Theorems · Theorem · measure theory

MeasureTheory.SimpleFunc.setToSimpleFunc_congr

∀ {α : Type u_1} {E : Type u_2} {F : Type u_3} [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),
  (∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0) →
    MeasureTheory.FinMeasAdditive μ T →
      ∀ {f g : MeasureTheory.SimpleFunc α E},
        MeasureTheory.Integrable (⇑f) μ →
          ⇑f =ᵐ[μ] ⇑g → MeasureTheory.SimpleFunc.setToSimpleFunc T f = MeasureTheory.SimpleFunc.setToSimpleFunc T g

Defined in

Mathlib.MeasureTheory.Integral.FinMeasAdditive

Cited by

10 results in Mathlib

Foundations

Depth 215 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace