| import Mathlib | |
| namespace OAI | |
| section | |
| open Set MeasureTheory Measure Filter Module | |
| open Set Filter MeasureTheory Measure ContinuousLinearMap | |
| open scoped Topology Convolution NNReal | |
| open Set Filter MeasureTheory Measure Metric | |
| open scoped Topology ContDiff | |
| open Set Filter Metric | |
| open Filter Set | |
| open Set Filter MeasureTheory TopologicalSpace | |
| open scoped Topology ENNReal | |
| open Set MeasureTheory | |
| open scoped RealInnerProductSpace | |
| open Matrix | |
| open scoped RealInnerProductSpace MatrixOrder | |
| open Set Filter MeasureTheory | |
| open scoped Topology ENNReal NNReal | |
| open MeasureTheory Filter Set Metric | |
| open scoped Topology Pointwise NNReal | |
| open scoped Topology NNReal | |
| open Set MeasureTheory Filter | |
| open scoped ENNReal NNReal Topology | |
| namespace SharpIntegralFillings.Slicing | |
| variable {X : Type*} [MeasurableSpace X] (μ : Measure X) [IsFiniteMeasure μ] | |
| {u : X → ℝ} | |
| noncomputable def stripMass (u : X → ℝ) (a : ℝ≥0) (t : ℝ) : ℝ≥0∞ := | |
| (a:ℝ≥0∞) * μ {x | t ≤ u x ∧ u x ≤ t+(a:ℝ)⁻¹} | |
| lemma measurable_stripMass (hu : Measurable u) (a : ℝ≥0) : | |
| Measurable (stripMass μ u a) := by | |
| have hset : MeasurableSet {p : ℝ × X | p.1 ≤ u p.2 ∧ u p.2 ≤ p.1+(a:ℝ)⁻¹} := | |
| (measurableSet_le measurable_fst (hu.comp measurable_snd)).inter | |
| (measurableSet_le (hu.comp measurable_snd) (measurable_fst.add_const _)) | |
| let f : ℝ → X → ℝ≥0∞ := fun t x => | |
| if t ≤ u x ∧ u x ≤ t+(a:ℝ)⁻¹ then (a:ℝ≥0∞) else 0 | |
| have hf : Measurable (Function.uncurry f) := by | |
| exact Measurable.ite hset measurable_const measurable_const | |
| have heq : stripMass μ u a = fun t => ∫⁻ x, f t x ∂μ := by | |
| funext t | |
| have h : MeasurableSet {x | t ≤ u x ∧ u x ≤ t+(a:ℝ)⁻¹} := | |
| (measurableSet_le measurable_const hu).inter | |
| (measurableSet_le hu measurable_const) | |
| simpa only [stripMass,Set.indicator_apply,Set.mem_ofPred_eq,lintegral_const, | |
| Measure.restrict_apply_univ,f] using | |
| (lintegral_indicator h (fun _ => (a:ℝ≥0∞))).symm | |
| rw [heq] | |
| exact hf.lintegral_prod_right | |
| lemma lintegral_stripMass (hu : Measurable u) {a : ℝ≥0} (ha : 0 < a) : | |
| ∫⁻ t : ℝ, stripMass μ u a t = μ univ := by | |
| have hset : MeasurableSet {p : ℝ × X | p.1 ≤ u p.2 ∧ u p.2 ≤ p.1+(a:ℝ)⁻¹} := | |
| (measurableSet_le measurable_fst (hu.comp measurable_snd)).inter | |
| (measurableSet_le (hu.comp measurable_snd) (measurable_fst.add_const _)) | |
| let f : ℝ → X → ℝ≥0∞ := fun t x => | |
| if t ≤ u x ∧ u x ≤ t+(a:ℝ)⁻¹ then (a:ℝ≥0∞) else 0 | |
| have hf : Measurable (Function.uncurry f) := | |
| Measurable.ite hset measurable_const measurable_const | |
