Theorems · Theorem · measure theory

MeasureTheory.SimpleFunc.setToSimpleFunc_eq_sum_of_subset

∀ {α : Type u_1} {F : Type u_3} {F' : Type u_4} [inst : NormedAddCommGroup F] [inst_1 : NormedSpace ℝ F]
  [inst_2 : NormedAddCommGroup F'] [inst_3 : NormedSpace ℝ F'] {m : MeasurableSpace α}
  [inst_4 : DecidablePred fun x => x ≠ 0] (T : Set α → F →L[ℝ] F'),
  T ∅ = 0 →
    ∀ {f : MeasureTheory.SimpleFunc α F} {s : Finset F},
      {x ∈ f.range | x ≠ 0} ⊆ s → MeasureTheory.SimpleFunc.setToSimpleFunc T f = ∑ x ∈ s, (T (⇑f ⁻¹' {x})) x

The setToSimpleFunc is equal to a sum over any set that includes f.range (except 0).

Defined in

Mathlib.MeasureTheory.Integral.FinMeasAdditive

Cited by

1 results in Mathlib

Foundations

Depth 159 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceDecidablePred