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b317718 | 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 | 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
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