Theory Vitali_Covering_Theorem

```(*  Title:      HOL/Analysis/Vitali_Covering_Theorem.thy
Authors:    LC Paulson, based on material from HOL Light
*)

section  ‹Vitali Covering Theorem and an Application to Negligibility›

theory Vitali_Covering_Theorem
imports
"HOL-Combinatorics.Permutations"
Equivalence_Lebesgue_Henstock_Integration
begin

lemma stretch_Galois:
fixes x :: "real^'n"
shows "(⋀k. m k ≠ 0) ⟹ ((y = (χ k. m k * x\$k)) ⟷ (χ k. y\$k / m k) = x)"
by auto

lemma lambda_swap_Galois:
"(x = (χ i. y \$ Transposition.transpose m n i) ⟷ (χ i. x \$ Transposition.transpose m n i) = y)"
by (auto; simp add: pointfree_idE vec_eq_iff)

fixes x :: "real^'n"
shows "m ≠ n ⟹ (x = (χ i. if i = m then y\$m + y\$n else y\$i) ⟷ (χ i. if i = m then x\$m - x\$n else x\$i) = y)"

lemma Vitali_covering_lemma_cballs_balls:
fixes a :: "'a ⇒ 'b::euclidean_space"
assumes "⋀i. i ∈ K ⟹ 0 < r i ∧ r i ≤ B"
obtains C where "countable C" "C ⊆ K"
"pairwise (λi j. disjnt (cball (a i) (r i)) (cball (a j) (r j))) C"
"⋀i. i ∈ K ⟹ ∃j. j ∈ C ∧
¬ disjnt (cball (a i) (r i)) (cball (a j) (r j)) ∧
cball (a i) (r i) ⊆ ball (a j) (5 * r j)"
proof (cases "K = {}")
case True
with that show ?thesis
by auto
next
case False
then have "B > 0"
using assms less_le_trans by auto
have rgt0[simp]: "⋀i. i ∈ K ⟹ 0 < r i"
using assms by auto
let ?djnt = "pairwise (λi j. disjnt (cball (a i) (r i)) (cball (a j) (r j)))"
have "∃C. ∀n. (C n ⊆ K ∧
(∀i ∈ C n. B/2 ^ n ≤ r i) ∧ ?djnt (C n) ∧
(∀i ∈ K. B/2 ^ n < r i
⟶ (∃j. j ∈ C n ∧
¬ disjnt (cball (a i) (r i)) (cball (a j) (r j)) ∧
cball (a i) (r i) ⊆ ball (a j) (5 * r j)))) ∧ (C n ⊆ C(Suc n))"
proof (rule dependent_nat_choice, safe)
fix C n
define D where "D ≡ {i ∈ K. B/2 ^ Suc n < r i ∧ (∀j∈C. disjnt (cball(a i)(r i)) (cball (a j) (r j)))}"
let ?cover_ar = "λi j. ¬ disjnt (cball (a i) (r i)) (cball (a j) (r j)) ∧
cball (a i) (r i) ⊆ ball (a j) (5 * r j)"
assume "C ⊆ K"
and Ble: "∀i∈C. B/2 ^ n ≤ r i"
and djntC: "?djnt C"
and cov_n: "∀i∈K. B/2 ^ n < r i ⟶ (∃j. j ∈ C ∧ ?cover_ar i j)"
have *: "∀C∈chains {C. C ⊆ D ∧ ?djnt C}. ⋃C ∈ {C. C ⊆ D ∧ ?djnt C}"
proof (clarsimp simp: chains_def)
fix C
assume C: "C ⊆ {C. C ⊆ D ∧ ?djnt C}" and "chain⇩⊆ C"
show "⋃C ⊆ D ∧ ?djnt (⋃C)"
unfolding pairwise_def
proof (intro ballI conjI impI)
show "⋃C ⊆ D"
using C by blast
next
fix x y
assume "x ∈ ⋃C" and "y ∈ ⋃C" and "x ≠ y"
then obtain X Y where XY: "x ∈ X" "X ∈ C" "y ∈ Y" "Y ∈ C"
by blast
then consider "X ⊆ Y" | "Y ⊆ X"
by (meson ‹chain⇩⊆ C› chain_subset_def)
then show "disjnt (cball (a x) (r x)) (cball (a y) (r y))"
proof cases
case 1
with C XY ‹x ≠ y› show ?thesis
unfolding pairwise_def by blast
next
case 2
with C XY ‹x ≠ y› show ?thesis
unfolding pairwise_def by blast
qed
qed
qed
obtain E where "E ⊆ D" and djntE: "?djnt E" and maximalE: "⋀X. ⟦X ⊆ D; ?djnt X; E ⊆ X⟧ ⟹ X = E"
using Zorn_Lemma [OF *] by safe blast
show "∃L. (L ⊆ K ∧
(∀i∈L. B/2 ^ Suc n ≤ r i) ∧ ?djnt L ∧
(∀i∈K. B/2 ^ Suc n < r i ⟶ (∃j. j ∈ L ∧ ?cover_ar i j))) ∧ C ⊆ L"
proof (intro exI conjI ballI)
show "C ∪ E ⊆ K"
using D_def ‹C ⊆ K› ‹E ⊆ D› by blast
show "B/2 ^ Suc n ≤ r i" if i: "i ∈ C ∪ E" for i
using i
proof
assume "i ∈ C"
have "B/2 ^ Suc n ≤ B/2 ^ n"
using ‹B > 0› by (simp add: field_split_simps)
also have "… ≤ r i"
using Ble ‹i ∈ C› by blast
finally show ?thesis .
