theorem extracted_formal_statement_19.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] (a b : ℝ) {X : Ω → ℝ} (hX : AEMeasurable X μ) (h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b)
  (h_original_goal : HasFiniteIntegral X μ) : ∀ᵐ (a_1 : Ω) ∂μ, ‖X a_1‖ ≤ ‖a‖ ⊔ ‖b‖ := sorry

State before:
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
a b : ℝ
X : Ω → ℝ
hX : AEMeasurable X μ
h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
⊢ HasFiniteIntegral X μ

Tactic: apply (hasFiniteIntegral_const (max ‖a‖ ‖b‖)).mono'

State after:
[Ω : Type u_1
 m : MeasurableSpace Ω
 μ : Measure Ω
 inst✝ : IsFiniteMeasure μ
 a b : ℝ
 X : Ω → ℝ
 hX : AEMeasurable X μ
 h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
 ⊢ ∀ᵐ (a_1 : Ω) ∂μ, ‖X a_1‖ ≤ ‖a‖ ⊔ ‖b‖]

Relevant lemmas:
MeasureTheory.hasFiniteIntegral_const : ∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : Measure α}
  [inst : NormedAddCommGroup β] [inst_1 : IsFiniteMeasure μ] (c : β), HasFiniteIntegral (fun x => c) μ