File: Mathlib/MeasureTheory/Function/LpSpace/ContinuousCompMeasurePreserving.lean

full_name: MeasureTheory.tendsto_measure_symmDiff_preimage_nhds_zero
code: /-- 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. -/
theorem tendsto_measure_symmDiff_preimage_nhds_zero
    {l : Filter α} {f : α → C(X, Y)} {g : C(X, Y)} {s : Set Y} (hfg : Tendsto f l (𝓝 g))
    (hf : ∀ᶠ a in l, MeasurePreserving (f a) μ ν) (hg : MeasurePreserving g μ ν)
    (hs : MeasurableSet s) (hνs : ν s ≠ ∞) :
    Tendsto (fun a ↦ μ ((f a ⁻¹' s) ∆ (g ⁻¹' s))) l (𝓝 0) 
goal: Filter.Tendsto (fun a => μ ((⇑(f a) ⁻¹' s) ∆ (⇑g ⁻¹' s))) l (𝓝 0)

full_name: MeasureTheory.Lp.compMeasurePreserving_continuous
code: /-- 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 `1 ≤ p < ∞` be an extended nonnegative real number.
Then the composition of a function `g : Lp E p ν`
and a measure preserving continuous function `f : C(X, Y)`
is continuous in both variables. -/
theorem compMeasurePreserving_continuous (hp : p ≠ ∞) :
    Continuous fun gf : Lp E p ν × {f : C(X, Y) // MeasurePreserving f μ ν} ↦
      compMeasurePreserving gf.2.1 gf.2.2 gf.1 
goal: Continuous fun gf => (MeasureTheory.Lp.compMeasurePreserving ⇑↑gf.2 ⋯) gf.1

full_name: Filter.Tendsto.compMeasurePreservingLp
code: theorem Filter.Tendsto.compMeasurePreservingLp {α : Type*} {l : Filter α}
    {f : α → Lp E p ν} {f₀ : Lp E p ν} {g : α → C(X, Y)} {g₀ : C(X, Y)}
    (hf : Tendsto f l (𝓝 f₀)) (hg : Tendsto g l (𝓝 g₀))
    (hgm : ∀ a, MeasurePreserving (g a) μ ν) (hgm₀ : MeasurePreserving g₀ μ ν) (hp : p ≠ ∞) :
    Tendsto (fun a ↦ Lp.compMeasurePreserving (g a) (hgm a) (f a)) l
      (𝓝 (Lp.compMeasurePreserving g₀ hgm₀ f₀)) 
goal: Filter.Tendsto (fun a => (MeasureTheory.Lp.compMeasurePreserving ⇑(g a) ⋯) (f a)) l
  (𝓝 ((MeasureTheory.Lp.compMeasurePreserving (⇑g₀) hgm₀) f₀))

full_name: ContinuousWithinAt.compMeasurePreservingLp
code: theorem ContinuousWithinAt.compMeasurePreservingLp (hf : ContinuousWithinAt f s z)
    (hg : ContinuousWithinAt g s z) (hgm : ∀ z, MeasurePreserving (g z) μ ν) (hp : p ≠ ∞) :
    ContinuousWithinAt (fun z ↦ Lp.compMeasurePreserving (g z) (hgm z) (f z)) s z 
goal: ContinuousWithinAt (fun z => (MeasureTheory.Lp.compMeasurePreserving ⇑(g z) ⋯) (f z)) s z

full_name: ContinuousAt.compMeasurePreservingLp
code: theorem ContinuousAt.compMeasurePreservingLp (hf : ContinuousAt f z)
    (hg : ContinuousAt g z) (hgm : ∀ z, MeasurePreserving (g z) μ ν) (hp : p ≠ ∞) :
    ContinuousAt (fun z ↦ Lp.compMeasurePreserving (g z) (hgm z) (f z)) z 
goal: ContinuousAt (fun z => (MeasureTheory.Lp.compMeasurePreserving ⇑(g z) ⋯) (f z)) z

full_name: ContinuousOn.compMeasurePreservingLp
code: theorem ContinuousOn.compMeasurePreservingLp (hf : ContinuousOn f s)
    (hg : ContinuousOn g s) (hgm : ∀ z, MeasurePreserving (g z) μ ν) (hp : p ≠ ∞) :
    ContinuousOn (fun z ↦ Lp.compMeasurePreserving (g z) (hgm z) (f z)) s 
goal: ContinuousOn (fun z => (MeasureTheory.Lp.compMeasurePreserving ⇑(g z) ⋯) (f z)) s

full_name: Continuous.compMeasurePreservingLp
code: theorem Continuous.compMeasurePreservingLp (hf : Continuous f) (hg : Continuous g)
    (hgm : ∀ z, MeasurePreserving (g z) μ ν) (hp : p ≠ ∞) :
    Continuous (fun z ↦ Lp.compMeasurePreserving (g z) (hgm z) (f z)) 
goal: Continuous fun z => (MeasureTheory.Lp.compMeasurePreserving ⇑(g z) ⋯) (f z)