theorem extracted_formal_statement_98.{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))
  (h_original_goal : 0 ≤ ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ) : 0 ≤ fun ω => rexp (t * X ω) := 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)
⊢ 0 ≤ ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ

Tactic: apply integral_nonneg

State after:
[case hf
 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)
 ⊢ 0 ≤ fun ω => rexp (t * X ω)]

Relevant lemmas:
MeasureTheory.integral_nonneg : ∀ {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {f : α → ℝ},
  0 ≤ f → 0 ≤ ∫ (a : α), f a ∂μ