| import Mathlib | |
| import OAI.GroupTheory.SimpleAmenable.RandomFields.ThresholdField | |
| namespace OAI | |
| section | |
| section | |
| open scoped symmDiff | |
| namespace SimpleAmenable | |
| open scoped commutatorElement | |
| open scoped commutatorElement | |
| section BitReindex | |
| open Classical MeasureTheory | |
| noncomputable def bitReindex {ι : Type*} (e : ι ≃ ι) (b : ι → Bool) : ι → Bool := | |
| fun z => b (e.symm z) | |
| theorem bitReindex_measurable {ι : Type*} (e : ι ≃ ι) : Measurable (bitReindex e) := | |
| Measurable.of_eval (fun z => measurable_pi_apply (e.symm z)) | |
| theorem thresholdField_transport {ι : Type*} (μ : Measure (ι → ℝ)) (e : ι ≃ ι) : | |
| (μ.map (fieldReindex e)).map thresholdField= | |
| (μ.map thresholdField).map (bitReindex e) := by | |
| rw [Measure.map_map thresholdField_measurable (fieldReindex_measurable e), | |
| Measure.map_map (bitReindex_measurable e) thresholdField_measurable] | |
| rfl | |
| theorem ObservableClose.threshold_transport {ι : Type*} (μ : Measure (ι → ℝ)) | |
| (e : ι ≃ ι) {ε : ℝ} (h : ObservableClose μ (μ.map (fieldReindex e)) ε) : | |
| ObservableClose (μ.map thresholdField) ((μ.map thresholdField).map (bitReindex e)) ε := by | |
| have hh := h.map thresholdField_measurable | |
| rw [thresholdField_transport] at hh | |
| exact hh | |
| end BitReindex | |
| end SimpleAmenable | |
| end | |
| end | |
| end OAI | |