Theorem: MeasureTheory.Measure.addHaar_eq_zero_of_disjoint_translates
File: Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean

Proof steps:
state_before: 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
u : ℕ → E
hu : Bornology.IsBounded (range u)
hs : Pairwise (Disjoint on fun n => {u n} + s)
h's : MeasurableSet s
⊢ μ s = 0
state_after: 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
u : ℕ → E
hu : Bornology.IsBounded (range u)
hs : Pairwise (Disjoint on fun n => {u n} + s)
h's : MeasurableSet s
⊢ ∀ (R : ℝ), μ (s ∩ closedBall 0 R) = 0
tactic: suffices H : ∀ R, μ (s ∩ closedBall 0 R) = 0 by
    apply le_antisymm _ (zero_le _)
    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]
state_before: 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
u : ℕ → E
hu : Bornology.IsBounded (range u)
hs : Pairwise (Disjoint on fun n => {u n} + s)
h's : MeasurableSet s
⊢ ∀ (R : ℝ), μ (s ∩ closedBall 0 R) = 0
state_after: 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
u : ℕ → E
hu : Bornology.IsBounded (range u)
hs : Pairwise (Disjoint on fun n => {u n} + s)
h's : MeasurableSet s
R : ℝ
⊢ μ (s ∩ closedBall 0 R) = 0
tactic: intro R
state_before: 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
u : ℕ → E
hu : Bornology.IsBounded (range u)
hs : Pairwise (Disjoint on fun n => {u n} + s)
h's : MeasurableSet s
R : ℝ
⊢ μ (s ∩ closedBall 0 R) = 0
state_after: 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
u : ℕ → E
hu : Bornology.IsBounded (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)
tactic: apply addHaar_eq_zero_of_disjoint_translates_aux μ u
    (isBounded_closedBall.subset inter_subset_right) hu _ (h's.inter measurableSet_closedBall)
state_before: 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
u : ℕ → E
hu : Bornology.IsBounded (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)
state_after: 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
u : ℕ → E
hu : Bornology.IsBounded (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
tactic: refine pairwise_disjoint_mono hs fun n => ?_
state_before: 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
u : ℕ → E
hu : Bornology.IsBounded (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
state_after: no goals
tactic: exact add_subset_add Subset.rfl inter_subset_left