Theorems · Theorem · measure theory

MeasureTheory.L1.SimpleFunc.setToL1S_nonneg

∀ {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {G'' : Type u_7} {G' : Type u_8}
  [inst : NormedAddCommGroup G'] [inst_1 : PartialOrder G'] [IsOrderedAddMonoid G'] [inst_3 : NormedSpace ℝ G']
  [inst_4 : NormedAddCommGroup G''] [inst_5 : PartialOrder G''] [inst_6 : NormedSpace ℝ G''] {T : Set α → G'' →L[ℝ] G'},
  (∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0) →
    MeasureTheory.FinMeasAdditive μ T →
      (∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : G''), 0 ≤ x → 0 ≤ (T s) x) →
        ∀ {f : ↥(α →₁ₛ[μ] G'')}, 0 ≤ f → 0 ≤ MeasureTheory.L1.SimpleFunc.setToL1S T f

Defined in

Mathlib.MeasureTheory.Integral.SetToL1

Cited by

3 results in Mathlib

Foundations

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

Assumes

NormedAddCommGroupPartialOrderIsOrderedAddMonoidNormedSpaceNormedAddCommGroupPartialOrderNormedSpace