr : r ≠ 0
⊢ AEMeasurable (rnDeriv ν (r • μ))

Next tactic:
exact (measurable_rnDeriv _ _).aemeasurable
/-- Radon-Nikodym derivative with respect to the scalar multiple of a measure.
See also `rnDeriv_smul_right'`, which requires sigma-finite `ν` and `μ`. -/
theorem rnDeriv_smul_right (ν μ : Measure α) [IsFiniteMeasure ν]
    [ν.HaveLebesgueDecomposition μ] {r : ℝ≥0} (hr : r ≠ 0) :
    ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ := by
  refine (absolutelyContinuous_smul $ ENNReal.coe_ne_zero.2 hr).ae_le
    (?_ : ν.rnDeriv (r • μ) =ᵐ[r • μ] r⁻¹ • ν.rnDeriv μ)
  rw [← withDensity_eq_iff]
  rotate_left
  · exact (measurable_rnDeriv _ _).aemeasurable
  · 

Proof state:
case hg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : HaveLebesgueDecomposition ν μ
r : ℝ≥0
hr : r ≠ 0
⊢ AEMeasurable (r⁻¹ • rnDeriv ν μ)

Next tactic:
exact (measurable_rnDeriv _ _).aemeasurable.const_smul _
/-- Radon-Nikodym derivative with respect to the scalar multiple of a measure.
See also `rnDeriv_smul_right'`, which requires sigma-finite `ν` and `μ`. -/
theorem rnDeriv_smul_right (ν μ : Measure α) [IsFiniteMeasure ν]
    [ν.HaveLebesgueDecomposition μ] {r : ℝ≥0} (hr : r ≠ 0) :
    ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ := by
  refine (absolutelyContinuous_smul $ ENNReal.coe_ne_zero.2 hr).ae_le
    (?_ : ν.rnDeriv (r • μ) =ᵐ[r • μ] r⁻¹ • ν.rnDeriv μ)
  rw [← withDensity_eq_iff]
  rotate_left
  · exact (measurable_rnDeriv _ _).aemeasurable
  · exact (measurable_rnDeriv _ _).aemeasurable.const_smul _
  · 

Proof state:
case hfi
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : HaveLebesgueDecomposition ν μ
r : ℝ≥0
hr : r ≠ 0
⊢ ∫⁻ (x : α), rnDeriv ν (r • μ) x ∂r • μ ≠ ⊤

Next tactic:
exact (lintegral_rnDeriv_lt_top ν _).ne
/-- Radon-Nikodym derivative with respect to the scalar multiple of a measure.
See also `rnDeriv_smul_right'`, which requires sigma-finite `ν` and `μ`. -/
theorem rnDeriv_smul_right (ν μ : Measure α) [IsFiniteMeasure ν]
    [ν.HaveLebesgueDecomposition μ] {r : ℝ≥0} (hr : r ≠ 0) :
    ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ := by
  refine (absolutelyContinuous_smul $ ENNReal.coe_ne_zero.2 hr).ae_le
    (?_ : ν.rnDeriv (r • μ) =ᵐ[r • μ] r⁻¹ • ν.rnDeriv μ)
  rw [← withDensity_eq_iff]
  rotate_left
  · exact (measurable_rnDeriv _ _).aemeasurable
  · exact (measurable_rnDeriv _ _).aemeasurable.const_smul _
  · exact (lintegral_rnDeriv_lt_top ν _).ne
  · 

Proof state:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : HaveLebesgueDecomposition ν μ
r : ℝ≥0
hr : r ≠ 0
⊢ withDensity (r • μ) (rnDeriv ν (r • μ)) = withDensity (r • μ) (r⁻¹ • rnDeriv ν μ)

Next tactic:
simp_rw [ENNReal.smul_def]
/-- Radon-Nikodym derivative with respect to the scalar multiple of a measure.
See also `rnDeriv_smul_right'`, which requires sigma-finite `ν` and `μ`. -/
theorem rnDeriv_smul_right (ν μ : Measure α) [IsFiniteMeasure ν]
    [ν.HaveLebesgueDecomposition μ] {r : ℝ≥0} (hr : r ≠ 0) :
    ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ := by
  refine (absolutelyContinuous_smul $ ENNReal.coe_ne_zero.2 hr).ae_le
    (?_ : ν.rnDeriv (r • μ) =ᵐ[r • μ] r⁻¹ • ν.rnDeriv μ)
  rw [← withDensity_eq_iff]
  rotate_left
  · exact (measurable_rnDeriv _ _).aemeasurable
  · exact (measurable_rnDeriv _ _).aemeasurable.const_smul _
  · exact (lintegral_rnDeriv_lt_top ν _).ne
  · simp_rw [ENNReal.smul_def]
    

