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

Proof steps:
state_before: 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 : MeasurableSet s
hνs : ν s ≠ ⊤
⊢ Tendsto (fun a => μ ((⇑(f a) ⁻¹' s) ∆ (⇑g ⁻¹' s))) l (𝓝 0)
state_after: 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 : MeasurableSet s
hνs : ν s ≠ ⊤
this : ν.InnerRegularCompactLTTop
⊢ Tendsto (fun a => μ ((⇑(f a) ⁻¹' s) ∆ (⇑g ⁻¹' s))) l (𝓝 0)
tactic: have : ν.InnerRegularCompactLTTop := by
    rw [← hg.map_eq]
    exact .map_of_continuous (map_continuous _)
state_before: 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 : MeasurableSet s
hνs : ν s ≠ ⊤
this : ν.InnerRegularCompactLTTop
⊢ Tendsto (fun a => μ ((⇑(f a) ⁻¹' s) ∆ (⇑g ⁻¹' s))) l (𝓝 0)
state_after: 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 : MeasurableSet s
hνs : ν s ≠ ⊤
this : ν.InnerRegularCompactLTTop
⊢ ∀ ε > 0, ∀ᶠ (x : α) in l, μ ((⇑(f x) ⁻¹' s) ∆ (⇑g ⁻¹' s)) ≤ ε
tactic: rw [ENNReal.tendsto_nhds_zero]
state_before: 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 : MeasurableSet s
hνs : ν s ≠ ⊤
this : ν.InnerRegularCompactLTTop
⊢ ∀ ε > 0, ∀ᶠ (x : α) in l, μ ((⇑(f x) ⁻¹' s) ∆ (⇑g ⁻¹' s)) ≤ ε
state_after: 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 : MeasurableSet s
hνs : ν s ≠ ⊤
this : ν.InnerRegularCompactLTTop
ε : ℝ≥0∞
hε : ε > 0
⊢ ∀ᶠ (x : α) in l, μ ((⇑(f x) ⁻¹' s) ∆ (⇑g ⁻¹' s)) ≤ ε
tactic: intro ε hε
state_before: 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 : MeasurableSet s
hνs : ν s ≠ ⊤
this : ν.InnerRegularCompactLTTop
ε : ℝ≥0∞
hε : ε > 0
⊢ ∀ᶠ (x : α) in l, μ ((⇑(f x) ⁻¹' s) ∆ (⇑g ⁻¹' s)) ≤ ε
state_after: case inr
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 : MeasurableSet s
hνs : ν s ≠ ⊤
this✝ : ν.InnerRegularCompactLTTop
ε : ℝ≥0∞
hε : ε > 0
this :
  ∀ {s : Set Y}, MeasurableSet s → ν s ≠ ⊤ → ∀ ε > 0, IsOpen s → ∀ᶠ (x : α) in l, μ ((⇑(f x) ⁻¹' s) ∆ (⇑g ⁻¹' s)) ≤ ε
hso : ¬IsOpen s
⊢ ∀ᶠ (x : α) in l, μ ((⇑(f x) ⁻¹' s) ∆ (⇑g ⁻¹' s)) ≤ ε

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 : MeasurableSet s
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: wlog hso : IsOpen s generalizing s ε
state_before: 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 : MeasurableSet s
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
state_after: 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 : MeasurableSet s
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
tactic: have hνs' : μ (g ⁻¹' s) ≠ ∞ := by rwa [hg.measure_preimage hs.nullMeasurableSet]
state_before: 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 : MeasurableSet s
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
state_after: case intro.intro.intro.intro
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 : MeasurableSet s
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
⊢ ∀ᶠ (x : α) in l, μ ((⇑(f x) ⁻¹' s) ∆ (⇑g ⁻¹' s)) ≤ ε
tactic: obtain ⟨K, hKg, hKco, hKcl, hKμ⟩ :
      ∃ K, MapsTo g K s ∧ IsCompact K ∧ IsClosed K ∧ μ (g ⁻¹' s \ K) < ε / 2 :=
    (hs.preimage hg.measurable).exists_isCompact_isClosed_diff_lt hνs' <| by simp [hε.ne']
state_before: case intro.intro.intro.intro
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 : MeasurableSet s
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
⊢ ∀ᶠ (x : α) in l, μ ((⇑(f x) ⁻¹' s) ∆ (⇑g ⁻¹' s)) ≤ ε
state_after: case intro.intro.intro.intro
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 : MeasurableSet s
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 : MeasurableSet K
⊢ ∀ᶠ (x : α) in l, μ ((⇑(f x) ⁻¹' s) ∆ (⇑g ⁻¹' s)) ≤ ε
tactic: have hKm : MeasurableSet K := hKcl.measurableSet
state_before: case intro.intro.intro.intro
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 : MeasurableSet s
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 : MeasurableSet K
⊢ ∀ᶠ (x : α) in l, μ ((⇑(f x) ⁻¹' s) ∆ (⇑g ⁻¹' s)) ≤ ε
state_after: 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 : MeasurableSet s
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 : MeasurableSet K
a : α
hfa : MeasurePreserving (⇑(f a)) μ ν
ha : MapsTo (⇑(f a)) K s
⊢ μ ((⇑(f a) ⁻¹' s) ∆ (⇑g ⁻¹' s)) ≤ ε
tactic: filter_upwards [hf, ContinuousMap.tendsto_nhds_compactOpen.mp hfg K hKco s hso hKg] with a hfa ha
state_before: 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 : MeasurableSet s
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 : MeasurableSet K
a : α
hfa : MeasurePreserving (⇑(f a)) μ ν
ha : MapsTo (⇑(f a)) K s
⊢ μ ((⇑(f a) ⁻¹' s) ∆ (⇑g ⁻¹' s)) ≤ ε
state_after: 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 : MeasurableSet s
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 : MeasurableSet K
a : α
hfa : MeasurePreserving (⇑(f a)) μ ν
ha : MapsTo (⇑(f a)) K s
⊢ μ ((⇑(f a) ⁻¹' s) ∆ (⇑g ⁻¹' s)) ≤ ε / 2 + ε / 2
tactic: rw [← ENNReal.add_halves ε]
state_before: 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 : MeasurableSet s
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 : MeasurableSet K
a : α
hfa : MeasurePreserving (⇑(f a)) μ ν
ha : MapsTo (⇑(f a)) K s
⊢ μ ((⇑(f a) ⁻¹' s) ∆ (⇑g ⁻¹' s)) ≤ ε / 2 + ε / 2
state_after: 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 : MeasurableSet s
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 : MeasurableSet K
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: refine (measure_symmDiff_le _ K _).trans ?_
state_before: 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 : MeasurableSet s
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 : MeasurableSet K
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
state_after: 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 : MeasurableSet s
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 : MeasurableSet K
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
tactic: rw [symmDiff_of_ge ha.subset_preimage, symmDiff_of_le hKg.subset_preimage]
state_before: 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 : MeasurableSet s
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 : MeasurableSet K
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
state_after: 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 : MeasurableSet s
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 : MeasurableSet K
a : α
hfa : MeasurePreserving (⇑(f a)) μ ν
ha : MapsTo (⇑(f a)) K s
⊢ μ (⇑(f a) ⁻¹' s \ K) ≤ ε / 2
tactic: gcongr
state_before: 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 : MeasurableSet s
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 : MeasurableSet K
a : α
hfa : MeasurePreserving (⇑(f a)) μ ν
ha : MapsTo (⇑(f a)) K s
⊢ μ (⇑(f a) ⁻¹' s \ K) ≤ ε / 2
state_after: 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 : MeasurableSet s
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 : MeasurableSet K
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: have hK' : μ K ≠ ∞ := ne_top_of_le_ne_top hνs' <| measure_mono hKg.subset_preimage
state_before: 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 : MeasurableSet s
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 : MeasurableSet K
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
state_after: 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 : MeasurableSet s
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 : MeasurableSet K
a : α
hfa : MeasurePreserving (⇑(f a)) μ ν
ha : MapsTo (⇑(f a)) K s
hK' : μ K ≠ ⊤
⊢ μ (⇑g ⁻¹' s \ K) ≤ ε / 2
tactic: rw [measure_diff_le_iff_le_add hKm ha.subset_preimage hK',
    hfa.measure_preimage hs.nullMeasurableSet, ← hg.measure_preimage hs.nullMeasurableSet,
    ← measure_diff_le_iff_le_add hKm hKg.subset_preimage hK']
state_before: 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 : MeasurableSet s
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 : MeasurableSet K
a : α
hfa : MeasurePreserving (⇑(f a)) μ ν
ha : MapsTo (⇑(f a)) K s
hK' : μ K ≠ ⊤
⊢ μ (⇑g ⁻¹' s \ K) ≤ ε / 2
state_after: no goals
tactic: exact hKμ.le