qed (use D_def ‹E ⊆ D› in auto)
show "?djnt (C ∪ E)"
using D_def ‹C ⊆ K› ‹E ⊆ D› djntC djntE
unfolding pairwise_def disjnt_def by blast
next
fix i
assume "i ∈ K"
show "B/2 ^ Suc n < r i ⟶ (∃j. j ∈ C ∪ E ∧ ?cover_ar i j)"
proof (cases "r i ≤ B/2^n")
case False
then show ?thesis
using cov_n ‹i ∈ K› by auto
next
case True
have "cball (a i) (r i) ⊆ ball (a j) (5 * r j)"
if less: "B/2 ^ Suc n < r i" and j: "j ∈ C ∪ E"
and nondis: "¬ disjnt (cball (a i) (r i)) (cball (a j) (r j))" for j
proof -
obtain x where x: "dist (a i) x ≤ r i" "dist (a j) x ≤ r j"
using nondis by (force simp: disjnt_def)
have "dist (a i) (a j) ≤ dist (a i) x + dist x (a j)"
also have "… ≤ r i + r j"
finally have aij: "dist (a i) (a j) + r i < 5 * r j" if "r i < 2 * r j"
using that by auto
show ?thesis
using j
proof
assume "j ∈ C"
have "B/2^n < 2 * r j"
using Ble True ‹j ∈ C› less by auto
with aij True show "cball (a i) (r i) ⊆ ball (a j) (5 * r j)"
next
assume "j ∈ E"
then have "B/2 ^ n < 2 * r j"
using D_def ‹E ⊆ D› by auto
with True have "r i < 2 * r j"
by auto
with aij show "cball (a i) (r i) ⊆ ball (a j) (5 * r j)"
qed
qed
moreover have "∃j. j ∈ C ∪ E ∧ ¬ disjnt (cball (a i) (r i)) (cball (a j) (r j))"
if "B/2 ^ Suc n < r i"
proof (rule classical)
assume NON: "¬ ?thesis"
show ?thesis
proof (cases "i ∈ D")
case True
have "insert i E = E"
proof (rule maximalE)
show "insert i E ⊆ D"
by (simp add: True ‹E ⊆ D›)
show "pairwise (λi j. disjnt (cball (a i) (r i)) (cball (a j) (r j))) (insert i E)"
using False NON by (auto simp: pairwise_insert djntE disjnt_sym)
qed auto
then show ?thesis
using ‹i ∈ K› assms by fastforce
next
case False
with that show ?thesis
by (auto simp: D_def disjnt_def ‹i ∈ K›)
qed
qed
ultimately
show "B/2 ^ Suc n < r i ⟶
(∃j. j ∈ C ∪ E ∧
¬ disjnt (cball (a i) (r i)) (cball (a j) (r j)) ∧
cball (a i) (r i) ⊆ ball (a j) (5 * r j))"
by blast
qed
qed auto
qed (use assms in force)
then obtain F where FK: "⋀n. F n ⊆ K"
and Fle: "⋀n i. i ∈ F n ⟹ B/2 ^ n ≤ r i"
and Fdjnt:  "⋀n. ?djnt (F n)"
and FF:  "⋀n i. ⟦i ∈ K; B/2 ^ n < r i⟧
⟹ ∃j. j ∈ F n ∧ ¬ disjnt (cball (a i) (r i)) (cball (a j) (r j)) ∧
cball (a i) (r i) ⊆ ball (a j) (5 * r j)"
and inc:  "⋀n. F n ⊆ F(Suc n)"
by (force simp: all_conj_distrib)
show thesis
proof
have *: "countable I"
if "I ⊆ K" and pw: "pairwise (λi j. disjnt (cball (a i) (r i)) (cball (a j) (r j))) I" for I
proof -
show ?thesis
proof (rule countable_image_inj_on [of "λi. cball(a i)(r i)"])
show "countable ((λi. cball (a i) (r i)) ` I)"
proof (rule countable_disjoint_nonempty_interior_subsets)
show "disjoint ((λi. cball (a i) (r i)) ` I)"
by (auto simp: dest: pairwiseD [OF pw] intro: pairwise_imageI)
show "⋀S. ⟦S ∈ (λi. cball (a i) (r i)) ` I; interior S = {}⟧ ⟹ S = {}"
using ‹I ⊆ K›
by (auto simp: not_less [symmetric])
qed
next
have "⋀x y. ⟦x ∈ I; y ∈ I; a x = a y; r x = r y⟧ ⟹ x = y"
using pw ‹I ⊆ K› assms
apply (clarsimp simp: pairwise_def disjnt_def)
by (metis assms centre_in_cball subsetD empty_iff inf.idem less_eq_real_def)
then show "inj_on (λi. cball (a i) (r i)) I"
using ‹I ⊆ K› by (fastforce simp: inj_on_def cball_eq_cball_iff dest: assms)
qed
qed
show "(Union(range F)) ⊆ K"
using FK by blast
moreover show "pairwise (λi j. disjnt (cball (a i) (r i)) (cball (a j) (r j))) (Union(range F))"
proof (rule pairwise_chain_Union)
show "chain⇩⊆ (range F)"
unfolding chain_subset_def by clarify (meson inc lift_Suc_mono_le linear subsetCE)
qed (use Fdjnt in blast)
ultimately show "countable (Union(range F))"
by (blast intro: *)
next
fix i assume "i ∈ K"
then obtain n where "(1/2) ^ n < r i / B"
using  ‹B > 0› assms real_arch_pow_inv by fastforce
then have B2: "B/2 ^ n < r i"
using ‹B > 0› by (simp add: field_split_simps)
have "0 < r i" "r i ≤ B"
by (auto simp: ‹i ∈ K› assms)
show "∃j. j ∈ (Union(range F)) ∧
¬ disjnt (cball (a i) (r i)) (cball (a j) (r j)) ∧
cball (a i) (r i) ⊆ ball (a j) (5 * r j)"
using FF [OF ‹i ∈ K› B2] by auto
qed
qed

subsection‹Vitali covering theorem›

lemma Vitali_covering_lemma_cballs:
fixes a :: "'a ⇒ 'b::euclidean_space"
assumes S: "S ⊆ (⋃i∈K. cball (a i) (r i))"
and r: "⋀i. i ∈ K ⟹ 0 < r i ∧ r i ≤ B"
obtains C where "countable C" "C ⊆ K"
"pairwise (λi j. disjnt (cball (a i) (r i)) (cball (a j) (r j))) C"
"S ⊆ (⋃i∈C. cball (a i) (5 * r i))"
proof -
obtain C where C: "countable C" "C ⊆ K"
"pairwise (λi j. disjnt (cball (a i) (r i)) (cball (a j) (r j))) C"
and cov: "⋀i. i ∈ K ⟹ ∃j. j ∈ C ∧ ¬ disjnt (cball (a i) (r i)) (cball (a j) (r j)) ∧
cball (a i) (r i) ⊆ ball (a j) (5 * r j)"
by (rule Vitali_covering_lemma_cballs_balls [OF r, where a=a]) (blast intro: that)+
show ?thesis
proof
have "(⋃i∈K. cball (a i) (r i)) ⊆ (⋃i∈C. cball (a i) (5 * r i))"
using cov subset_iff by fastforce
with S show "S ⊆ (⋃i∈C. cball (a i) (5 * r i))"
by blast
qed (use C in auto)
qed

lemma Vitali_covering_lemma_balls:
fixes a :: "'a ⇒ 'b::euclidean_space"
assumes S: "S ⊆ (⋃i∈K. ball (a i) (r i))"
and r: "⋀i. i ∈ K ⟹ 0 < r i ∧ r i ≤ B"
obtains C where "countable C" "C ⊆ K"
"pairwise (λi j. disjnt (ball (a i) (r i)) (ball (a j) (r j))) C"
"S ⊆ (⋃i∈C. ball (a i) (5 * r i))"
proof -
obtain C where C: "countable C" "C ⊆ K"
and pw:  "pairwise (λi j. disjnt (cball (a i) (r i)) (cball (a j) (r j))) C"
and cov: "⋀i. i ∈ K ⟹ ∃j. j ∈ C ∧ ¬ disjnt (cball (a i) (r i)) (cball (a j) (r j)) ∧
cball (a i) (r i) ⊆ ball (a j) (5 * r j)"
by (rule Vitali_covering_lemma_cballs_balls [OF r, where a=a]) (blast intro: that)+
show ?thesis
proof
have "(⋃i∈K. ball (a i) (r i)) ⊆ (⋃i∈C. ball (a i) (5 * r i))"
using cov subset_iff
by clarsimp (meson less_imp_le mem_ball mem_cball subset_eq)
with S show "S ⊆ (⋃i∈C. ball (a i) (5 * r i))"
by blast
show "pairwise (λi j. disjnt (ball (a i) (r i)) (ball (a j) (r j))) C"
