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a2ee19a | 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 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 | 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
/-- Only nonloops can contribute to a feasible sum. -/
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)
/-- Nonloop moments follow from the assumed finite offline optimum moment. -/
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
/-- Conversely, finite moments only on the nonloops suffice for finite E[OPT]. -/
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
/-- An exact criterion: loop moments are deliberately absent on the right. -/
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
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