-- Mathlib/MeasureTheory/Measure/ContinuousPreimage.lean
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 : NullMeasurableSet s ν) (hνs : ν s ≠ ∞) :
    Tendsto (fun a ↦ μ ((f a ⁻¹' s) ∆ (g ⁻¹' s))) l (𝓝 0)

Proof state:
case h.h₁
α : Type u_1
X : Type u_2
Y : Type u_3
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace X
inst✝⁷ : BorelSpace X
inst✝⁶ : R1Space X
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : MeasurableSpace Y
inst✝³ : BorelSpace Y
inst✝² : R1Space Y
μ : Measure X
ν : Measure Y
inst✝¹ : μ.InnerRegularCompactLTTop
inst✝ : IsLocallyFiniteMeasure ν
l : Filter α
f : α → C(X, Y)
g : C(X, Y)
hfg : Tendsto f l (𝓝 g)
hf : ∀ᶠ (a : α) in l, MeasurePreserving (⇑(f a)) μ ν
hg : MeasurePreserving (⇑g) μ ν
this : ν.InnerRegularCompactLTTop
s : Set Y
hs : NullMeasurableSet s ν
hνs : ν s ≠ ⊤
ε : ℝ≥0∞
hε : ε > 0
hso : IsOpen s
hνs' : μ (⇑g ⁻¹' s) ≠ ⊤
K : Set X
hKg : MapsTo (⇑g) K s
hKco : IsCompact K
hKcl : IsClosed K
hKμ : μ (⇑g ⁻¹' s \ K) < ε / 2
hKm : NullMeasurableSet K μ
a : α
hfa : MeasurePreserving (⇑(f a)) μ ν
ha : MapsTo (⇑(f a)) K s
hK' : μ K ≠ ⊤
⊢ μ (⇑(f a) ⁻¹' s \ K) ≤ ε / 2

Tactic:
rw [measure_diff_le_iff_le_add hKm ha.subset_preimage hK', hfa.measure_preimage hs,
  ← hg.measure_preimage hs, ← measure_diff_le_iff_le_add hKm hKg.subset_preimage hK']

Result:
case h.h₁
α : Type u_1
X : Type u_2
Y : Type u_3
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace X
inst✝⁷ : BorelSpace X
inst✝⁶ : R1Space X
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : MeasurableSpace Y
inst✝³ : BorelSpace Y
inst✝² : R1Space Y
μ : Measure X
ν : Measure Y
inst✝¹ : μ.InnerRegularCompactLTTop
inst✝ : IsLocallyFiniteMeasure ν
l : Filter α
f : α → C(X, Y)
g : C(X, Y)
hfg : Tendsto f l (𝓝 g)
hf : ∀ᶠ (a : α) in l, MeasurePreserving (⇑(f a)) μ ν
hg : MeasurePreserving (⇑g) μ ν
this : ν.InnerRegularCompactLTTop
s : Set Y
hs : NullMeasurableSet s ν
hνs : ν s ≠ ⊤
ε : ℝ≥0∞
hε : ε > 0
hso : IsOpen s
hνs' : μ (⇑g ⁻¹' s) ≠ ⊤
K : Set X
hKg : MapsTo (⇑g) K s
hKco : IsCompact K
hKcl : IsClosed K
hKμ : μ (⇑g ⁻¹' s \ K) < ε / 2
hKm : NullMeasurableSet K μ
a : α
hfa : MeasurePreserving (⇑(f a)) μ ν
ha : MapsTo (⇑(f a)) K s
hK' : μ K ≠ ⊤
⊢ μ (⇑g ⁻¹' s \ K) ≤ ε / 2

