openAI-math-dataset / lean /OAI /NumberTheory /DukePrimeDegree /Entropy /ConditionalBaseEmbedding.lean
| 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 | |