/-- As an intermediate step to show that `μ.withDensity (v.limRatioMeas hρ) = ρ`, we show here
that `ρ ≤ t μ.withDensity (v.limRatioMeas hρ)` for any `t > 1`. -/
theorem le_mul_withDensity {s : Set α} (hs : MeasurableSet s) {t : ℝ≥0} (ht : 1 < t) :
    ρ s ≤ t * μ.withDensity (v.limRatioMeas hρ) s := by
  /- We cut `s` into the sets where `v.limRatioMeas hρ = 0`, where `v.limRatioMeas hρ = ∞`, and
    where `v.limRatioMeas hρ ∈ [t^n, t^(n+1))` for `n : ℤ`. The first and second have measure `0`.
    For the latter, since `v.limRatioMeas hρ` fluctuates by at most `t` on this slice, we can use
    `measure_le_mul_of_subset_limRatioMeas_lt` and `mul_measure_le_of_subset_lt_limRatioMeas` to
    show that the two measures are comparable up to `t`. -/
  have t_ne_zero' : t ≠ 0 := (zero_lt_one.trans ht).ne'
  have t_ne_zero : (t : ℝ≥0∞) ≠ 0 := by simpa only [ENNReal.coe_eq_zero, Ne.def] using t_ne_zero'
  let ν := μ.withDensity (v.limRatioMeas hρ)
  let f := v.limRatioMeas hρ
  have f_meas : Measurable f := v.limRatioMeas_measurable hρ
  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
  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
  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']
      _ = ∫⁻ _ 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
  

Proof state:
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : 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 : 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

Next tactic:
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