Theorems · Theorem · measure theory

MeasureTheory.SimpleFunc.integral_eq_sum_of_subset

∀ {α : Type u_1} {F : Type u_3} [inst : NormedAddCommGroup F] [inst_1 : NormedSpace ℝ F] {m : MeasurableSpace α}
  {μ : MeasureTheory.Measure α} [inst_2 : DecidablePred fun x => x ≠ 0] {f : MeasureTheory.SimpleFunc α F}
  {s : Finset F}, {x ∈ f.range | x ≠ 0} ⊆ s → MeasureTheory.SimpleFunc.integral μ f = ∑ x ∈ s, μ.real (⇑f ⁻¹' {x}) • x

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

Defined in

Mathlib.MeasureTheory.Integral.Bochner.L1

Cited by

3 results in Mathlib

Foundations

Depth 174 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceDecidablePred