| import OAI.Combinatorics.DiscreteConvexity.FiniteProbability | |
| namespace OAI | |
| /-! | |
| # Extracting outcomes from a finite probability law | |
| Normalization guarantees a positive-weight outcome. It also guarantees an | |
| outcome below the expectation and a witness to every positive-probability | |
| event; strict positivity of every individual mass is not required. | |
| -/ | |
| namespace TalagrandDiscreteConvexity.FiniteLaw | |
| open scoped BigOperators | |
| variable {Ω : Type*} [Fintype Ω] | |
| theorem exists_weight_pos (μ : FiniteLaw Ω) : ∃ ω, 0 < μ.weight ω := by | |
| classical | |
| by_contra h | |
| have hz (ω : Ω) : μ.weight ω = 0 := | |
| le_antisymm (le_of_not_gt (fun hp => h ⟨ω, hp⟩)) (μ.nonneg ω) | |
| have htotal : (∑ ω, μ.weight ω) = 0 := by simp [hz] | |
| rw [μ.total] at htotal | |
| exact one_ne_zero htotal | |
| theorem exists_le_expectation (μ : FiniteLaw Ω) (f : Ω → ℝ) : | |
| ∃ ω, f ω ≤ μ.expectation f := by | |
| classical | |
| obtain ⟨ω₀, hω₀⟩ := μ.exists_weight_pos | |
| by_contra h | |
| have hlt (ω : Ω) : μ.expectation f < f ω := | |
| lt_of_not_ge (fun hle => h ⟨ω, hle⟩) | |
| have hsum : μ.expectation (fun _ => μ.expectation f) < μ.expectation f := by | |
| apply Finset.sum_lt_sum | |
| · intro ω _ | |
| exact mul_le_mul_of_nonneg_left (hlt ω).le (μ.nonneg ω) | |
| · exact ⟨ω₀, Finset.mem_univ ω₀, | |
| mul_lt_mul_of_pos_left (hlt ω₀) hω₀⟩ | |
| rw [expectation_const] at hsum | |
| exact (lt_irrefl _) hsum | |
| theorem exists_of_prob_pos (μ : FiniteLaw Ω) {P : Ω → Prop} | |
| (hP : 0 < μ.prob P) : ∃ ω, P ω := by | |
| classical | |
| by_contra h | |
| have hz : μ.prob P = 0 := by | |
| unfold prob | |
| apply Finset.sum_eq_zero | |
| intro ω _ | |
| exact ite_eq_right (fun hp => h ⟨ω, hp⟩) | |
| exact (ne_of_gt hP) hz | |
| end TalagrandDiscreteConvexity.FiniteLaw | |
| end OAI | |