Theorems · Theorem · measure theory

MeasureTheory.VectorMeasure.variation_withDensity

∀ {X : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} {mX : MeasurableSpace X} [inst : NormedAddCommGroup E]
  [inst_1 : NormedSpace ℝ E] [inst_2 : NormedAddCommGroup F] [inst_3 : NormedSpace ℝ F] [inst_4 : NormedAddCommGroup G]
  [inst_5 : NormedSpace ℝ G] {μ : MeasureTheory.VectorMeasure X F} {f : X → E} {B : E →L[ℝ] F →L[ℝ] G}
  [CompleteSpace G],
  μ.Integrable f →
    (∀ (x : E) (y : F), ‖(B x) y‖₊ = ‖x‖₊ * ‖y‖₊) →
      (μ.withDensity f B).variation = (μ.transpose B).variation.withDensity fun x => ‖f x‖ₑ

If ‖B x y‖ = ‖x‖ * ‖y‖ for all x, y, then the variation of a vector measure with density f wrt μ is the measure with density ‖f‖ₑ with respect to the variation of μ. The condition on B is necessary: for a counterexample without it, let B be the scalar product in ℝ² and f x everywhere horizontal and μ s everywhere vertical. Then μ.withDensity f B = 0 so its variation is zero, while the integral of ‖f‖ₑ is not.

Defined in

Mathlib.MeasureTheory.VectorMeasure.WithDensityVec

Cited by

1 results in Mathlib

Foundations

Depth 258 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpace