Theorems · Theorem · measure theory

MeasureTheory.setToFun_simpleFunc

∀ {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} {C : ℝ} [CompleteSpace F] (hT : MeasureTheory.DominatedFinMeasAdditive μ T C)
  (f : MeasureTheory.SimpleFunc α E),
  MeasureTheory.Integrable (⇑f) μ → MeasureTheory.setToFun μ T hT ⇑f = ∑ x ∈ f.range, (T (⇑f ⁻¹' {x})) x

Defined in

Mathlib.MeasureTheory.Integral.SetToL1

Cited by

2 results in Mathlib

Foundations

Depth 243 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpace