| 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 |
| 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 |
| (Function.eval ⟨j,hji⟩ : ({k : I |
| ext v |
| simp [eq_comm] |
| change law {v : {k : I |
| rw [he] |
| exact hz |
| have hp : ∀ᵐ z : ℝ × ({k : I |
| ∂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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|
