author paulson Tue, 17 Apr 2018 12:09:23 +0100 changeset 67996 6a9d1b31a7c5 parent 67991 53ab458395a8 child 67997 ae76012879c6
Vitali covering theorem
```--- a/src/HOL/Analysis/Analysis.thy	Tue Apr 17 10:22:42 2018 +0100
+++ b/src/HOL/Analysis/Analysis.thy	Tue Apr 17 12:09:23 2018 +0100
@@ -22,7 +22,7 @@
FPS_Convergence
Generalised_Binomial_Theorem
Gamma_Function
-  Ball_Volume
+  Vitali_Covering_Theorem
Lipschitz
begin
```
```--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Analysis/Vitali_Covering_Theorem.thy	Tue Apr 17 12:09:23 2018 +0100
@@ -0,0 +1,652 @@
+theory Vitali_Covering_Theorem
+  imports Ball_Volume "HOL-Library.Permutations"
+
+begin
+
+lemma stretch_Galois:
+  fixes x :: "real^'n"
+  shows "(\<And>k. m k \<noteq> 0) \<Longrightarrow> ((y = (\<chi> k. m k * x\$k)) \<longleftrightarrow> (\<chi> k. y\$k / m k) = x)"
+  by auto
+
+lemma lambda_swap_Galois:
+   "(x = (\<chi> i. y \$ Fun.swap m n id i) \<longleftrightarrow> (\<chi> i. x \$ Fun.swap m n id i) = y)"
+  by (auto; simp add: pointfree_idE vec_eq_iff)
+
+  fixes x :: "real^'n"
+  shows "m \<noteq> n \<Longrightarrow> (x = (\<chi> i. if i = m then y\$m + y\$n else y\$i) \<longleftrightarrow> (\<chi> i. if i = m then x\$m - x\$n else x\$i) = y)"
+  by (safe; simp add: vec_eq_iff)
+
+
+lemma Vitali_covering_lemma_cballs_balls:
+  fixes a :: "'a \<Rightarrow> 'b::euclidean_space"
+  assumes "\<And>i. i \<in> K \<Longrightarrow> 0 < r i \<and> r i \<le> B"
+  obtains C where "countable C" "C \<subseteq> K"
+     "pairwise (\<lambda>i j. disjnt (cball (a i) (r i)) (cball (a j) (r j))) C"
+     "\<And>i. i \<in> K \<Longrightarrow> \<exists>j. j \<in> C \<and>
+                        \<not> disjnt (cball (a i) (r i)) (cball (a j) (r j)) \<and>
+                        cball (a i) (r i) \<subseteq> 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]: "\<And>i. i \<in> K \<Longrightarrow> 0 < r i"
+    using assms by auto
+  let ?djnt = "pairwise (\<lambda>i j. disjnt (cball (a i) (r i)) (cball (a j) (r j)))"
+  have "\<exists>C. \<forall>n. (C n \<subseteq> K \<and>
+             (\<forall>i \<in> C n. B/2 ^ n \<le> r i) \<and> ?djnt (C n) \<and>
+             (\<forall>i \<in> K. B/2 ^ n < r i
+                 \<longrightarrow> (\<exists>j. j \<in> C n \<and>
+                         \<not> disjnt (cball (a i) (r i)) (cball (a j) (r j)) \<and>
+                         cball (a i) (r i) \<subseteq> ball (a j) (5 * r j)))) \<and> (C n \<subseteq> C(Suc n))"
+  proof (rule dependent_nat_choice, safe)
+    fix C n
+    define D where "D \<equiv> {i \<in> K. B/2 ^ Suc n < r i \<and> (\<forall>j\<in>C. disjnt (cball(a i)(r i)) (cball (a j) (r j)))}"
+    let ?cover_ar = "\<lambda>i j. \<not> disjnt (cball (a i) (r i)) (cball (a j) (r j)) \<and>
+                             cball (a i) (r i) \<subseteq> ball (a j) (5 * r j)"
+    assume "C \<subseteq> K"
+      and Ble: "\<forall>i\<in>C. B/2 ^ n \<le> r i"
+      and djntC: "?djnt C"
+      and cov_n: "\<forall>i\<in>K. B/2 ^ n < r i \<longrightarrow> (\<exists>j. j \<in> C \<and> ?cover_ar i j)"
+    have *: "\<forall>C\<in>chains {C. C \<subseteq> D \<and> ?djnt C}. \<Union>C \<in> {C. C \<subseteq> D \<and> ?djnt C}"
+    proof (clarsimp simp: chains_def)
+      fix C
+      assume C: "C \<subseteq> {C. C \<subseteq> D \<and> ?djnt C}" and "chain\<^sub>\<subseteq> C"
+      show "\<Union>C \<subseteq> D \<and> ?djnt (\<Union>C)"
+        unfolding pairwise_def
+      proof (intro ballI conjI impI)
+        show "\<Union>C \<subseteq> D"
+          using C by blast
+      next
+        fix x y
+        assume "x \<in> \<Union>C" and "y \<in> \<Union>C" and "x \<noteq> y"
+        then obtain X Y where XY: "x \<in> X" "X \<in> C" "y \<in> Y" "Y \<in> C"
+          by blast
+        then consider "X \<subseteq> Y" | "Y \<subseteq> X"
