| import Mathlib.MeasureTheory.Integral.Pi | |
| import Mathlib.MeasureTheory.Integral.Prod | |
| import Mathlib.MeasureTheory.Integral.Bochner.SumMeasure | |
| namespace OAI | |
| /-! | |
| # Precommitted finite conditional-randomness tables | |
| The secretary construction draws all conditional random choices before the | |
| random arrival permutation. A table indexed by every possible branch/prefix | |
| pair does this without concealing later randomness. Its law is the ordinary | |
| finite product measure `Measure.pi κ`. Evaluating the table after an independent | |
| prefix experiment has exactly the corresponding conditional law. | |
| These lemmas concern only this sampling construction, not the source theorem's | |
| competitive guarantee. In particular an application must separately prove the | |
| prefix law and the concrete source conditional-kernel laws. | |
| -/ | |
| namespace MatroidProphet.Secretary | |
| open MeasureTheory | |
| open scoped BigOperators | |
| variable {ι α Ω : Type*} [Fintype ι] [Fintype α] [Fintype Ω] | |
| [MeasurableSpace α] [MeasurableSingletonClass α] | |
| [MeasurableSpace Ω] [MeasurableSingletonClass Ω] | |
| /-- All conditional samples can be drawn initially and independently. Evaluating | |
| one coordinate after an independent finite experiment has the prescribed law. -/ | |
| theorem integral_precommittedTable | |
| (κ : ι → Measure α) [∀ i, IsProbabilityMeasure (κ i)] | |
| (μ : Measure Ω) [IsFiniteMeasure μ] (index : Ω → ι) (f : ι → α → ℝ) : | |
| (∫ z : Ω × (ι → α), f (index z.1) (z.2 (index z.1)) | |
| ∂μ.prod (Measure.pi κ)) = | |
| ∫ ω, ∫ r, f (index ω) r ∂κ (index ω) ∂μ := by | |
| rw [integral_prod _ Integrable.of_finite] | |
| apply integral_congr_ae | |
| exact ae_of_all _ fun ω => by | |
| exact integral_comp_eval (μ := κ) (i := index ω) | |
| (f := f (index ω)) | |
| (measurable_of_countable (f (index ω))).aestronglyMeasurable | |
| /-- The same identity with the independent experiment explicitly summed. It | |
| retains the complete table, including all coordinates unused in this run. -/ | |
| theorem integral_precommittedTable_eq_sum | |
| (κ : ι → Measure α) [∀ i, IsProbabilityMeasure (κ i)] | |
| (μ : Measure Ω) [IsFiniteMeasure μ] (index : Ω → ι) (f : ι → α → ℝ) : | |
| (∫ z : Ω × (ι → α), f (index z.1) (z.2 (index z.1)) | |
| ∂μ.prod (Measure.pi κ)) = | |
| ∑ ω, μ.real {ω} * (∫ r, f (index ω) r ∂κ (index ω)) := by | |
| rw [integral_precommittedTable] | |
| simpa only [smul_eq_mul, measureReal_def] using | |
| (integral_fintype (μ := μ) | |
| (f := fun ω => ∫ r, f (index ω) r ∂κ (index ω)) Integrable.of_finite) | |
| end MatroidProphet.Secretary | |
| end OAI | |