File size: 2,625 Bytes
b317718 | 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 | import OAI.Probability.GaussianReplacement.StreamVolume
import OAI.Probability.GaussianRegression.Sphere
namespace OAI
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology
noncomputable section
universe u
namespace NoiselessRegression
open InnerProductSpace
section SphereBounds
open scoped Pointwise
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
[FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
end SphereBounds
open scoped Pointwise
section MoreSphere
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
[FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
end MoreSphere
section FubiniAux
variable {X Y Z : Type*} [MeasurableSpace X] [MeasurableSpace Y] [MeasurableSpace Z]
{μ : Measure X} {ν : Measure Y} [SFinite μ] [SFinite ν]
lemma ae_fiber_aemeasurable {f : X × Y → Z} (hf : AEMeasurable f (μ.prod ν)) :
∀ᵐ y ∂ν, AEMeasurable (fun x => f (x,y)) μ := by
have he := Measure.ae_ae_of_ae_prod hf.prod_swap.ae_eq_mk
filter_upwards [he] with y hy
have hm : Measurable (fun x => hf.prod_swap.mk _ (y,x)) :=
hf.prod_swap.measurable_mk.comp (measurable_const.prodMk measurable_id)
exact hm.aemeasurable.congr (Filter.EventuallyEq.symm hy)
lemma prod_null_set_bound {S : Set (X × Y)} (hS : NullMeasurableSet S (μ.prod ν))
{c : ℝ≥0∞} (h : ∀ᵐ y ∂ν, μ {x | (x,y) ∈ S} ≤ c) :
(μ.prod ν) S ≤ c * ν univ := by
have hm : AEMeasurable (S.indicator (fun _ => (1 : ℝ≥0∞))) (μ.prod ν) :=
aemeasurable_const.indicator₀ hS
rw [← one_mul ((μ.prod ν) S), ← lintegral_indicator_const₀ hS 1, lintegral_prod_symm _ hm]
apply (lintegral_mono_ae ?_).trans_eq (lintegral_const c)
filter_upwards [h, ae_fiber_aemeasurable hm] with y hy hmy
have hs : NullMeasurableSet {x | (x,y) ∈ S} μ := by
have heq : {x | (x,y) ∈ S} = {x | S.indicator (fun _ => (1 : ℝ≥0∞)) (x,y) = 1} := by
ext x
by_cases hx : (x,y) ∈ S <;> simp [hx]
rw [heq]
exact nullMeasurableSet_eq_fun hmy aemeasurable_const
change (∫⁻ x, ({x | (x,y) ∈ S}).indicator (fun _ => (1 : ℝ≥0∞)) x ∂μ) ≤ c
simpa [lintegral_indicator_const₀ hs] using hy
end FubiniAux
instance sampleLaw_probability (d T : ℕ) : IsProbabilityMeasure (sampleLaw d T) := by
unfold sampleLaw
infer_instance
instance uniformSphere_probability (d : ℕ) [NeZero d] : IsProbabilityMeasure (uniformSphere d) := by
change IsProbabilityMeasure (sphereLaw (E := Vector d))
infer_instance
end NoiselessRegression
end
end OAI
|