Theorem: VitaliFamily.withDensity_le_mul
File: Mathlib/MeasureTheory/Covering/Differentiation.lean

tactic: have t_ne_zero' : t ≠ 0 := (zero_lt_one.trans ht).ne'
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
⊢ ↑↑(withDensity μ (limRatioMeas v hρ)) s ≤ ↑t ^ 2 * ↑↑ρ s

tactic: have t_ne_zero : (t : ℝ≥0∞) ≠ 0 := by simpa only [ENNReal.coe_eq_zero, Ne.def] using t_ne_zero'
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
⊢ ↑↑(withDensity μ (limRatioMeas v hρ)) s ≤ ↑t ^ 2 * ↑↑ρ s

tactic: let ν := μ.withDensity (v.limRatioMeas hρ)
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
⊢ ↑↑(withDensity μ (limRatioMeas v hρ)) s ≤ ↑t ^ 2 * ↑↑ρ s

tactic: let f := v.limRatioMeas hρ
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)
⊢ ↑↑(withDensity μ (limRatioMeas v hρ)) s ≤ ↑t ^ 2 * ↑↑ρ s

tactic: have f_meas : Measurable f := v.limRatioMeas_measurable hρ
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
⊢ ↑↑(withDensity μ (limRatioMeas v hρ)) s ≤ ↑t ^ 2 * ↑↑ρ s

tactic: have A : ν (s ∩ f ⁻¹' {0}) ≤ ((t : ℝ≥0∞) ^ 2 • ρ) (s ∩ f ⁻¹' {0}) := by
  apply le_trans _ (zero_le _)
  have M : MeasurableSet (s ∩ f ⁻¹' {0}) := hs.inter (f_meas (measurableSet_singleton _))
  simp only [nonpos_iff_eq_zero, M, withDensity_apply, lintegral_eq_zero_iff f_meas]
  apply (ae_restrict_iff' M).2
  exact eventually_of_forall fun x hx => hx.2
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
⊢ ↑↑(withDensity μ (limRatioMeas v hρ)) s ≤ ↑t ^ 2 * ↑↑ρ s

tactic: have B : ν (s ∩ f ⁻¹' {∞}) ≤ ((t : ℝ≥0∞) ^ 2 • ρ) (s ∩ f ⁻¹' {∞}) := by
  apply le_trans (le_of_eq _) (zero_le _)
  apply withDensity_absolutelyContinuous μ _
  rw [← nonpos_iff_eq_zero]
  exact (measure_mono (inter_subset_right _ _)).trans (v.measure_limRatioMeas_top hρ).le
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
A : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})
⊢ ↑↑(withDensity μ (limRatioMeas v hρ)) s ≤ ↑t ^ 2 * ↑↑ρ s

tactic: have C :
  ∀ n : ℤ,
    ν (s ∩ f ⁻¹' Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))) ≤
      ((t : ℝ≥0∞) ^ 2 • ρ) (s ∩ f ⁻¹' Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))) := by
  intro n
  let I := Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))
  have M : MeasurableSet (s ∩ f ⁻¹' I) := hs.inter (f_meas measurableSet_Ico)
  simp only [M, withDensity_apply, coe_nnreal_smul_apply]
  calc
    (∫⁻ x in s ∩ f ⁻¹' I, f x ∂μ) ≤ ∫⁻ x in s ∩ f ⁻¹' I, (t : ℝ≥0∞) ^ (n + 1) ∂μ :=
      lintegral_mono_ae ((ae_restrict_iff' M).2 (eventually_of_forall fun x hx => hx.2.2.le))
    _ = (t : ℝ≥0∞) ^ (n + 1) * μ (s ∩ f ⁻¹' I) := by
      simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter]
    _ = (t : ℝ≥0∞) ^ (2 : ℤ) * ((t : ℝ≥0∞) ^ (n - 1) * μ (s ∩ f ⁻¹' I)) := by
      rw [← mul_assoc, ← ENNReal.zpow_add t_ne_zero ENNReal.coe_ne_top]
      congr 2
      abel
    _ ≤ (t : ℝ≥0∞) ^ 2 * ρ (s ∩ f ⁻¹' I) := by
      refine' mul_le_mul_left' _ _
      rw [← ENNReal.coe_zpow (zero_lt_one.trans ht).ne']
      apply v.mul_measure_le_of_subset_lt_limRatioMeas hρ
      intro x hx
      apply lt_of_lt_of_le _ hx.2.1
      rw [← ENNReal.coe_zpow (zero_lt_one.trans ht).ne', ENNReal.coe_lt_coe, sub_eq_add_neg,
        zpow_add₀ t_ne_zero']
      conv_rhs => rw [← mul_one (t ^ n)]
      refine' mul_lt_mul' le_rfl _ (zero_le _) (NNReal.zpow_pos t_ne_zero' _)
      rw [zpow_neg_one]
      exact inv_lt_one ht
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
A : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})
B : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})
⊢ ↑↑(withDensity μ (limRatioMeas v hρ)) s ≤ ↑t ^ 2 * ↑↑ρ s

