| import OAI.Probability.InvariantIsing.Arrays.PathSymbolAEUniqueness | |
| namespace OAI | |
| /-! Interval integrals and symbols depend only on the path almost everywhere. | |
| The endpoints have zero Lebesgue measure. -/ | |
| noncomputable section | |
| open MeasureTheory Set | |
| namespace InvariantIsing | |
| lemma intervalIntegral_congr_path_ae {a b : ℝ → ℝ} (he : a =ᵐ[pathMeasure] b) | |
| {s t : ℝ} (hs : 0 ≤ s) (ht : t ≤ 1) (hst : s ≤ t) : | |
| (∫ u in s..t, a u) = ∫ u in s..t, b u := by | |
| have hev : ∀ᵐ u ∂volume, u ∈ Ioo (0 : ℝ) 1 → a u = b u := | |
| (ae_restrict_iff' measurableSet_Ioo).mp he | |
| apply intervalIntegral.integral_congr_ae | |
| filter_upwards [hev, volume.ae_ne (0 : ℝ), volume.ae_ne (1 : ℝ)] with u hu hu0 hu1 humem | |
| rw [uIoc_of_le hst] at humem | |
| exact hu ⟨hs.trans_lt humem.1, lt_of_le_of_ne (humem.2.trans ht) hu1⟩ | |
| lemma pathSymbol_congr_ae {a b : ℝ → ℝ} (d : ℝ) (he : a =ᵐ[pathMeasure] b) : | |
| (fun s => pathSymbol d a s) =ᵐ[pathMeasure] fun s => pathSymbol d b s := by | |
| filter_upwards [he, ae_restrict_mem measurableSet_Ioo] with s hs hsmem | |
| unfold pathSymbol | |
| rw [hs, intervalIntegral_congr_path_ae he hsmem.1.le le_rfl hsmem.2.le] | |
| end InvariantIsing | |
| end | |
| end OAI | |