Theorems · Theorem · measure theory

MeasureTheory.SimpleFunc.setToSimpleFunc_smul

∀ {α : Type u_1} {F : Type u_3} {𝕜 : Type u_6} [inst : NormedAddCommGroup F] [inst_1 : NormedSpace ℝ F]
  {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_7} [inst_2 : NormedAddCommGroup E]
  [inst_3 : SMulZeroClass 𝕜 E] [inst_4 : NormedSpace ℝ E] [inst_5 : DistribSMul 𝕜 F] (T : Set α → E →L[ℝ] F),
  MeasureTheory.FinMeasAdditive μ T →
    (∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c • x) = c • (T s) x) →
      ∀ (c : 𝕜) {f : MeasureTheory.SimpleFunc α E},
        MeasureTheory.Integrable (⇑f) μ →
          MeasureTheory.SimpleFunc.setToSimpleFunc T (c • f) = c • MeasureTheory.SimpleFunc.setToSimpleFunc T f

Defined in

Mathlib.MeasureTheory.Integral.FinMeasAdditive

Cited by

2 results in Mathlib

Foundations

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

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupSMulZeroClassNormedSpaceDistribSMul