Module: Mathlib.MeasureTheory.Covering.Differentiation

Goal state before:
pp: α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
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
⊢ (μ.withDensity (v.limRatioMeas hρ)) s ≤ ↑t ^ 2 * ρ s

Tactic:
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` (in fact `t^2` for technical reasons of
    strict inequalities). -/
  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] 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
  -- Note(kmill): smul elaborator when used for CoeFun fails to get CoeFun instance to trigger
  -- unless you use the `(... :)` notation. Another fix is using `(2 : Nat)`, so this appears
  -- to be an unpleasant interaction with default instances.
  have A : ν (s ∩ f ⁻¹' {0}) ≤ ((t : ℝ≥0∞) ^ 2 • ρ :) (s ∩ f ⁻¹' {0}) := by
    apply zero_le.trans'
    have M : MeasurableSet (s ∩ f ⁻¹' {0}) := hs.inter (f_meas (measurableSet_singleton _))
    simp only [f, ν, 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
  have B : ν (s ∩ f ⁻¹' {∞}) ≤ ((t : ℝ≥0∞) ^ 2 • ρ :) (s ∩ f ⁻¹' {∞}) := by
    rw [withDensity_absolutelyContinuous μ]
    · exact zero_le
    · rw [← nonpos_iff_eq_zero]
      exact (measure_mono inter_subset_right).trans (v.measure_limRatioMeas_top hρ).le
  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 [ν, I, M, withDensity_apply]
    calc
      (∫⁻ x in s ∩ f ⁻¹' I, f x ∂μ) ≤ ∫⁻ _ 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 : ℤ) * ρ (s ∩ f ⁻¹' I) := by
        gcongr
        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)]
        gcongr
        rw [zpow_neg_one]
        exact inv_lt_one_of_one_lt₀ ht
  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 ^ n) (t ^ (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

Goal state after: