Theorems · Theorem · measure theory

MeasureTheory.VectorMeasure.setIntegral_eq_integral_of_ae_compl_eq_zero

∀ {X : Type u_2} {E : Type u_3} {F : Type u_4} {G : Type u_5} {mX : MeasurableSpace X} [inst : NormedAddCommGroup E]
  [inst_1 : NormedAddCommGroup F] [inst_2 : NormedAddCommGroup G] {μ : MeasureTheory.VectorMeasure X F} {f : X → E}
  {s : Set X} [inst_3 : NormedSpace ℝ E] [inst_4 : NormedSpace ℝ F] [inst_5 : NormedSpace ℝ G] {B : E →L[ℝ] F →L[ℝ] G},
  MeasurableSet s → (∀ᵐ (x : X) ∂μ.variation, x ∉ s → f x = 0) → ∫ᵛ (x : X) in s, f x ∂[B; μ] = ∫ᵛ (x : X), f x ∂[B; μ]

If a function vanishes almost everywhere on sᶜ, then its integral on s coincides with its integral on the whole space.

Defined in

Mathlib.MeasureTheory.VectorMeasure.SetIntegral

Cited by

1 results in Mathlib

Foundations

Depth 254 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedAddCommGroupNormedAddCommGroupNormedSpaceNormedSpaceNormedSpace