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caec230 | 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 | import OAI.MeasureTheory.DyadicAvoidance.ExceptionalExpectation
namespace OAI
universe u_A u_K
noncomputable section
open MeasureTheory
namespace Problem310.RandomSetDensity
/-- A selector outcome chooses the terminal address read at each point. -/
def selectedSet {A : Type u_A} {K : Type u_K} (addr : A → ℝ → K) (z : A × (K → Bool)) : Set ℝ :=
{x | z.2 (addr z.1 x) = true}
/-- Finite terminal tables preserve spatial measurability of the address map. -/
theorem measurableSet_selectedSet
{A : Type u_A} {K : Type u_K} [Fintype K] [MeasurableSpace K] [MeasurableSingletonClass K]
(addr : A → ℝ → K) (haddr : ∀ a, Measurable (addr a))
(z : A × (K → Bool)) : MeasurableSet (selectedSet addr z) := by
exact ((measurable_of_countable z.2).comp (haddr z.1)) (measurableSet_singleton true)
/-- Periodicity of terminal addresses gives periodicity of every realization. -/
theorem periodic_selectedSet
{A : Type u_A} {K : Type u_K} (addr : A → ℝ → K)
(haddr : ∀ a x, addr a (x + 1) = addr a x) (z : A × (K → Bool)) :
∀ x : ℝ, x + 1 ∈ selectedSet addr z ↔ x ∈ selectedSet addr z := by
intro x
simp only [selectedSet, Set.mem_ofPred_eq, haddr]
/-- Independence of terminal labels from selectors gives the same membership
probability at every spatial point, even with adaptively selected addresses. -/
theorem pointwise_probability_selectedSet
{A : Type u_A} {K : Type u_K} [Fintype A] [MeasurableSpace A] [MeasurableSingletonClass A]
[Fintype K] (P : Measure A) [IsProbabilityMeasure P]
(ν : Measure Bool) [IsProbabilityMeasure ν]
(addr : A → ℝ → K) (x : ℝ) :
(P.prod (Measure.pi (fun _ : K => ν))) {z | x ∈ selectedSet addr z} =
ν {true} := by
classical
have hmeas : MeasurableSet {z : A × (K → Bool) | x ∈ selectedSet addr z} :=
(Set.toFinite _).measurableSet
rw [Measure.prod_apply hmeas]
have hsection (a : A) :
(Measure.pi (fun _ : K => ν)) {ω : K → Bool | ω (addr a x) = true} = ν {true} := by
exact (measurePreserving_eval (fun _ : K => ν) (addr a x)).measure_preimage
(measurableSet_singleton true).nullMeasurableSet
change (∫⁻ a, (Measure.pi (fun _ : K => ν))
{ω : K → Bool | ω (addr a x) = true} ∂P) = ν {true}
simp only [hsection]
simp
/-- Exact first-moment density identity for the random routed set. -/
theorem expected_density_selectedSet
{A : Type u_A} {K : Type u_K} [Fintype A] [MeasurableSpace A] [MeasurableSingletonClass A]
[Fintype K] [MeasurableSpace K] [MeasurableSingletonClass K]
(P : Measure A) [IsProbabilityMeasure P]
(ν : Measure Bool) [IsProbabilityMeasure ν]
(addr : A → ℝ → K) (haddr : ∀ a, Measurable (addr a)) :
(∫⁻ z, volume (selectedSet addr z ∩ Set.Icc (0 : ℝ) 1)
∂(P.prod (Measure.pi (fun _ : K => ν)))) = ν {true} := by
classical
rw [Problem310.Auxiliary.expected_density_eq_pointwise_probability
(P.prod (Measure.pi (fun _ : K => ν))) (selectedSet addr)
(measurableSet_selectedSet addr haddr)]
simp only [pointwise_probability_selectedSet P ν addr]
simp
/-- The standard Boolean Bernoulli law has the expected real-parameter masses. -/
theorem bernoulli_true_mass (p : unitInterval) :
ProbabilityTheory.bernoulliMeasure true false p {true} =
ENNReal.ofReal (p : ℝ) := by
rw [ProbabilityTheory.bernoulliMeasure_apply_of_mem_of_notMem p
(measurableSet_singleton true) (by simp) (by simp)]
exact (ENNReal.ofReal_eq_coe_nnreal p.property.1).symm
theorem bernoulli_false_mass (p : unitInterval) :
ProbabilityTheory.bernoulliMeasure true false p {false} =
ENNReal.ofReal (1 - (p : ℝ)) := by
rw [ProbabilityTheory.bernoulliMeasure_apply_of_notMem_of_mem p
(measurableSet_singleton false) (by simp) (by simp)]
exact (ENNReal.ofReal_eq_coe_nnreal (sub_nonneg.mpr p.property.2)).symm
/-- Bernoulli specialization with exactly the `ofReal p` density budget. -/
theorem expected_density_bernoulli
{A : Type u_A} {K : Type u_K} [Fintype A] [MeasurableSpace A] [MeasurableSingletonClass A]
[Fintype K] [MeasurableSpace K] [MeasurableSingletonClass K]
(P : Measure A) [IsProbabilityMeasure P] (p : unitInterval)
(addr : A → ℝ → K) (haddr : ∀ a, Measurable (addr a)) :
(∫⁻ z, volume (selectedSet addr z ∩ Set.Icc (0 : ℝ) 1)
∂(P.prod (Measure.pi (fun _ : K =>
ProbabilityTheory.bernoulliMeasure true false p)))) = ENNReal.ofReal (p : ℝ) := by
rw [expected_density_selectedSet P _ addr haddr, bernoulli_true_mass]
end Problem310.RandomSetDensity
end
end OAI
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