+          by (meson \<open>chain\<^sub>\<subseteq> C\<close> chain_subset_def)
+        then show "disjnt (cball (a x) (r x)) (cball (a y) (r y))"
+        proof cases
+          case 1
+          with C XY \<open>x \<noteq> y\<close> show ?thesis
+            unfolding pairwise_def by blast
+        next
+          case 2
+          with C XY \<open>x \<noteq> y\<close> show ?thesis
+            unfolding pairwise_def by blast
+        qed
+      qed
+    qed
+    obtain E where "E \<subseteq> D" and djntE: "?djnt E" and maximalE: "\<And>X. \<lbrakk>X \<subseteq> D; ?djnt X; E \<subseteq> X\<rbrakk> \<Longrightarrow> X = E"
+      using Zorn_Lemma [OF *] by safe blast
+    show "\<exists>L. (L \<subseteq> K \<and>
+               (\<forall>i\<in>L. B/2 ^ Suc n \<le> r i) \<and> ?djnt L \<and>
+               (\<forall>i\<in>K. B/2 ^ Suc n < r i \<longrightarrow> (\<exists>j. j \<in> L \<and> ?cover_ar i j))) \<and> C \<subseteq> L"
+    proof (intro exI conjI ballI)
+      show "C \<union> E \<subseteq> K"
+        using D_def \<open>C \<subseteq> K\<close> \<open>E \<subseteq> D\<close> by blast
+      show "B/2 ^ Suc n \<le> r i" if i: "i \<in> C \<union> E" for i
+        using i
+      proof
+        assume "i \<in> C"
+        have "B/2 ^ Suc n \<le> B/2 ^ n"
+          using \<open>B > 0\<close> by (simp add: divide_simps)
+        also have "\<dots> \<le> r i"
+          using Ble \<open>i \<in> C\<close> by blast
+        finally show ?thesis .
+      qed (use D_def \<open>E \<subseteq> D\<close> in auto)
+      show "?djnt (C \<union> E)"
+        using D_def \<open>C \<subseteq> K\<close> \<open>E \<subseteq> D\<close> djntC djntE
+        unfolding pairwise_def disjnt_def by blast
+    next
+      fix i
+      assume "i \<in> K"
+      show "B/2 ^ Suc n < r i \<longrightarrow> (\<exists>j. j \<in> C \<union> E \<and> ?cover_ar i j)"
+      proof (cases "r i \<le> B/2^n")
+        case False
+        then show ?thesis
+          using cov_n \<open>i \<in> K\<close> by auto
+      next
+        case True
+        have "cball (a i) (r i) \<subseteq> ball (a j) (5 * r j)"
+          if less: "B/2 ^ Suc n < r i" and j: "j \<in> C \<union> E"
+            and nondis: "\<not> disjnt (cball (a i) (r i)) (cball (a j) (r j))" for j
+        proof -
+          obtain x where x: "dist (a i) x \<le> r i" "dist (a j) x \<le> r j"
+            using nondis by (force simp: disjnt_def)
+          have "dist (a i) (a j) \<le> dist (a i) x + dist x (a j)"
+            by (simp add: dist_triangle)
+          also have "\<dots> \<le> r i + r j"
+            by (metis add_mono_thms_linordered_semiring(1) dist_commute x)
+          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 \<in> C"
+            have "B/2^n < 2 * r j"
+              using Ble True \<open>j \<in> C\<close> less by auto
+            with aij True show "cball (a i) (r i) \<subseteq> ball (a j) (5 * r j)"
+              by (simp add: cball_subset_ball_iff)
+          next
+            assume "j \<in> E"
+            then have "B/2 ^ n < 2 * r j"
+              using D_def \<open>E \<subseteq> D\<close> by auto
+            with True have "r i < 2 * r j"
+              by auto
+            with aij show "cball (a i) (r i) \<subseteq> ball (a j) (5 * r j)"
+              by (simp add: cball_subset_ball_iff)
+          qed
+        qed
+      moreover have "\<exists>j. j \<in> C \<union> E \<and> \<not> disjnt (cball (a i) (r i)) (cball (a j) (r j))"
+        if "B/2 ^ Suc n < r i"
+      proof (rule classical)
+        assume NON: "\<not> ?thesis"
+        show ?thesis
+        proof (cases "i \<in> D")
+          case True
+          have "insert i E = E"
+          proof (rule maximalE)
+            show "insert i E \<subseteq> D"
