Theorems · Theorem · measure theory

MeasureTheory.setToFun_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}
  {T : Set α → E →L[ℝ] F} {C : ℝ} [inst_4 : NormedDivisionRing 𝕜] [inst_5 : Module 𝕜 E] [NormSMulClass 𝕜 E]
  [inst_7 : Module 𝕜 F] [NormSMulClass 𝕜 F] (hT : MeasureTheory.DominatedFinMeasAdditive μ T C),
  (∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c • x) = c • (T s) x) →
    ∀ (c : 𝕜) (f : α → E), MeasureTheory.setToFun μ T hT (c • f) = c • MeasureTheory.setToFun μ T hT f

Defined in

Mathlib.MeasureTheory.Integral.SetToL1

Cited by

3 results in Mathlib

Foundations

Depth 245 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedDivisionRingModuleNormSMulClassModuleNormSMulClass