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a89d282 | 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 | import OAI.Probability.InvariantIsing.Spectral.ScaledSpectralLaw
namespace OAI
/-! Spectral support under a scalar of either sign. -/
noncomputable section
open MeasureTheory Set
namespace InvariantIsing
lemma signed_scaled_spectral_support (ν : ProbabilityMeasure ℝ) (a b c : ℝ)
(hbound : (ν : Measure ℝ).support ⊆ Icc a b) :
(scaledSpectralLaw ν c : Measure ℝ).support ⊆
Icc (min (c*a) (c*b)) (max (c*a) (c*b)) := by
apply Measure.support_subset_of_isClosed isClosed_Icc
change ∀ᵐ y ∂((ν : Measure ℝ).map (fun x => c*x)),
y∈Icc (min (c*a) (c*b)) (max (c*a) (c*b))
apply (ae_map_iff (measurable_const.mul measurable_id).aemeasurable measurableSet_Icc).mpr
filter_upwards [(ν : Measure ℝ).support_mem_ae] with x hx
by_cases hc : 0 ≤ c
· exact ⟨(min_le_left _ _).trans (mul_le_mul_of_nonneg_left (hbound hx).1 hc),
(mul_le_mul_of_nonneg_left (hbound hx).2 hc).trans (le_max_right _ _)⟩
· exact ⟨(min_le_right _ _).trans (mul_le_mul_of_nonpos_left (hbound hx).2 (le_of_not_ge hc)),
(mul_le_mul_of_nonpos_left (hbound hx).1 (le_of_not_ge hc)).trans (le_max_left _ _)⟩
lemma signed_scaled_spectral_endpoints (ν : ProbabilityMeasure ℝ) (a b c : ℝ)
(ha : a∈(ν : Measure ℝ).support) (hb : b∈(ν : Measure ℝ).support) :
min (c*a) (c*b)∈(scaledSpectralLaw ν c : Measure ℝ).support ∧
max (c*a) (c*b)∈(scaledSpectralLaw ν c : Measure ℝ).support := by
by_cases h : c*a ≤ c*b
· simpa only [min_eq_left h,max_eq_right h] using
And.intro (scaled_spectral_support_mem ν c a ha) (scaled_spectral_support_mem ν c b hb)
· simpa only [min_eq_right (le_of_not_ge h),max_eq_left (le_of_not_ge h)] using
And.intro (scaled_spectral_support_mem ν c b hb) (scaled_spectral_support_mem ν c a ha)
end InvariantIsing
end
end OAI
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