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dd14c26 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 | 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
end LowerProof
end
end
end
end OAI
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