α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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
⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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
⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s
have t_ne_zero' : t ≠ 0 := (zero_lt_one.trans ht).ne'
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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
⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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
⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s
have t_ne_zero : (t : ℝ≥0∞) ≠ 0 := by simpa only [ENNReal.coe_eq_zero, Ne.def] using t_ne_zero'
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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)
⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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
⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s
let ν := μ.withDensity (v.limRatioMeas hρ)
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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
⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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)
⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s
let f := v.limRatioMeas hρ
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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
⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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
⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s
have f_meas : Measurable f := v.limRatioMeas_measurable hρ
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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
⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s
have A : ρ (s ∩ f ⁻¹' {0}) ≤ (t • ν) (s ∩ f ⁻¹' {0}) := by
  refine' le_trans (measure_mono (inter_subset_right _ _)) (le_trans (le_of_eq _) (zero_le _))
  exact v.measure_limRatioMeas_zero hρ
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})
⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s
have B : ρ (s ∩ f ⁻¹' {∞}) ≤ (t • ν) (s ∩ f ⁻¹' {∞}) := by
  apply le_trans (le_of_eq _) (zero_le _)
  apply hρ
  rw [← nonpos_iff_eq_zero]
  exact (measure_mono (inter_subset_right _ _)).trans (v.measure_limRatioMeas_top hρ).le
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})
C : ∀ (n : ℤ), ↑↑ρ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))
⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})
⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s
have C :
  ∀ n : ℤ,
    ρ (s ∩ f ⁻¹' Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))) ≤
      (t • ν) (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
    ρ (s ∩ f ⁻¹' I) ≤ (t : ℝ≥0∞) ^ (n + 1) * μ (s ∩ f ⁻¹' I) := by
      rw [← ENNReal.coe_zpow t_ne_zero']
      apply v.measure_le_mul_of_subset_limRatioMeas_lt hρ
      intro x hx
      apply hx.2.2.trans_le (le_of_eq _)
      rw [ENNReal.coe_zpow t_ne_zero']
    _ = ∫⁻ x in s ∩ f ⁻¹' I, (t : ℝ≥0∞) ^ (n + 1) ∂μ := by
      simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter]
    _ ≤ ∫⁻ x in s ∩ f ⁻¹' I, t * f x ∂μ := by
      apply lintegral_mono_ae ((ae_restrict_iff' M).2 (eventually_of_forall fun x hx => ?_))
      rw [add_comm, ENNReal.zpow_add t_ne_zero ENNReal.coe_ne_top, zpow_one]
      exact mul_le_mul_left' hx.2.1 _
    _ = t * ∫⁻ x in s ∩ f ⁻¹' I, f x ∂μ := lintegral_const_mul _ f_meas
no goals
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})
C : ∀ (n : ℤ), ↑↑ρ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))
⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s
calc
  ρ s =
    ρ (s ∩ f ⁻¹' {0}) + ρ (s ∩ f ⁻¹' {∞}) +
      ∑' n : ℤ, ρ (s ∩ f ⁻¹' Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))) :=
    measure_eq_measure_preimage_add_measure_tsum_Ico_zpow ρ f_meas hs ht
  _ ≤
      (t • ν) (s ∩ f ⁻¹' {0}) + (t • ν) (s ∩ f ⁻¹' {∞}) +
        ∑' n : ℤ, (t • ν) (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 • ν) s :=
    (measure_eq_measure_preimage_add_measure_tsum_Ico_zpow (t • ν) f_meas hs ht).symm
no goals
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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
simpa only [ENNReal.coe_eq_zero, Ne.def] using t_ne_zero'
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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
⊢ ↑↑ρ (f ⁻¹' {0}) = 0
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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
⊢ ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
refine' le_trans (measure_mono (inter_subset_right _ _)) (le_trans (le_of_eq _) (zero_le _))
no goals
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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
⊢ ↑↑ρ (f ⁻¹' {0}) = 0
exact v.measure_limRatioMeas_zero hρ
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
⊢ ↑↑ρ (s ∩ f ⁻¹' {⊤}) = 0
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
⊢ ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})
apply le_trans (le_of_eq _) (zero_le _)
case a
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
⊢ ↑↑μ (s ∩ f ⁻¹' {⊤}) = 0
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
⊢ ↑↑ρ (s ∩ f ⁻¹' {⊤}) = 0
apply hρ
case a
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
⊢ ↑↑μ (s ∩ f ⁻¹' {⊤}) ≤ 0
case a
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
⊢ ↑↑μ (s ∩ f ⁻¹' {⊤}) = 0
rw [← nonpos_iff_eq_zero]
no goals
case a
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
