| import Mathlib |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open MeasureTheory Filter |
| open scoped ENNReal NNReal Topology |
|
|
| section LowerProof |
| open Matrix Topology TopologicalSpace ProbabilityTheory Classical WithLp |
| open scoped Matrix.Norms.Elementwise |
|
|
| namespace LogConcaveSampling |
|
|
| lemma map_prod_eq_of_fiber_law {A B C : Type*} |
| [MeasurableSpace A] [MeasurableSpace B] [MeasurableSpace C] |
| {μ : Measure A} {ν : Measure B} [SFinite μ] [SFinite ν] |
| {f g : A × B → C} (hf : Measurable f) (hg : Measurable g) |
| (h : ∀ b, μ.map (fun a => f (a,b)) = μ.map (fun a => g (a,b))) : |
| (μ.prod ν).map f = (μ.prod ν).map g := by |
| ext s hs |
| rw [Measure.map_apply hf hs, Measure.map_apply hg hs, |
| Measure.prod_apply_symm (hs.preimage hf), Measure.prod_apply_symm (hs.preimage hg)] |
| apply lintegral_congr |
| intro b |
| have hb := congrArg (fun κ : Measure C => κ s) (h b) |
| rw [Measure.map_apply (f := fun a => f (a,b)) (by fun_prop) hs, |
| Measure.map_apply (f := fun a => g (a,b)) (by fun_prop) hs] at hb |
| exact hb |
|
|
| lemma map_prod_eq_of_constant_fiber_law {A B C : Type*} |
| [MeasurableSpace A] [MeasurableSpace B] [MeasurableSpace C] |
| {μ : Measure A} {ν : Measure B} [IsProbabilityMeasure μ] [SFinite ν] |
| {η : Measure C} {f : A × B → C} (hf : Measurable f) |
| (h : ∀ a, ν.map (fun b => f (a,b)) = η) : (μ.prod ν).map f = η := by |
| ext s hs |
| rw [Measure.map_apply hf hs, Measure.prod_apply (hs.preimage hf)] |
| have hh (a : A) : ν (Prod.mk a ⁻¹' (f ⁻¹' s)) = η s := by |
| have ha := congrArg (fun κ : Measure C => κ s) (h a) |
| rw [Measure.map_apply (f := fun b => f (a,b)) (by fun_prop) hs] at ha |
| exact ha |
| simp_rw [hh] |
| simp |
|
|
| end LogConcaveSampling |
|
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| end LowerProof |
| end |
| end |
| end |
|
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| end OAI |
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