noncomputable def TalagrandGeneral.weightedHammingDist {n : ℕ} {Ω : Fin n → Type u_1} [(i : Fin n) → DecidableEq (Ω i)] (α : Fin n → ℝ) (x y : (i : Fin n) → Ω i) : ℝ

The weighted Hamming distance between $x$ and $y$ with weights $α$: $d_α(x,y) = \sum_{i : x_i \neq y_i} α_i$.

noncomputable def TalagrandGeneral.weightedHammingDistSet {n : ℕ} {Ω : Fin n → Type u_1} [(i : Fin n) → DecidableEq (Ω i)] (α : Fin n → ℝ) (x : (i : Fin n) → Ω i) (A : Set ((i : Fin n) → Ω i)) : ℝ

The weighted Hamming distance from $x$ to a set $A$: $d_α(x,A) = \inf_{y \in A} d_α(x,y)$.

noncomputable def TalagrandGeneral.talagrandConvexDist {n : ℕ} {Ω : Fin n → Type u_1} [(i : Fin n) → DecidableEq (Ω i)] (x : (i : Fin n) → Ω i) (A : Set ((i : Fin n) → Ω i)) : ℝ

Talagrand's convex distance from $x$ to $A$: $d_T(x,A) = \sup_{α \geq 0,\ \|α\|_2 = 1} d_α(x,A)$.

theorem talagrand_inequality_general {n : ℕ} {Ω : Fin n → Type u_1} [(i : Fin n) → DecidableEq (Ω i)] [(i : Fin n) → MeasurableSpace (Ω i)] (μ : (i : Fin n) → MeasureTheory.Measure (Ω i)) [∀ (i : Fin n), MeasureTheory.IsProbabilityMeasure (μ i)] (A : Set ((i : Fin n) → Ω i)) (hA : MeasurableSet A) (t : ℝ) (ht : 0 ≤ t) : (MeasureTheory.Measure.pi μ) A * (MeasureTheory.Measure.pi μ) {x : (i : Fin n) → Ω i | t ≤ TalagrandGeneral.talagrandConvexDist x A} ≤ ENNReal.ofReal (Real.exp (-(t ^ 2 / 4)))

Talagrand's inequality (general form, Theorem 9.5.11): for a product probability measure $\mu$ and any measurable set $A$, $\mu(A) \cdot \mu(\{x : d_T(x, A) \geq t\}) \leq e^{-t^2/4}$.