+              by (simp add: True \<open>E \<subseteq> D\<close>)
+            show "pairwise (\<lambda>i 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 \<open>i \<in> K\<close> assms by fastforce
+        next
+          case False
+          with that show ?thesis
+            by (auto simp: D_def disjnt_def \<open>i \<in> K\<close>)
+        qed
+      qed
+      ultimately
+      show "B/2 ^ Suc n < r i \<longrightarrow>
+            (\<exists>j. j \<in> C \<union> E \<and>
+                 \<not> disjnt (cball (a i) (r i)) (cball (a j) (r j)) \<and>
+                 cball (a i) (r i) \<subseteq> ball (a j) (5 * r j))"
+        by blast
+      qed
+    qed auto
+  qed (use assms in force)
+  then obtain F where FK: "\<And>n. F n \<subseteq> K"
+               and Fle: "\<And>n i. i \<in> F n \<Longrightarrow> B/2 ^ n \<le> r i"
+               and Fdjnt:  "\<And>n. ?djnt (F n)"
+               and FF:  "\<And>n i. \<lbrakk>i \<in> K; B/2 ^ n < r i\<rbrakk>
+                        \<Longrightarrow> \<exists>j. j \<in> F n \<and> \<not> disjnt (cball (a i) (r i)) (cball (a j) (r j)) \<and>
+                                cball (a i) (r i) \<subseteq> ball (a j) (5 * r j)"
+               and inc:  "\<And>n. F n \<subseteq> F(Suc n)"
+    by (force simp: all_conj_distrib)
+  show thesis
+  proof
+    have *: "countable I"
+      if "I \<subseteq> K" and pw: "pairwise (\<lambda>i 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 "\<lambda>i. cball(a i)(r i)"])
+        show "countable ((\<lambda>i. cball (a i) (r i)) ` I)"
+        proof (rule countable_disjoint_nonempty_interior_subsets)
+          show "disjoint ((\<lambda>i. cball (a i) (r i)) ` I)"
+            by (auto simp: dest: pairwiseD [OF pw] intro: pairwise_imageI)
+          show "\<And>S. \<lbrakk>S \<in> (\<lambda>i. cball (a i) (r i)) ` I; interior S = {}\<rbrakk> \<Longrightarrow> S = {}"
+            using \<open>I \<subseteq> K\<close>
+            by (auto simp: not_less [symmetric])
+        qed
+      next
+        have "\<And>x y. \<lbrakk>x \<in> I; y \<in> I; a x = a y; r x = r y\<rbrakk> \<Longrightarrow> x = y"
+          using pw \<open>I \<subseteq> K\<close> 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 (\<lambda>i. cball (a i) (r i)) I"
+          using \<open>I \<subseteq> K\<close> by (fastforce simp: inj_on_def cball_eq_cball_iff dest: assms)
+      qed
+    qed
+    show "(Union(range F)) \<subseteq> K"
+      using FK by blast
+    moreover show "pairwise (\<lambda>i j. disjnt (cball (a i) (r i)) (cball (a j) (r j))) (Union(range F))"
+    proof (rule pairwise_chain_Union)
+      show "chain\<^sub>\<subseteq> (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 \<in> K"
+    then obtain n where "(1/2) ^ n < r i / B"
+      using  \<open>B > 0\<close> assms real_arch_pow_inv by fastforce
+    then have B2: "B/2 ^ n < r i"
+      using \<open>B > 0\<close> by (simp add: divide_simps)
+    have "0 < r i" "r i \<le> B"
+      by (auto simp: \<open>i \<in> K\<close> assms)
+    show "\<exists>j. j \<in> (Union(range F)) \<and>
+          \<not> disjnt (cball (a i) (r i)) (cball (a j) (r j)) \<and>
+          cball (a i) (r i) \<subseteq> ball (a j) (5 * r j)"
+      using FF [OF \<open>i \<in> K\<close> B2] by auto
+  qed
+qed
+
+subsection\<open>Vitali covering theorem\<close>
+
+lemma Vitali_covering_lemma_cballs:
+  fixes a :: "'a \<Rightarrow> 'b::euclidean_space"
+  assumes S: "S \<subseteq> (\<Union>i\<in>K. cball (a i) (r i))"
+      and r: "\<And>i. i \<in> K \<Longrightarrow> 0 < r i \<and> r i \<le> B"
+  obtains C where "countable C" "C \<subseteq> K"
+     "pairwise (\<lambda>i j. disjnt (cball (a i) (r i)) (cball (a j) (r j))) C"
+     "S \<subseteq> (\<Union>i\<in>C. cball (a i) (5 * r i))"
+proof -
+  obtain C where C: "countable C" "C \<subseteq> K"
+                    "pairwise (\<lambda>i j. disjnt (cball (a i) (r i)) (cball (a j) (r j))) C"
+           and cov: "\<And>i. i \<in> K \<Longrightarrow> \<exists>j. j \<in> C \<and> \<not> disjnt (cball (a i) (r i)) (cball (a j) (r j)) \<and>
+                        cball (a i) (r i) \<subseteq> 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 "(\<Union>i\<in>K. cball (a i) (r i)) \<subseteq> (\<Union>i\<in>C. cball (a i) (5 * r i))"
+      using cov subset_iff by fastforce
+    with S show "S \<subseteq> (\<Union>i\<in>C. cball (a i) (5 * r i))"
+      by blast
+  qed (use C in auto)
+qed
+
+lemma Vitali_covering_lemma_balls:
+  fixes a :: "'a \<Rightarrow> 'b::euclidean_space"
+  assumes S: "S \<subseteq> (\<Union>i\<in>K. ball (a i) (r i))"
+      and r: "\<And>i. i \<in> K \<Longrightarrow> 0 < r i \<and> r i \<le> B"
+  obtains C where "countable C" "C \<subseteq> K"
+     "pairwise (\<lambda>i j. disjnt (ball (a i) (r i)) (ball (a j) (r j))) C"
+     "S \<subseteq> (\<Union>i\<in>C. ball (a i) (5 * r i))"
+proof -
+  obtain C where C: "countable C" "C \<subseteq> K"
+           and pw:  "pairwise (\<lambda>i j. disjnt (cball (a i) (r i)) (cball (a j) (r j))) C"
+           and cov: "\<And>i. i \<in> K \<Longrightarrow> \<exists>j. j \<in> C \<and> \<not> disjnt (cball (a i) (r i)) (cball (a j) (r j)) \<and>
+                        cball (a i) (r i) \<subseteq> 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 "(\<Union>i\<in>K. ball (a i) (r i)) \<subseteq> (\<Union>i\<in>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 \<subseteq> (\<Union>i\<in>C. ball (a i) (5 * r i))"
+      by blast
+    show "pairwise (\<lambda>i 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
+
+
+proposition Vitali_covering_theorem_cballs:
+  fixes a :: "'a \<Rightarrow> 'n::euclidean_space"
+  assumes r: "\<And>i. i \<in> K \<Longrightarrow> 0 < r i"
+      and S: "\<And>x d. \<lbrakk>x \<in> S; 0 < d\<rbrakk>
+                     \<Longrightarrow> \<exists>i. i \<in> K \<and> x \<in> cball (a i) (r i) \<and> r i < d"
+  obtains C where "countable C" "C \<subseteq> K"
+     "pairwise (\<lambda>i j. disjnt (cball (a i) (r i)) (cball (a j) (r j))) C"
+     "negligible(S - (\<Union>i \<in> C. cball (a i) (r i)))"
+proof -
+  let ?\<mu> = "measure lebesgue"
+  have *: "\<exists>C. countable C \<and> C \<subseteq> K \<and>
+            pairwise (\<lambda>i j. disjnt (cball (a i) (r i)) (cball (a j) (r j))) C \<and>
+            negligible(S - (\<Union>i \<in> C. cball (a i) (r i)))"
+    if r01: "\<And>i. i \<in> K \<Longrightarrow> 0 < r i \<and> r i \<le> 1"
+       and Sd: "\<And>x d. \<lbrakk>x \<in> S; 0 < d\<rbrakk> \<Longrightarrow> \<exists>i. i \<in> K \<and> x \<in> cball (a i) (r i) \<and> r i < d"
+     for K r and a :: "'a \<Rightarrow> 'n"
+  proof -
+    obtain C where C: "countable C" "C \<subseteq> K"
+      and pwC: "pairwise (\<lambda>i j. disjnt (cball (a i) (r i)) (cball (a j) (r j))) C"
+      and cov: "\<And>i. i \<in> K \<Longrightarrow> \<exists>j. j \<in> C \<and> \<not> disjnt (cball (a i) (r i)) (cball (a j) (r j)) \<and>
+                        cball (a i) (r i) \<subseteq> ball (a j) (5 * r j)"
+      by (rule Vitali_covering_lemma_cballs_balls [of K r 1 a]) (auto simp: r01)
+    have ar_injective: "\<And>x y. \<lbrakk>x \<in> C; y \<in> C; a x = a y; r x = r y\<rbrakk> \<Longrightarrow> x = y"
+      using \<open>C \<subseteq> K\<close> pwC cov
+      by (force simp: pairwise_def disjnt_def)
+    show ?thesis
+    proof (intro exI conjI)
+      show "negligible (S - (\<Union>i\<in>C. cball (a i) (r i)))"
+      proof (clarsimp simp: negligible_on_intervals [of "S-T" for T])
+        fix l u
+        show "negligible ((S - (\<Union>i\<in>C. cball (a i) (r i))) \<inter> cbox l u)"
