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✝ : IsAddHaarMeasure μ
s : Set E
u : ℕ → E
hu : Metric.Bounded (range u)
hs : Pairwise (Disjoint on fun n => {u n} + s)
h's : MeasurableSet s
H : ∀ (R : ℝ), ↑↑μ (s ∩ closedBall 0 R) = 0
⊢ ↑↑μ s = 0

case 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✝ : IsAddHaarMeasure μ
s : Set E
u : ℕ → E
hu : Metric.Bounded (range u)
hs : Pairwise (Disjoint on fun n => {u n} + s)
h's : MeasurableSet s
⊢ ∀ (R : ℝ), ↑↑μ (s ∩ closedBall 0 R) = 0
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✝ : IsAddHaarMeasure μ
s : Set E
u : ℕ → E
hu : Metric.Bounded (range u)
hs : Pairwise (Disjoint on fun n => {u n} + s)
h's : MeasurableSet s
⊢ ↑↑μ s = 0
suffices H : ∀ R, μ (s ∩ closedBall 0 R) = 0
case 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✝ : IsAddHaarMeasure μ
s : Set E
u : ℕ → E
hu : Metric.Bounded (range u)
hs : Pairwise (Disjoint on fun n => {u n} + s)
h's : MeasurableSet s
R : ℝ
⊢ ↑↑μ (s ∩ closedBall 0 R) = 0
case 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✝ : IsAddHaarMeasure μ
s : Set E
u : ℕ → E
hu : Metric.Bounded (range u)
hs : Pairwise (Disjoint on fun n => {u n} + s)
h's : MeasurableSet s
⊢ ∀ (R : ℝ), ↑↑μ (s ∩ closedBall 0 R) = 0
intro R
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✝ : IsAddHaarMeasure μ
s : Set E
u : ℕ → E
hu : Metric.Bounded (range u)
hs : Pairwise (Disjoint on fun n => {u n} + s)
h's : MeasurableSet s
R : ℝ
⊢ Pairwise (Disjoint on fun n => {u n} + s ∩ closedBall 0 R)
case 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✝ : IsAddHaarMeasure μ
s : Set E
u : ℕ → E
hu : Metric.Bounded (range u)
hs : Pairwise (Disjoint on fun n => {u n} + s)
h's : MeasurableSet s
R : ℝ
⊢ ↑↑μ (s ∩ closedBall 0 R) = 0
apply add_haar_eq_zero_of_disjoint_translates_aux μ u
  (bounded_closedBall.mono (inter_subset_right _ _)) hu _ (h's.inter measurableSet_closedBall)
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✝ : IsAddHaarMeasure μ
s : Set E
u : ℕ → E
hu : Metric.Bounded (range u)
hs : Pairwise (Disjoint on fun n => {u n} + s)
h's : MeasurableSet s
R : ℝ
n : ℕ
⊢ {u n} + s ∩ closedBall 0 R ≤ {u n} + s
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✝ : IsAddHaarMeasure μ
s : Set E
u : ℕ → E
hu : Metric.Bounded (range u)
hs : Pairwise (Disjoint on fun n => {u n} + s)
h's : MeasurableSet s
R : ℝ
⊢ Pairwise (Disjoint on fun n => {u n} + s ∩ closedBall 0 R)
refine pairwise_disjoint_mono hs fun n => ?_
no goals
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✝ : IsAddHaarMeasure μ
s : Set E
u : ℕ → E
hu : Metric.Bounded (range u)
hs : Pairwise (Disjoint on fun n => {u n} + s)
h's : MeasurableSet s
R : ℝ
n : ℕ
⊢ {u n} + s ∩ closedBall 0 R ≤ {u n} + s
exact add_subset_add Subset.rfl (inter_subset_left _ _)
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✝ : IsAddHaarMeasure μ
s : Set E
u : ℕ → E
hu : Metric.Bounded (range u)
hs : Pairwise (Disjoint on fun n => {u n} + s)
h's : MeasurableSet s
H : ∀ (R : ℝ), ↑↑μ (s ∩ closedBall 0 R) = 0
⊢ ↑↑μ s ≤ 0
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✝ : IsAddHaarMeasure μ
s : Set E
u : ℕ → E
hu : Metric.Bounded (range u)
hs : Pairwise (Disjoint on fun n => {u n} + s)
h's : MeasurableSet s
H : ∀ (R : ℝ), ↑↑μ (s ∩ closedBall 0 R) = 0
⊢ ↑↑μ s = 0
apply le_antisymm _ (zero_le _)
no goals
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✝ : IsAddHaarMeasure μ
s : Set E
u : ℕ → E
hu : Metric.Bounded (range u)
hs : Pairwise (Disjoint on fun n => {u n} + s)
h's : MeasurableSet s
H : ∀ (R : ℝ), ↑↑μ (s ∩ closedBall 0 R) = 0
⊢ ↑↑μ s ≤ 0
calc
  μ s ≤ ∑' n : ℕ, μ (s ∩ closedBall 0 n) := by
    conv_lhs => rw [← iUnion_inter_closedBall_nat s 0]
    exact measure_iUnion_le _
  _ = 0 := by simp only [H, tsum_zero]
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✝ : IsAddHaarMeasure μ
s : Set E
u : ℕ → E
hu : Metric.Bounded (range u)
hs : Pairwise (Disjoint on fun n => {u n} + s)
h's : MeasurableSet s
H : ∀ (R : ℝ), ↑↑μ (s ∩ closedBall 0 R) = 0
⊢ ↑↑μ (⋃ (n : ℕ), s ∩ closedBall 0 ↑n) ≤ ∑' (n : ℕ), ↑↑μ (s ∩ closedBall 0 ↑n)
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✝ : IsAddHaarMeasure μ
s : Set E
u : ℕ → E
hu : Metric.Bounded (range u)
hs : Pairwise (Disjoint on fun n => {u n} + s)
h's : MeasurableSet s
H : ∀ (R : ℝ), ↑↑μ (s ∩ closedBall 0 R) = 0
⊢ ↑↑μ s ≤ ∑' (n : ℕ), ↑↑μ (s ∩ closedBall 0 ↑n)
conv_lhs => rw [← iUnion_inter_closedBall_nat s 0]
no goals
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✝ : IsAddHaarMeasure μ
s : Set E
u : ℕ → E
hu : Metric.Bounded (range u)
hs : Pairwise (Disjoint on fun n => {u n} + s)
h's : MeasurableSet s
H : ∀ (R : ℝ), ↑↑μ (s ∩ closedBall 0 R) = 0
⊢ ↑↑μ (⋃ (n : ℕ), s ∩ closedBall 0 ↑n) ≤ ∑' (n : ℕ), ↑↑μ (s ∩ closedBall 0 ↑n)
exact measure_iUnion_le _
no goals
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✝ : IsAddHaarMeasure μ
s : Set E
u : ℕ → E
hu : Metric.Bounded (range u)
hs : Pairwise (Disjoint on fun n => {u n} + s)
h's : MeasurableSet s
H : ∀ (R : ℝ), ↑↑μ (s ∩ closedBall 0 R) = 0
⊢ (∑' (n : ℕ), ↑↑μ (s ∩ closedBall 0 ↑n)) = 0
simp only [H, tsum_zero]