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f205175 | 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 82 83 84 85 86 87 88 89 90 | import Mathlib
import OAI.Probability.CriticalPercolation.Entropy.FreshSet
namespace OAI
section
section
open Set MeasureTheory ProbabilityTheory
open Real
open Set Real
open scoped Topology
open scoped BigOperators
open Set
open Set MeasureTheory
open scoped Classical
open Set MeasureTheory Real
open scoped Interval
namespace CriticalPercolation.Priority
variable {I : Type*} [Fintype I]
instance : NullSingletonClass uniform01 := by
unfold uniform01
infer_instance
lemma ae_eval_ne (i j : I) (hji : j ≠ i) :
∀ᵐ u : I → ℝ ∂law, u i ≠ u j := by
have hj : ∀ t : ℝ, ∀ᵐ v : {k : I // k ≠ i} → ℝ ∂law,
t ≠ v ⟨j,hji⟩ := by
intro t
apply ae_iff.mpr
have hz := Measure.pi_eval_preimage_null
(fun _ : {k : I // k ≠ i} => uniform01)
(i := ⟨j,hji⟩) (measure_singleton t)
have he : {v : {k : I // k ≠ i} → ℝ | ¬t ≠ v ⟨j,hji⟩} =
(Function.eval ⟨j,hji⟩ : ({k : I // k ≠ i} → ℝ) → ℝ) ⁻¹' {t} := by
ext v
simp [eq_comm]
change law {v : {k : I // k ≠ i} → ℝ | ¬t ≠ v ⟨j,hji⟩} = 0
rw [he]
exact hz
have hp : ∀ᵐ z : ℝ × ({k : I // k ≠ i} → ℝ)
∂uniform01.prod law, z.1 ≠ z.2 ⟨j,hji⟩ := by
apply (Measure.ae_prod_iff_ae_ae (by measurability)).mpr
exact Filter.Eventually.of_forall hj
exact (split_preserving i).quasiMeasurePreserving.ae hp
lemma ae_injective : ∀ᵐ u : I → ℝ ∂law, Function.Injective u := by
have h : ∀ i j : I, ∀ᵐ u : I → ℝ ∂law, u i = u j → i = j := by
intro i j
by_cases hji : j = i
· subst j; exact Filter.Eventually.of_forall (fun _ _ => rfl)
· filter_upwards [ae_eval_ne i j hji] with u hu
exact fun he => False.elim (hu he)
change ∀ᵐ u : I → ℝ ∂law, ∀ i j, u i = u j → i = j
rw [ae_all_iff]
intro i
rw [ae_all_iff]
exact h i
lemma ae_exists_cell : ∀ᵐ u : I → ℝ ∂law, ∃ o : Order I, u ∈ cell o := by
filter_upwards [ae_injective] with u hu
exact exists_cell u hu
lemma sum_cell_indicator (f : (I → ℝ) → ℝ) :
(fun u => ∑ o : Order I, (cell o).indicator f u) =ᵐ[law] f := by
filter_upwards [ae_exists_cell] with u hu
obtain ⟨o,ho⟩ := hu
rw [Finset.sum_eq_single o]
· exact Set.indicator_of_mem ho f
· intro b _ hb
apply Set.indicator_of_notMem
exact fun hub => Set.disjoint_left.mp (cell_disjoint hb) hub ho
· simp
lemma sum_integral_cells (f : (I → ℝ) → ℝ) (hf : Integrable f law) :
(∑ o : Order I, ∫ u in cell o, f u ∂law) = ∫ u, f u ∂law := by
simp_rw [← integral_indicator (measurableSet_cell _)]
rw [← integral_finsetSum]
· exact integral_congr_ae (sum_cell_indicator f)
· intro o _
exact hf.indicator (measurableSet_cell o)
end CriticalPercolation.Priority
end
end
end OAI
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