| have heq : ∀ t, stripMass μ u a t = ∫⁻ x, f t x ∂μ := by | |
| intro t | |
| have h : MeasurableSet {x | t ≤ u x ∧ u x ≤ t+(a:ℝ)⁻¹} := | |
| (measurableSet_le measurable_const hu).inter | |
| (measurableSet_le hu measurable_const) | |
| simpa only [stripMass,Set.indicator_apply,Set.mem_ofPred_eq,lintegral_const, | |
| Measure.restrict_apply_univ,f] using | |
| (lintegral_indicator h (fun _ => (a:ℝ≥0∞))).symm | |
| simp_rw [heq] | |
| rw [lintegral_lintegral_swap hf.aemeasurable] | |
| have ht : ∀ x, ∫⁻ t : ℝ, f t x = 1 := by | |
| intro x | |
| have heq' : (fun t => f t x) = (Icc (u x-(a:ℝ)⁻¹) (u x)).indicator (fun _ => (a:ℝ≥0∞)) := by | |
| funext t | |
| have hequiv : (t ≤ u x ∧ u x ≤ t+(a:ℝ)⁻¹) ↔ | |
| t ∈ Icc (u x-(a:ℝ)⁻¹) (u x) := by | |
| simp only [mem_Icc]; constructor <;> intro h <;> constructor <;> linarith [h.1,h.2] | |
| simp only [f,Set.indicator_apply,hequiv] | |
| rw [heq',lintegral_indicator measurableSet_Icc,lintegral_const,Measure.restrict_apply_univ, | |
| Real.volume_Icc] | |
| simp only [sub_sub_cancel] | |
| rw [ENNReal.ofReal_inv_of_pos (show 0 < (a:ℝ) from ha),ENNReal.ofReal_coe_nnreal] | |
| exact ENNReal.mul_inv_cancel (by exact_mod_cast ne_of_gt ha) ENNReal.coe_ne_top | |
| simp_rw [ht] | |
| simp | |
| lemma ae_finite_strip_liminf (hu : Measurable u) : | |
| ∀ᵐ t : ℝ, liminf (fun n : ℕ => stripMass μ u ((n:ℝ≥0)+1) t) atTop < (⊤ : ℝ≥0∞) := by | |
| have hmeas := fun n : ℕ => measurable_stripMass μ hu ((n:ℝ≥0)+1) | |
| have hbound : (∫⁻ t : ℝ, liminf (fun n : ℕ => stripMass μ u ((n:ℝ≥0)+1) t) atTop) ≤ μ univ := by | |
| have h := lintegral_liminf_le (μ := volume) (u := atTop) hmeas | |
| have hi : ∀ n : ℕ, ∫⁻ t : ℝ, stripMass μ u ((n:ℝ≥0)+1) t = μ univ := | |
| fun n => lintegral_stripMass μ hu (by positivity) | |
| simpa only [hi,liminf_const] using h | |
| exact ae_lt_top (Measurable.liminf hmeas) (ne_top_of_le_ne_top (measure_ne_top μ univ) hbound) | |
| lemma ae_bounded_strip_subsequence (hu : Measurable u) : | |
| ∀ᵐ t : ℝ, ∃ (j : ℕ → ℕ) (M : ℝ≥0), Tendsto j atTop atTop ∧ | |
| ∀ n, stripMass μ u ((j n:ℝ≥0)+1) t ≤ M := by | |
| filter_upwards [ae_finite_strip_liminf μ hu] with t ht | |
| let r := liminf (fun n : ℕ => stripMass μ u ((n:ℝ≥0)+1) t) atTop | |
| let M : ℝ≥0 := r.toNNReal+1 | |
| have hr : r < (M:ℝ≥0∞) := by | |
| change r < ((r.toNNReal+1:ℝ≥0):ℝ≥0∞) | |
| rw [ENNReal.coe_add,ENNReal.coe_toNNReal ht.ne] | |
| exact ENNReal.lt_add_right ht.ne (by simp) | |
| have hf := frequently_lt_of_liminf_lt (u := fun n : ℕ => stripMass μ u ((n:ℝ≥0)+1) t) (h := hr) | |
| choose j hj hbound using frequently_atTop.mp hf | |
| refine ⟨j,M,?_,fun n => (hbound n).le⟩ | |
| exact tendsto_atTop_mono hj tendsto_id | |
| end SharpIntegralFillings.Slicing | |
| end | |
| end OAI | |