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a44267b | 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 | import OAI.NumberTheory.DukePrimeDegree.Entropy.ConditionalRectangleProducts
namespace OAI
/-! # Including the base of an independent conditional rectangle -/
noncomputable section
open MeasureTheory ProbabilityTheory
namespace DukePrimeDegree
open PrimeDegreePackets
theorem conditional_product_map_base_embedding
{S T B : Type*} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace B]
[StandardBorelSpace B] [Nonempty B]
(ρ : Measure (S × (B × ℝ))) [IsFiniteMeasure ρ]
(e : S → T) (he : MeasurableEmbedding e)
(hprod : ∀ᵐ s ∂ρ.fst,
ρ.condKernel s = (ρ.condKernel s).fst.prod (ρ.condKernel s).snd) :
let σ := ρ.map (Prod.map e id)
∀ᵐ t ∂σ.fst, σ.condKernel t = (σ.condKernel t).fst.prod (σ.condKernel t).snd := by
intro σ
have hfst : σ.fst = ρ.fst.map e :=
fst_map_fibers_of_measurable ρ e he.measurable Prod.snd measurable_snd
have hc : ∀ᵐ s ∂ρ.fst, σ.condKernel (e s) = ρ.condKernel s := by
have H := conditional_map_fibers_embedding ρ e he Prod.snd measurable_snd
filter_upwards [H] with s hs
change σ.condKernel (e s) = (ρ.condKernel s).map id at hs
simpa only [Measure.map_id] using hs
rw [hfst]
apply he.ae_map_iff.mpr
filter_upwards [hc,hprod] with s hs hp
simpa only [hs] using hp
end DukePrimeDegree
end
end OAI
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