| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Gradient.ClosedLinear |
| import OAI.Geometry.CAT0Fillings.Compactness.ChartDensity |
| import OAI.Geometry.CAT0Fillings.Compactness.Tails |
| import OAI.Geometry.CAT0Fillings.Compactness.Borel |
|
|
| namespace OAI |
|
|
| section |
|
|
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal BigOperators |
|
|
| namespace CAT0Fillings.JointBV |
| variable {β : Type*} [MeasurableSpace β] (σ : ℕ → Measure β) |
| (μ : Measure (ℕ × β)) [IsFiniteMeasure μ] |
| (hμ : μ = Measure.sum (fun i => (σ i).map (Prod.mk i))) |
|
|
| include hμ |
| omit [IsFiniteMeasure μ] in |
| lemma integral_prefix [IsFiniteMeasure μ] {f : ℕ × β → ℝ} (hf : Integrable f μ) (N : ℕ) : |
| (∫ w, (if w.1 < N then f w else 0) ∂μ) = |
| ∑ i ∈ Finset.range N, ∫ z, f (i,z) ∂σ i := by |
| have hs : MeasurableSet {w : ℕ × β | w.1 < N} := measurableSet_lt measurable_fst measurable_const |
| have hi : Integrable (fun w : ℕ × β => if w.1 < N then f w else 0) μ := by |
| apply (hf.indicator hs).congr |
| filter_upwards [] with w |
| by_cases hw : w.1 < N <;> simp [hw] |
| rw [hμ] at hi ⊢ |
| rw [integral_sum_measure hi] |
| have he (i : ℕ) : (∫ w : ℕ × β, (if w.1 < N then f w else 0) ∂(σ i).map (Prod.mk i)) = |
| if i < N then ∫ z, f (i,z) ∂σ i else 0 := by |
| rw [(measurableEmbedding_prodMk_left i).integral_map] |
| by_cases hi : i < N <;> simp [hi] |
| simp_rw [he] |
| rw [tsum_eq_sum (s := Finset.range N)] |
| · apply Finset.sum_congr rfl |
| intro i hi |
| simp only [Finset.mem_range] at hi |
| rw [ite_eq_left hi] |
| · intro i hi |
| simp only [Finset.mem_range] at hi |
| rw [ite_eq_right hi] |
|
|
| lemma integral_abs_le_finite_parts_add_tail {f : ℕ × β → ℝ} |
| (hf : MemLp f 2 μ) (N : ℕ) : |
| (∫ w, |f w| ∂μ) ≤ (∑ i ∈ Finset.range N, ∫ z, |f (i,z)| ∂σ i) + |
| lpNorm f 2 μ*lpNorm (indexTail Prod.fst N) 2 μ := by |
| have hi := (hf.integrable (by norm_num)).abs |
| have hs : MeasurableSet {w : ℕ × β | w.1 < N} := measurableSet_lt measurable_fst measurable_const |
| have hp : Integrable (fun w : ℕ × β => if w.1 < N then |f w| else 0) μ := by |
| apply (hi.indicator hs).congr |
| filter_upwards [] with w |
| by_cases hw : w.1 < N <;> simp [hw] |
| have ht : Integrable (fun w : ℕ × β => |f w| *|indexTail Prod.fst N w|) μ := by |
| apply hi.mul_bdd (c := 1) (measurable_indexTail Prod.fst measurable_fst N).abs.aestronglyMeasurable |
| exact Eventually.of_forall fun w => by |
| simpa only [Real.norm_eq_abs,abs_abs] using indexTail_bound Prod.fst N w |
| have he : (fun w => |f w|) = fun w => (if w.1 < N then |f w| else 0)+|f w| *|indexTail Prod.fst N w| := by |
| ext w |
| by_cases hw : w.1 < N |
| · simp [indexTail,hw,Nat.not_le.mpr hw] |
| · simp [indexTail,hw,Nat.le_of_not_gt hw] |
| calc |
| _ = (∫ w, (if w.1 < N then |f w| else 0) ∂μ)+(∫ w, |f w| *|indexTail Prod.fst N w| ∂μ) := by |
| rw [←integral_add hp ht,←he] |
| _ ≤ (∑ i ∈ Finset.range N, ∫ z, |f (i,z)| ∂σ i) +lpNorm f 2 μ*lpNorm (indexTail Prod.fst N) 2 μ := by |