+          unfolding negligible_outer_le
+        proof (intro allI impI)
+          fix e::real
+          assume "e > 0"
+          define D where "D \<equiv> {i \<in> C. \<not> disjnt (ball(a i) (5 * r i)) (cbox l u)}"
+          then have "D \<subseteq> C"
+            by auto
+          have "countable D"
+            unfolding D_def using \<open>countable C\<close> by simp
+          have UD: "(\<Union>i\<in>D. cball (a i) (r i)) \<in> lmeasurable"
+          proof (rule fmeasurableI2)
+            show "cbox (l - 6 *\<^sub>R One) (u + 6 *\<^sub>R One) \<in> lmeasurable"
+              by blast
+            have "y \<in> cbox (l - 6 *\<^sub>R One) (u + 6 *\<^sub>R One)"
+              if "i \<in> C" and x: "x \<in> cbox l u" and ai: "dist (a i) y \<le> 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 \<in> Basis"
+                then have xyj: "\<bar>x \<bullet> j - y \<bullet> j\<bar> \<le> dist y x"
+                  by (metis Basis_le_norm dist_commute dist_norm inner_diff_left)
+                have "l \<bullet> j \<le> x \<bullet> j"
+                  using \<open>j \<in> Basis\<close> mem_box \<open>x \<in> cbox l u\<close> by blast
+                also have "\<dots> \<le> y \<bullet> j + 6 * r i"
+                  using d6 xyj by (auto simp: algebra_simps)
+                also have "\<dots> \<le> y \<bullet> j + 6"
+                  using r01 [of i] \<open>C \<subseteq> K\<close> \<open>i \<in> C\<close> by auto
+                finally have l: "l \<bullet> j \<le> y \<bullet> j + 6" .
+                have "y \<bullet> j \<le> x \<bullet> j + 6 * r i"
+                  using d6 xyj by (auto simp: algebra_simps)
+                also have "\<dots> \<le> u \<bullet> j + 6 * r i"
+                  using j  x by (auto simp: mem_box)
+                also have "\<dots> \<le> u \<bullet> j + 6"
+                  using r01 [of i] \<open>C \<subseteq> K\<close> \<open>i \<in> C\<close> by auto
+                finally have u: "y \<bullet> j \<le> u \<bullet> j + 6" .
+                show "l \<bullet> j \<le> y \<bullet> j + 6 \<and> y \<bullet> j \<le> u \<bullet> j + 6"
+                  using l u by blast
+              qed
+            qed
+            then show "(\<Union>i\<in>D. cball (a i) (r i)) \<subseteq> cbox (l - 6 *\<^sub>R One) (u + 6 *\<^sub>R One)"
+              by (force simp: D_def disjnt_def)
+            show "(\<Union>i\<in>D. cball (a i) (r i)) \<in> sets lebesgue"
+              using \<open>countable D\<close> by auto
+          qed
+          obtain D1 where "D1 \<subseteq> D" "finite D1"
+            and measD1: "?\<mu> (\<Union>i\<in>D. cball (a i) (r i)) - e / 5 ^ DIM('n) < ?\<mu> (\<Union>i\<in>D1. cball (a i) (r i))"
+          proof (rule measure_countable_Union_approachable [where e = "e / 5 ^ (DIM('n))"])
+            show "countable ((\<lambda>i. cball (a i) (r i)) ` D)"
+              using \<open>countable D\<close> by auto
+            show "\<And>d. d \<in> (\<lambda>i. cball (a i) (r i)) ` D \<Longrightarrow> d \<in> lmeasurable"
+              by auto
+            show "\<And>D'. \<lbrakk>D' \<subseteq> (\<lambda>i. cball (a i) (r i)) ` D; finite D'\<rbrakk> \<Longrightarrow> ?\<mu> (\<Union>D') \<le> ?\<mu> (\<Union>i\<in>D. cball (a i) (r i))"
+              by (fastforce simp add: intro!: measure_mono_fmeasurable UD)
+          qed (use \<open>e > 0\<close> in \<open>auto dest: finite_subset_image\<close>)
+          show "\<exists>T. (S - (\<Union>i\<in>C. cball (a i) (r i))) \<inter>
+                    cbox l u \<subseteq> T \<and> T \<in> lmeasurable \<and> ?\<mu> T \<le> e"
+          proof (intro exI conjI)
+            show "(S - (\<Union>i\<in>C. cball (a i) (r i))) \<inter> cbox l u \<subseteq> (\<Union>i\<in>D - D1. ball (a i) (5 * r i))"
+            proof clarify
