Theorems · Theorem · measure theory

MeasureTheory.integral_tendsto_of_tendsto_of_monotone

∀ {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ℕ → α → ℝ} {F : α → ℝ},
  (∀ (n : ℕ), MeasureTheory.Integrable (f n) μ) →
    MeasureTheory.Integrable F μ →
      (∀ᵐ (x : α) ∂μ, Monotone fun n => f n x) →
        (∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun n => f n x) Filter.atTop (nhds (F x))) →
          Filter.Tendsto (fun n => ∫ (x : α), f n x ∂μ) Filter.atTop (nhds (∫ (x : α), F x ∂μ))

Monotone convergence theorem for real-valued functions and Bochner integrals

Defined in

Mathlib.MeasureTheory.Integral.Bochner.Basic

Cited by

1 results in Mathlib

Foundations

Depth 257 from the axioms · uses propext, Classical.choice, Quot.sound