| import OAI.NumberTheory.DukePrimeDegree.Dynamics.KernelStabilizerMeasurable | |
| namespace OAI | |
| /-! # A common full-measure graph for Dirac fiber conditionals -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Set | |
| namespace DukePrimeDegree | |
| variable {Z U : Type*} [MeasurableSpace Z] [MeasurableSpace U] | |
| [StandardBorelSpace U] | |
| def kernelDiracGraph (κ : Kernel Z U) : Set (Z × U) := | |
| {z | κ z.1=Measure.dirac z.2} | |
| lemma measurableSet_kernelDiracGraph (κ : Kernel Z U) [IsFiniteKernel κ] : | |
| MeasurableSet (kernelDiracGraph κ) := by | |
| have h := measurableSet_finite_kernel_eq (κ.comap Prod.fst measurable_fst) | |
| (Kernel.deterministic Prod.snd measurable_snd) | |
| simpa only [kernelDiracGraph,Kernel.comap_apply,Kernel.deterministic_apply] using h | |
| lemma kernelDiracGraph_fiber_unique (κ : Kernel Z U) | |
| {x y : Z × U} (hx : x ∈ kernelDiracGraph κ) | |
| (hy : y ∈ kernelDiracGraph κ) (hbase : x.1=y.1) : x.2=y.2 := by | |
| apply MeasureTheory.injective_dirac | |
| exact hx.symm.trans ((congrArg κ hbase).trans hy) | |
| theorem exists_full_measure_unique_fiber_graph | |
| (ρ : Measure (Z × U)) (κ : Kernel Z U) [IsFiniteKernel κ] | |
| (hκ : ∀ᵐ z ∂ρ, κ z.1=Measure.dirac z.2) : | |
| ∃ E : Set (Z × U), MeasurableSet E ∧ (∀ᵐ z ∂ρ, z ∈ E) ∧ | |
| ∀ x ∈ E, ∀ y ∈ E, x.1=y.1 → x.2=y.2 := | |
| ⟨kernelDiracGraph κ,measurableSet_kernelDiracGraph κ,hκ, | |
| fun _ hx _ hy he => kernelDiracGraph_fiber_unique κ hx hy he⟩ | |
| end DukePrimeDegree | |
| end | |
| end OAI | |