Theorems · Theorem · measure theory
MeasureTheory.integrable_norm_iff
∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : NormedAddCommGroup β]
{f : α → β},
MeasureTheory.AEStronglyMeasurable f μ → (MeasureTheory.Integrable (fun a => ‖f a‖) μ ↔ MeasureTheory.Integrable f μ)- Cited by
- 19 results in Mathlib
- Foundations
- Depth 184 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- NormedAddCommGroup
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement · cited by 25,697
- NormedAddCommGroupstatement and proof · cited by 15,752
- MeasurableSpacestatement and proof · cited by 13,106
- MeasureTheory.Measurestatement and proof · cited by 10,939
- Norm.normstatement and proof · cited by 5,413
- MeasureTheory.Integrablestatement · cited by 1,367
- MeasureTheory.AEStronglyMeasurablestatement and proof · cited by 755
- MeasureTheory.HasFiniteIntegralproof · cited by 120
- MeasureTheory.AEStronglyMeasurable.normproof · cited by 30
Cited by19
Results whose statement or proof uses this declaration.
- VectorFourier.fourierIntegral_convergent_iffproof · cited by 9
- integrableOn_exp_mul_complex_Ioiproof · cited by 4
- integrableOn_Ioi_cpow_of_ltproof · cited by 3
- ProbabilityTheory.integrable_rpow_mul_exp_of_integrable_exp_mulproof · cited by 3
- hasDerivAt_integral_of_dominated_loc_of_lipproof · cited by 2
- InformationTheory.integrable_llr_compProd_iffproof · cited by 2
- IntervalIntegrable.intervalIntegrable_norm_iffproof · cited by 2
- MeasureTheory.tendstoInDistribution_of_tendstoInMeasure_subproof · cited by 2
- ProbabilityTheory.integrable_rpow_abs_mul_exp_add_of_integrable_exp_mulproof · cited by 2
- MeasureTheory.Integrable.integral_condDistrib_mapproof · cited by 1
- ProbabilityTheory.integrable_cexp_mul_of_re_mem_integrableExpSetproof · cited by 1
- InformationTheory.integral_llr_compProd_eq_addproof · cited by 1