Theorems · Theorem · measure theory

MeasureTheory.L1.SimpleFunc.setToL1S_congr_measure

∀ {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 μ' : MeasureTheory.Measure α}
  (T : Set α → E →L[ℝ] F),
  (∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0) →
    MeasureTheory.FinMeasAdditive μ T →
      μ.AbsolutelyContinuous μ' →
        ∀ (f : ↥(α →₁ₛ[μ] E)) (f' : ↥(α →₁ₛ[μ'] E)),
          ↑↑↑f =ᵐ[μ] ↑↑↑f' → MeasureTheory.L1.SimpleFunc.setToL1S T f = MeasureTheory.L1.SimpleFunc.setToL1S T f'

setToL1S does not change if we replace the measure μ by μ' with μ ≪ μ'. The statement uses two functions f and f' because they have to belong to different types, but morally these are the same function (we have f =ᵐ[μ] f').

Defined in

Mathlib.MeasureTheory.Integral.SetToL1

Cited by

1 results in Mathlib

Foundations

Depth 228 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace