| import OAI.Probability.FreeIsing.Gaussian.LinearSignalTilt |
| import OAI.Probability.FreeIsing.Spins.Kernel |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory |
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| namespace Problem367 |
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| theorem condExp_spin_eq_of_restricted_density |
| {Ω Y : Type*} {mΩ : MeasurableSpace Ω} {mY : MeasurableSpace Y} |
| {mu : Measure Ω} [IsFiniteMeasure mu] |
| {S : Ω → IsingSpin} (hS : Measurable S) |
| {X : Ω → Y} (hX : Measurable X) {q : Y → ℝ} (hq : Measurable q) |
| (hqb : ∀ᵐ y ∂Measure.map X mu, |q y| ≤ 1) |
| (hdensity : Measure.map X (mu.restrict {x | S x = IsingSpin.pos}) = |
| (Measure.map X mu).withDensity (fun y => ENNReal.ofReal ((1 + q y) / 2))) : |
| mu[fun x => spinSign (S x) | mY.comap X] =ᵐ[mu] fun x => q (X x) := by |
| let E : Set Ω := {x | S x = IsingSpin.pos} |
| have hE : MeasurableSet E := |
| (show MeasurableSet ({IsingSpin.pos} : Set IsingSpin) from trivial).preimage hS |
| let I : Ω → ℝ := E.indicator (fun _ => 1) |
| have hI : Integrable I mu := (integrable_const (1 : ℝ)).indicator hE |
| have hsrc : (fun x => ((mu.restrict E).rnDeriv mu x).toReal) =ᵐ[mu] I := by |
| filter_upwards [Measure.rnDeriv_restrict_self mu hE] with x hx |
| rw [hx] |
| by_cases hxE : x ∈ E <;> simp [I, hxE] |
| have hRN := toReal_rnDeriv_map (Measure.absolutelyContinuous_restrict (μ := mu) (s := E)) hX |
| have hpost : |
| (fun x => ((Measure.map X (mu.restrict E)).rnDeriv (Measure.map X mu) (X x)).toReal) |
| =ᵐ[mu] mu[I | mY.comap X] := |
| hRN.trans (condExp_congr_ae hsrc) |
| have htarg : |
| (fun x => ((Measure.map X (mu.restrict E)).rnDeriv (Measure.map X mu) (X x)).toReal) |
| =ᵐ[mu] fun x => (1 + q (X x)) / 2 := by |
| have hderiv := Measure.rnDeriv_withDensity (Measure.map X mu) |
| (show Measurable (fun y => ENNReal.ofReal ((1 + q y) / 2)) by fun_prop) |
| rw [← hdensity] at hderiv |
| filter_upwards [ae_of_ae_map hX.aemeasurable hderiv, |
| ae_of_ae_map hX.aemeasurable hqb] with x hx hxb |
| rw [hx, ENNReal.toReal_ofReal] |
| have := (abs_le.mp hxb).1 |
| linarith |
| have hIp : mu[I | mY.comap X] =ᵐ[mu] fun x => (1 + q (X x)) / 2 := |
| hpost.symm.trans htarg |
| have hsrepr : (fun x => spinSign (S x)) = (2 : ℝ) • I - (fun _ => 1) := by |
| funext x |
| cases hs : S x <;> norm_num [I, E, hs, spinSign] |
| rw [hsrepr] |
| have hsub := condExp_sub (hI.smul (2 : ℝ)) (integrable_const (1 : ℝ)) (mY.comap X) |
| have hmul := condExp_smul (μ := mu) (2 : ℝ) I (mY.comap X) |
| filter_upwards [hsub, hmul, hIp] with x hxsub hxmul hxI |
| rw [hxsub, Pi.sub_apply, hxmul, Pi.smul_apply, smul_eq_mul, |
| condExp_const hX.comap_le, hxI] |
| ring |
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| theorem condExp_spin_eq_of_conditional_linearTilt |
| {Ω Y : Type*} {mΩ : MeasurableSpace Ω} {mY : MeasurableSpace Y} |
| {mu : Measure Ω} [IsFiniteMeasure mu] |
| {S : Ω → IsingSpin} (hS : Measurable S) |
| {X : Ω → Y} (hX : Measurable X) {q : Y → ℝ} (hq : Measurable q) |
| (hqb : ∀ᵐ y ∂Measure.map X mu, |q y| ≤ 1) |
| (hfair : mu {x | S x = IsingSpin.pos} = (1 / 2 : ENNReal)) |
| (hcond : Measure.map X (ProbabilityTheory.cond mu {x | S x = IsingSpin.pos}) = |
| linearSignalTilt (Measure.map X mu) q 1) : |
| mu[fun x => spinSign (S x) | mY.comap X] =ᵐ[mu] fun x => q (X x) := by |
| apply condExp_spin_eq_of_restricted_density hS hX hq hqb |
| have hscaled := congrArg (fun nu : Measure Y => (1 / 2 : ENNReal) • nu) hcond |
| rw [ProbabilityTheory.cond, Measure.map_smul _ hX.aemeasurable, hfair, smul_smul] at hscaled |
| norm_num at hscaled |
| rw [ENNReal.inv_mul_cancel (by norm_num : (2 : ENNReal) ≠ 0) |
| (by norm_num : (2 : ENNReal) ≠ ⊤), one_smul] at hscaled |
| rw [hscaled, linearSignalTilt, ← withDensity_smul (2⁻¹) (by fun_prop)] |
| apply withDensity_congr_ae |
| apply Filter.Eventually.of_forall |
| intro y |
| simp only [Pi.smul_apply, smul_eq_mul, one_mul] |
| rw [ENNReal.ofReal_div_of_pos (by norm_num : (0 : ℝ) < 2)] |
| simp [div_eq_mul_inv, mul_comm] |
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| end Problem367 |
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| end |
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| end OAI |
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