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67aee1d | 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 | import Mathlib
namespace OAI
section
section
noncomputable section
open MeasureTheory Set
open scoped ENNReal
namespace HardSphere.Comparison
variable {X : Type*} [MeasurableSpace X]
def subtypeConditional (P : Measure X) (S : Set X) : Measure S :=
(P S)⁻¹ • P.comap Subtype.val
lemma subtypeConditional_apply (P : Measure X) {S : Set X} (hS : MeasurableSet S) (A : Set S) :
subtypeConditional P S A=(P S)⁻¹*P (Subtype.val '' A) := by
rw [subtypeConditional,Measure.smul_apply,smul_eq_mul,comap_subtype_coe_apply hS]
lemma subtypeConditional_probability (P : Measure X) [IsFiniteMeasure P]
{S : Set X} (hS : MeasurableSet S) (hP : P S≠0) : IsProbabilityMeasure (subtypeConditional P S) := by
constructor
rw [subtypeConditional_apply P hS]
simp only [Set.image_univ,Subtype.range_coe]
exact ENNReal.inv_mul_cancel hP (measure_ne_top _ _)
lemma subtypeConditional_map (P : Measure X) {S : Set X} (hS : MeasurableSet S) :
(subtypeConditional P S).map (Subtype.val:S→X)=(P S)⁻¹ • P.restrict S := by
rw [subtypeConditional,Measure.map_smul, map_comap_subtype_coe hS]
exact measurable_subtype_coe.aemeasurable
lemma subtypeConditional_preimage (P : Measure X) {S : Set X} (hS : MeasurableSet S) (A : Set X) :
subtypeConditional P S (Subtype.val ⁻¹' A)=(P S)⁻¹*P (A∩S) := by
rw [subtypeConditional_apply P hS]
have he : Subtype.val '' (Subtype.val ⁻¹' A : Set S) = A∩S := by
ext x
simp only [Set.mem_image,Set.mem_preimage,Set.mem_inter_iff]
constructor
· rintro ⟨y,hy,rfl⟩
exact ⟨hy,y.property⟩
· rintro ⟨hx,hs⟩
exact ⟨⟨x,hs⟩,hx,rfl⟩
rw [he]
lemma subtypeConditional_le {S : Set X} (P : Measure X) (hS : MeasurableSet S)
(A : Set X) {a b : ℝ≥0∞} (ha : a≤P S) (hb : P (A∩S)≤b) :
subtypeConditional P S (Subtype.val ⁻¹' A)≤a⁻¹*b := by
rw [subtypeConditional_preimage P hS]
exact mul_le_mul' (ENNReal.inv_le_inv.mpr ha) hb
end HardSphere.Comparison
end
end
end
end OAI
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