| rw [integral_prefix σ μ hμ hi N] |
| exact add_le_add le_rfl (integral_abs_mul_le_lpNorm_two hf (memLp_indexTail μ Prod.fst measurable_fst N)) |
|
|
| end CAT0Fillings.JointBV |
| end |
|
|
| section |
|
|
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal BigOperators |
|
|
| namespace CAT0Fillings.ChartGeometry |
| open JointBV Foundations |
|
|
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] {k : ℕ} {T : Functional X (k+1)} |
| {hT : IsMetricCurrent T} (q : ChartGeometry hT) |
|
|
| noncomputable def value1 {u : X → ℝ} {K : ℝ≥0} (hu : LipschitzWith K u) : |
| Lp ℝ 1 (MassMeasure.currentMassMeasure hT) := |
| ((boundedLip_of_lipschitz hu).memLp 1).toLp u |
|
|
| omit [Nonempty X] in |
| lemma dist_value1 [Nonempty X] {u v : X → ℝ} {K J : ℝ≥0} |
| (hu : LipschitzWith K u) (hv : LipschitzWith J v) : |
| dist (value1 (hT := hT) hu) (value1 hv) = |
| ∫ x, |u x-v x| ∂MassMeasure.currentMassMeasure hT := by |
| rw [dist_eq_norm,L1.norm_eq_integral_norm] |
| apply integral_congr_ae |
| filter_upwards [Lp.coeFn_sub (value1 (hT := hT) hu) (value1 hv), |
| ((boundedLip_of_lipschitz hu).memLp (μ := MassMeasure.currentMassMeasure hT) 1).coeFn_toLp, |
| ((boundedLip_of_lipschitz hv).memLp (μ := MassMeasure.currentMassMeasure hT) 1).coeFn_toLp] with x hx hux hvx |
| rw [hx] |
| change ‖(((boundedLip_of_lipschitz hu).memLp 1).toLp u) x- |
| (((boundedLip_of_lipschitz hv).memLp 1).toLp v) x‖ = _ |
| rw [hux,hvx,Real.norm_eq_abs] |
|
|
| lemma dist_coordinateL1_eq (i : ℕ) {u v : X → ℝ} (hu : Measurable u) (hv : Measurable v) |
| (U V : ℝ≥0) (hU : ∀ x, |u x| ≤ U) (hV : ∀ x, |v x| ≤ V) : |
| dist (q.coordinateL1 i hu U hU) (q.coordinateL1 i hv V hV) = |
| ∫ z, |u ((q.chart i).paramExtended z)-v ((q.chart i).paramExtended z)| ∂q.coordinateMeasure i := by |
| rw [q.dist_coordinateL1,coordinateMeasure,integral_withDensity_eq_integral_toReal_smul₀ |
| (q.density_integrable i).aestronglyMeasurable.aemeasurable.ennreal_ofReal |
| (Eventually.of_forall fun _ => ENNReal.ofReal_lt_top)] |
| simp only [ENNReal.toReal_ofReal (q.density_nonneg i _),smul_eq_mul] |
|
|
| lemma dist_value1_le_prefix {u v : X → ℝ} {K J : ℝ≥0} |
| (hu : LipschitzWith K u) (hv : LipschitzWith J v) |
| (U V : ℝ≥0) (hU : ∀ x, |u x| ≤ U) (hV : ∀ x, |v x| ≤ V) (N : ℕ) : |
| dist (value1 (hT := hT) hu) (value1 hv) ≤ |
| (∑ i ∈ Finset.range N, dist (q.coordinateL1 i hu.continuous.measurable U hU) |
| (q.coordinateL1 i hv.continuous.measurable V hV)) + |
| (‖value (hT := hT) hu‖+‖value (hT := hT) hv‖)*lpNorm (indexTail Prod.fst N) 2 q.atlasMeasure := by |
| let μ := MassMeasure.currentMassMeasure hT |
| have hmp : MeasurePreserving q.atlasParam q.atlasMeasure μ := ⟨q.measurable_atlasParam,q.map_atlasMeasure⟩ |
| have hmem : MemLp (fun x => u x-v x) 2 μ := ((boundedLip_of_lipschitz hu).memLp 2).sub ((boundedLip_of_lipschitz hv).memLp 2) |
| have hma := hmem.comp_measurePreserving hmp |
