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e63eb7f | 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 | 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
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