theorem extracted_formal_statement_101.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] (t : ℝ) {X : Ω → ℝ} (hX : AEMeasurable X μ) (ω : Ω)
  (w : (∂μ.tilted fun x => t * X x/∂μ) ω = ENNReal.ofReal (rexp (t * X ω) / ∫ (x : Ω), rexp (t * X x) ∂μ)) (i : Ω) :
  0 ≤ rexp (t * X i) := sorry

State before:
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
t : ℝ
f : ℝ → ℝ
X : Ω → ℝ
hX : AEMeasurable X μ
ω : Ω
w : (∂μ.tilted fun x => t * X x/∂μ) ω = ENNReal.ofReal (rexp (t * X ω) / ∫ (x : Ω), rexp (t * X x) ∂μ)
i : Ω
⊢ 0 ≤ rexp (t * X i)

Tactic: exact exp_nonneg (t * X i)

State after:

Relevant lemmas:
Real.exp_nonneg : ∀ (x : ℝ), 0 ≤ rexp x