Integrals with exponential decay at ∞

As easy special cases of general theorems in the library, we prove the following test for integrability:

• integrable_of_isBigO_exp_neg: If f is continuous on [a,∞), for some a ∈ ℝ, and there exists b > 0 such that f(x) = O(exp(-b x)) as x → ∞, then f is integrable on (a, ∞).

theorem exp_neg_integrableOn_Ioi (a : ℝ) {b : ℝ} (h : 0 < b) : MeasureTheory.IntegrableOn (fun (x : ℝ) => Real.exp (-b * x)) (Set.Ioi a) MeasureTheory.volume

exp (-b * x) is integrable on (a, ∞).

theorem integrable_of_isBigO_exp_neg {f : ℝ → ℝ} {a : ℝ} {b : ℝ} (h0 : 0 < b) (h1 : ContinuousOn f (Set.Ici a)) (h2 : f =O[Filter.atTop] fun (x : ℝ) => Real.exp (-b * x)) : MeasureTheory.IntegrableOn f (Set.Ioi a) MeasureTheory.volume

If f is continuous on [a, ∞), and is O (exp (-b * x)) at ∞ for some b > 0, then f is integrable on (a, ∞).