Proof state:
case h
α : Type u_1
X : Type u_2
Y : Type u_3
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace X
inst✝⁷ : BorelSpace X
inst✝⁶ : R1Space X
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : MeasurableSpace Y
inst✝³ : BorelSpace Y
inst✝² : R1Space Y
μ : Measure X
ν : Measure Y
inst✝¹ : μ.InnerRegularCompactLTTop
inst✝ : IsLocallyFiniteMeasure ν
l : Filter α
f : α → C(X, Y)
g : C(X, Y)
hfg : Tendsto f l (𝓝 g)
hf : ∀ᶠ (a : α) in l, MeasurePreserving (⇑(f a)) μ ν
hg : MeasurePreserving (⇑g) μ ν
this : ν.InnerRegularCompactLTTop
s : Set Y
hs : NullMeasurableSet s ν
hνs : ν s ≠ ⊤
ε : ℝ≥0∞
hε : ε > 0
hso : IsOpen s
hνs' : μ (⇑g ⁻¹' s) ≠ ⊤
K : Set X
hKg : MapsTo (⇑g) K s
hKco : IsCompact K
hKcl : IsClosed K
hKμ : μ (⇑g ⁻¹' s \ K) < ε / 2
hKm : NullMeasurableSet K μ
a : α
hfa : MeasurePreserving (⇑(f a)) μ ν
ha : MapsTo (⇑(f a)) K s
⊢ μ ((⇑(f a) ⁻¹' s) ∆ K) + μ (K ∆ (⇑g ⁻¹' s)) ≤ ε / 2 + ε / 2

Tactic:
rw [symmDiff_of_ge ha.subset_preimage, symmDiff_of_le hKg.subset_preimage]

Result:
case h
α : Type u_1
X : Type u_2
Y : Type u_3
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace X
inst✝⁷ : BorelSpace X
inst✝⁶ : R1Space X
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : MeasurableSpace Y
inst✝³ : BorelSpace Y
inst✝² : R1Space Y
μ : Measure X
ν : Measure Y
inst✝¹ : μ.InnerRegularCompactLTTop
inst✝ : IsLocallyFiniteMeasure ν
l : Filter α
f : α → C(X, Y)
g : C(X, Y)
hfg : Tendsto f l (𝓝 g)
hf : ∀ᶠ (a : α) in l, MeasurePreserving (⇑(f a)) μ ν
hg : MeasurePreserving (⇑g) μ ν
this : ν.InnerRegularCompactLTTop
s : Set Y
hs : NullMeasurableSet s ν
hνs : ν s ≠ ⊤
ε : ℝ≥0∞
hε : ε > 0
hso : IsOpen s
hνs' : μ (⇑g ⁻¹' s) ≠ ⊤
K : Set X
hKg : MapsTo (⇑g) K s
hKco : IsCompact K
hKcl : IsClosed K
hKμ : μ (⇑g ⁻¹' s \ K) < ε / 2
hKm : NullMeasurableSet K μ
a : α
hfa : MeasurePreserving (⇑(f a)) μ ν
ha : MapsTo (⇑(f a)) K s
⊢ μ (⇑(f a) ⁻¹' s \ K) + μ (⇑g ⁻¹' s \ K) ≤ ε / 2 + ε / 2

Proof state:
α : Type u_1
X : Type u_2
Y : Type u_3
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace X
inst✝⁷ : BorelSpace X
inst✝⁶ : R1Space X
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : MeasurableSpace Y
inst✝³ : BorelSpace Y
inst✝² : R1Space Y
μ : Measure X
ν : Measure Y
inst✝¹ : μ.InnerRegularCompactLTTop
inst✝ : IsLocallyFiniteMeasure ν
l : Filter α
f : α → C(X, Y)
g : C(X, Y)
s : Set Y
hfg : Tendsto f l (𝓝 g)
hf : ∀ᶠ (a : α) in l, MeasurePreserving (⇑(f a)) μ ν
hg : MeasurePreserving (⇑g) μ ν
hs : NullMeasurableSet s ν
hνs : ν s ≠ ⊤
⊢ ν.InnerRegularCompactLTTop

Tactic:
rw [← hg.map_eq]

Result:
α : Type u_1
X : Type u_2
Y : Type u_3
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace X
inst✝⁷ : BorelSpace X
inst✝⁶ : R1Space X
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : MeasurableSpace Y
inst✝³ : BorelSpace Y
inst✝² : R1Space Y
μ : Measure X
ν : Measure Y
inst✝¹ : μ.InnerRegularCompactLTTop
inst✝ : IsLocallyFiniteMeasure ν
l : Filter α
f : α → C(X, Y)
g : C(X, Y)
s : Set Y
hfg : Tendsto f l (𝓝 g)
hf : ∀ᶠ (a : α) in l, MeasurePreserving (⇑(f a)) μ ν
hg : MeasurePreserving (⇑g) μ ν
hs : NullMeasurableSet s ν
hνs : ν s ≠ ⊤
⊢ (Measure.map (⇑g) μ).InnerRegularCompactLTTop

Proof state:
case bc
α : Type u_1
X : Type u_2
Y : Type u_3
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace X
inst✝⁷ : BorelSpace X
inst✝⁶ : R1Space X
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : MeasurableSpace Y
inst✝³ : BorelSpace Y
inst✝² : R1Space Y
μ : Measure X
ν : Measure Y
inst✝¹ : μ.InnerRegularCompactLTTop
inst✝ : IsLocallyFiniteMeasure ν
l : Filter α
f : α → C(X, Y)
g : C(X, Y)
s : Set Y
hfg : Tendsto f l (𝓝 g)
hf : ∀ᶠ (a : α) in l, MeasurePreserving (⇑(f a)) μ ν
hg : MeasurePreserving (⇑g) μ ν
hs : NullMeasurableSet s ν
hνs : ν s ≠ ⊤
this✝ : ν.InnerRegularCompactLTTop
ε : ℝ≥0∞
hε : ε > 0
this :
  ∀ {s : Set Y},
    NullMeasurableSet s ν → ν s ≠ ⊤ → ∀ ε > 0, IsOpen s → ∀ᶠ (x : α) in l, μ ((⇑(f x) ⁻¹' s) ∆ (⇑g ⁻¹' s)) ≤ ε
hso : ¬IsOpen s
H : 0 < ε / 3
U : Set Y
hUo : IsOpen U
hU : ν U < ⊤
hmU : NullMeasurableSet U ν
hUs : ν (U ∆ s) ≤ ε / 3
a : α
hfa : MeasurePreserving (⇑(f a)) μ ν
ha : μ ((⇑(f a) ⁻¹' U) ∆ (⇑g ⁻¹' U)) ≤ ε / 3
⊢ μ ((⇑(f a) ⁻¹' s) ∆ (⇑g ⁻¹' U)) ≤ μ ((⇑(f a) ⁻¹' s) ∆ (⇑(f a) ⁻¹' U)) + μ ((⇑(f a) ⁻¹' U) ∆ (⇑g ⁻¹' U))

Tactic:
apply measure_symmDiff_le

Result:
no goals

Proof state:
α : Type u_1
X : Type u_2
Y : Type u_3
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace X
inst✝⁷ : BorelSpace X
inst✝⁶ : R1Space X
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : MeasurableSpace Y
inst✝³ : BorelSpace Y
inst✝² : R1Space Y
μ : Measure X
ν : Measure Y
inst✝¹ : μ.InnerRegularCompactLTTop
inst✝ : IsLocallyFiniteMeasure ν
l : Filter α
f : α → C(X, Y)
g : C(X, Y)
hfg : Tendsto f l (𝓝 g)
hf : ∀ᶠ (a : α) in l, MeasurePreserving (⇑(f a)) μ ν
hg : MeasurePreserving (⇑g) μ ν
this : ν.InnerRegularCompactLTTop
s : Set Y
hs : NullMeasurableSet s ν
hνs : ν s ≠ ⊤
ε : ℝ≥0∞
hε : ε > 0
hso : IsOpen s
⊢ ∀ᶠ (x : α) in l, μ ((⇑(f x) ⁻¹' s) ∆ (⇑g ⁻¹' s)) ≤ ε

Tactic:
have hνs' : μ (g ⁻¹' s) ≠ ∞ := by rwa [hg.measure_preimage hs]

Result:
α : Type u_1
X : Type u_2
Y : Type u_3
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace X
inst✝⁷ : BorelSpace X
inst✝⁶ : R1Space X
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : MeasurableSpace Y
inst✝³ : BorelSpace Y
inst✝² : R1Space Y
μ : Measure X
ν : Measure Y
inst✝¹ : μ.InnerRegularCompactLTTop
inst✝ : IsLocallyFiniteMeasure ν
l : Filter α
f : α → C(X, Y)
g : C(X, Y)
hfg : Tendsto f l (𝓝 g)
hf : ∀ᶠ (a : α) in l, MeasurePreserving (⇑(f a)) μ ν
hg : MeasurePreserving (⇑g) μ ν
this : ν.InnerRegularCompactLTTop
s : Set Y
hs : NullMeasurableSet s ν
hνs : ν s ≠ ⊤
ε : ℝ≥0∞
hε : ε > 0
hso : IsOpen s
hνs' : μ (⇑g ⁻¹' s) ≠ ⊤
⊢ ∀ᶠ (x : α) in l, μ ((⇑(f x) ⁻¹' s) ∆ (⇑g ⁻¹' s)) ≤ ε