Proof state:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : HaveLebesgueDecomposition ν μ
r : ℝ≥0
hr : r ≠ 0
⊢ withDensity (↑r • μ) (rnDeriv ν (↑r • μ)) = withDensity (↑r • μ) (↑r⁻¹ • rnDeriv ν μ)

Next tactic:
rw [withDensity_smul _ (measurable_rnDeriv _ _)]
/-- Radon-Nikodym derivative with respect to the scalar multiple of a measure.
See also `rnDeriv_smul_right'`, which requires sigma-finite `ν` and `μ`. -/
theorem rnDeriv_smul_right (ν μ : Measure α) [IsFiniteMeasure ν]
    [ν.HaveLebesgueDecomposition μ] {r : ℝ≥0} (hr : r ≠ 0) :
    ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ := by
  refine (absolutelyContinuous_smul $ ENNReal.coe_ne_zero.2 hr).ae_le
    (?_ : ν.rnDeriv (r • μ) =ᵐ[r • μ] r⁻¹ • ν.rnDeriv μ)
  rw [← withDensity_eq_iff]
  rotate_left
  · exact (measurable_rnDeriv _ _).aemeasurable
  · exact (measurable_rnDeriv _ _).aemeasurable.const_smul _
  · exact (lintegral_rnDeriv_lt_top ν _).ne
  · simp_rw [ENNReal.smul_def]
    rw [withDensity_smul _ (measurable_rnDeriv _ _)]
    

Proof state:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : HaveLebesgueDecomposition ν μ
r : ℝ≥0
hr : r ≠ 0
⊢ withDensity (↑r • μ) (rnDeriv ν (↑r • μ)) = ↑r⁻¹ • withDensity (↑r • μ) (rnDeriv ν μ)

Next tactic:
suffices ν.singularPart (r • μ) + withDensity (r • μ) (rnDeriv ν (r • μ))
        = ν.singularPart (r • μ) + r⁻¹ • withDensity (r • μ) (rnDeriv ν μ) by
      rwa [add_right_inj] at this
/-- Radon-Nikodym derivative with respect to the scalar multiple of a measure.
See also `rnDeriv_smul_right'`, which requires sigma-finite `ν` and `μ`. -/
theorem rnDeriv_smul_right (ν μ : Measure α) [IsFiniteMeasure ν]
    [ν.HaveLebesgueDecomposition μ] {r : ℝ≥0} (hr : r ≠ 0) :
    ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ := by
  refine (absolutelyContinuous_smul $ ENNReal.coe_ne_zero.2 hr).ae_le
    (?_ : ν.rnDeriv (r • μ) =ᵐ[r • μ] r⁻¹ • ν.rnDeriv μ)
  rw [← withDensity_eq_iff]
  rotate_left
  · exact (measurable_rnDeriv _ _).aemeasurable
  · exact (measurable_rnDeriv _ _).aemeasurable.const_smul _
  · exact (lintegral_rnDeriv_lt_top ν _).ne
  · simp_rw [ENNReal.smul_def]
    rw [withDensity_smul _ (measurable_rnDeriv _ _)]
    suffices ν.singularPart (r • μ) + withDensity (r • μ) (rnDeriv ν (r • μ))
        = ν.singularPart (r • μ) + r⁻¹ • withDensity (r • μ) (rnDeriv ν μ) by
      

Proof state:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : HaveLebesgueDecomposition ν μ
r : ℝ≥0
hr : r ≠ 0
this :
  singularPart ν (r • μ) + withDensity (r • μ) (rnDeriv ν (r • μ)) =
    singularPart ν (r • μ) + r⁻¹ • withDensity (r • μ) (rnDeriv ν μ)
⊢ withDensity (↑r • μ) (rnDeriv ν (↑r • μ)) = ↑r⁻¹ • withDensity (↑r • μ) (rnDeriv ν μ)

Next tactic:
rwa [add_right_inj] at this
/-- Radon-Nikodym derivative with respect to the scalar multiple of a measure.
See also `rnDeriv_smul_right'`, which requires sigma-finite `ν` and `μ`. -/
theorem rnDeriv_smul_right (ν μ : Measure α) [IsFiniteMeasure ν]
    [ν.HaveLebesgueDecomposition μ] {r : ℝ≥0} (hr : r ≠ 0) :
    ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ := by
  refine (absolutelyContinuous_smul $ ENNReal.coe_ne_zero.2 hr).ae_le
    (?_ : ν.rnDeriv (r • μ) =ᵐ[r • μ] r⁻¹ • ν.rnDeriv μ)
  rw [← withDensity_eq_iff]
  rotate_left
  · exact (measurable_rnDeriv _ _).aemeasurable
  · exact (measurable_rnDeriv _ _).aemeasurable.const_smul _
  · exact (lintegral_rnDeriv_lt_top ν _).ne
  · simp_rw [ENNReal.smul_def]
    rw [withDensity_smul _ (measurable_rnDeriv _ _)]
    suffices ν.singularPart (r • μ) + withDensity (r • μ) (rnDeriv ν (r • μ))
        = ν.singularPart (r • μ) + r⁻¹ • withDensity (r • μ) (rnDeriv ν μ) by
      rwa [add_right_inj] at this
    

Proof state:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : HaveLebesgueDecomposition ν μ
r : ℝ≥0
hr : r ≠ 0
⊢ singularPart ν (r • μ) + withDensity (r • μ) (rnDeriv ν (r • μ)) =
    singularPart ν (r • μ) + r⁻¹ • withDensity (r • μ) (rnDeriv ν μ)

Next tactic:
rw [← ν.haveLebesgueDecomposition_add (r • μ), singularPart_smul_right _ _ _ hr,
      ENNReal.smul_def r, withDensity_smul_measure, ← ENNReal.smul_def, ← smul_assoc,
      smul_eq_mul, inv_mul_cancel hr, one_smul]
/-- Radon-Nikodym derivative with respect to the scalar multiple of a measure.
See also `rnDeriv_smul_right'`, which requires sigma-finite `ν` and `μ`. -/
theorem rnDeriv_smul_right (ν μ : Measure α) [IsFiniteMeasure ν]
    [ν.HaveLebesgueDecomposition μ] {r : ℝ≥0} (hr : r ≠ 0) :
    ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ := by
  refine (absolutelyContinuous_smul $ ENNReal.coe_ne_zero.2 hr).ae_le
    (?_ : ν.rnDeriv (r • μ) =ᵐ[r • μ] r⁻¹ • ν.rnDeriv μ)
  rw [← withDensity_eq_iff]
  rotate_left
  · exact (measurable_rnDeriv _ _).aemeasurable
  · exact (measurable_rnDeriv _ _).aemeasurable.const_smul _
  · exact (lintegral_rnDeriv_lt_top ν _).ne
  · simp_rw [ENNReal.smul_def]
    rw [withDensity_smul _ (measurable_rnDeriv _ _)]
    suffices ν.singularPart (r • μ) + withDensity (r • μ) (rnDeriv ν (r • μ))
        = ν.singularPart (r • μ) + r⁻¹ • withDensity (r • μ) (rnDeriv ν μ) by
      rwa [add_right_inj] at this
    rw [← ν.haveLebesgueDecomposition_add (r • μ), singularPart_smul_right _ _ _ hr,
      ENNReal.smul_def r, withDensity_smul_measure, ← ENNReal.smul_def, ← smul_assoc,
      smul_eq_mul, inv_mul_cancel hr, one_smul]
    

Proof state:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : HaveLebesgueDecomposition ν μ
r : ℝ≥0
hr : r ≠ 0
⊢ ν = singularPart ν μ + withDensity μ (rnDeriv ν μ)

Next tactic:
exact ν.haveLebesgueDecomposition_add μ
/-- Radon-Nikodym derivative with respect to the scalar multiple of a measure.
See also `rnDeriv_smul_right_of_ne_top'`, which requires sigma-finite `ν` and `μ`. -/
theorem rnDeriv_smul_right_of_ne_top (ν μ : Measure α) [IsFiniteMeasure ν]
    [ν.HaveLebesgueDecomposition μ] {r : ℝ≥0∞} (hr : r ≠ 0) (hr_ne_top : r ≠ ∞) :
    ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ := by
  

Proof state:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : HaveLebesgueDecomposition ν μ
r : ℝ≥0∞
hr : r ≠ 0
hr_ne_top : r ≠ ⊤
⊢ rnDeriv ν (r • μ) =ᶠ[ae μ] r⁻¹ • rnDeriv ν μ

