Theorems · Theorem · functional analysis

MeasureTheory.eLpNorm_smul_le_eLpNorm_mul_eLpNorm_top

∀ {𝕜 : Type u_1} {α : Type u_2} {E : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α}
  [inst : NormedRing 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : MulActionWithZero 𝕜 E] [IsBoundedSMul 𝕜 E]
  (p : ENNReal) (f : α → E) {φ : α → 𝕜},
  MeasureTheory.AEStronglyMeasurable φ μ →
    MeasureTheory.eLpNorm (φ • f) p μ ≤ MeasureTheory.eLpNorm φ p μ * MeasureTheory.eLpNorm f ⊤ μ

Defined in

Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Cited by

1 results in Mathlib

Foundations

Depth 213 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedRingNormedAddCommGroupMulActionWithZeroIsBoundedSMul