using pw
by (clarsimp simp: pairwise_def) (meson ball_subset_cball disjnt_subset1 disjnt_subset2)
qed (use C in auto)
qed

theorem Vitali_covering_theorem_cballs:
fixes a :: "'a ⇒ 'n::euclidean_space"
assumes r: "⋀i. i ∈ K ⟹ 0 < r i"
and S: "⋀x d. ⟦x ∈ S; 0 < d⟧
⟹ ∃i. i ∈ K ∧ x ∈ cball (a i) (r i) ∧ r i < d"
obtains C where "countable C" "C ⊆ K"
"pairwise (λi j. disjnt (cball (a i) (r i)) (cball (a j) (r j))) C"
"negligible(S - (⋃i ∈ C. cball (a i) (r i)))"
proof -
let ?μ = "measure lebesgue"
have *: "∃C. countable C ∧ C ⊆ K ∧
pairwise (λi j. disjnt (cball (a i) (r i)) (cball (a j) (r j))) C ∧
negligible(S - (⋃i ∈ C. cball (a i) (r i)))"
if r01: "⋀i. i ∈ K ⟹ 0 < r i ∧ r i ≤ 1"
and Sd: "⋀x d. ⟦x ∈ S; 0 < d⟧ ⟹ ∃i. i ∈ K ∧ x ∈ cball (a i) (r i) ∧ r i < d"
for K r and a :: "'a ⇒ 'n"
proof -
obtain C where C: "countable C" "C ⊆ K"
and pwC: "pairwise (λi j. disjnt (cball (a i) (r i)) (cball (a j) (r j))) C"
and cov: "⋀i. i ∈ K ⟹ ∃j. j ∈ C ∧ ¬ disjnt (cball (a i) (r i)) (cball (a j) (r j)) ∧
cball (a i) (r i) ⊆ ball (a j) (5 * r j)"
by (rule Vitali_covering_lemma_cballs_balls [of K r 1 a]) (auto simp: r01)
have ar_injective: "⋀x y. ⟦x ∈ C; y ∈ C; a x = a y; r x = r y⟧ ⟹ x = y"
using ‹C ⊆ K› pwC cov
by (force simp: pairwise_def disjnt_def)
show ?thesis
proof (intro exI conjI)
show "negligible (S - (⋃i∈C. cball (a i) (r i)))"
proof (clarsimp simp: negligible_on_intervals [of "S-T" for T])
fix l u
show "negligible ((S - (⋃i∈C. cball (a i) (r i))) ∩ cbox l u)"
unfolding negligible_outer_le
proof (intro allI impI)
fix e::real
assume "e > 0"
define D where "D ≡ {i ∈ C. ¬ disjnt (ball(a i) (5 * r i)) (cbox l u)}"
then have "D ⊆ C"
by auto
have "countable D"
unfolding D_def using ‹countable C› by simp
have UD: "(⋃i∈D. cball (a i) (r i)) ∈ lmeasurable"
proof (rule fmeasurableI2)
show "cbox (l - 6 *⇩R One) (u + 6 *⇩R One) ∈ lmeasurable"
by blast
have "y ∈ cbox (l - 6 *⇩R One) (u + 6 *⇩R One)"
if "i ∈ C" and x: "x ∈ cbox l u" and ai: "dist (a i) y ≤ r i" "dist (a i) x < 5 * r i"
for i x y
proof -
have d6: "dist y x < 6 * r i"
using dist_triangle3 [of y x "a i"] that by linarith
show ?thesis
proof (clarsimp simp: mem_box algebra_simps)
fix j::'n
assume j: "j ∈ Basis"
then have xyj: "¦x ∙ j - y ∙ j¦ ≤ dist y x"
by (metis Basis_le_norm dist_commute dist_norm inner_diff_left)
have "l ∙ j ≤ x ∙ j"
using ‹j ∈ Basis› mem_box ‹x ∈ cbox l u› by blast
also have "… ≤ y ∙ j + 6 * r i"
using d6 xyj by (auto simp: algebra_simps)
also have "… ≤ y ∙ j + 6"
using r01 [of i] ‹C ⊆ K› ‹i ∈ C› by auto
finally have l: "l ∙ j ≤ y ∙ j + 6" .
have "y ∙ j ≤ x ∙ j + 6 * r i"
using d6 xyj by (auto simp: algebra_simps)
also have "… ≤ u ∙ j + 6 * r i"
using j  x by (auto simp: mem_box)
also have "… ≤ u ∙ j + 6"
using r01 [of i] ‹C ⊆ K› ‹i ∈ C› by auto
finally have u: "y ∙ j ≤ u ∙ j + 6" .
