Theorems · Theorem · probability
ProbabilityTheory.integrable_exp_mul_of_le_of_le
∀ {Ω : Type u_1} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : MeasureTheory.Measure Ω} {t a b : ℝ},
MeasureTheory.Integrable (fun ω => Real.exp (a * X ω)) μ →
MeasureTheory.Integrable (fun ω => Real.exp (b * X ω)) μ →
a ≤ t → t ≤ b → MeasureTheory.Integrable (fun ω => Real.exp (t * X ω)) μ- Cited by
- 7 results in Mathlib
- Foundations
- Depth 208 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites29
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- MeasurableSpacestatement and proof · cited by 13,106
- MeasureTheory.Measurestatement and proof · cited by 10,939
- le_antisymmproof · cited by 2,068
- absproof · cited by 1,814
- MeasureTheory.Integrablestatement and proof · cited by 1,367
- le_of_ltproof · cited by 1,175
- Real.expstatement and proof · cited by 871
- MeasureTheory.AEStronglyMeasurableproof · cited by 755
- mul_le_mul_of_nonneg_rightproof · cited by 301
- le_totalproof · cited by 294
- abs_of_nonnegproof · cited by 279
Cited by7
Results whose statement or proof uses this declaration.
- ProbabilityTheory.IsGaussian.memLp_idproof · cited by 4
- ProbabilityTheory.Kernel.HasSubgaussianMGF.ae_forall_integrable_exp_mulproof · cited by 4
- ProbabilityTheory.Kernel.HasSubgaussianMGF.of_ratproof · cited by 1
- ProbabilityTheory.convex_integrableExpSetproof · cited by 1
- ProbabilityTheory.integrable_exp_mul_of_nonneg_of_leproof · cited by 0
- ProbabilityTheory.integrable_exp_mul_of_nonpos_of_geproof · cited by 0
- ProbabilityTheory.integrable_exp_mul_of_abs_leproof · cited by 0