File size: 2,721 Bytes
2fca478 | 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.Geometry.PinnedDistances.FiniteLaw
namespace OAI
universe u1 u2 u3 u4 u5
section
noncomputable section
open MeasureTheory Filter
open scoped Topology
namespace WeakPinned.Bounded
lemma integrable_one_bound {X : Type u1} [MeasurableSpace X] (μ : Measure X) [IsFiniteMeasure μ]
(f : X → ℝ) (hf : Measurable f) (hbound : ∀ x,‖f x‖≤1) : Integrable f μ := by
exact (integrable_const (1:ℝ)).mono' hf.aestronglyMeasurable (Eventually.of_forall hbound)
variable {Ω : Type u2} {E : Type u3} {X : Type u4} {Y : Type u5} [MeasurableSpace Ω] [MeasurableSpace E]
[MeasurableSpace X] [MeasurableSpace Y]
lemma integral_fiber_indicator (Q : Measure E) (μ : Measure X)
[IsProbabilityMeasure Q] [IsProbabilityMeasure μ]
(f : E × X → ℝ) (hf : Measurable f) (hbound : ∀ p,‖f p‖≤1)
(s : Set E) (hs : MeasurableSet s) :
(∫ e in s, ∫ x,f (e,x) ∂ μ ∂ Q)=
∫ p,(s ×ˢ Set.univ).indicator f p ∂ Q.prod μ := by
have hi := integrable_one_bound (Q.prod μ) f hf hbound
rw [integral_prod _ (hi.indicator (hs.prod MeasurableSet.univ))]
rw [← integral_indicator hs]
apply integral_congr_ae
filter_upwards [] with e
by_cases he : e∈s
· simp [Set.indicator_of_mem,he]
· simp [Set.indicator_of_notMem,he]
lemma conditional_integral_eq
(R : Measure Ω) (Q : Measure E) (μ : Measure X) (ν : Measure Y)
[IsProbabilityMeasure R] [IsProbabilityMeasure Q]
[IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
(e : Ω → E) (z : Ω → X) (w : Ω → Y)
(he : Measurable e) (hz : Measurable z) (hw : Measurable w)
(hmz : R.map (fun r => (e r,z r))=Q.prod μ)
(hmw : R.map (fun r => (e r,w r))=Q.prod ν)
(f : E × X → ℝ) (g : E × Y → ℝ)
(hf : Measurable f) (hg : Measurable g)
(hfb : ∀ p,‖f p‖≤1) (hgb : ∀ p,‖g p‖≤1)
(heq : ∀ᵐ r ∂ R,f (e r,z r)=g (e r,w r)) :
∀ᵐ a ∂ Q,(∫ x,f (a,x) ∂ μ)=(∫ y,g (a,y) ∂ ν) := by
apply Integrable.ae_eq_of_forall_setIntegral_eq
· exact (integrable_one_bound (Q.prod μ) f hf hfb).integral_prod_left
· exact (integrable_one_bound (Q.prod ν) g hg hgb).integral_prod_left
· intro s hs _
rw [integral_fiber_indicator Q μ f hf hfb s hs,
integral_fiber_indicator Q ν g hg hgb s hs,← hmz,← hmw]
rw [integral_map_of_stronglyMeasurable (he.prodMk hz)
(hf.indicator (hs.prod MeasurableSet.univ)).stronglyMeasurable,
integral_map_of_stronglyMeasurable (he.prodMk hw)
(hg.indicator (hs.prod MeasurableSet.univ)).stronglyMeasurable]
apply integral_congr_ae
filter_upwards [heq] with r hr
by_cases h : e r∈s <;> simp [h,hr]
end WeakPinned.Bounded
end
end
end OAI
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