tactic: calc
  ν s =
    ν (s ∩ f ⁻¹' {0}) + ν (s ∩ f ⁻¹' {∞}) +
      ∑' n : ℤ, ν (s ∩ f ⁻¹' Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))) :=
    measure_eq_measure_preimage_add_measure_tsum_Ico_zpow ν f_meas hs ht
  _ ≤
      ((t : ℝ≥0∞) ^ 2 • ρ) (s ∩ f ⁻¹' {0}) + ((t : ℝ≥0∞) ^ 2 • ρ) (s ∩ f ⁻¹' {∞}) +
        ∑' n : ℤ, ((t : ℝ≥0∞) ^ 2 • ρ) (s ∩ f ⁻¹' Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))) :=
    (add_le_add (add_le_add A B) (ENNReal.tsum_le_tsum C))
  _ = ((t : ℝ≥0∞) ^ 2 • ρ) s :=
    (measure_eq_measure_preimage_add_measure_tsum_Ico_zpow ((t : ℝ≥0∞) ^ 2 • ρ) f_meas hs ht).symm
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
A : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})
B : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})
C : ∀ (n : ℤ), ↑↑ν (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))
⊢ ↑↑(withDensity μ (limRatioMeas v hρ)) s ≤ ↑t ^ 2 * ↑↑ρ s

tactic: simpa only [ENNReal.coe_eq_zero, Ne.def] using t_ne_zero'
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
⊢ ↑t ≠ 0

tactic: apply le_trans _ (zero_le _)
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
⊢ ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})

tactic: have M : MeasurableSet (s ∩ f ⁻¹' {0}) := hs.inter (f_meas (measurableSet_singleton _))
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
⊢ ↑↑ν (s ∩ f ⁻¹' {0}) ≤ 0

tactic: simp only [nonpos_iff_eq_zero, M, withDensity_apply, lintegral_eq_zero_iff f_meas]
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
M : MeasurableSet (s ∩ f ⁻¹' {0})
⊢ ↑↑ν (s ∩ f ⁻¹' {0}) ≤ 0

tactic: apply (ae_restrict_iff' M).2
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
M : MeasurableSet (s ∩ f ⁻¹' {0})
⊢ limRatioMeas v hρ =ᶠ[ae (Measure.restrict μ (s ∩ limRatioMeas v hρ ⁻¹' {0}))] 0

tactic: exact eventually_of_forall fun x hx => hx.2
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
M : MeasurableSet (s ∩ f ⁻¹' {0})
⊢ ∀ᵐ (x : α) ∂μ, x ∈ s ∩ f ⁻¹' {0} → limRatioMeas v hρ x = OfNat.ofNat 0 x

tactic: apply le_trans (le_of_eq _) (zero_le _)
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
A : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})
⊢ ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})

tactic: apply withDensity_absolutelyContinuous μ _
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
A : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})
⊢ ↑↑ν (s ∩ f ⁻¹' {⊤}) = 0

tactic: rw [← nonpos_iff_eq_zero]
state_before: case a
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
A : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})
⊢ ↑↑μ (s ∩ f ⁻¹' {⊤}) = 0