show "l ∙ j ≤ y ∙ j + 6 ∧ y ∙ j ≤ u ∙ j + 6"
using l u by blast
qed
qed
then show "(⋃i∈D. cball (a i) (r i)) ⊆ cbox (l - 6 *⇩R One) (u + 6 *⇩R One)"
by (force simp: D_def disjnt_def)
show "(⋃i∈D. cball (a i) (r i)) ∈ sets lebesgue"
using ‹countable D› by auto
qed
obtain D1 where "D1 ⊆ D" "finite D1"
and measD1: "?μ (⋃i∈D. cball (a i) (r i)) - e / 5 ^ DIM('n) < ?μ (⋃i∈D1. cball (a i) (r i))"
proof (rule measure_countable_Union_approachable [where e = "e / 5 ^ (DIM('n))"])
show "countable ((λi. cball (a i) (r i)) ` D)"
using ‹countable D› by auto
show "⋀d. d ∈ (λi. cball (a i) (r i)) ` D ⟹ d ∈ lmeasurable"
by auto
show "⋀D'. ⟦D' ⊆ (λi. cball (a i) (r i)) ` D; finite D'⟧ ⟹ ?μ (⋃D') ≤ ?μ (⋃i∈D. cball (a i) (r i))"
by (fastforce simp add: intro!: measure_mono_fmeasurable UD)
qed (use ‹e > 0› in ‹auto dest: finite_subset_image›)
show "∃T. (S - (⋃i∈C. cball (a i) (r i))) ∩
cbox l u ⊆ T ∧ T ∈ lmeasurable ∧ ?μ T ≤ e"
proof (intro exI conjI)
show "(S - (⋃i∈C. cball (a i) (r i))) ∩ cbox l u ⊆ (⋃i∈D - D1. ball (a i) (5 * r i))"
proof clarify
fix x
assume x: "x ∈ cbox l u" "x ∈ S" "x ∉ (⋃i∈C. cball (a i) (r i))"
have "closed (⋃i∈D1. cball (a i) (r i))"
using ‹finite D1› by blast
moreover have "x ∉ (⋃j∈D1. cball (a j) (r j))"
using x ‹D1 ⊆ D› unfolding D_def by blast
ultimately obtain q where "q > 0" and q: "ball x q ⊆ - (⋃i∈D1. cball (a i) (r i))"
by (metis (no_types, lifting) ComplI open_contains_ball closed_def)
obtain i where "i ∈ K" and xi: "x ∈ cball (a i) (r i)" and ri: "r i < q/2"
using Sd [OF ‹x ∈ S›] ‹q > 0› half_gt_zero by blast
then obtain j where "j ∈ C"
and nondisj: "¬ disjnt (cball (a i) (r i)) (cball (a j) (r j))"
and sub5j:  "cball (a i) (r i) ⊆ ball (a j) (5 * r j)"
using cov [OF ‹i ∈ K›] by metis
show "x ∈ (⋃i∈D - D1. ball (a i) (5 * r i))"
proof
show "j ∈ D - D1"
proof
show "j ∈ D"
using ‹j ∈ C› sub5j ‹x ∈ cbox l u› xi by (auto simp: D_def disjnt_def)
obtain y where yi: "dist (a i) y ≤ r i" and yj: "dist (a j) y ≤ r j"
using disjnt_def nondisj by fastforce
have "dist x y ≤ r i + r i"
by (metis add_mono dist_commute dist_triangle_le mem_cball xi yi)
also have "… < q"
using ri by linarith
finally have "y ∈ ball x q"
by simp
with yj q show "j ∉ D1"
by (auto simp: disjoint_UN_iff)
qed
show "x ∈ ball (a j) (5 * r j)"
using xi sub5j by blast
qed
qed
have 3: "?μ (⋃i∈D2. ball (a i) (5 * r i)) ≤ e"
if D2: "D2 ⊆ D - D1" and "finite D2" for D2
proof -
have rgt0: "0 < r i" if "i ∈ D2" for i
using ‹C ⊆ K› D_def ‹i ∈ D2› D2 r01
then have inj: "inj_on (λi. ball (a i) (5 * r i)) D2"
using ‹C ⊆ K› D2 by (fastforce simp: inj_on_def D_def ball_eq_ball_iff intro: ar_injective)
have "?μ (⋃i∈D2. ball (a i) (5 * r i)) ≤ sum (?μ) ((λi. ball (a i) (5 * r i)) ` D2)"
using that by (force intro: measure_Union_le)
also have "…  = (∑i∈D2. ?μ (ball (a i) (5 * r i)))"
also have "… = (∑i∈D2. 5 ^ DIM('n) * ?μ (ball (a i) (r i)))"
proof (rule sum.cong [OF refl])
fix i assume "i ∈ D2"
thus "?μ (ball (a i) (5 * r i)) = 5 ^ DIM('n) * ?μ (ball (a i) (r i))"
using content_ball_conv_unit_ball[of "5 * r i" "a i"]
content_ball_conv_unit_ball[of "r i" "a i"] rgt0[of i] by auto
qed
also have "… = (∑i∈D2. ?μ (ball (a i) (r i))) * 5 ^ DIM('n)"
finally have "?μ (⋃i∈D2. ball (a i) (5 * r i)) ≤ (∑i∈D2. ?μ (ball (a i) (r i))) * 5 ^ DIM('n)" .
moreover have "(∑i∈D2. ?μ (ball (a i) (r i))) ≤ e / 5 ^ DIM('n)"
proof -
have D12_dis: "((⋃x∈D1. cball (a x) (r x)) ∩ (⋃x∈D2. cball (a x) (r x))) ≤ {}"
proof clarify
fix w d1 d2
assume "d1 ∈ D1" "w d1 d2 ∈ cball (a d1) (r d1)" "d2 ∈ D2" "w d1 d2 ∈ cball (a d2) (r d2)"
then show "w d1 d2 ∈ {}"
by (metis DiffE disjnt_iff subsetCE D2 ‹D1 ⊆ D› ‹D ⊆ C› pairwiseD [OF pwC, of d1 d2])
qed
have inj: "inj_on (λi. cball (a i) (r i)) D2"
using rgt0 D2 ‹D ⊆ C› by (force simp: inj_on_def cball_eq_cball_iff intro!: ar_injective)
have ds: "disjoint ((λi. cball (a i) (r i)) ` D2)"
using D2 ‹D ⊆ C› by (auto intro: pairwiseI pairwiseD [OF pwC])
have "(∑i∈D2. ?μ (ball (a i) (r i))) = (∑i∈D2. ?μ (cball (a i) (r i)))"
also have "… = sum ?μ ((λi. cball (a i) (r i)) ` D2)"
also have "… = ?μ (⋃i∈D2. cball (a i) (r i))"
by (auto intro: measure_Union' [symmetric] ds simp add: ‹finite D2›)
finally have "?μ (⋃i∈D1. cball (a i) (r i)) + (∑i∈D2. ?μ (ball (a i) (r i))) =
?μ (⋃i∈D1. cball (a i) (r i)) + ?μ (⋃i∈D2. cball (a i) (r i))"
by simp
also have "… = ?μ (⋃i ∈ D1 ∪ D2. cball (a i) (r i))"
using D12_dis by (simp add: measure_Un3 ‹finite D1› ‹finite D2› fmeasurable.finite_UN)
also have "… ≤ ?μ (⋃i∈D. cball (a i) (r i))"
using D2 ‹D1 ⊆ D› by (fastforce intro!: measure_mono_fmeasurable [OF _ _ UD] ‹finite D1› ‹finite D2›)
finally have "?μ (⋃i∈D1. cball (a i) (r i)) + (∑i∈D2. ?μ (ball (a i) (r i))) ≤ ?μ (⋃i∈D. cball (a i) (r i))" .
with measD1 show ?thesis
by simp
qed
ultimately show ?thesis
qed
have co: "countable (D - D1)"
show "(⋃i∈D - D1. ball (a i) (5 * r i)) ∈ lmeasurable"
using ‹e > 0› by (auto simp: fmeasurable_UN_bound [OF co _ 3])
show "?μ (⋃i∈D - D1. ball (a i) (5 * r i)) ≤ e"
using ‹e > 0› by (auto simp: measure_UN_bound [OF co _ 3])
qed
qed
qed
qed (use C pwC in auto)
qed
define K' where "K' ≡ {i ∈ K. r i ≤ 1}"
have 1: "⋀i. i ∈ K' ⟹ 0 < r i ∧ r i ≤ 1"
using K'_def r by auto
have 2: "∃i. i ∈ K' ∧ x ∈ cball (a i) (r i) ∧ r i < d"
if "x ∈ S ∧ 0 < d" for x d
using that by (auto simp: K'_def dest!: S [where d = "min d 1"])
have "K' ⊆ K"
using K'_def by auto
then show thesis
using * [OF 1 2] that by fastforce
qed

theorem Vitali_covering_theorem_balls:
fixes a :: "'a ⇒ 'b::euclidean_space"
assumes S: "⋀x d. ⟦x ∈ S; 0 < d⟧ ⟹ ∃i. i ∈ K ∧ x ∈ ball (a i) (r i) ∧ r i < d"
obtains C where "countable C" "C ⊆ K"
"pairwise (λi j. disjnt (ball (a i) (r i)) (ball (a j) (r j))) C"
"negligible(S - (⋃i ∈ C. ball (a i) (r i)))"
proof -
have 1: "∃i. i ∈ {i ∈ K. 0 < r i} ∧ x ∈ cball (a i) (r i) ∧ r i < d"
if xd: "x ∈ S" "d > 0" for x d
by (metis (mono_tags, lifting) assms ball_eq_empty less_eq_real_def mem_Collect_eq mem_ball mem_cball not_le xd(1) xd(2))
obtain C where C: "countable C" "C ⊆ K"
and pw: "pairwise (λi j. disjnt (cball (a i) (r i)) (cball (a j) (r j))) C"
and neg: "negligible(S - (⋃i ∈ C. cball (a i) (r i)))"
by (rule Vitali_covering_theorem_cballs [of "{i ∈ K. 0 < r i}" r S a, OF _ 1]) auto
show thesis
proof
show "pairwise (λi j. disjnt (ball (a i) (r i)) (ball (a j) (r j))) C"
apply (rule pairwise_mono [OF pw])
apply (auto simp: disjnt_def)
by (meson disjoint_iff_not_equal less_imp_le mem_cball)
have "negligible (⋃i∈C. sphere (a i) (r i))"
by (auto intro: negligible_sphere ‹countable C›)
then have "negligible (S - (⋃i ∈ C. cball(a i)(r i)) ∪ (⋃i ∈ C. sphere (a i) (r i)))"
by (rule negligible_Un [OF neg])
then show "negligible (S - (⋃i∈C. ball (a i) (r i)))"
by (rule negligible_subset) force
qed (use C in auto)
qed

lemma negligible_eq_zero_density_alt:
"negligible S ⟷
(∀x ∈ S. ∀e > 0.
∃d U. 0 < d ∧ d ≤ e ∧ S ∩ ball x d ⊆ U ∧
U ∈ lmeasurable ∧ measure lebesgue U < e * measure lebesgue (ball x d))"
(is "_ = (∀x ∈ S. ∀e > 0. ?Q x e)")
proof (intro iffI ballI allI impI)
fix x and e :: real
assume "negligible S" and "x ∈ S" and "e > 0"
then
show "∃d U. 0 < d ∧ d ≤ e ∧ S ∩ ball x d ⊆ U ∧ U ∈ lmeasurable ∧
measure lebesgue U < e * measure lebesgue (ball x d)"
apply (rule_tac x=e in exI)
apply (rule_tac x="S ∩ ball x e" in exI)
apply (auto simp: negligible_imp_measurable negligible_Int negligible_imp_measure0 zero_less_measure_iff
intro: mult_pos_pos content_ball_pos)
done
next
assume R [rule_format]: "∀x ∈ S. ∀e > 0. ?Q x e"
let ?μ = "measure lebesgue"
have "∃U. openin (top_of_set S) U ∧ z ∈ U ∧ negligible U"
if "z ∈ S" for z
proof (intro exI conjI)
show "openin (top_of_set S) (S ∩ ball z 1)"
show "z ∈ S ∩ ball z 1"
using ‹z ∈ S› by auto
show "negligible (S ∩ ball z 1)"
proof (clarsimp simp: negligible_outer_le)
fix e :: "real"
assume "e > 0"
let ?K = "{(x,d). x ∈ S ∧ 0 < d ∧ ball x d ⊆ ball z 1 ∧
(∃U. S ∩ ball x d ⊆ U ∧ U ∈ lmeasurable ∧
?μ U < e / ?μ (ball z 1) * ?μ (ball x d))}"
obtain C where "countable C" and Csub: "C ⊆ ?K"
and pwC: "pairwise (λi j. disjnt (ball (fst i) (snd i)) (ball (fst j) (snd j))) C"
and negC: "negligible((S ∩ ball z 1) - (⋃i ∈ C. ball (fst i) (snd i)))"
proof (rule Vitali_covering_theorem_balls [of "S ∩ ball z 1" ?K fst snd])
fix x and d :: "real"
assume x: "x ∈ S ∩ ball z 1" and "d > 0"
obtain k where "k > 0" and k: "ball x k ⊆ ball z 1"
by (meson Int_iff open_ball openE x)
let ?ε = "min (e / ?μ (ball z 1) / 2) (min (d / 2) k)"
obtain r U where r: "r > 0" "r ≤ ?ε" and U: "S ∩ ball x r ⊆ U" "U ∈ lmeasurable"
and mU: "?μ U < ?ε * ?μ (ball x r)"
using R [of x ?ε] ‹d > 0› ‹e > 0› ‹k > 0› x by (auto simp: content_ball_pos)
show "∃i. i ∈ ?K ∧ x ∈ ball (fst i) (snd i) ∧ snd i < d"
proof (rule exI [of _ "(x,r)"], simp, intro conjI exI)
have "ball x r ⊆ ball x k"
using r by (simp add: ball_subset_ball_iff)
also have "… ⊆ ball z 1"
using ball_subset_ball_iff k by auto
finally show "ball x r ⊆ ball z 1" .
have "?ε * ?μ (ball x r) ≤ e * content (ball x r) / content (ball z 1)"
using r ‹e > 0› by (simp add: ord_class.min_def field_split_simps content_ball_pos)
with mU show "?μ U < e * content (ball x r) / content (ball z 1)"
by auto
qed (use r U x in auto)
qed
have "∃U. case p of (x,d) ⇒ S ∩ ball x d ⊆ U ∧
U ∈ lmeasurable ∧ ?μ U < e / ?μ (ball z 1) * ?μ (ball x d)"
if "p ∈ C" for p
using that Csub unfolding case_prod_unfold by blast
then obtain U where U:
"⋀p. p ∈ C ⟹
case p of (x,d) ⇒ S ∩ ball x d ⊆ U p ∧
U p ∈ lmeasurable ∧ ?μ (U p) < e / ?μ (ball z 1) * ?μ (ball x d)"
by (rule that [OF someI_ex])
let ?T = "((S ∩ ball z 1) - (⋃(x,d)∈C. ball x d)) ∪ ⋃(U ` C)"
show "∃T. S ∩ ball z 1 ⊆ T ∧ T ∈ lmeasurable ∧ ?μ T ≤ e"
proof (intro exI conjI)
show "S ∩ ball z 1 ⊆ ?T"
using U by fastforce
{ have Um: "U i ∈ lmeasurable" if "i ∈ C" for i
using that U by blast
have lee: "?μ (⋃i∈I. U i) ≤ e" if "I ⊆ C" "finite I" for I
proof -
have "?μ (⋃(x,d)∈I. ball x d) ≤ ?μ (ball z 1)"
apply (rule measure_mono_fmeasurable)
using ‹I ⊆ C› ‹finite I› Csub by (force simp: prod.case_eq_if sets.finite_UN)+
then have le1: "(?μ (⋃(x,d)∈I. ball x d) / ?μ (ball z 1)) ≤ 1"
have "?μ (⋃i∈I. U i) ≤ (∑i∈I. ?μ (U i))"
using that U by (blast intro: measure_UNION_le)
also have "… ≤ (∑(x,r)∈I. e / ?μ (ball z 1) * ?μ (ball x r))"
by (rule sum_mono) (use ‹I ⊆ C› U in force)
also have "… = (e / ?μ (ball z 1)) * (∑(x,r)∈I. ?μ (ball x r))"
by (simp add: case_prod_app prod.case_distrib sum_distrib_left)
also have "… = e * (?μ (⋃(x,r)∈I. ball x r) / ?μ (ball z 1))"
apply (subst measure_UNION')
using that pwC by (auto simp: case_prod_unfold elim: pairwise_mono)
also have "… ≤ e"
by (metis mult.commute mult.left_neutral mult_le_cancel_right_pos ‹e > 0› le1)
finally show ?thesis .
qed
have "⋃(U ` C) ∈ lmeasurable" "?μ (⋃(U ` C)) ≤ e"
using ‹e > 0› Um lee
by(auto intro!: fmeasurable_UN_bound [OF ‹countable C›] measure_UN_bound [OF ‹countable C›])
}
moreover have "?μ ?T = ?μ (⋃(U ` C))"
proof (rule measure_negligible_symdiff [OF ‹⋃(U ` C) ∈ lmeasurable›])
show "negligible((⋃(U ` C) - ?T) ∪ (?T - ⋃(U ` C)))"
by (force intro!: negligible_subset [OF negC])
qed
ultimately show "?T ∈ lmeasurable"  "?μ ?T ≤ e"
by (simp_all add: fmeasurable.Un negC negligible_imp_measurable split_def)
qed
qed
qed
with locally_negligible_alt show "negligible S"
by metis
qed

proposition negligible_eq_zero_density:
"negligible S ⟷
(∀x∈S. ∀r>0. ∀e>0. ∃d. 0 < d ∧ d ≤ r ∧
(∃U. S ∩ ball x d ⊆ U ∧ U ∈ lmeasurable ∧ measure lebesgue U < e * measure lebesgue (ball x d)))"
proof -
let ?Q = "λx d e. ∃U. S ∩ ball x d ⊆ U ∧ U ∈ lmeasurable ∧ measure lebesgue U < e * content (ball x d)"
have "(∀e>0. ∃d>0. d ≤ e ∧ ?Q x d e) = (∀r>0. ∀e>0. ∃d>0. d ≤ r ∧ ?Q x d e)"
if "x ∈ S" for x
proof (intro iffI allI impI)
fix r :: "real" and e :: "real"
assume L [rule_format]: "∀e>0. ∃d>0. d ≤ e ∧ ?Q x d e" and "r > 0" "e > 0"
show "∃d>0. d ≤ r ∧ ?Q x d e"
using L [of "min r e"] apply (rule ex_forward)
using ‹r > 0› ‹e > 0›  by (auto intro: less_le_trans elim!: ex_forward simp: content_ball_pos)
qed auto
then show ?thesis
by (force simp: negligible_eq_zero_density_alt)
qed

end
```