| import Mathlib |
|
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| namespace OAI |
|
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| open MeasureTheory ProbabilityTheory Set |
| open scoped BigOperators ENNReal |
|
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| namespace RepeatedLocalization |
|
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| namespace FiniteEntropy |
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|
| lemma lintegral_countable_comp {α ι : Type*} [MeasurableSpace α] [MeasurableSpace ι] |
| [Countable ι] [MeasurableSingletonClass ι] (μ : Measure α) (L : α → ι) |
| (hL : Measurable L) (F : ι → ℝ≥0∞) : |
| (∫⁻ x,F (L x) ∂μ) = ∑' i,F i * μ {x | L x = i} := by |
| rw [← lintegral_map (measurable_of_countable F) hL,lintegral_countable'] |
| congr 1 |
| funext i |
| rw [Measure.map_apply hL (measurableSet_singleton i)] |
| rfl |
|
|
| lemma fiber_union {α ι : Type*} (L : α → ι) : (⋃ i,{x | L x = i}) = Set.univ := by |
| ext x |
| simp only [Set.mem_iUnion,Set.mem_ofPred_eq,Set.mem_univ,iff_true] |
| exact ⟨L x,rfl⟩ |
|
|
| lemma fiber_pairwise {α ι : Type*} (L : α → ι) : |
| Pairwise (fun i j => Disjoint {x | L x = i} {x | L x = j}) := by |
| intro i j hij |
| exact Set.disjoint_left.mpr (fun x hi hj => hij (hi.symm.trans hj)) |
|
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| lemma hasSum_integral_fibers {α ι : Type*} [MeasurableSpace α] [MeasurableSpace ι] |
| [Countable ι] [MeasurableSingletonClass ι] (μ : Measure α) {L : α → ι} |
| (hL : Measurable L) {g : α → ℝ} (hg : Integrable g μ) : |
| HasSum (fun i => ∫ x in {x | L x = i},g x ∂μ) (∫ x,g x ∂μ) := by |
| simpa only [fiber_union,Measure.restrict_univ] using hasSum_integral_iUnion (s := fun i => {x | L x = i}) |
| (fun i => hL (measurableSet_singleton i)) (fiber_pairwise L) hg.integrableOn |
|
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| lemma integral_comp_fiber {α ι : Type*} [MeasurableSpace α] (μ : Measure α) |
| (L : α → ι) (F : ι → ℝ) (i : ι) (hi : MeasurableSet {x | L x = i}) : |
| (∫ x in {x | L x = i},F (L x) ∂μ) = (μ {x | L x = i}).toReal * F i := by |
| rw [setIntegral_congr_fun hi (fun x hx => congrArg F hx)] |
| simp only [integral_const,Measure.real,Measure.restrict_apply_univ,smul_eq_mul] |
|
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| lemma integrableOn_comp_fiber {α ι : Type*} [MeasurableSpace α] (μ : Measure α) [IsFiniteMeasure μ] |
| (L : α → ι) (F : ι → ℝ) (i : ι) (hi : MeasurableSet {x | L x = i}) : |
| IntegrableOn (fun x => F (L x)) {x | L x = i} μ := |
| (integrableOn_const (C := F i) (s := {x | L x = i})).congr_fun (fun x hx => (congrArg F hx).symm) hi |
|
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| lemma countable_cost_bound {α ι : Type*} [MeasurableSpace α] [MeasurableSpace ι] |
| [Countable ι] [MeasurableSingletonClass ι] (μ : Measure α) [IsFiniteMeasure μ] |
| (L : α → ι) (hL : Measurable L) (F : ι → ℝ) (hF : ∀ i,0 ≤ F i) |
| {g : α → ℝ} (hg : Integrable g μ) |
| (hb : ∀ i,(μ {x | L x = i}).toReal * F i ≤ ∫ x in {x | L x = i},g x ∂μ) : |
| Integrable (fun x => F (L x)) μ ∧ (∫ x,F (L x) ∂μ) ≤ ∫ x,g x ∂μ := by |
| have hsumG := hasSum_integral_fibers μ hL hg |
| have hnon (i : ι) : 0 ≤ (μ {x | L x = i}).toReal * F i := mul_nonneg ENNReal.toReal_nonneg (hF i) |
| have hsum : Summable (fun i => (μ {x | L x = i}).toReal * F i) := |
| Summable.of_nonneg_of_le hnon hb hsumG.summable |
| have hInt : Integrable (fun x => F (L x)) μ := by |
| have hs := integrableOn_iUnion_of_summable_integral_norm |
| (fun i => integrableOn_comp_fiber μ L F i (hL (measurableSet_singleton i))) |
| apply (integrableOn_univ.mp ?_) |
| rw [← fiber_union L] |
| apply hs |
| have he (i : ι) : (∫ x in {x | L x = i},‖F (L x)‖ ∂μ) = |
| (μ {x | L x = i}).toReal * F i := by |
| rw [integral_comp_fiber μ L (fun j => ‖F j‖) i (hL (measurableSet_singleton i))] |
| rw [Real.norm_eq_abs,abs_of_nonneg (hF i)] |
| simpa only [he] using hsum |
| refine ⟨hInt,?_⟩ |
| have hsumF := hasSum_integral_fibers μ hL hInt |
| rw [← hsumF.tsum_eq,← hsumG.tsum_eq] |
| apply Summable.tsum_le_tsum _ hsumF.summable hsumG.summable |
| intro i |
| rw [integral_comp_fiber μ L F i (hL (measurableSet_singleton i))] |
| exact hb i |
|
|
| lemma partition_integral_bound {α ι : Type*} [MeasurableSpace α] [MeasurableSpace ι] |
| [Countable ι] [MeasurableSingletonClass ι] (μ : Measure α) |
| (L : α → ι) (hL : Measurable L) {f g : α → ℝ} (hf0 : ∀ x,0 ≤ f x) |
| (hf : ∀ i,IntegrableOn f {x | L x = i} μ) (hg : Integrable g μ) |
| (hb : ∀ i,(∫ x in {x | L x = i},f x ∂μ) ≤ ∫ x in {x | L x = i},g x ∂μ) : |
| Integrable f μ ∧ (∫ x,f x ∂μ) ≤ ∫ x,g x ∂μ := by |
| have hsumG := hasSum_integral_fibers μ hL hg |
| have hsum : Summable (fun i => ∫ x in {x | L x = i},f x ∂μ) := |
| Summable.of_nonneg_of_le (fun i => integral_nonneg hf0) hb hsumG.summable |
| have hInt : Integrable f μ := by |
| have hs : IntegrableOn f (⋃ i,{x | L x = i}) μ := |
| integrableOn_iUnion_of_summable_integral_norm hf (by |
| simpa only [Real.norm_eq_abs,abs_of_nonneg (hf0 _)] using hsum) |
| simpa only [fiber_union,integrableOn_univ] using hs |
| refine ⟨hInt,?_⟩ |
| have hsumF := hasSum_integral_fibers μ hL hInt |
| rw [← hsumF.tsum_eq,← hsumG.tsum_eq] |
| exact Summable.tsum_le_tsum hb hsumF.summable hsumG.summable |
|
|
| end FiniteEntropy |
|
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| end RepeatedLocalization |
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| end OAI |
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