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a2ee19a | 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 | import OAI.Probability.SATComputability.IncidentMasks
import OAI.Probability.SATComputability.PoissonCounts
namespace OAI
namespace FixedClauseThreshold.Computability
open DilutedSpinGlass _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
open scoped BigOperators Classical
theorem finite_indicator_integrable {X Y : Type*} [MeasurableSpace X]
[Fintype Y] [MeasurableSpace Y] [MeasurableSingletonClass Y]
(μ : Measure X) [IsProbabilityMeasure μ] (g : X → Y) (hg : Measurable g) (y : Y) :
Integrable (fun x => if g x = y then (1 : ℝ) else 0) μ := by
apply Integrable.of_bound
(((measurable_of_countable (fun z : Y => if z = y then (1 : ℝ) else 0)).comp hg).aestronglyMeasurable) 1
filter_upwards [] with x
dsimp only [Function.comp_def]
split_ifs <;> norm_num
noncomputable def finitePushforward {X Y : Type*} [MeasurableSpace X]
[Fintype Y] [MeasurableSpace Y] [MeasurableSingletonClass Y]
(μ : Measure X) [IsProbabilityMeasure μ] (g : X → Y) (hg : Measurable g) : FiniteLaw Y where
weight y := ∫ x, if g x = y then 1 else 0 ∂μ
nonneg y := integral_nonneg (fun x => by split_ifs <;> norm_num)
total := by
rw [← integral_finsetSum _ (fun y _ => finite_indicator_integrable μ g hg y)]
simp
theorem finitePushforward_expect {X Y : Type*} [MeasurableSpace X]
[Fintype Y] [MeasurableSpace Y] [MeasurableSingletonClass Y]
(μ : Measure X) [IsProbabilityMeasure μ] (g : X → Y) (hg : Measurable g) (f : Y → ℝ) :
(finitePushforward μ g hg).expect f = ∫ x, f (g x) ∂μ := by
simp only [FiniteLaw.expect, finitePushforward]
simp_rw [← integral_mul_const]
rw [← integral_finsetSum _ (fun y _ => (finite_indicator_integrable μ g hg y).mul_const (f y))]
apply integral_congr_ae
filter_upwards [] with x
simp
theorem bounded_finite_observable {X Y : Type*} [MeasurableSpace X]
[Fintype Y] [MeasurableSpace Y] [MeasurableSingletonClass Y]
(μ : Measure X) [IsProbabilityMeasure μ] (g : X → Y) (hg : Measurable g) (f : Y → ℝ) :
Integrable (fun x => f (g x)) μ := by
apply Integrable.of_bound ((measurable_of_countable f).comp hg).aestronglyMeasurable
(∑ y, |f y|)
filter_upwards [] with x
rw [Real.norm_eq_abs]
exact Finset.single_le_sum (fun y _ => abs_nonneg (f y)) (Finset.mem_univ (g x))
end FixedClauseThreshold.Computability
end OAI
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