Next tactic:
have h : ν.rnDeriv (r.toNNReal • μ) =ᵐ[μ] r.toNNReal⁻¹ • ν.rnDeriv μ := by
    refine rnDeriv_smul_right ν μ ?_
    rw [ne_eq, ENNReal.toNNReal_eq_zero_iff]
    simp [hr, hr_ne_top]
/-- Radon-Nikodym derivative with respect to the scalar multiple of a measure.
See also `rnDeriv_smul_right_of_ne_top'`, which requires sigma-finite `ν` and `μ`. -/
theorem rnDeriv_smul_right_of_ne_top (ν μ : Measure α) [IsFiniteMeasure ν]
    [ν.HaveLebesgueDecomposition μ] {r : ℝ≥0∞} (hr : r ≠ 0) (hr_ne_top : r ≠ ∞) :
    ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ := by
  have h : ν.rnDeriv (r.toNNReal • μ) =ᵐ[μ] r.toNNReal⁻¹ • ν.rnDeriv μ := by
    

Proof state:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : HaveLebesgueDecomposition ν μ
r : ℝ≥0∞
hr : r ≠ 0
hr_ne_top : r ≠ ⊤
⊢ rnDeriv ν (ENNReal.toNNReal r • μ) =ᶠ[ae μ] (ENNReal.toNNReal r)⁻¹ • rnDeriv ν μ

Next tactic:
refine rnDeriv_smul_right ν μ ?_
/-- Radon-Nikodym derivative with respect to the scalar multiple of a measure.
See also `rnDeriv_smul_right_of_ne_top'`, which requires sigma-finite `ν` and `μ`. -/
theorem rnDeriv_smul_right_of_ne_top (ν μ : Measure α) [IsFiniteMeasure ν]
    [ν.HaveLebesgueDecomposition μ] {r : ℝ≥0∞} (hr : r ≠ 0) (hr_ne_top : r ≠ ∞) :
    ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ := by
  have h : ν.rnDeriv (r.toNNReal • μ) =ᵐ[μ] r.toNNReal⁻¹ • ν.rnDeriv μ := by
    refine rnDeriv_smul_right ν μ ?_
    

Proof state:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : HaveLebesgueDecomposition ν μ
r : ℝ≥0∞
hr : r ≠ 0
hr_ne_top : r ≠ ⊤
⊢ ENNReal.toNNReal r ≠ 0

Next tactic:
rw [ne_eq, ENNReal.toNNReal_eq_zero_iff]
/-- Radon-Nikodym derivative with respect to the scalar multiple of a measure.
See also `rnDeriv_smul_right_of_ne_top'`, which requires sigma-finite `ν` and `μ`. -/
theorem rnDeriv_smul_right_of_ne_top (ν μ : Measure α) [IsFiniteMeasure ν]
    [ν.HaveLebesgueDecomposition μ] {r : ℝ≥0∞} (hr : r ≠ 0) (hr_ne_top : r ≠ ∞) :
    ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ := by
  have h : ν.rnDeriv (r.toNNReal • μ) =ᵐ[μ] r.toNNReal⁻¹ • ν.rnDeriv μ := by
    refine rnDeriv_smul_right ν μ ?_
    rw [ne_eq, ENNReal.toNNReal_eq_zero_iff]
    

Proof state:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : HaveLebesgueDecomposition ν μ
r : ℝ≥0∞
hr : r ≠ 0
hr_ne_top : r ≠ ⊤
⊢ ¬(r = 0 ∨ r = ⊤)

Next tactic:
simp [hr, hr_ne_top]
/-- Radon-Nikodym derivative with respect to the scalar multiple of a measure.
See also `rnDeriv_smul_right_of_ne_top'`, which requires sigma-finite `ν` and `μ`. -/
theorem rnDeriv_smul_right_of_ne_top (ν μ : Measure α) [IsFiniteMeasure ν]
    [ν.HaveLebesgueDecomposition μ] {r : ℝ≥0∞} (hr : r ≠ 0) (hr_ne_top : r ≠ ∞) :
    ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ := by
  have h : ν.rnDeriv (r.toNNReal • μ) =ᵐ[μ] r.toNNReal⁻¹ • ν.rnDeriv μ := by
    refine rnDeriv_smul_right ν μ ?_
    rw [ne_eq, ENNReal.toNNReal_eq_zero_iff]
    simp [hr, hr_ne_top]
  

