File size: 3,472 Bytes
f205175 | 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 | import Mathlib
import OAI.Probability.CriticalPercolation.ProductMeasure.Density
namespace OAI
section
section
open scoped BigOperators
open Set MeasureTheory ProbabilityTheory
open scoped Classical ENNReal
namespace CriticalPercolation
namespace BernoulliProduct
noncomputable def bitMeasure (p : unitInterval) : Measure Prop :=
unitInterval.toNNReal p • Measure.dirac True +
unitInterval.toNNReal (unitInterval.symm p) • Measure.dirac False
instance bitMeasure_probability (p : unitInterval) : IsProbabilityMeasure (bitMeasure p) := by
constructor
simp [bitMeasure]
noncomputable def bitWeight (p : unitInterval) (b : Prop) : ℝ :=
if b then p else 1-p
lemma bitWeight_sum (p : unitInterval) : ∑ b, bitWeight p b = 1 := by
simp [bitWeight]
lemma bitWeight_nonneg (p : unitInterval) (b : Prop) : 0 ≤ bitWeight p b := by
unfold bitWeight
split_ifs
· exact p.property.1
· exact sub_nonneg.mpr p.property.2
lemma bitMeasure_real_singleton (p : unitInterval) (b : Prop) :
(bitMeasure p).real {b} = bitWeight p b := by
by_cases hb : b
· have he : b = True := propext (iff_true_intro hb)
subst b
simp [bitMeasure, bitWeight, measureReal_def]
· have he : b = False := propext (iff_false_intro hb)
subst b
simp [bitMeasure, bitWeight, measureReal_def]
variable {E : Type*}
noncomputable def law (q : E → unitInterval) : Measure (Set E) :=
Measure.map MeasurableEquiv.setOfPred (Measure.infinitePi fun e => bitMeasure (q e))
instance law_probability (q : E → unitInterval) : IsProbabilityMeasure (law q) := by
unfold law
infer_instance
lemma law_const (p : unitInterval) :
law (fun _ : E => p) = setBernoulli Set.univ p := by
rw [law, setBernoulli_eq_map]
rfl
lemma integral_bits_eq_expect [Fintype E] (q : E → unitInterval) (f : (E → Prop) → ℝ) :
(∫ ω, f ω ∂(Measure.infinitePi fun e => bitMeasure (q e))) =
FiniteProduct.expect (fun e => bitWeight (q e)) f := by
rw [Measure.infinitePi_eq_pi, integral_fintype Integrable.of_finite]
unfold FiniteProduct.expect FiniteProduct.density
apply Finset.sum_congr rfl
intro ω _
rw [smul_eq_mul]
congr 1
simp only [measureReal_def, Measure.pi_singleton, ENNReal.toReal_prod]
exact Finset.prod_congr rfl (fun e _ => bitMeasure_real_singleton (q e) (ω e))
lemma integral_law_eq_expect [Fintype E] (q : E → unitInterval) (f : Set E → ℝ) :
(∫ ω, f ω ∂law q) =
FiniteProduct.expect (fun e => bitWeight (q e)) (fun ω => f {e | ω e}) := by
rw [law, integral_map MeasurableEquiv.setOfPred.measurable.aemeasurable
Integrable.of_finite.aestronglyMeasurable, integral_bits_eq_expect]
rfl
lemma law_map_restrict (q : E → unitInterval) (s : Set E) :
(law q).map (fun ω : Set E => Subtype.val ⁻¹' ω : Set E → Set s) =
law (fun e : s => q e) := by
have hmeas : Measurable (fun ω : Set E => (Subtype.val ⁻¹' ω : Set s)) :=
measurable_set_iff.mpr (fun bond => measurable_set_mem bond.val)
unfold law
rw [Measure.map_map hmeas MeasurableEquiv.setOfPred.measurable]
have hcomp : (fun ω : Set E => Subtype.val ⁻¹' ω : Set E → Set s) ∘
MeasurableEquiv.setOfPred =
MeasurableEquiv.setOfPred ∘ s.domRestrict := rfl
rw [hcomp, ← Measure.map_map MeasurableEquiv.setOfPred.measurable (by fun_prop),
Measure.infinitePi_map_restrict']
end BernoulliProduct
end CriticalPercolation
open Set
open scoped Classical
end
end
end OAI
|