lebesgue_differentiation

theorem lebesgue_differentiation {f : X → ℂ} (hf : BoundedFiniteSupport f) :    ∀ᵐ x ∂volume, ∃ (c : ℕ → X) (r : ℕ → ℝ),    Tendsto (fun i ↦ ⨍ y in ball (c i) (r i), f y ∂volume) atTop (𝓝 (f x)) ∧    Tendsto r atTop (𝓝[>] 0) ∧    ∀ i, x ∈ ball (c i) (r i) := by  -- By the Vitali covering theorem, the conclusion of the theorem is true for closed balls.  have ineq (x : X) {r : ℝ} (hr : r > 0) :      volume (closedBall x (3 * r)) ≤ (defaultA a) ^ 2 * volume (closedBall x r) := calc    _ ≤ volume (ball x (2 ^ 2 * (0.9 * r))) := measure_mono (closedBall_subset_ball (by linarith))    _ ≤ (defaultA a) ^ 2 * volume (ball x (0.9 * r)) := measure_ball_two_le_same_iterate _ _ 2    _ ≤ (defaultA a) ^ 2 * volume (closedBall x r) := by      gcongr; exact ball_subset_closedBall.trans <| closedBall_subset_closedBall <| by linarith  let v : VitaliFamily volume := Vitali.vitaliFamily volume _    (fun x ↦ eventually_nhdsWithin_of_forall (s := Ioi 0) (fun r ↦ ineq x) |>.frequently)  refine (v.ae_tendsto_average hf.integrable.locallyIntegrable).mono (fun x hx ↦ ?_)  have tendsto_closedBall : Tendsto (closedBall x) (𝓝[>] 0) (v.filterAt x) := by    rw [v.tendsto_filterAt_iff]    refine ⟨eventually_nhdsWithin_iff.mpr (Eventually.of_forall fun r hr ↦ ?_), fun ε hε ↦ ?_⟩    · exact ⟨isClosed_closedBall, ⟨x, mem_interior.mpr ⟨ball x r, ball_subset_closedBall,        isOpen_ball, mem_ball_self hr⟩⟩, r, by tauto, ineq x hr⟩    · rw [eventually_nhdsWithin_iff, _root_.eventually_nhds_iff]      exact ⟨Iio ε, fun y hy _ ↦ closedBall_subset_closedBall hy.le, ⟨isOpen_Iio, hε⟩⟩  -- We prove a stronger result: we can use any balls centered at x with radii decreasing to 0  have ⟨r, _, hr0, hr⟩ := exists_seq_strictAnti_tendsto_nhdsWithin (0 : ℝ)  refine ⟨fun _ ↦ x, r, ?_, hr, (mem_ball_self <| hr0 ·)⟩  suffices Tendsto (⨍ y in ball x ·, f y) (𝓝[>] 0) (𝓝 (f x)) from this.comp hr  -- Now we translate the known result about closed balls to the desired result about open balls,  -- by approximating the average over the open ball by an average over a closed ball within it.  refine Metric.tendsto_nhds.mpr (fun ε hε ↦ ?_)  have := Metric.tendsto_nhds.mp (hx.comp tendsto_closedBall) (ε / 2) (half_pos hε)  rw [eventually_nhdsWithin_iff, Metric.eventually_nhds_iff] at this ⊢  have ⟨δ, δ0, hδ⟩ := this  refine ⟨δ, δ0, fun r hr hr0 ↦ ?_⟩  have ⟨ρ, ρ_mono, ρ_lt_r, tendsto_ρ_r⟩ := exists_seq_strictMono_tendsto r  let cB (n : ℕ) := closedBall x (ρ n)  suffices ∀ᶠ n in atTop, ρ n > 0 ∧ dist (⨍ y in ball x r, f y) (⨍ y in cB n, f y) < ε / 2 by    have ⟨n, hn0, hn⟩ := this.exists    apply lt_of_le_of_lt <| dist_triangle (⨍ y in ball x r, f y) (⨍ y in cB n, f y) (f x)    rw [← add_halves ε]    refine add_lt_add hn (hδ ?_ hn0)    have r_lt_δ : r < δ := by simpa [abs_eq_self.mpr (mem_Ioi.mp hr0).le] using hr    rw [dist_zero_right, Real.norm_eq_abs, abs_lt]    exact ⟨lt_trans (neg_neg_iff_pos.mpr δ0) hn0, lt_trans (ρ_lt_r n) r_lt_δ⟩  apply Eventually.and (tendsto_ρ_r.eventually_const_lt hr0)  -- It remains to confirm that `⨍ y in cB n, f y` estimates `⨍ y in ball x r, f y` for large `n`:  suffices Tendsto (⨍ y in cB ·, f y) atTop (𝓝 (⨍ y in ball x r, f y)) by    have := (continuous_dist.uncurry_left (⨍ y in ball x r, f y)).continuousAt.tendsto.comp this    simpa using Filter.eventually_atTop.mpr (Metric.tendsto_atTop.mp this (ε / 2) (half_pos hε))  -- We first check that `∫ y in cB n, f y` estimates `∫ y in ball x r, f y`:  have hsm (n : ℕ) : MeasurableSet (cB n) := measurableSet_closedBall  have h_mono : Monotone cB := fun m n hmn ↦ closedBall_subset_closedBall (ρ_mono.le_iff_le.mpr hmn)  have := MeasureTheory.tendsto_setIntegral_of_monotone hsm h_mono hf.integrable.integrableOn  have iUnion_cB : ⋃ n, cB n = ball x r := by    refine subset_antisymm (iUnion_subset (closedBall_subset_ball <| ρ_lt_r ·)) (fun y hy ↦ ?_)    have ⟨n, hn⟩ := (tendsto_ρ_r.eventually_const_lt hy).exists    use closedBall x (ρ n), ⟨n, rfl⟩, hn.le  -- Finally, we check that this estimate works for averages as well as integrals.  simp_rw [average, integral_smul_measure]  refine Tendsto.smul ?_ (iUnion_cB ▸ this)  simp only [MeasurableSet.univ, Measure.restrict_apply, univ_inter, toReal_inv]  refine (tendsto_inv₀ ?_).comp ?_  · exact ENNReal.toReal_ne_zero.mpr ⟨(measure_ball_pos volume x hr0).ne', measure_ball_ne_top⟩  · apply (ENNReal.tendsto_toReal measure_ball_ne_top).comp    exact iUnion_cB ▸ tendsto_measure_iUnion_atTop h_mono