Proof state:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : HaveLebesgueDecomposition ν μ
r : ℝ≥0∞
hr : r ≠ 0
hr_ne_top : r ≠ ⊤
h : rnDeriv ν (ENNReal.toNNReal r • μ) =ᶠ[ae μ] (ENNReal.toNNReal r)⁻¹ • rnDeriv ν μ
⊢ rnDeriv ν (r • μ) =ᶠ[ae μ] r⁻¹ • rnDeriv ν μ

Next tactic:
have : (r.toNNReal)⁻¹ • rnDeriv ν μ = r⁻¹ • rnDeriv ν μ := by
    ext x
    simp only [Pi.smul_apply, ENNReal.smul_def, ne_eq, smul_eq_mul]
    rw [ENNReal.coe_inv, ENNReal.coe_toNNReal hr_ne_top]
    rw [ne_eq, ENNReal.toNNReal_eq_zero_iff]
    simp [hr, hr_ne_top]
/-- Radon-Nikodym derivative with respect to the scalar multiple of a measure.
See also `rnDeriv_smul_right_of_ne_top'`, which requires sigma-finite `ν` and `μ`. -/
theorem rnDeriv_smul_right_of_ne_top (ν μ : Measure α) [IsFiniteMeasure ν]
    [ν.HaveLebesgueDecomposition μ] {r : ℝ≥0∞} (hr : r ≠ 0) (hr_ne_top : r ≠ ∞) :
    ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ := by
  have h : ν.rnDeriv (r.toNNReal • μ) =ᵐ[μ] r.toNNReal⁻¹ • ν.rnDeriv μ := by
    refine rnDeriv_smul_right ν μ ?_
    rw [ne_eq, ENNReal.toNNReal_eq_zero_iff]
    simp [hr, hr_ne_top]
  have : (r.toNNReal)⁻¹ • rnDeriv ν μ = r⁻¹ • rnDeriv ν μ := by
    

Proof state:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : HaveLebesgueDecomposition ν μ
r : ℝ≥0∞
hr : r ≠ 0
hr_ne_top : r ≠ ⊤
h : rnDeriv ν (ENNReal.toNNReal r • μ) =ᶠ[ae μ] (ENNReal.toNNReal r)⁻¹ • rnDeriv ν μ
⊢ (ENNReal.toNNReal r)⁻¹ • rnDeriv ν μ = r⁻¹ • rnDeriv ν μ

Next tactic:
ext x
/-- Radon-Nikodym derivative with respect to the scalar multiple of a measure.
See also `rnDeriv_smul_right_of_ne_top'`, which requires sigma-finite `ν` and `μ`. -/
theorem rnDeriv_smul_right_of_ne_top (ν μ : Measure α) [IsFiniteMeasure ν]
    [ν.HaveLebesgueDecomposition μ] {r : ℝ≥0∞} (hr : r ≠ 0) (hr_ne_top : r ≠ ∞) :
    ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ := by
  have h : ν.rnDeriv (r.toNNReal • μ) =ᵐ[μ] r.toNNReal⁻¹ • ν.rnDeriv μ := by
    refine rnDeriv_smul_right ν μ ?_
    rw [ne_eq, ENNReal.toNNReal_eq_zero_iff]
    simp [hr, hr_ne_top]
  have : (r.toNNReal)⁻¹ • rnDeriv ν μ = r⁻¹ • rnDeriv ν μ := by
    ext x
    

Proof state:
case h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : HaveLebesgueDecomposition ν μ
r : ℝ≥0∞
hr : r ≠ 0
hr_ne_top : r ≠ ⊤
h : rnDeriv ν (ENNReal.toNNReal r • μ) =ᶠ[ae μ] (ENNReal.toNNReal r)⁻¹ • rnDeriv ν μ
x : α
⊢ ((ENNReal.toNNReal r)⁻¹ • rnDeriv ν μ) x = (r⁻¹ • rnDeriv ν μ) x

Next tactic:
simp only [Pi.smul_apply, ENNReal.smul_def, ne_eq, smul_eq_mul]
/-- Radon-Nikodym derivative with respect to the scalar multiple of a measure.
See also `rnDeriv_smul_right_of_ne_top'`, which requires sigma-finite `ν` and `μ`. -/
theorem rnDeriv_smul_right_of_ne_top (ν μ : Measure α) [IsFiniteMeasure ν]
    [ν.HaveLebesgueDecomposition μ] {r : ℝ≥0∞} (hr : r ≠ 0) (hr_ne_top : r ≠ ∞) :
    ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ := by
  have h : ν.rnDeriv (r.toNNReal • μ) =ᵐ[μ] r.toNNReal⁻¹ • ν.rnDeriv μ := by
    refine rnDeriv_smul_right ν μ ?_
    rw [ne_eq, ENNReal.toNNReal_eq_zero_iff]
    simp [hr, hr_ne_top]
  have : (r.toNNReal)⁻¹ • rnDeriv ν μ = r⁻¹ • rnDeriv ν μ := by
    ext x
    simp only [Pi.smul_apply, ENNReal.smul_def, ne_eq, smul_eq_mul]
    

