Theorems · Theorem · measure theory

MeasureTheory.VectorMeasure.integral_eq_setToFun_transpose

∀ {X : Type u_2} {E : Type u_4} {F : Type u_5} {G : Type u_6} {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} {B : E →L[ℝ] F →L[ℝ] G} {f : X → E},
  μ.Integrable f → ∫ᵛ (x : X), f x ∂[B; μ] = MeasureTheory.setToFun (μ.transpose B).variation ⇑(μ.transpose B) ⋯ f

Defined in

Mathlib.MeasureTheory.VectorMeasure.Integral

Cited by

1 results in Mathlib

Foundations

Depth 246 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace