theorem tendsto_preCdf_atBot_zero (ρ : Measure (α × ℝ)) [IsFiniteMeasure ρ] :
    ∀ᵐ a ∂ρ.fst, Tendsto (fun r => preCdf ρ r a) atBot (𝓝 0) := by
  -- We show first that `preCdf` has a limit in ℝ≥0∞ almost everywhere.
  -- We then show that the integral of `pre_cdf` tends to 0, and that it also tends
  -- to the integral of the limit. Since the limit has integral 0, it is equal to 0 a.e.
  suffices ∀ᵐ a ∂ρ.fst, Tendsto (fun r => preCdf ρ (-r) a) atTop (𝓝 0) by
    filter_upwards [this] with a ha
    have h_eq_neg : (fun r : ℚ => preCdf ρ r a) = fun r : ℚ => preCdf ρ (- -r) a := by
      simp_rw [neg_neg]
    rw [h_eq_neg]
    exact ha.comp tendsto_neg_atBot_atTop
  have h_exists : ∀ᵐ a ∂ρ.fst, ∃ l, Tendsto (fun r => preCdf ρ (-r) a) atTop (𝓝 l) := by
    filter_upwards [monotone_preCdf ρ] with a ha
    have h_anti : Antitone fun r => preCdf ρ (-r) a := fun p q hpq => ha (neg_le_neg hpq)
    have h_tendsto :
      Tendsto (fun r => preCdf ρ (-r) a) atTop atBot ∨
        ∃ l, Tendsto (fun r => preCdf ρ (-r) a) atTop (𝓝 l) :=
      tendsto_of_antitone h_anti
    cases' h_tendsto with h_bot h_tendsto
    · exact ⟨0, Tendsto.mono_right h_bot atBot_le_nhds_bot⟩
    · exact h_tendsto
  classical
  let F : α → ℝ≥0∞ := fun a =>
    if h : ∃ l, Tendsto (fun r => preCdf ρ (-r) a) atTop (𝓝 l) then h.choose else 0
  have h_tendsto : ∀ᵐ a ∂ρ.fst, Tendsto (fun r => preCdf ρ (-r) a) atTop (𝓝 (F a)) := by
    filter_upwards [h_exists] with a ha
    simp_rw [dif_pos ha]
    exact ha.choose_spec
  suffices h_lintegral_eq : ∫⁻ a, F a ∂ρ.fst = 0
  · have hF_ae_meas : AEMeasurable F ρ.fst := by
      refine' aemeasurable_of_tendsto_metrizable_ae _ (fun n => _) h_tendsto
      exact measurable_preCdf.aemeasurable
    rw [lintegral_eq_zero_iff' hF_ae_meas] at h_lintegral_eq
    filter_upwards [h_tendsto, h_lintegral_eq] with a ha_tendsto ha_eq
    rwa [ha_eq] at ha_tendsto
  have h_lintegral :
    Tendsto (fun r => ∫⁻ a, preCdf ρ (-r) a ∂ρ.fst) atTop (𝓝 (∫⁻ a, F a ∂ρ.fst)) := by
    refine'
      tendsto_lintegral_filter_of_dominated_convergence (fun _ => 1)
        (eventually_of_forall fun _ => measurable_preCdf) (eventually_of_forall fun _ => _) _
        h_tendsto
    · filter_upwards [preCdf_le_one ρ] with a ha using ha _
    · rw [lintegral_one]
      exact measure_ne_top _ _
  have h_lintegral' : Tendsto (fun r => ∫⁻ a, preCdf ρ (-r) a ∂ρ.fst) atTop (𝓝 0) := by
    have h_lintegral_eq :
      (fun r => ∫⁻ a, preCdf ρ (-r) a ∂ρ.fst) = fun r : ℚ => ρ (univ ×ˢ Iic (-r : ℝ)) := by
      ext1 n
      rw [← set_lintegral_univ, set_lintegral_preCdf_fst ρ _ MeasurableSet.univ,
        Measure.IicSnd_univ]
      norm_cast
    rw [h_lintegral_eq]
    

Proof state:
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
h_exists : ∀ᵐ (a : α) ∂Measure.fst ρ, ∃ l, Tendsto (fun r => preCdf ρ (-r) a) atTop (𝓝 l)
F : α → ℝ≥0∞ := fun a => if h : ∃ l, Tendsto (fun r => preCdf ρ (-r) a) atTop (𝓝 l) then Exists.choose h else 0
h_tendsto : ∀ᵐ (a : α) ∂Measure.fst ρ, Tendsto (fun r => preCdf ρ (-r) a) atTop (𝓝 (F a))
h_lintegral : Tendsto (fun r => ∫⁻ (a : α), preCdf ρ (-r) a ∂Measure.fst ρ) atTop (𝓝 (∫⁻ (a : α), F a ∂Measure.fst ρ))
h_lintegral_eq : (fun r => ∫⁻ (a : α), preCdf ρ (-r) a ∂Measure.fst ρ) = fun r => ↑↑ρ (univ ×ˢ Iic (-↑r))
⊢ Tendsto (fun r => ↑↑ρ (univ ×ˢ Iic (-↑r))) atTop (𝓝 0)

Next tactic:
have h_zero_eq_measure_iInter : (0 : ℝ≥0∞) = ρ (⋂ r : ℚ, univ ×ˢ Iic (-r : ℝ)) := by
      suffices ⋂ r : ℚ, Iic (-(r : ℝ)) = ∅ by rw [← prod_iInter, this, prod_empty, measure_empty]
      ext1 x
      simp only [mem_iInter, mem_Iic, mem_empty_iff_false, iff_false_iff, not_forall, not_le]
      simp_rw [neg_lt]
      exact exists_rat_gt _