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c4cdbbd | 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 69 70 71 72 73 74 75 76 77 78 79 80 81 | import Mathlib.MeasureTheory.Measure.Prod
namespace OAI
/-! Joint orbit laws from measure-preserving rotations and pointwise Haar
kernels. The proof avoids measurable eigenvector selections. -/
noncomputable section
open MeasureTheory
namespace InvariantIsing
lemma paired_map_eq_of_comp {G D : Type*} [MeasurableSpace G] [MeasurableSpace D]
(H : Measure G) (d : D) (f Y : G → ℝ) (T : G → G)
(hf : Measurable f) (hT : Measurable T) (hmap : H.map T=H)
(hY : ∀ u, Y u=f (T u)) :
H.map (fun u => (d,Y u))=H.map (fun u => (d,f u)) := by
have he : (fun u => (d,Y u))=(fun u => (d,f u)) ∘ T := by
funext u
exact Prod.ext rfl (hY u)
rw [he,← Measure.map_map (measurable_const.prodMk hf) hT,hmap]
lemma map_prod_eq_of_fiber_maps {X G E : Type*}
[MeasurableSpace X] [MeasurableSpace G] [MeasurableSpace E]
(P : Measure X) (H : Measure G) [SFinite P] [SFinite H]
(f g : X × G → E) (hf : Measurable f) (hg : Measurable g)
(h : ∀ x, H.map (fun u => f (x,u))=H.map (fun u => g (x,u))) :
(P.prod H).map f=(P.prod H).map g := by
ext s hs
rw [Measure.map_apply hf hs,Measure.map_apply hg hs,
Measure.prod_apply (hf hs),Measure.prod_apply (hg hs)]
apply lintegral_congr_ae
exact Filter.Eventually.of_forall fun x => by
have hh := congrArg (fun Q : Measure E => Q s) (h x)
have hfx : Measurable (fun u => f (x,u)) := hf.comp (measurable_const.prodMk measurable_id)
have hgx : Measurable (fun u => g (x,u)) := hg.comp (measurable_const.prodMk measurable_id)
rw [Measure.map_apply hfx hs,Measure.map_apply hgx hs] at hh
exact hh
theorem invariant_joint_orbit_law {X G D : Type*}
[MeasurableSpace X] [MeasurableSpace G] [MeasurableSpace D]
(P : Measure X) [IsProbabilityMeasure P] (H : Measure G) [IsProbabilityMeasure H]
(act : G → X → X) (hact : Measurable (Function.uncurry act))
(hpres : ∀ u, P.map (act u)=P)
(d : X → D) (Y : X → ℝ) (hd : Measurable d) (hY : Measurable Y)
(hinv : ∀ u x, d (act u x)=d x)
(orbit : D → G → ℝ) (horbit : Measurable (fun z : D × G => orbit z.1 z.2))
(hkernel : ∀ x, H.map (fun u => (d x,Y (act u x)))=
H.map (fun u => (d x,orbit (d x) u))) :
P.map (fun x => (d x,Y x))=
((P.map d).prod H).map (fun z => (z.1,orbit z.1 z.2)) := by
have hs := (MeasurePreserving.id H).skew_product hact
(Filter.Eventually.of_forall hpres)
have hp := ((show MeasurePreserving Prod.snd (H.prod P) P from measurePreserving_snd).comp hs).comp
(show MeasurePreserving Prod.swap (P.prod H) (H.prod P) from Measure.measurePreserving_swap)
have hf : Measurable (fun x => (d x,Y x)) := hd.prodMk hY
have he := congrArg (fun Q : Measure X => Q.map (fun x => (d x,Y x))) hp.map_eq
rw [Measure.map_map hf hp.measurable] at he
have hF : Measurable (fun z : X × G => (d z.1,Y (act z.2 z.1))) :=
(hd.comp measurable_fst).prodMk (hY.comp (hact.comp measurable_swap))
have hB : Measurable (fun z : X × G => (d z.1,orbit (d z.1) z.2)) :=
(hd.comp measurable_fst).prodMk
(horbit.comp ((hd.comp measurable_fst).prodMk measurable_snd))
have havg : P.map (fun x => (d x,Y x))=
(P.prod H).map (fun z : X × G => (d z.1,Y (act z.2 z.1))) := by
refine he.symm.trans ?_
congr 1
funext z
exact Prod.ext (hinv z.2 z.1) rfl
rw [havg,map_prod_eq_of_fiber_maps P H _ _ hF hB hkernel]
have hdp : MeasurePreserving d P (P.map d) := ⟨hd,rfl⟩
have hprod := hdp.prod (MeasurePreserving.id H)
have ht := congrArg (fun Q : Measure (D × G) =>
Q.map (fun z => (z.1,orbit z.1 z.2))) hprod.map_eq
rw [Measure.map_map (measurable_fst.prodMk horbit) hprod.measurable] at ht
exact ht
end InvariantIsing
end
end OAI
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