Theorems · Theorem · measure theory

MeasureTheory.SimpleFunc.setToSimpleFunc_mono

∀ {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'] [IsOrderedAddMonoid G']
  {T : Set α → G' →L[ℝ] G''},
  MeasureTheory.FinMeasAdditive μ T →
    (∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x) →
      ∀ {f g : MeasureTheory.SimpleFunc α G'},
        MeasureTheory.Integrable (⇑f) μ →
          MeasureTheory.Integrable (⇑g) μ →
            f ≤ g → MeasureTheory.SimpleFunc.setToSimpleFunc T f ≤ MeasureTheory.SimpleFunc.setToSimpleFunc T g

Defined in

Mathlib.MeasureTheory.Integral.FinMeasAdditive

Cited by

0 results in Mathlib

Foundations

Depth 216 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupPartialOrderIsOrderedAddMonoidNormedSpaceNormedAddCommGroupPartialOrderNormedSpaceIsOrderedAddMonoid