Theorems · Theorem · measure theory

MeasureTheory.L1.SimpleFunc.setToL1S_smul

∀ {α : Type u_1} {E : Type u_2} {F : Type u_3} {𝕜 : Type u_6} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E]
  [inst_2 : NormedAddCommGroup F] [inst_3 : NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α}
  [inst_4 : NormedRing 𝕜] [inst_5 : Module 𝕜 E] [inst_6 : IsBoundedSMul 𝕜 E] [inst_7 : DistribSMul 𝕜 F]
  (T : Set α → E →L[ℝ] F),
  (∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0) →
    MeasureTheory.FinMeasAdditive μ T →
      (∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c • x) = c • (T s) x) →
        ∀ (c : 𝕜) (f : ↥(α →₁ₛ[μ] E)),
          MeasureTheory.L1.SimpleFunc.setToL1S T (c • f) = c • MeasureTheory.L1.SimpleFunc.setToL1S T f

Defined in

Mathlib.MeasureTheory.Integral.SetToL1

Cited by

1 results in Mathlib

Foundations

Depth 236 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedRingModuleIsBoundedSMulDistribSMul