Theorems · Theorem · measure theory

MeasureTheory.extend_union

∀ {α : Type u_1} {P : Set α → Prop} {m : (s : Set α) → P s → ENNReal} (P0 : P ∅),
  m ∅ P0 = 0 →
    ∀ (PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)),
      (∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),
          Pairwise (Function.onFun Disjoint f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯) →
        ∀ {s₁ s₂ : Set α},
          Disjoint s₁ s₂ →
            P s₁ → P s₂ → MeasureTheory.extend m (s₁ ∪ s₂) = MeasureTheory.extend m s₁ + MeasureTheory.extend m s₂

Defined in

Mathlib.MeasureTheory.OuterMeasure.Induced

Cited by

1 results in Mathlib

Foundations

Depth 130 from the axioms · uses propext, Classical.choice, Quot.sound