Theorems · Theorem · probability

ProbabilityTheory.integrable_exp_mul_abs

∀ {Ω : Type u_1} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : MeasureTheory.Measure Ω} {t : ℝ},
  MeasureTheory.Integrable (fun ω => Real.exp (t * X ω)) μ →
    MeasureTheory.Integrable (fun ω => Real.exp (-t * X ω)) μ →
      MeasureTheory.Integrable (fun ω => Real.exp (t * |X ω|)) μ

If ω ↦ exp (t * X ω) is integrable at t and -t, then ω ↦ exp (t * |X ω|) is integrable.

Defined in

Mathlib.Probability.Moments.IntegrableExpMul

Cited by

1 results in Mathlib

Foundations

Depth 209 from the axioms · uses propext, Classical.choice, Quot.sound