Theorems · Theorem · probability

MeasureTheory.TendstoInDistribution.continuous_comp

• 1000+ list: Continuous mapping theorem

∀ {ι : Type u_1} {E : Type u_2} {Ω' : Type u_3} {Ω : ι → Type u_5} {m : (i : ι) → MeasurableSpace (Ω i)}
  {μ : (i : ι) → MeasureTheory.Measure (Ω i)} [inst : ∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μ i)]
  {m' : MeasurableSpace Ω'} {μ' : MeasureTheory.Measure Ω'} [inst_1 : MeasureTheory.IsProbabilityMeasure μ']
  {mE : MeasurableSpace E} {X : (i : ι) → Ω i → E} {Z : Ω' → E} {l : Filter ι} [inst_2 : TopologicalSpace E]
  {F : Type u_6} [inst_3 : OpensMeasurableSpace E] [inst_4 : TopologicalSpace F] [inst_5 : MeasurableSpace F]
  [inst_6 : BorelSpace F] {g : E → F},
  Continuous g →
    MeasureTheory.TendstoInDistribution X l Z μ μ' →
      MeasureTheory.TendstoInDistribution (fun n => g ∘ X n) l (g ∘ Z) μ μ'

Continuous mapping theorem: if X n tends to Z in distribution and g is continuous, then g ∘ X n tends to g ∘ Z in distribution.

Defined in

Mathlib.MeasureTheory.Function.ConvergenceInDistribution

Cited by

3 results in Mathlib

Foundations

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

Assumes

MeasureTheory.IsProbabilityMeasureMeasureTheory.IsProbabilityMeasureTopologicalSpaceOpensMeasurableSpaceTopologicalSpaceMeasurableSpaceBorelSpace