integrable_exp_neg_add_mul_abs

Plain-English statement

- One-dimensional Laplace tails are Lebesgue-integrable.

Scope guard. This card records a compiled local declaration. Its mathematical scope is exactly the Lean statement below; the Registry note and source correspondence may describe motivation but do not strengthen it.

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Lean statement

theorem integrable_exp_neg_add_mul_abs {a b : ℝ} (ha : 0 < a) :
    Integrable (fun x : ℝ => Real.exp (-(a * |x| + b))) volume := by
  have hright0 : IntegrableOn (fun x : ℝ => Real.exp ((-a) * x)) (Ioi 0) := by
    exact integrableOn_exp_mul_Ioi (a := -a) (by linarith) 0
  have hright_base :
      IntegrableOn (fun x : ℝ => Real.exp (-b) * Real.exp ((-a) * x)) (Ioi 0) :=
    hright0.const_mul (Real.exp (-b))
  have hright : IntegrableOn (fun x : ℝ => Real.exp (-(a * |x| + b))) (Ioi 0) := by
    refine (integrableOn_congr_fun ?_ measurableSet_Ioi).mp hright_base
    intro x hx
    have hxnonneg : 0 ≤ x := le_of_lt hx
    change Real.exp (-b) * Real.exp (-a * x) = Real.exp (-(a * |x| + b))
    rw [abs_of_nonneg hxnonneg]
    rw [show -(a * x + b) = -b + (-a) * x by ring]
    rw [Real.exp_add]
  have hleft0 : IntegrableOn (fun x : ℝ => Real.exp (a * x)) (Iic 0) := by
    exact integrableOn_exp_mul_Iic (a := a) ha 0
  have hleft_base :
      IntegrableOn (fun x : ℝ => Real.exp (-b) * Real.exp (a * x)) (Iic 0) :=
    hleft0.const_mul (Real.exp (-b))
  have hleft : IntegrableOn (fun x : ℝ => Real.exp (-(a * |x| + b))) (Iic 0) := by
    refine (integrableOn_congr_fun ?_ measurableSet_Iic).mp hleft_base
    intro x hx
    have hxnonpos : x ≤ 0 := hx
    change Real.exp (-b) * Real.exp (a * x) = Real.exp (-(a * |x| + b))
    rw [abs_of_nonpos hxnonpos]
    rw [show -(a * -x + b) = -b + a * x by ring]
    rw [Real.exp_add]
  rw [← integrableOn_univ]
  rw [← Iic_union_Ioi (a := (0 : ℝ)), integrableOn_union]
  exact ⟨hleft, hright⟩

/-- The `ℝ≥0∞` integral of a one-dimensional Laplace tail is finite. -/

Open AutoSamplingTheory/TechnicalLemmas/Analysis/Integrability.lean:97published source at 7bcd37294df1

Proof architecture

Chewi DENS/CONV root: integrability of one-dimensional Laplace tails `exp (-(a|x|+b))` for log-concave non-strongly-convex examples

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Hidden assumptions and non-claims

Measurability is represented explicitly or must be supplied by a dependency.
Integrability is an input or proved output; a displayed integral alone does not supply it.