theorem extracted_formal_statement_3.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] {X : Ω → ℝ} (t b : ℝ) (ht : t > 0) (hX : AEMeasurable X μ) (hb : ∀ᵐ (ω : Ω) ∂μ, X ω ≤ b)
  (b' : ∀ᵐ (ω : Ω) ∂μ, rexp (t * X ω) ≤ rexp (t * b)) (p : ∀ᵐ (ω : Ω) ∂μ, ↑‖rexp (t * X ω)‖₊ ≤ rexp (t * b))
  (h_original_goal : ∫⁻ (ω : Ω), ↑‖rexp (t * X ω)‖₊ ∂μ ≤ ∫⁻ (x : Ω), ↑‖rexp (t * b)‖₊ ∂μ) :
  ∀ᵐ (a : Ω) ∂μ, ↑‖rexp (t * X a)‖₊ ≤ ↑‖rexp (t * b)‖₊ := sorry

State before:
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : Ω → ℝ
t b : ℝ
ht : t > 0
hX : AEMeasurable X μ
hb : ∀ᵐ (ω : Ω) ∂μ, X ω ≤ b
b' : ∀ᵐ (ω : Ω) ∂μ, rexp (t * X ω) ≤ rexp (t * b)
p : ∀ᵐ (ω : Ω) ∂μ, ↑‖rexp (t * X ω)‖₊ ≤ rexp (t * b)
⊢ ∫⁻ (ω : Ω), ↑‖rexp (t * X ω)‖₊ ∂μ ≤ ∫⁻ (x : Ω), ↑‖rexp (t * b)‖₊ ∂μ

Tactic: apply lintegral_mono_ae

State after:
[case h
 Ω : Type u_1
 m : MeasurableSpace Ω
 μ : Measure Ω
 inst✝ : IsFiniteMeasure μ
 X : Ω → ℝ
 t b : ℝ
 ht : t > 0
 hX : AEMeasurable X μ
 hb : ∀ᵐ (ω : Ω) ∂μ, X ω ≤ b
 b' : ∀ᵐ (ω : Ω) ∂μ, rexp (t * X ω) ≤ rexp (t * b)
 p : ∀ᵐ (ω : Ω) ∂μ, ↑‖rexp (t * X ω)‖₊ ≤ rexp (t * b)
 ⊢ ∀ᵐ (a : Ω) ∂μ, ↑‖rexp (t * X a)‖₊ ≤ ↑‖rexp (t * b)‖₊]

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