| import Mathlib |
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| namespace OAI |
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| section |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
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| namespace SharpIntegralFillings |
| open MeasureTheory |
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| noncomputable def densityPush {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] |
| (μ : Measure α) (φ : α → β) (ρ : α → ℝ) : Measure β := by |
| classical |
| exact if AEMeasurable φ (μ.withDensity (fun x => ENNReal.ofReal (ρ x))) then |
| Measure.map φ (μ.withDensity (fun x => ENNReal.ofReal (ρ x))) else 0 |
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| lemma densityPush_eq_map {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] |
| (μ : Measure α) {φ : α → β} (hφ : Measurable φ) (ρ : α → ℝ) : |
| densityPush μ φ ρ = Measure.map φ (μ.withDensity (fun x => ENNReal.ofReal (ρ x))) := by |
| exact ite_eq_left hφ.aemeasurable |
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| lemma densityPush_finite {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] |
| (μ : Measure α) (φ : α → β) {ρ : α → ℝ} (hρ : Integrable ρ μ) : |
| IsFiniteMeasure (densityPush μ φ ρ) := by |
| let := isFiniteMeasure_withDensity_ofReal hρ.hasFiniteIntegral |
| unfold densityPush |
| split |
| · exact MeasureTheory.Measure.isFiniteMeasure_map _ _ |
| · infer_instance |
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| lemma integral_densityPush {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] |
| [TopologicalSpace β] [BorelSpace β] |
| (μ : Measure α) {φ : α → β} (hφ : Measurable φ) {ρ : α → ℝ} |
| (hρ : Integrable ρ μ) (hρ0 : 0 ≤ᵐ[μ] ρ) |
| {b : β → ℝ} (hb : Continuous b) : |
| (∫ x, b x ∂densityPush μ φ ρ) = ∫ z, ρ z*b (φ z) ∂μ := by |
| rw [densityPush_eq_map μ hφ,integral_map hφ.aemeasurable hb.aestronglyMeasurable, |
| integral_withDensity_eq_integral_toReal_smul₀ |
| hρ.aestronglyMeasurable.aemeasurable.ennreal_ofReal |
| (Eventually.of_forall fun _ => ENNReal.ofReal_lt_top)] |
| apply integral_congr_ae |
| filter_upwards [hρ0] with z hz |
| simp only [ENNReal.toReal_ofReal hz,smul_eq_mul] |
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| lemma integral_densityPush_measurable {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] |
| (μ : Measure α) {φ : α → β} (hφ : Measurable φ) {ρ : α → ℝ} |
| (hρ : Integrable ρ μ) (hρ0 : 0 ≤ᵐ[μ] ρ) |
| {b : β → ℝ} (hb : Measurable b) : |
| (∫ x, b x ∂densityPush μ φ ρ) = ∫ z, ρ z*b (φ z) ∂μ := by |
| rw [densityPush_eq_map μ hφ,integral_map hφ.aemeasurable hb.aestronglyMeasurable, |
| integral_withDensity_eq_integral_toReal_smul₀ |
| hρ.aestronglyMeasurable.aemeasurable.ennreal_ofReal |
| (Eventually.of_forall fun _ => ENNReal.ofReal_lt_top)] |
| apply integral_congr_ae |
| filter_upwards [hρ0] with z hz |
| simp only [ENNReal.toReal_ofReal hz,smul_eq_mul] |
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| section BorelChartTests |
| variable {Z X : Type*} [MeasurableSpace Z] [MetricSpace X] [CompactSpace X] |
| [MeasurableSpace X] [BorelSpace X] |
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| omit [MetricSpace X] [CompactSpace X] [BorelSpace X] in |
| lemma integrable_weighted_comp (μ : Measure Z) {φ : Z → X} (hφ : Measurable φ) |
| {w : Z → ℝ} (hw : Integrable w μ) {b : X → ℝ} |
| (hb : Measurable b) (hB : ∃ M : ℝ, ∀ x, |b x| ≤ M) : |
| Integrable (fun z => w z * b (φ z)) μ := by |
| obtain ⟨M,hM⟩ := hB |
| exact hw.mul_bdd (hb.comp hφ).aestronglyMeasurable |
| (Eventually.of_forall fun z => by simpa only [Real.norm_eq_abs] using hM (φ z)) |
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| end BorelChartTests |
| end SharpIntegralFillings |
| end |
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| end OAI |
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