theorem extracted_formal_statement_18.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsProbabilityMeasure μ] (a b t : ℝ) {X : Ω → ℝ} (h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b) (hX : AEMeasurable X μ)
  (x : IsProbabilityMeasure (μ.tilted fun ω => t * X ω)) :
  Var[X; μ.tilted fun ω => t * X ω] ≤ ((b - a) / 2) ^ 2 := sorry

State before:
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
a b t : ℝ
X : Ω → ℝ
h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
hX : AEMeasurable X μ
x✝ : IsProbabilityMeasure (μ.tilted fun ω => t * X ω)
⊢ Var[X; μ.tilted fun ω => t * X ω] ≤ ((b - a) / 2) ^ 2

Tactic: exact
  variance_le_sq_of_bounded ((tilted_absolutelyContinuous μ fun ω ↦ t * X ω) h)
    (hX.mono_ac (tilted_absolutelyContinuous μ fun ω ↦ t * X ω))

State after:

Relevant lemmas:
MeasureTheory.tilted_absolutelyContinuous : ∀ {α : Type u_1} {mα : MeasurableSpace α} (μ : Measure α) (f : α → ℝ),
  μ.tilted f ≪ μ