| have hh := integral_abs_le_finite_parts_add_tail q.coordinateMeasure q.atlasMeasure rfl hma N |
| have hleft : (∫ w, |u (q.atlasParam w)-v (q.atlasParam w)| ∂q.atlasMeasure) = |
| dist (value1 (hT := hT) hu) (value1 hv) := by |
| rw [dist_value1,←q.map_atlasMeasure] |
| symm |
| simpa only [Pi.sub_apply] using integral_map q.measurable_atlasParam.aemeasurable |
| (hu.continuous.sub hv.continuous).abs.aestronglyMeasurable |
| change (∫ w, |u (q.atlasParam w)-v (q.atlasParam w)| ∂q.atlasMeasure) ≤ _ at hh |
| rw [hleft] at hh |
| have hnorm : lpNorm ((fun x => u x-v x) ∘ q.atlasParam) 2 q.atlasMeasure ≤ |
| ‖value (hT := hT) hu‖+‖value (hT := hT) hv‖ := by |
| rw [←toReal_eLpNorm,eLpNorm_comp_measurePreserving hmem.aestronglyMeasurable hmp] |
| have he : hmem.toLp _ = value (hT := hT) hu-value hv := |
| MemLp.toLp_sub ((boundedLip_of_lipschitz hu).memLp 2) ((boundedLip_of_lipschitz hv).memLp 2) |
| rw [←Lp.norm_toLp (fun x => u x-v x) hmem,he] |
| exact norm_sub_le _ _ |
| apply hh.trans |
| apply add_le_add |
| · apply Finset.sum_le_sum |
| intro index _ |
| rw [q.dist_coordinateL1_eq] |
| rfl |
| · exact mul_le_mul_of_nonneg_right hnorm lpNorm_nonneg |
|
|
| theorem totallyBounded_value1 {ι : Type*} |
| (h : Slicing.NormalApprox (k+1) T) (hz : boundarySucc T = 0) (hX : IsCAT0 X) |
| (u : ι → X → ℝ) (L U : ι → ℝ≥0) (hu : ∀ j, LipschitzWith (L j) (u j)) |
| (hU : ∀ j x, |u j x| ≤ U j) |
| (M : ℝ≥0) (hMv : ∀ j, ‖value (hT := hT) (hu j)‖ ≤ M) |
| (hMg : ∀ j, ‖q.gradient (hu j)‖ ≤ M) : |
| TotallyBounded (range fun j => value1 (hT := hT) (hu j)) := by |
| apply totallyBounded_range_of_finite_coordinate_control |
| (q.totallyBounded_coordinate h hz hX u L U hu hU M hMv hMg) |
| intro ε hε |
| have ht := (tendsto_lpNorm_indexTail q.atlasMeasure Prod.fst measurable_fst).const_mul (2*(M:ℝ)) |
| rw [mul_zero] at ht |
| obtain ⟨N,hN⟩ := ((tendsto_order.1 ht).2 (ε/2) (half_pos hε)).exists |
| refine ⟨N,ε/(2*((N:ℝ)+1)),div_pos hε (by positivity),?_⟩ |
| intro a b hab |
| have hh := q.dist_value1_le_prefix (hu a) (hu b) (U a) (U b) (hU a) (hU b) N |
| have hs : (∑ i ∈ Finset.range N, dist (q.coordinateL1 i (hu a).continuous.measurable (U a) (hU a)) |
| (q.coordinateL1 i (hu b).continuous.measurable (U b) (hU b))) ≤ (N:ℝ)*(ε/(2*((N:ℝ)+1))) := by |
| calc _ ≤ ∑ _i ∈ Finset.range N, ε/(2*((N:ℝ)+1)) := by |
| apply Finset.sum_le_sum |
| intro i hi |
| exact (hab ⟨i,Finset.mem_range.mp hi⟩).le |
| _ = _ := by simp |
| have hp := mul_le_mul_of_nonneg_right (add_le_add (hMv a) (hMv b)) |
| (show 0 ≤ lpNorm (indexTail Prod.fst N) 2 q.atlasMeasure from lpNorm_nonneg) |
| have hd : (N:ℝ)*(ε/(2*((N:ℝ)+1))) < ε/2 := by |
| have he : (N:ℝ)*(ε/(2*((N:ℝ)+1))) = (ε/2)*((N:ℝ)/((N:ℝ)+1)) := by field_simp |
| rw [he] |
| apply mul_lt_of_lt_one_right (half_pos hε) |
| exact (div_lt_one (by positivity)).2 (by linarith) |
| have : (M:ℝ)+(M:ℝ) = 2*(M:ℝ) := by ring |
| rw [this] at hp |
| linarith |
|
|
| end CAT0Fillings.ChartGeometry |
| end |
|
|
| end OAI |
|
|