+              fix x
+              assume x: "x \<in> cbox l u" "x \<in> S" "x \<notin> (\<Union>i\<in>C. cball (a i) (r i))"
+              have "closed (\<Union>i\<in>D1. cball (a i) (r i))"
+                using \<open>finite D1\<close> by blast
+              moreover have "x \<notin> (\<Union>j\<in>D1. cball (a j) (r j))"
+                using x \<open>D1 \<subseteq> D\<close> unfolding D_def by blast
+              ultimately obtain q where "q > 0" and q: "ball x q \<subseteq> - (\<Union>i\<in>D1. cball (a i) (r i))"
+                by (metis (no_types, lifting) ComplI open_contains_ball closed_def)
+              obtain i where "i \<in> K" and xi: "x \<in> cball (a i) (r i)" and ri: "r i < q/2"
+                using Sd [OF \<open>x \<in> S\<close>] \<open>q > 0\<close> half_gt_zero by blast
+              then obtain j where "j \<in> C"
+                             and nondisj: "\<not> disjnt (cball (a i) (r i)) (cball (a j) (r j))"
+                             and sub5j:  "cball (a i) (r i) \<subseteq> ball (a j) (5 * r j)"
+                using cov [OF \<open>i \<in> K\<close>] by metis
+              show "x \<in> (\<Union>i\<in>D - D1. ball (a i) (5 * r i))"
+              proof
+                show "j \<in> D - D1"
+                proof
+                  show "j \<in> D"
+                    using \<open>j \<in> C\<close> sub5j \<open>x \<in> cbox l u\<close> xi by (auto simp: D_def disjnt_def)
+                  obtain y where yi: "dist (a i) y \<le> r i" and yj: "dist (a j) y \<le> r j"
+                    using disjnt_def nondisj by fastforce
+                  have "dist x y \<le> r i + r i"
+                    by (metis add_mono dist_commute dist_triangle_le mem_cball xi yi)
+                  also have "\<dots> < q"
+                    using ri by linarith
+                  finally have "y \<in> ball x q"
+                    by simp
+                  with yj q show "j \<notin> D1"
+                    by (auto simp: disjoint_UN_iff)
+                qed
+                show "x \<in> ball (a j) (5 * r j)"
+                  using xi sub5j by blast
+              qed
+            qed
+            have 3: "?\<mu> (\<Union>i\<in>D2. ball (a i) (5 * r i)) \<le> e"
+              if D2: "D2 \<subseteq> D - D1" and "finite D2" for D2
+            proof -
+              have rgt0: "0 < r i" if "i \<in> D2" for i
+                using \<open>C \<subseteq> K\<close> D_def \<open>i \<in> D2\<close> D2 r01
+                by (simp add: subset_iff)
+              then have inj: "inj_on (\<lambda>i. ball (a i) (5 * r i)) D2"
+                using \<open>C \<subseteq> K\<close> D2 by (fastforce simp: inj_on_def D_def ball_eq_ball_iff intro: ar_injective)
+              have "?\<mu> (\<Union>i\<in>D2. ball (a i) (5 * r i)) \<le> sum (?\<mu>) ((\<lambda>i. ball (a i) (5 * r i)) ` D2)"
+                using that by (force intro: measure_Union_le)
+              also have "\<dots>  = (\<Sum>i\<in>D2. ?\<mu> (ball (a i) (5 * r i)))"
+                by (simp add: comm_monoid_add_class.sum.reindex [OF inj])
+              also have "\<dots> = (\<Sum>i\<in>D2. 5 ^ DIM('n) * ?\<mu> (ball (a i) (r i)))"
+              proof (rule sum.cong [OF refl])
+                fix i
+                assume "i \<in> D2"
+                show "?\<mu> (ball (a i) (5 * r i)) = 5 ^ DIM('n) * ?\<mu> (ball (a i) (r i))"
+                  using rgt0 [OF \<open>i \<in> D2\<close>] by (simp add: content_ball)
+              qed
+              also have "\<dots> = (\<Sum>i\<in>D2. ?\<mu> (ball (a i) (r i))) * 5 ^ DIM('n)"
+                by (simp add: sum_distrib_left mult.commute)
+              finally have "?\<mu> (\<Union>i\<in>D2. ball (a i) (5 * r i)) \<le> (\<Sum>i\<in>D2. ?\<mu> (ball (a i) (r i))) * 5 ^ DIM('n)" .