⊢ ↑↑μ (s ∩ f ⁻¹' {⊤}) ≤ 0
exact (measure_mono (inter_subset_right _ _)).trans (v.measure_limRatioMeas_top hρ).le
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})
n : ℤ
⊢ ↑↑ρ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})
⊢ ∀ (n : ℤ), ↑↑ρ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))
intro n
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})
n : ℤ
I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))
⊢ ↑↑ρ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})
n : ℤ
⊢ ↑↑ρ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))
let I := Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})
n : ℤ
I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))
M : MeasurableSet (s ∩ f ⁻¹' I)
⊢ ↑↑ρ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})
n : ℤ
I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))
⊢ ↑↑ρ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))
have M : MeasurableSet (s ∩ f ⁻¹' I) := hs.inter (f_meas measurableSet_Ico)
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})
n : ℤ
I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))
M : MeasurableSet (s ∩ f ⁻¹' I)
⊢ ↑↑ρ (s ∩ limRatioMeas v hρ ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤
    ↑t * ∫⁻ (a : α) in s ∩ limRatioMeas v hρ ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)), limRatioMeas v hρ a ∂μ
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})
n : ℤ
I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))
M : MeasurableSet (s ∩ f ⁻¹' I)
⊢ ↑↑ρ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))
simp only [M, withDensity_apply, coe_nnreal_smul_apply]
no goals
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})
n : ℤ
I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))
M : MeasurableSet (s ∩ f ⁻¹' I)
⊢ ↑↑ρ (s ∩ limRatioMeas v hρ ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤
    ↑t * ∫⁻ (a : α) in s ∩ limRatioMeas v hρ ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)), limRatioMeas v hρ a ∂μ
calc
  ρ (s ∩ f ⁻¹' I) ≤ (t : ℝ≥0∞) ^ (n + 1) * μ (s ∩ f ⁻¹' I) := by
    rw [← ENNReal.coe_zpow t_ne_zero']
    apply v.measure_le_mul_of_subset_limRatioMeas_lt hρ
    intro x hx
    apply hx.2.2.trans_le (le_of_eq _)
    rw [ENNReal.coe_zpow t_ne_zero']
  _ = ∫⁻ x in s ∩ f ⁻¹' I, (t : ℝ≥0∞) ^ (n + 1) ∂μ := by
    simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter]
  _ ≤ ∫⁻ x in s ∩ f ⁻¹' I, t * f x ∂μ := by
    apply lintegral_mono_ae ((ae_restrict_iff' M).2 (eventually_of_forall fun x hx => ?_))
    rw [add_comm, ENNReal.zpow_add t_ne_zero ENNReal.coe_ne_top, zpow_one]
    exact mul_le_mul_left' hx.2.1 _
  _ = t * ∫⁻ x in s ∩ f ⁻¹' I, f x ∂μ := lintegral_const_mul _ f_meas
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})
n : ℤ
I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))
M : MeasurableSet (s ∩ f ⁻¹' I)
⊢ ↑↑ρ (s ∩ f ⁻¹' I) ≤ ↑(t ^ (n + 1)) * ↑↑μ (s ∩ f ⁻¹' I)
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})
n : ℤ
I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))
M : MeasurableSet (s ∩ f ⁻¹' I)
⊢ ↑↑ρ (s ∩ f ⁻¹' I) ≤ ↑t ^ (n + 1) * ↑↑μ (s ∩ f ⁻¹' I)
rw [← ENNReal.coe_zpow t_ne_zero']
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (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 | limRatioMeas v hρ x < ↑(t ^ (n + 1))}
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤})
n : ℤ
I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))
M : MeasurableSet (s ∩ f ⁻¹' I)
⊢ ↑↑ρ (s ∩ f ⁻¹' I) ≤ ↑(t ^ (n + 1)) * ↑↑μ (s ∩ f ⁻¹' I)
apply v.measure_le_mul_of_subset_limRatioMeas_lt hρ
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (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 | limRatioMeas v hρ x < ↑(t ^ (n + 1))}
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (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 | limRatioMeas v hρ x < ↑(t ^ (n + 1))}
intro x hx
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (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 + 1))
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (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 | limRatioMeas v hρ x < ↑(t ^ (n + 1))}
apply hx.2.2.trans_le (le_of_eq _)
no goals
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (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 + 1))
rw [ENNReal.coe_zpow t_ne_zero']
no goals
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (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) = ∫⁻ (x : α) in s ∩ f ⁻¹' I, ↑t ^ (n + 1) ∂μ
simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter]
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (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 * f x
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (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) ≤ ∫⁻ (x : α) in s ∩ f ⁻¹' I, ↑t * f x ∂μ
apply lintegral_mono_ae ((ae_restrict_iff' M).2 (eventually_of_forall fun x hx => ?_))
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (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 * ↑t ^ n ≤ ↑t * f x
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (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 * f x
rw [add_comm, ENNReal.zpow_add t_ne_zero ENNReal.coe_ne_top, zpow_one]
no goals
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type ?u.4937781
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 : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0})
B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (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 * ↑t ^ n ≤ ↑t * f x
exact mul_le_mul_left' hx.2.1 _