Theorems · Theorem · measure theory

MeasureTheory.FiniteMeasure.tendsto_iff_forall_integral_tendsto

∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] [inst_1 : TopologicalSpace Ω] [inst_2 : OpensMeasurableSpace Ω]
  {γ : Type u_2} {F : Filter γ} {μs : γ → MeasureTheory.FiniteMeasure Ω} {μ : MeasureTheory.FiniteMeasure Ω},
  Filter.Tendsto μs F (nhds μ) ↔
    ∀ (f : BoundedContinuousFunction Ω ℝ),
      Filter.Tendsto (fun i => ∫ (x : Ω), f x ∂↑(μs i)) F (nhds (∫ (x : Ω), f x ∂↑μ))

A characterization of weak convergence in terms of integrals of bounded continuous real-valued functions.

Defined in

Mathlib.MeasureTheory.Measure.FiniteMeasure

Cited by

3 results in Mathlib

Foundations

Depth 255 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceTopologicalSpaceOpensMeasurableSpace