tactic: exact (measure_mono (inter_subset_right _ _)).trans (v.measure_limRatioMeas_top hρ).le
state_before: case a
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
A : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})
⊢ ↑↑μ (s ∩ f ⁻¹' {⊤}) ≤ 0

tactic: intro n
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
A : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})
B : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})
⊢ ∀ (n : ℤ), ↑↑ν (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))

tactic: let I := Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
A : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})
B : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})
n : ℤ
⊢ ↑↑ν (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))

tactic: have M : MeasurableSet (s ∩ f ⁻¹' I) := hs.inter (f_meas measurableSet_Ico)
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
A : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})
B : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})
n : ℤ
I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))
⊢ ↑↑ν (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))

tactic: simp only [M, withDensity_apply, coe_nnreal_smul_apply]
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
A : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})
B : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})
n : ℤ
I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))
M : MeasurableSet (s ∩ f ⁻¹' I)
⊢ ↑↑ν (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))

tactic: calc
  (∫⁻ x in s ∩ f ⁻¹' I, f x ∂μ) ≤ ∫⁻ x in s ∩ f ⁻¹' I, (t : ℝ≥0∞) ^ (n + 1) ∂μ :=
    lintegral_mono_ae ((ae_restrict_iff' M).2 (eventually_of_forall fun x hx => hx.2.2.le))
  _ = (t : ℝ≥0∞) ^ (n + 1) * μ (s ∩ f ⁻¹' I) := by
    simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter]
  _ = (t : ℝ≥0∞) ^ (2 : ℤ) * ((t : ℝ≥0∞) ^ (n - 1) * μ (s ∩ f ⁻¹' I)) := by
    rw [← mul_assoc, ← ENNReal.zpow_add t_ne_zero ENNReal.coe_ne_top]
    congr 2
    abel
  _ ≤ (t : ℝ≥0∞) ^ 2 * ρ (s ∩ f ⁻¹' I) := by
    refine' mul_le_mul_left' _ _
    rw [← ENNReal.coe_zpow (zero_lt_one.trans ht).ne']
    apply v.mul_measure_le_of_subset_lt_limRatioMeas hρ
    intro x hx
    apply lt_of_lt_of_le _ hx.2.1
    rw [← ENNReal.coe_zpow (zero_lt_one.trans ht).ne', ENNReal.coe_lt_coe, sub_eq_add_neg,
      zpow_add₀ t_ne_zero']
    conv_rhs => rw [← mul_one (t ^ n)]
    refine' mul_lt_mul' le_rfl _ (zero_le _) (NNReal.zpow_pos t_ne_zero' _)
    rw [zpow_neg_one]
    exact inv_lt_one ht
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
A : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})
B : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})
n : ℤ
I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))
M : MeasurableSet (s ∩ f ⁻¹' I)
⊢ (∫⁻ (a : α) in s ∩ limRatioMeas v hρ ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)), limRatioMeas v hρ a ∂μ) ≤
    ↑↑(↑t ^ 2 • ρ) (s ∩ limRatioMeas v hρ ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))

tactic: simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter]
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
A : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})
B : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})
n : ℤ
I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))
M : MeasurableSet (s ∩ f ⁻¹' I)
⊢ (∫⁻ (x : α) in s ∩ f ⁻¹' I, ↑t ^ (n + 1) ∂μ) = ↑t ^ (n + 1) * ↑↑μ (s ∩ f ⁻¹' I)

tactic: rw [← mul_assoc, ← ENNReal.zpow_add t_ne_zero ENNReal.coe_ne_top]
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
A : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})
B : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})
n : ℤ
I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))
M : MeasurableSet (s ∩ f ⁻¹' I)
⊢ ↑t ^ (n + 1) * ↑↑μ (s ∩ f ⁻¹' I) = ↑t ^ 2 * (↑t ^ (n - 1) * ↑↑μ (s ∩ f ⁻¹' I))

tactic: congr 2
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
A : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})
B : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})
n : ℤ
I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))
M : MeasurableSet (s ∩ f ⁻¹' I)
⊢ ↑t ^ (n + 1) * ↑↑μ (s ∩ f ⁻¹' I) = ↑t ^ (2 + (n - 1)) * ↑↑μ (s ∩ f ⁻¹' I)