+              moreover have "(\<Sum>i\<in>D2. ?\<mu> (ball (a i) (r i))) \<le> e / 5 ^ DIM('n)"
+              proof -
+                have D12_dis: "((\<Union>x\<in>D1. cball (a x) (r x)) \<inter> (\<Union>x\<in>D2. cball (a x) (r x))) \<le> {}"
+                proof clarify
+                  fix w d1 d2
+                  assume "d1 \<in> D1" "w d1 d2 \<in> cball (a d1) (r d1)" "d2 \<in> D2" "w d1 d2 \<in> cball (a d2) (r d2)"
+                  then show "w d1 d2 \<in> {}"
+                    by (metis DiffE disjnt_iff subsetCE D2 \<open>D1 \<subseteq> D\<close> \<open>D \<subseteq> C\<close> pairwiseD [OF pwC, of d1 d2])
+                qed
+                have inj: "inj_on (\<lambda>i. cball (a i) (r i)) D2"
+                  using rgt0 D2 \<open>D \<subseteq> C\<close> by (force simp: inj_on_def cball_eq_cball_iff intro!: ar_injective)
+                have ds: "disjoint ((\<lambda>i. cball (a i) (r i)) ` D2)"
+                  using D2 \<open>D \<subseteq> C\<close> by (auto intro: pairwiseI pairwiseD [OF pwC])
+                have "(\<Sum>i\<in>D2. ?\<mu> (ball (a i) (r i))) = (\<Sum>i\<in>D2. ?\<mu> (cball (a i) (r i)))"
+                  using rgt0 by (simp add: content_ball content_cball less_eq_real_def)
+                also have "\<dots> = sum ?\<mu> ((\<lambda>i. cball (a i) (r i)) ` D2)"
+                  by (simp add: comm_monoid_add_class.sum.reindex [OF inj])
+                also have "\<dots> = ?\<mu> (\<Union>i\<in>D2. cball (a i) (r i))"
+                  by (auto intro: measure_Union' [symmetric] ds simp add: \<open>finite D2\<close>)
+                finally have "?\<mu> (\<Union>i\<in>D1. cball (a i) (r i)) + (\<Sum>i\<in>D2. ?\<mu> (ball (a i) (r i))) =
+                              ?\<mu> (\<Union>i\<in>D1. cball (a i) (r i)) + ?\<mu> (\<Union>i\<in>D2. cball (a i) (r i))"
+                  by simp
+                also have "\<dots> = ?\<mu> (\<Union>i \<in> D1 \<union> D2. cball (a i) (r i))"
+                  using D12_dis by (simp add: measure_Un3 \<open>finite D1\<close> \<open>finite D2\<close> fmeasurable.finite_UN)
+                also have "\<dots> \<le> ?\<mu> (\<Union>i\<in>D. cball (a i) (r i))"
+                  using D2 \<open>D1 \<subseteq> D\<close> by (fastforce intro!: measure_mono_fmeasurable [OF _ _ UD] \<open>finite D1\<close> \<open>finite D2\<close>)
+                finally have "?\<mu> (\<Union>i\<in>D1. cball (a i) (r i)) + (\<Sum>i\<in>D2. ?\<mu> (ball (a i) (r i))) \<le> ?\<mu> (\<Union>i\<in>D. cball (a i) (r i))" .
+                with measD1 show ?thesis
+                  by simp
+              qed
+                ultimately show ?thesis
+                  by (simp add: divide_simps)
+            qed
+            have co: "countable (D - D1)"
+              by (simp add: \<open>countable D\<close>)
+            show "(\<Union>i\<in>D - D1. ball (a i) (5 * r i)) \<in> lmeasurable"
+              using \<open>e > 0\<close> by (auto simp: fmeasurable_UN_bound [OF co _ 3])
+            show "?\<mu> (\<Union>i\<in>D - D1. ball (a i) (5 * r i)) \<le> e"
+              using \<open>e > 0\<close> by (auto simp: measure_UN_bound [OF co _ 3])
+          qed
+        qed
+      qed
+    qed (use C pwC in auto)
+  qed
+  define K' where "K' \<equiv> {i \<in> K. r i \<le> 1}"
+  have 1: "\<And>i. i \<in> K' \<Longrightarrow> 0 < r i \<and> r i \<le> 1"
+    using K'_def r by auto
+  have 2: "\<exists>i. i \<in> K' \<and> x \<in> cball (a i) (r i) \<and> r i < d"
+    if "x \<in> S \<and> 0 < d" for x d
+    using that by (auto simp: K'_def dest!: S [where d = "min d 1"])
+  have "K' \<subseteq> K"
+    using K'_def by auto
+  then show thesis
+    using * [OF 1 2] that by fastforce
+qed
+
+
+proposition Vitali_covering_theorem_balls:
+  fixes a :: "'a \<Rightarrow> 'b::euclidean_space"
+  assumes S: "\<And>x d. \<lbrakk>x \<in> S; 0 < d\<rbrakk> \<Longrightarrow> \<exists>i. i \<in> K \<and> x \<in> ball (a i) (r i) \<and> r i < d"
+  obtains C where "countable C" "C \<subseteq> K"
+     "pairwise (\<lambda>i j. disjnt (ball (a i) (r i)) (ball (a j) (r j))) C"
+     "negligible(S - (\<Union>i \<in> C. ball (a i) (r i)))"
+proof -
+  have 1: "\<exists>i. i \<in> {i \<in> K. 0 < r i} \<and> x \<in> cball (a i) (r i) \<and> r i < d"
+         if xd: "x \<in> 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 \<subseteq> K"
+             and pw: "pairwise (\<lambda>i j. disjnt (cball (a i) (r i)) (cball (a j) (r j))) C"
+             and neg: "negligible(S - (\<Union>i \<in> C. cball (a i) (r i)))"
+    by (rule Vitali_covering_theorem_cballs [of "{i \<in> K. 0 < r i}" r S a, OF _ 1]) auto
+  show thesis
+  proof
+    show "pairwise (\<lambda>i 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 (\<Union>i\<in>C. sphere (a i) (r i))"
+      by (auto intro: negligible_sphere \<open>countable C\<close>)
+    then have "negligible (S - (\<Union>i \<in> C. cball(a i)(r i)) \<union> (\<Union>i \<in> C. sphere (a i) (r i)))"
+      by (rule negligible_Un [OF neg])
+    then show "negligible (S - (\<Union>i\<in>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 \<longleftrightarrow>
+      (\<forall>x \<in> S. \<forall>e > 0.
