Theorem: Filter.Tendsto.compMeasurePreservingLp
File: Mathlib/MeasureTheory/Function/LpSpace/ContinuousCompMeasurePreserving.lean

Proof steps:
state_before: α✝ : Type u_1
X : Type u_2
Y : Type u_3
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : BorelSpace X
inst✝⁸ : R1Space X
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : BorelSpace Y
inst✝⁴ : R1Space Y
μ : Measure X
ν : Measure Y
inst✝³ : μ.InnerRegularCompactLTTop
inst✝² : IsLocallyFiniteMeasure ν
E : Type u_4
inst✝¹ : NormedAddCommGroup E
p : ℝ≥0∞
inst✝ : Fact (1 ≤ p)
α : Type u_5
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 : α), MeasurePreserving (⇑(g a)) μ ν
hgm₀ : MeasurePreserving (⇑g₀) μ ν
hp : p ≠ ⊤
⊢ Tendsto (fun a => (Lp.compMeasurePreserving ⇑(g a) ⋯) (f a)) l (𝓝 ((Lp.compMeasurePreserving (⇑g₀) hgm₀) f₀))
state_after: α✝ : Type u_1
X : Type u_2
Y : Type u_3
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : BorelSpace X
inst✝⁸ : R1Space X
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : BorelSpace Y
inst✝⁴ : R1Space Y
μ : Measure X
ν : Measure Y
inst✝³ : μ.InnerRegularCompactLTTop
inst✝² : IsLocallyFiniteMeasure ν
E : Type u_4
inst✝¹ : NormedAddCommGroup E
p : ℝ≥0∞
inst✝ : Fact (1 ≤ p)
α : Type u_5
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 : α), MeasurePreserving (⇑(g a)) μ ν
hgm₀ : MeasurePreserving (⇑g₀) μ ν
hp : p ≠ ⊤
this :
  Tendsto (fun gf => (Lp.compMeasurePreserving ⇑↑gf.2 ⋯) gf.1) (𝓝 (f₀, ⟨g₀, hgm₀⟩))
    (𝓝 ((Lp.compMeasurePreserving ⇑↑(f₀, ⟨g₀, hgm₀⟩).2 ⋯) (f₀, ⟨g₀, hgm₀⟩).1))
⊢ Tendsto (fun a => (Lp.compMeasurePreserving ⇑(g a) ⋯) (f a)) l (𝓝 ((Lp.compMeasurePreserving (⇑g₀) hgm₀) f₀))
tactic: have := (Lp.compMeasurePreserving_continuous μ ν E hp).tendsto ⟨f₀, g₀, hgm₀⟩
state_before: α✝ : Type u_1
X : Type u_2
Y : Type u_3
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : BorelSpace X
inst✝⁸ : R1Space X
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : BorelSpace Y
inst✝⁴ : R1Space Y
μ : Measure X
ν : Measure Y
inst✝³ : μ.InnerRegularCompactLTTop
inst✝² : IsLocallyFiniteMeasure ν
E : Type u_4
inst✝¹ : NormedAddCommGroup E
p : ℝ≥0∞
inst✝ : Fact (1 ≤ p)
α : Type u_5
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 : α), MeasurePreserving (⇑(g a)) μ ν
hgm₀ : MeasurePreserving (⇑g₀) μ ν
hp : p ≠ ⊤
this :
  Tendsto (fun gf => (Lp.compMeasurePreserving ⇑↑gf.2 ⋯) gf.1) (𝓝 (f₀, ⟨g₀, hgm₀⟩))
    (𝓝 ((Lp.compMeasurePreserving ⇑↑(f₀, ⟨g₀, hgm₀⟩).2 ⋯) (f₀, ⟨g₀, hgm₀⟩).1))
⊢ Tendsto (fun a => (Lp.compMeasurePreserving ⇑(g a) ⋯) (f a)) l (𝓝 ((Lp.compMeasurePreserving (⇑g₀) hgm₀) f₀))
state_after: α✝ : Type u_1
X : Type u_2
Y : Type u_3
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : BorelSpace X
inst✝⁸ : R1Space X
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : BorelSpace Y
inst✝⁴ : R1Space Y
μ : Measure X
ν : Measure Y
inst✝³ : μ.InnerRegularCompactLTTop
inst✝² : IsLocallyFiniteMeasure ν
E : Type u_4
inst✝¹ : NormedAddCommGroup E
p : ℝ≥0∞
inst✝ : Fact (1 ≤ p)
α : Type u_5
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)
hgm : ∀ (a : α), MeasurePreserving (⇑(g a)) μ ν
hgm₀ : MeasurePreserving (⇑g₀) μ ν
hp : p ≠ ⊤
this :
  Tendsto (fun gf => (Lp.compMeasurePreserving ⇑↑gf.2 ⋯) gf.1) (𝓝 (f₀, ⟨g₀, hgm₀⟩))
    (𝓝 ((Lp.compMeasurePreserving ⇑↑(f₀, ⟨g₀, hgm₀⟩).2 ⋯) (f₀, ⟨g₀, hgm₀⟩).1))
hg : Tendsto (fun a => ⟨g a, ⋯⟩) l (𝓝 ⟨g₀, hgm₀⟩)
⊢ Tendsto (fun a => (Lp.compMeasurePreserving ⇑(g a) ⋯) (f a)) l (𝓝 ((Lp.compMeasurePreserving (⇑g₀) hgm₀) f₀))
tactic: replace hg : Tendsto (fun a ↦ ⟨g a, hgm a⟩ : α → {g : C(X, Y) // MeasurePreserving g μ ν})
      l (𝓝 ⟨g₀, hgm₀⟩) :=
    tendsto_subtype_rng.2 hg
state_before: α✝ : Type u_1
X : Type u_2
Y : Type u_3
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : BorelSpace X
inst✝⁸ : R1Space X
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : BorelSpace Y
inst✝⁴ : R1Space Y
μ : Measure X
ν : Measure Y
inst✝³ : μ.InnerRegularCompactLTTop
inst✝² : IsLocallyFiniteMeasure ν
E : Type u_4
inst✝¹ : NormedAddCommGroup E
p : ℝ≥0∞
inst✝ : Fact (1 ≤ p)
α : Type u_5
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)
hgm : ∀ (a : α), MeasurePreserving (⇑(g a)) μ ν
hgm₀ : MeasurePreserving (⇑g₀) μ ν
hp : p ≠ ⊤
this :
  Tendsto (fun gf => (Lp.compMeasurePreserving ⇑↑gf.2 ⋯) gf.1) (𝓝 (f₀, ⟨g₀, hgm₀⟩))
    (𝓝 ((Lp.compMeasurePreserving ⇑↑(f₀, ⟨g₀, hgm₀⟩).2 ⋯) (f₀, ⟨g₀, hgm₀⟩).1))
hg : Tendsto (fun a => ⟨g a, ⋯⟩) l (𝓝 ⟨g₀, hgm₀⟩)
⊢ Tendsto (fun a => (Lp.compMeasurePreserving ⇑(g a) ⋯) (f a)) l (𝓝 ((Lp.compMeasurePreserving (⇑g₀) hgm₀) f₀))
state_after: no goals
tactic: convert this.comp (hf.prod_mk_nhds hg)