| import OAI.NumberTheory.PrimePackets.ZeroSeparation |
|
|
| namespace OAI |
|
|
| /-! |
| # Product structure from equal normalized real conditionals |
|
|
| This is the finite-rectangle step of Einsiedler |
| A common conditional measure, even if initially specified only up to a |
| scalar on each fiber, forces a genuine product measure after normalization. |
| -/ |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped ENNReal |
|
|
| namespace DukePrimeDegree |
| open PrimeDegreePackets |
|
|
| / |
| theorem eq_prod_of_constant_real_conditionals {B : Type*} [MeasurableSpace B] |
| (μ : Measure (B × ℝ)) [IsFiniteMeasure μ] |
| (ν : Measure ℝ) [IsFiniteMeasure ν] |
| (h : ∀ᵐ x ∂μ.fst, μ.condKernel x = ν) : |
| μ = μ.fst.prod ν := by |
| have h' : μ.condKernel =ᵐ[μ.fst] Kernel.const B ν := h |
| calc |
| μ = μ.fst ⊗ₘ μ.condKernel := (Measure.disintegrate μ μ.condKernel).symm |
| _ = μ.fst ⊗ₘ Kernel.const B ν := Measure.compProd_congr h' |
| _ = μ.fst.prod ν := Measure.compProd_const |
|
|
| / |
| normalizing their common model. The only scalar left is the marginal. -/ |
| theorem eq_prod_of_projective_real_conditionals {B : Type*} [MeasurableSpace B] |
| (μ : Measure (B × ℝ)) [IsFiniteMeasure μ] |
| (ν : Measure ℝ) [IsFiniteMeasure ν] (hν : ν univ ≠ 0) |
| (h : ∀ᵐ x ∂μ.fst, ∃ c : ℝ≥0∞, μ.condKernel x = c • ν) : |
| μ = μ.fst.prod (anchorNormalize ν univ) := by |
| let : IsFiniteMeasure (anchorNormalize ν univ) := by |
| constructor |
| change (ν univ)⁻¹ * ν univ < ∞ |
| rw [ENNReal.inv_mul_cancel hν (measure_ne_top ν univ)] |
| exact ENNReal.one_lt_top |
| apply eq_prod_of_constant_real_conditionals μ (anchorNormalize ν univ) |
| filter_upwards [h] with x hx |
| obtain ⟨c, hc⟩ := hx |
| have hmass : c * ν univ = 1 := by |
| have H := congrArg (fun ρ : Measure ℝ => ρ univ) hc |
| simpa only [Measure.smul_apply, smul_eq_mul, measure_univ] using H.symm |
| have heq : c = (ν univ)⁻¹ := by |
| calc |
| c = c * (ν univ * (ν univ)⁻¹) := by |
| rw [ENNReal.mul_inv_cancel hν (measure_ne_top ν univ), mul_one] |
| _ = (c * ν univ) * (ν univ)⁻¹ := (mul_assoc _ _ _).symm |
| _ = (ν univ)⁻¹ := by rw [hmass, one_mul] |
| simpa only [heq, anchorNormalize] using hc |
|
|
| end DukePrimeDegree |
|
|
| end |
|
|
| end OAI |
|
|
