Theorems · Theorem · measure theory

MeasureTheory.tendsto_measure_symmDiff_preimage_nhds_zero

∀ {α : Type u_1} {X : Type u_2} {Y : Type u_3} [inst : TopologicalSpace X] [inst_1 : MeasurableSpace X] [BorelSpace X]
  [R1Space X] [inst_4 : TopologicalSpace Y] [inst_5 : MeasurableSpace Y] [BorelSpace Y] [R1Space Y]
  {μ : MeasureTheory.Measure X} {ν : MeasureTheory.Measure Y} [μ.InnerRegularCompactLTTop]
  [MeasureTheory.IsLocallyFiniteMeasure ν] {l : Filter α} {f : α → C(X, Y)} {g : C(X, Y)} {s : Set Y},
  Filter.Tendsto f l (nhds g) →
    (∀ᶠ (a : α) in l, MeasureTheory.MeasurePreserving (⇑(f a)) μ ν) →
      MeasureTheory.MeasurePreserving (⇑g) μ ν →
        MeasureTheory.NullMeasurableSet s ν →
          ν s ≠ ⊤ → Filter.Tendsto (fun a => μ (symmDiff (⇑(f a) ⁻¹' s) (⇑g ⁻¹' s))) l (nhds 0)

Let X and Y be R₁ topological spaces with Borel σ-algebras and measures μ and ν, respectively. Suppose that μ is inner regular for finite measure sets with respect to compact sets and ν is a locally finite measure. Let f : α → C(X, Y) be a family of continuous maps that converges to a continuous map g : C(X, Y) in the compact-open topology along a filter l. Suppose that g is a measure-preserving map and f a is a measure-preserving map eventually along l. Then for any finite measure measurable set s, the preimages f a ⁻¹' s tend to the preimage g ⁻¹' s in measure. More precisely, the measure of the symmetric difference of these two sets tends to zero.

Defined in

Mathlib.MeasureTheory.Measure.ContinuousPreimage

Cited by

2 results in Mathlib

Foundations

Depth 204 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceMeasurableSpaceBorelSpaceR1SpaceTopologicalSpaceMeasurableSpaceBorelSpaceR1SpaceMeasureTheory.Measure.InnerRegularCompactLTTopMeasureTheory.IsLocallyFiniteMeasure