openAI-math-dataset / lean /OAI /Geometry /CylinderCovering /Convexity /CompactOuterConvergence.lean
| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter MeasureTheory Metric | |
| open scoped Topology | |
| namespace CylinderCovering.CompactOuterConvergence | |
| variable {E : Type*} [MetricSpace E] [MeasurableSpace E] | |
| [OpensMeasurableSpace E] [ProperSpace E] | |
| {μ : Measure E} [IsFiniteMeasureOnCompacts μ] | |
| theorem cthickening_excess_tendsto {D : Set E} (hD : IsCompact D) : | |
| Tendsto (fun r : ℝ => μ (cthickening r D \ D)) (𝓝 0) (𝓝 0) := by | |
| have h := (ENNReal.continuous_sub_right (μ D)).tendsto (μ D) |>.comp | |
| (tendsto_measure_cthickening_of_isCompact (μ := μ) hD) | |
| have heq : ∀ r : ℝ, μ (cthickening r D \ D) = μ (cthickening r D) - μ D := | |
| fun _ => measure_sdiff (self_subset_cthickening D) | |
| hD.measurableSet.nullMeasurableSet hD.measure_lt_top.ne | |
| simpa only [heq, tsub_self, Function.comp_def] using h | |
| theorem excess_tendsto_of_eventually_subset_cthickening | |
| {ι : Type*} {l : Filter ι} {D : Set E} (hD : IsCompact D) | |
| {U : ι → Set E} {r : ι → ℝ} (hr : Tendsto r l (𝓝 0)) | |
| (hU : ∀ᶠ i in l, U i ⊆ cthickening (r i) D) : | |
| Tendsto (fun i => μ (U i \ D)) l (𝓝 0) := by | |
| apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds | |
| ((cthickening_excess_tendsto (μ := μ) hD).comp hr) | |
| · exact Eventually.of_forall (fun _ => bot_le) | |
| · filter_upwards [hU] with i hi | |
| exact measure_mono (sdiff_subset_sdiff_left hi) | |
| theorem real_excess_tendsto_of_eventually_subset_cthickening | |
| {ι : Type*} {l : Filter ι} {D : Set E} (hD : IsCompact D) | |
| {U : ι → Set E} {r : ι → ℝ} (hr : Tendsto r l (𝓝 0)) | |
| (hU : ∀ᶠ i in l, U i ⊆ cthickening (r i) D) : | |
| Tendsto (fun i => μ.real (U i \ D)) l (𝓝 0) := by | |
| simpa only [measureReal_def, ENNReal.toReal_zero, Function.comp_def] using | |
| (ENNReal.tendsto_toReal ENNReal.zero_ne_top).comp | |
| (excess_tendsto_of_eventually_subset_cthickening hD hr hU) | |
| theorem linear_mesh_real_excess_tendsto {D : Set E} (hD : IsCompact D) | |
| (C : ℝ) {U : ℝ → Set E} | |
| (hU : ∀ δ : ℝ, 0 < δ → U δ ⊆ cthickening (C * δ) D) : | |
| Tendsto (fun δ : ℝ => μ.real (U δ \ D)) (𝓝[>] 0) (𝓝 0) := by | |
| apply real_excess_tendsto_of_eventually_subset_cthickening hD | |
| (r := fun δ => C * δ) | |
| · simpa using (tendsto_const_nhds.mul | |
| (tendsto_id.mono_left nhdsWithin_le_nhds) : | |
| Tendsto (fun δ : ℝ => C * δ) (𝓝[>] 0) (𝓝 (C * 0))) | |
| · filter_upwards [self_mem_nhdsWithin] with δ hδ | |
| exact hU δ hδ | |
| end CylinderCovering.CompactOuterConvergence | |
| end | |
| end OAI | |