-- Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean
theorem tendsto_addHaar_inter_smul_zero_of_density_zero (s : Set E) (x : E)
    (h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)) (t : Set E)
    (ht : MeasurableSet t) (h''t : μ t ≠ ∞) :
    Tendsto (fun r : ℝ => μ (s ∩ ({x} + r • t)) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 0)

Proof state:
case inl
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t : Set E
ht : MeasurableSet t
h''t : μ t ≠ ⊤
ε : ℝ≥0∞
εpos : 0 < ε
h't : μ t = 0
⊢ ∀ᶠ (b : ℝ) in 𝓝[>] 0, μ (s ∩ ({x} + b • t)) / μ ({x} + b • t) < ε

Tactic:
filter_upwards with r

Result:
case inl.h
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t : Set E
ht : MeasurableSet t
h''t : μ t ≠ ⊤
ε : ℝ≥0∞
εpos : 0 < ε
h't : μ t = 0
r : ℝ
⊢ μ (s ∩ ({x} + r • t)) / μ ({x} + r • t) < ε

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t : Set E
ht : MeasurableSet t
h''t : μ t ≠ ⊤
ε : ℝ≥0∞
εpos : 0 < ε
h't : μ t ≠ 0
n : ℕ
npos : 0 < n
hn : μ (t \ closedBall 0 ↑n) < ε / 2 * μ t
L : Tendsto (fun r => μ (s ∩ ({x} + r • (t ∩ closedBall 0 ↑n))) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 0)
r : ℝ
hr : μ (s ∩ ({x} + r • (t ∩ closedBall 0 ↑n))) / μ ({x} + r • t) < ε / 2
rpos : 0 < r
I : μ (s ∩ ({x} + r • t)) ≤ μ (s ∩ ({x} + r • (t ∩ closedBall 0 ↑n))) + μ ({x} + r • (t \ closedBall 0 ↑n))
⊢ μ (s ∩ ({x} + r • (t ∩ closedBall 0 ↑n))) / μ ({x} + r • t) + μ ({x} + r • (t \ closedBall 0 ↑n)) / μ ({x} + r • t) <
    ε / 2 + ε / 2

Tactic:
apply ENNReal.add_lt_add hr _

Result:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t : Set E
ht : MeasurableSet t
h''t : μ t ≠ ⊤
ε : ℝ≥0∞
εpos : 0 < ε
h't : μ t ≠ 0
n : ℕ
npos : 0 < n
hn : μ (t \ closedBall 0 ↑n) < ε / 2 * μ t
L : Tendsto (fun r => μ (s ∩ ({x} + r • (t ∩ closedBall 0 ↑n))) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 0)
r : ℝ
hr : μ (s ∩ ({x} + r • (t ∩ closedBall 0 ↑n))) / μ ({x} + r • t) < ε / 2
rpos : 0 < r
I : μ (s ∩ ({x} + r • t)) ≤ μ (s ∩ ({x} + r • (t ∩ closedBall 0 ↑n))) + μ ({x} + r • (t \ closedBall 0 ↑n))
⊢ μ ({x} + r • (t \ closedBall 0 ↑n)) / μ ({x} + r • t) < ε / 2

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t : Set E
ht : MeasurableSet t
h''t : μ t ≠ ⊤
ε : ℝ≥0∞
εpos : 0 < ε
h't : μ t ≠ 0
n : ℕ
npos : 0 < n
hn : μ (t \ closedBall 0 ↑n) < ε / 2 * μ t
L : Tendsto (fun r => μ (s ∩ ({x} + r • (t ∩ closedBall 0 ↑n))) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 0)
r : ℝ
hr : μ (s ∩ ({x} + r • (t ∩ closedBall 0 ↑n))) / μ ({x} + r • t) < ε / 2
rpos : 0 < r
⊢ μ (s ∩ ({x} + r • (t ∩ closedBall 0 ↑n))) + μ (s ∩ ({x} + r • (t \ closedBall 0 ↑n))) ≤
    μ (s ∩ ({x} + r • (t ∩ closedBall 0 ↑n))) + μ ({x} + r • (t \ closedBall 0 ↑n))

Tactic:
gcongr

Result:
case bc.h
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t : Set E
ht : MeasurableSet t
h''t : μ t ≠ ⊤
ε : ℝ≥0∞
εpos : 0 < ε
h't : μ t ≠ 0
n : ℕ
npos : 0 < n
hn : μ (t \ closedBall 0 ↑n) < ε / 2 * μ t
L : Tendsto (fun r => μ (s ∩ ({x} + r • (t ∩ closedBall 0 ↑n))) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 0)
r : ℝ
hr : μ (s ∩ ({x} + r • (t ∩ closedBall 0 ↑n))) / μ ({x} + r • t) < ε / 2
rpos : 0 < r
⊢ s ∩ ({x} + r • (t \ closedBall 0 ↑n)) ⊆ {x} + r • (t \ closedBall 0 ↑n)

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t : Set E
ht : MeasurableSet t
h''t : μ t ≠ ⊤
ε : ℝ≥0∞
εpos : 0 < ε
h't : μ t ≠ 0
this : ⋂ n, t \ closedBall 0 ↑n = ∅
A : Tendsto (fun n => μ (t \ closedBall 0 ↑n)) atTop (𝓝 0)
I : 0 < ε / 2 * μ t
⊢ ∃ n, 0 < n ∧ μ (t \ closedBall 0 ↑n) < ε / 2 * μ t

Tactic:
exact (Eventually.and (Ioi_mem_atTop 0) ((tendsto_order.1 A).2 _ I)).exists

Result:
no goals

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t : Set E
ht : MeasurableSet t
h''t : μ t ≠ ⊤
ε : ℝ≥0∞
εpos : 0 < ε
h't : μ t ≠ 0
⊢ Tendsto (fun n => μ (t \ closedBall 0 ↑n)) atTop (𝓝 (μ (⋂ n, t \ closedBall 0 ↑n)))

Tactic:
have N : ∃ n : ℕ, μ (t \ closedBall 0 n) ≠ ∞ :=
  ⟨0, ((measure_mono diff_subset).trans_lt h''t.lt_top).ne⟩

Result:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t : Set E
ht : MeasurableSet t
h''t : μ t ≠ ⊤
ε : ℝ≥0∞
εpos : 0 < ε
h't : μ t ≠ 0
N : ∃ n, μ (t \ closedBall 0 ↑n) ≠ ⊤
⊢ Tendsto (fun n => μ (t \ closedBall 0 ↑n)) atTop (𝓝 (μ (⋂ n, t \ closedBall 0 ↑n)))