Theorems · Theorem · measure theory

MeasureTheory.SimpleFunc.integral_eq_sum

∀ {α : Type u_1} {E : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {m : MeasurableSpace α}
  {μ : MeasureTheory.Measure α} [CompleteSpace E] (f : MeasureTheory.SimpleFunc α E),
  MeasureTheory.Integrable (⇑f) μ → ∫ (x : α), f x ∂μ = ∑ x ∈ f.range, μ.real (⇑f ⁻¹' {x}) • x

Defined in

Mathlib.MeasureTheory.Integral.Bochner.Basic

Cited by

1 results in Mathlib

Foundations

Depth 253 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceCompleteSpace