| import OAI.Probability.MatroidSecretary.Weights.DegenerateCases |
| import OAI.Probability.MatroidProphet.Main |
| import Mathlib.MeasureTheory.Integral.Bochner.Basic |
|
|
| namespace OAI |
|
|
| /-! |
| # The exact finite-moment requirement |
|
|
| For a finite matroid and nonnegative measurable weights, integrability of the |
| actual offline optimum is equivalent to integrability of the nonloop |
| coordinates. No loop-coordinate moments are imposed. This is a supporting |
| scope check for introduction.tex, Theorem `thm:main`, and setup.tex's |
| one-sample simulation argument; it does not replace its competitive claim. |
| -/ |
|
|
| namespace MatroidProphet.DegenerateCases |
|
|
| open MeasureTheory Classical |
| open scoped BigOperators |
|
|
| / |
| theorem optimum_le_sum_nonloops {n : ℕ} (M : Matroid (Fin n)) |
| (w : Weights n) (hw : ∀ e, 0 ≤ w e) : |
| optimum M w ≤ ∑ e : Fin n, if M.IsNonloop e then w e else 0 := by |
| classical |
| have hn (e : Fin n) : 0 ≤ if M.IsNonloop e then w e else 0 := by |
| split_ifs |
| · exact hw e |
| · exact le_rfl |
| apply optimum_le_of_independent_bound M w _ (Finset.sum_nonneg fun e _ => hn e) |
| intro I hI |
| calc |
| (∑ e ∈ I, w e) = ∑ e ∈ I, if M.IsNonloop e then w e else 0 := by |
| apply Finset.sum_congr rfl |
| intro e he |
| simp [hI.isNonloop_of_mem he] |
| _ ≤ ∑ e : Fin n, if M.IsNonloop e then w e else 0 := by |
| exact Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ I) |
| (fun e _ _ => hn e) |
|
|
| / |
| theorem integrable_nonloop_of_integrable_optimum {n : ℕ} |
| (M : Matroid (Fin n)) {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) (V : Ω → Weights n) (hV : Measurable V) |
| (hVN : ∀ᵐ ω ∂μ, ∀ e, 0 ≤ V ω e) |
| (hopt : Integrable (fun ω => optimum M (V ω)) μ) |
| (e : Fin n) (he : M.IsNonloop e) : Integrable (fun ω => V ω e) μ := by |
| apply hopt.mono' ((measurable_pi_apply e).comp hV).aestronglyMeasurable |
| filter_upwards [hVN] with ω hω |
| change ‖V ω e‖ ≤ optimum M (V ω) |
| rw [Real.norm_eq_abs, abs_of_nonneg (hω e)] |
| exact nonloop_weight_le_optimum M (V ω) e he |
|
|
| / |
| theorem integrable_optimum_of_integrable_nonloops {n : ℕ} |
| (M : Matroid (Fin n)) {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) (V : Ω → Weights n) (hV : Measurable V) |
| (hVN : ∀ᵐ ω ∂μ, ∀ e, 0 ≤ V ω e) |
| (hmom : ∀ e, M.IsNonloop e → Integrable (fun ω => V ω e) μ) : |
| Integrable (fun ω => optimum M (V ω)) μ := by |
| classical |
| have hsum : Integrable (fun ω => |
| ∑ e : Fin n, if M.IsNonloop e then V ω e else 0) μ := by |
| apply integrable_finsetSum |
| intro e _ |
| by_cases he : M.IsNonloop e |
| · simpa only [ite_eq_left he] using hmom e he |
| · simp only [ite_eq_right he] |
| exact integrable_zero _ _ _ |
| apply hsum.mono' ((measurable_optimum M).comp hV).aestronglyMeasurable |
| filter_upwards [hVN] with ω hω |
| change ‖optimum M (V ω)‖ ≤ ∑ e : Fin n, if M.IsNonloop e then V ω e else 0 |
| rw [Real.norm_eq_abs, abs_of_nonneg (optimum_nonneg M (V ω))] |
| exact optimum_le_sum_nonloops M (V ω) hω |
|
|
| / |
| theorem integrable_optimum_iff {n : ℕ} (M : Matroid (Fin n)) |
| {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) |
| (V : Ω → Weights n) (hV : Measurable V) |
| (hVN : ∀ᵐ ω ∂μ, ∀ e, 0 ≤ V ω e) : |
| Integrable (fun ω => optimum M (V ω)) μ ↔ |
| ∀ e, M.IsNonloop e → Integrable (fun ω => V ω e) μ := by |
| constructor |
| · exact integrable_nonloop_of_integrable_optimum M μ V hV hVN |
| · exact integrable_optimum_of_integrable_nonloops M μ V hV hVN |
|
|
| end MatroidProphet.DegenerateCases |
|
|
| end OAI |
|
|
