| import OAI.NumberTheory.PrimePackets.ObservationSigmaAlgebra |
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| namespace OAI |
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| /-! # Identification with genuine conditional expectation -/ |
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| noncomputable section |
| open MeasureTheory Set |
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| namespace PrimeDegreePackets |
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| variable {Y α β : Type*} [MeasurableSpace Y] [Fintype β] |
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| omit [Fintype β] in |
| lemma setIntegral_observationConditionalProbability |
| (μ : Measure Y) [IsFiniteMeasure μ] (P : Y → α) (Q : Y → β) |
| (hQ : ∀ b, MeasurableSet {y | Q y = b}) (a : α) (b : β) : |
| (∫ y in {y | Q y = b}, observationConditionalProbability μ P Q a y ∂μ) = |
| μ.real {y | (Q y,P y) = (b,a)} := by |
| have heq : (fun y => observationConditionalProbability μ P Q a y) =ᵐ[μ.restrict {y | Q y = b}] |
| (fun _ => μ.real {y | (Q y,P y) = (b,a)} / μ.real {y | Q y = b}) := by |
| filter_upwards [ae_restrict_mem (hQ b)] with y hy |
| simp only [observationConditionalProbability, hy] |
| rw [integral_congr_ae heq, integral_const, measureReal_restrict_apply_univ, smul_eq_mul] |
| by_cases hb : μ.real {y | Q y = b} = 0 |
| · have hz : μ.real {y | (Q y,P y) = (b,a)} = 0 := by |
| apply le_antisymm _ measureReal_nonneg |
| have hsub : {y | (Q y,P y) = (b,a)} ⊆ {y | Q y = b} := |
| fun y hy => congrArg Prod.fst hy |
| calc |
| _ ≤ μ.real {y | Q y = b} := |
| measureReal_mono hsub (measure_ne_top _ _) |
| _ = 0 := hb |
| rw [hb,hz] |
| simp only [zero_div,mul_zero] |
| · exact mul_div_cancel₀ _ hb |
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| theorem observationConditionalProbability_ae_eq_condExp |
| (μ : Measure Y) [IsFiniteMeasure μ] (P : Y → α) (Q : Y → β) |
| (hP : ∀ a, MeasurableSet {y | P y = a}) |
| (hQ : ∀ b, MeasurableSet {y | Q y = b}) (a : α) : |
| observationConditionalProbability μ P Q a =ᵐ[μ] |
| μ[({y | P y = a}.indicator (fun _ => (1 : ℝ))) | observationSigma Q] := by |
| apply observation_values_ae_eq_condExp μ Q hQ _ ((integrable_const (1 : ℝ)).indicator (hP a)) |
| (fun b => μ.real {y | (Q y,P y) = (b,a)} / μ.real {y | Q y = b}) |
| intro b |
| change (∫ y in {y | Q y = b}, observationConditionalProbability μ P Q a y ∂μ) = _ |
| rw [setIntegral_observationConditionalProbability μ P Q hQ a b, |
| integral_indicator (hP a), integral_const] |
| simp only [measureReal_restrict_apply_univ, measureReal_restrict_apply (hP a), smul_eq_mul, mul_one] |
| apply congrArg μ.real |
| ext y |
| simp only [Prod.mk.injEq, Set.mem_ofPred_eq, Set.mem_inter_iff] |
| tauto |
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| end PrimeDegreePackets |
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| end |
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| end OAI |
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