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dd14c26 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 | import Mathlib
namespace OAI
open MeasureTheory ProbabilityTheory Set
open scoped BigOperators ENNReal
namespace RepeatedLocalization
namespace FiniteEntropy
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))
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
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]
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
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
end RepeatedLocalization
end OAI
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