Proof state:
case h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : HaveLebesgueDecomposition ν μ
r : ℝ≥0∞
hr : r ≠ 0
hr_ne_top : r ≠ ⊤
h : rnDeriv ν (ENNReal.toNNReal r • μ) =ᶠ[ae μ] (ENNReal.toNNReal r)⁻¹ • rnDeriv ν μ
x : α
⊢ ↑(ENNReal.toNNReal r)⁻¹ * rnDeriv ν μ x = r⁻¹ * rnDeriv ν μ x

Next tactic:
rw [ENNReal.coe_inv, ENNReal.coe_toNNReal hr_ne_top]
/-- Radon-Nikodym derivative with respect to the scalar multiple of a measure.
See also `rnDeriv_smul_right_of_ne_top'`, which requires sigma-finite `ν` and `μ`. -/
theorem rnDeriv_smul_right_of_ne_top (ν μ : Measure α) [IsFiniteMeasure ν]
    [ν.HaveLebesgueDecomposition μ] {r : ℝ≥0∞} (hr : r ≠ 0) (hr_ne_top : r ≠ ∞) :
    ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ := by
  have h : ν.rnDeriv (r.toNNReal • μ) =ᵐ[μ] r.toNNReal⁻¹ • ν.rnDeriv μ := by
    refine rnDeriv_smul_right ν μ ?_
    rw [ne_eq, ENNReal.toNNReal_eq_zero_iff]
    simp [hr, hr_ne_top]
  have : (r.toNNReal)⁻¹ • rnDeriv ν μ = r⁻¹ • rnDeriv ν μ := by
    ext x
    simp only [Pi.smul_apply, ENNReal.smul_def, ne_eq, smul_eq_mul]
    rw [ENNReal.coe_inv, ENNReal.coe_toNNReal hr_ne_top]
    

Proof state:
case h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : HaveLebesgueDecomposition ν μ
r : ℝ≥0∞
hr : r ≠ 0
hr_ne_top : r ≠ ⊤
h : rnDeriv ν (ENNReal.toNNReal r • μ) =ᶠ[ae μ] (ENNReal.toNNReal r)⁻¹ • rnDeriv ν μ
x : α
⊢ ENNReal.toNNReal r ≠ 0

Next tactic:
rw [ne_eq, ENNReal.toNNReal_eq_zero_iff]
/-- Radon-Nikodym derivative with respect to the scalar multiple of a measure.
See also `rnDeriv_smul_right_of_ne_top'`, which requires sigma-finite `ν` and `μ`. -/
theorem rnDeriv_smul_right_of_ne_top (ν μ : Measure α) [IsFiniteMeasure ν]
    [ν.HaveLebesgueDecomposition μ] {r : ℝ≥0∞} (hr : r ≠ 0) (hr_ne_top : r ≠ ∞) :
    ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ := by
  have h : ν.rnDeriv (r.toNNReal • μ) =ᵐ[μ] r.toNNReal⁻¹ • ν.rnDeriv μ := by
    refine rnDeriv_smul_right ν μ ?_
    rw [ne_eq, ENNReal.toNNReal_eq_zero_iff]
    simp [hr, hr_ne_top]
  have : (r.toNNReal)⁻¹ • rnDeriv ν μ = r⁻¹ • rnDeriv ν μ := by
    ext x
    simp only [Pi.smul_apply, ENNReal.smul_def, ne_eq, smul_eq_mul]
    rw [ENNReal.coe_inv, ENNReal.coe_toNNReal hr_ne_top]
    rw [ne_eq, ENNReal.toNNReal_eq_zero_iff]
    

Proof state:
case h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : HaveLebesgueDecomposition ν μ
r : ℝ≥0∞
hr : r ≠ 0
hr_ne_top : r ≠ ⊤
h : rnDeriv ν (ENNReal.toNNReal r • μ) =ᶠ[ae μ] (ENNReal.toNNReal r)⁻¹ • rnDeriv ν μ
x : α
⊢ ¬(r = 0 ∨ r = ⊤)

Next tactic:
simp [hr, hr_ne_top]