theorem extracted_formal_statement_0.{u} {Ω : Type u} [inst : MeasurableSpace Ω] (μ : Measure Ω := by volume_tac)
  [inst_1 : IsProbabilityMeasure μ] {X : Ω → ℝ} (h0 : ∫ (x : Ω), X x ∂μ = 0)
  (h_original_goal :
    let f' := fun t => ∫ (x : Ω), X x ∂Measure.tilted μ fun ω => t * X ω;
    f' 0 = 0) :
  ∫ (x : Ω), X x ∂μ = 0 := sorry

State before:
Ω : Type u
inst✝¹ : MeasurableSpace Ω
μ : autoParam (Measure Ω) _auto✝
inst✝ : IsProbabilityMeasure μ
X : Ω → ℝ
h0 : ∫ (x : Ω), X x ∂μ = 0
⊢ let f' := fun t => ∫ (x : Ω), X x ∂Measure.tilted μ fun ω => t * X ω;
  f' 0 = 0

Tactic: simp only [zero_mul, tilted_const', measure_univ, inv_one, one_smul]

State after:
[Ω : Type u
 inst✝¹ : MeasurableSpace Ω
 μ : autoParam (Measure Ω) _auto✝
 inst✝ : IsProbabilityMeasure μ
 X : Ω → ℝ
 h0 : ∫ (x : Ω), X x ∂μ = 0
 ⊢ ∫ (x : Ω), X x ∂μ = 0]

Relevant lemmas:
MeasureTheory.IsProbabilityMeasure.measure_univ : ∀ {α : Type u_1} {m0 : MeasurableSpace α} {μ : Measure α}
  [self : IsProbabilityMeasure μ], μ Set.univ = 1
inv_one : ∀ {G : Type u_1} [inst : InvOneClass G], 1⁻¹ = 1
one_smul : ∀ (M : Type u_1) {α : Type u_5} [inst : Monoid M] [inst_2 : MulAction M α] (b : α), 1 • b = b
MulZeroClass.zero_mul : ∀ {M₀ : Type u} [self : MulZeroClass M₀] (a : M₀), 0 * a = 0
MeasureTheory.tilted_const' : ∀ {α : Type u_1} {mα : MeasurableSpace α} (μ : Measure α) (c : ℝ),
  (μ.tilted fun x => c) = (μ Set.univ)⁻¹ • μ