Theorems · Theorem · measure theory

MeasureTheory.lintegral_iSup_directed

∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable β]
  {f : β → α → ENNReal},
  (∀ (b : β), AEMeasurable (f b) μ) →
    Directed (fun x1 x2 => x1 ≤ x2) f → ∫⁻ (a : α), ⨆ b, f b a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ

Monotone convergence theorem for a supremum over a directed family and indexed by a countable type.

Defined in

Mathlib.MeasureTheory.Integral.Lebesgue.Add

Cited by

2 results in Mathlib

Foundations

Depth 202 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

Countable