theorem extracted_formal_statement_4.{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)) :
  0 ≤ ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ := 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
    apply Pi.le_def.mpr _
    intro i
    simp only [Pi.zero_apply]
    exact exp_nonneg (t * X i)

State after:

Relevant lemmas:
Real.exp_nonneg : ∀ (x : ℝ), 0 ≤ rexp x
Pi.zero_apply : ∀ {I : Type u} {f : I → Type v₁} (i : I) [inst : (i : I) → Zero (f i)], 0 i = 0
MeasureTheory.integral_nonneg : ∀ {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {f : α → ℝ},
  0 ≤ f → 0 ≤ ∫ (a : α), f a ∂μ