Theorems · Theorem · measure theory

MeasureTheory.setToFun_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'] [ClosedIciTopology G'']
  {T : Set α → G' →L[ℝ] G''} {C : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C),
  (∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x) →
    ∀ {f : α → G'}, 0 ≤ᵐ[μ] f → 0 ≤ MeasureTheory.setToFun μ T hT f

Defined in

Mathlib.MeasureTheory.Integral.SetToL1

Cited by

2 results in Mathlib

Foundations

Depth 243 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupPartialOrderIsOrderedAddMonoidNormedSpaceNormedAddCommGroupPartialOrderNormedSpaceClosedIciTopology