tactic: abel
state_before: case e_a.e_a
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
A : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})
B : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})
n : ℤ
I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))
M : MeasurableSet (s ∩ f ⁻¹' I)
⊢ n + 1 = 2 + (n - 1)

tactic: refine' mul_le_mul_left' _ _
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
A : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})
B : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})
n : ℤ
I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))
M : MeasurableSet (s ∩ f ⁻¹' I)
⊢ ↑t ^ 2 * (↑t ^ (n - 1) * ↑↑μ (s ∩ f ⁻¹' I)) ≤ ↑t ^ 2 * ↑↑ρ (s ∩ f ⁻¹' I)

tactic: rw [← ENNReal.coe_zpow (zero_lt_one.trans ht).ne']
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
A : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})
B : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})
n : ℤ
I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))
M : MeasurableSet (s ∩ f ⁻¹' I)
⊢ ↑t ^ (n - 1) * ↑↑μ (s ∩ f ⁻¹' I) ≤ ↑↑ρ (s ∩ f ⁻¹' I)

tactic: apply v.mul_measure_le_of_subset_lt_limRatioMeas hρ
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
A : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})
B : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})
n : ℤ
I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))
M : MeasurableSet (s ∩ f ⁻¹' I)
⊢ ↑(t ^ (n - 1)) * ↑↑μ (s ∩ f ⁻¹' I) ≤ ↑↑ρ (s ∩ f ⁻¹' I)

tactic: intro x hx
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
A : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})
B : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})
n : ℤ
I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))
M : MeasurableSet (s ∩ f ⁻¹' I)
⊢ s ∩ f ⁻¹' I ⊆ {x | ↑(t ^ (n - 1)) < limRatioMeas v hρ x}

tactic: apply lt_of_lt_of_le _ hx.2.1
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
A : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})
B : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})
n : ℤ
I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))
M : MeasurableSet (s ∩ f ⁻¹' I)
x : α
hx : x ∈ s ∩ f ⁻¹' I
⊢ x ∈ {x | ↑(t ^ (n - 1)) < limRatioMeas v hρ x}

tactic: rw [← ENNReal.coe_zpow (zero_lt_one.trans ht).ne', ENNReal.coe_lt_coe, sub_eq_add_neg,
  zpow_add₀ t_ne_zero']
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
A : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})
B : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})
n : ℤ
I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))
M : MeasurableSet (s ∩ f ⁻¹' I)
x : α
hx : x ∈ s ∩ f ⁻¹' I
⊢ ↑(t ^ (n - 1)) < ↑t ^ n

tactic: conv_rhs => rw [← mul_one (t ^ n)]
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
A : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})
B : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})
n : ℤ
I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))
M : MeasurableSet (s ∩ f ⁻¹' I)
x : α
hx : x ∈ s ∩ f ⁻¹' I
⊢ t ^ n * t ^ (-1) < t ^ n

tactic: refine' mul_lt_mul' le_rfl _ (zero_le _) (NNReal.zpow_pos t_ne_zero' _)
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
A : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})
B : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})
n : ℤ
I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))
M : MeasurableSet (s ∩ f ⁻¹' I)
x : α
hx : x ∈ s ∩ f ⁻¹' I
⊢ t ^ n * t ^ (-1) < t ^ n * 1

tactic: rw [zpow_neg_one]
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
A : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})
B : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})
n : ℤ
I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))
M : MeasurableSet (s ∩ f ⁻¹' I)
x : α
hx : x ∈ s ∩ f ⁻¹' I
⊢ t ^ (-1) < 1

tactic: exact inv_lt_one ht
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4684926
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
t_ne_zero' : t ≠ 0
t_ne_zero : ↑t ≠ 0
ν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)
f : α → ℝ≥0∞ := limRatioMeas v hρ
f_meas : Measurable f
A : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})
B : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})
n : ℤ
I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))
M : MeasurableSet (s ∩ f ⁻¹' I)
x : α
hx : x ∈ s ∩ f ⁻¹' I
⊢ t⁻¹ < 1