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b797a5b | 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 | import OAI.NumberTheory.PrimePackets.ObservationSigmaAlgebra
namespace OAI
/-! # Identification with genuine conditional expectation -/
noncomputable section
open MeasureTheory Set
namespace PrimeDegreePackets
variable {Y α β : Type*} [MeasurableSpace Y] [Fintype β]
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
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
end PrimeDegreePackets
end
end OAI
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