Theorems · Theorem · measure theory

MeasureTheory.StronglyMeasurable.setToFun_prod_right

∀ {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} {β : Type u_7} {mβ : MeasurableSpace β} [MeasureTheory.SFinite μ]
  (hT : MeasureTheory.DominatedFinMeasAdditive μ T C),
  (∀ (s : Set (β × α)), MeasurableSet s → MeasureTheory.StronglyMeasurable fun x => T (Prod.mk x ⁻¹' s)) →
    ∀ ⦃f : β → α → E⦄,
      MeasureTheory.StronglyMeasurable (Function.uncurry f) →
        MeasureTheory.StronglyMeasurable fun x => MeasureTheory.setToFun μ T hT (f x)

The setToFun operation is measurable. This shows that the integrand of (the right-hand-side of) Fubini's theorem is measurable. This version has f in curried form.

Defined in

Mathlib.MeasureTheory.Integral.SetToL1

Cited by

1 results in Mathlib

Foundations

Depth 245 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceMeasureTheory.SFinite