Theorems · Theorem · measure theory

MeasureTheory.integral_fn_integral_sub

∀ {α : Type u_1} {β : Type u_2} {E : Type u_3} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β]
  {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [inst_2 : NormedAddCommGroup E] [MeasureTheory.SFinite ν]
  [inst_4 : NormedSpace ℝ E] [MeasureTheory.SFinite μ] {E' : Type u_4} [inst_6 : NormedAddCommGroup E']
  [inst_7 : NormedSpace ℝ E'] ⦃f g : α × β → E⦄ (F : E → E'),
  MeasureTheory.Integrable f (μ.prod ν) →
    MeasureTheory.Integrable g (μ.prod ν) →
      ∫ (x : α), F (∫ (y : β), f (x, y) - g (x, y) ∂ν) ∂μ =
        ∫ (x : α), F (∫ (y : β), f (x, y) ∂ν - ∫ (y : β), g (x, y) ∂ν) ∂μ

Integrals commute with subtraction inside another integral. F can be any measurable function.

Defined in

Mathlib.MeasureTheory.Integral.Prod

Cited by

1 results in Mathlib

Foundations

Depth 261 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceMeasurableSpaceNormedAddCommGroupMeasureTheory.SFiniteNormedSpaceMeasureTheory.SFiniteNormedAddCommGroupNormedSpace