Theorems · Theorem · measure theory

MeasureTheory.Lp.ae_eq_of_forall_setIntegral_eq

∀ {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : NormedAddCommGroup E]
  [inst_1 : NormedSpace ℝ E] [CompleteSpace E] {p : ENNReal} (f g : ↥(MeasureTheory.Lp E p μ)),
  p ≠ 0 →
    p ≠ ⊤ →
      (∀ (s : Set α), MeasurableSet s → μ s < ⊤ → MeasureTheory.IntegrableOn (↑↑f) s μ) →
        (∀ (s : Set α), MeasurableSet s → μ s < ⊤ → MeasureTheory.IntegrableOn (↑↑g) s μ) →
          (∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, ↑↑f x ∂μ = ∫ (x : α) in s, ↑↑g x ∂μ) →
            ↑↑f =ᵐ[μ] ↑↑g

Defined in

Mathlib.MeasureTheory.Function.AEEqOfIntegral

Cited by

0 results in Mathlib

Foundations

Depth 264 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceCompleteSpace