| 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_second_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 : ∀ a, ν.map (fun b => f (a,b)) = ν.map (fun b => 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 (hs.preimage hf), Measure.prod_apply (hs.preimage hg)] | |
| apply lintegral_congr | |
| intro a | |
| 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, | |
| Measure.map_apply (f := fun b => g (a,b)) (by fun_prop) hs] at ha | |
| exact ha | |
| end LogConcaveSampling | |
| end LowerProof | |
| end | |
| end | |
| end | |
| end OAI | |