Theorems · Theorem · measure theory

MeasureTheory.integrable_norm_rpow_iff

∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : NormedAddCommGroup β]
  {f : α → β} {p : ENNReal},
  MeasureTheory.AEStronglyMeasurable f μ →
    p ≠ 0 → p ≠ ⊤ → (MeasureTheory.Integrable (fun x => ‖f x‖ ^ p.toReal) μ ↔ MeasureTheory.MemLp f p μ)

Defined in: Mathlib.MeasureTheory.Function.L1Space.Integrable
Cited by: 5 results in Mathlib
Foundations: Depth 213 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: NormedAddCommGroup

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Cites14

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
ENNRealstatement and proof · cited by 9,879
Top.topstatement and proof · cited by 9,680
Norm.normstatement and proof · cited by 5,413
MeasureTheory.Integrablestatement and proof · cited by 1,367
ENNReal.toRealstatement and proof · cited by 859
MeasureTheory.AEStronglyMeasurablestatement and proof · cited by 755
MeasureTheory.MemLpstatement and proof · cited by 457
MeasureTheory.memLp_one_iff_integrableproof · cited by 73

Cited by5

Results whose statement or proof uses this declaration.

MeasureTheory.MemLp.condExpproof · cited by 3
ProbabilityTheory.memLp_of_mem_interior_integrableExpSetproof · cited by 3
MeasureTheory.MemLp.lpNorm_condExp_le_lpNormproof · cited by 1
MeasureTheory.integrable_norm_rpow_of_leproof · cited by 1
MeasureTheory.memLp_id_of_self_sub_integralproof · cited by 1