+        \<exists>d U. 0 < d \<and> d \<le> e \<and> S \<inter> ball x d \<subseteq> U \<and>
+              U \<in> lmeasurable \<and> measure lebesgue U < e * measure lebesgue (ball x d))"
+     (is "_ = (\<forall>x \<in> S. \<forall>e > 0. ?Q x e)")
+proof (intro iffI ballI allI impI)
+  fix x and e :: real
+  assume "negligible S" and "x \<in> S" and "e > 0"
+  then
+  show "\<exists>d U. 0 < d \<and> d \<le> e \<and> S \<inter> ball x d \<subseteq> U \<and> U \<in> lmeasurable \<and>
+              measure lebesgue U < e * measure lebesgue (ball x d)"
+    apply (rule_tac x=e in exI)
+    apply (rule_tac x="S \<inter> ball x e" in exI)
+    apply (auto simp: negligible_imp_measurable negligible_Int negligible_imp_measure0 zero_less_measure_iff)
+    done
+next
+  assume R [rule_format]: "\<forall>x \<in> S. \<forall>e > 0. ?Q x e"
+  let ?\<mu> = "measure lebesgue"
+  have "\<exists>U. openin (subtopology euclidean S) U \<and> z \<in> U \<and> negligible U"
+    if "z \<in> S" for z
+  proof (intro exI conjI)
+    show "openin (subtopology euclidean S) (S \<inter> ball z 1)"
+      by (simp add: openin_open_Int)
+    show "z \<in> S \<inter> ball z 1"
+      using \<open>z \<in> S\<close> by auto
+    show "negligible (S \<inter> ball z 1)"
+    proof (clarsimp simp: negligible_outer_le)
+      fix e :: "real"
+      assume "e > 0"
+      let ?K = "{(x,d). x \<in> S \<and> 0 < d \<and> ball x d \<subseteq> ball z 1 \<and>
+                     (\<exists>U. S \<inter> ball x d \<subseteq> U \<and> U \<in> lmeasurable \<and>
+                         ?\<mu> U < e / ?\<mu> (ball z 1) * ?\<mu> (ball x d))}"
+      obtain C where "countable C" and Csub: "C \<subseteq> ?K"
+        and pwC: "pairwise (\<lambda>i j. disjnt (ball (fst i) (snd i)) (ball (fst j) (snd j))) C"
+        and negC: "negligible((S \<inter> ball z 1) - (\<Union>i \<in> C. ball (fst i) (snd i)))"
+      proof (rule Vitali_covering_theorem_balls [of "S \<inter> ball z 1" ?K fst snd])
+        fix x and d :: "real"
+        assume x: "x \<in> S \<inter> ball z 1" and "d > 0"
+        obtain k where "k > 0" and k: "ball x k \<subseteq> ball z 1"
+          by (meson Int_iff open_ball openE x)
+        let ?\<epsilon> = "min (e / ?\<mu> (ball z 1) / 2) (min (d / 2) k)"
+        obtain r U where r: "r > 0" "r \<le> ?\<epsilon>" and U: "S \<inter> ball x r \<subseteq> U" "U \<in> lmeasurable"
+          and mU: "?\<mu> U < ?\<epsilon> * ?\<mu> (ball x r)"
+          using R [of x ?\<epsilon>] \<open>d > 0\<close> \<open>e > 0\<close> \<open>k > 0\<close> x by auto
+        show "\<exists>i. i \<in> ?K \<and> x \<in> ball (fst i) (snd i) \<and> snd i < d"
+        proof (rule exI [of _ "(x,r)"], simp, intro conjI exI)
+          have "ball x r \<subseteq> ball x k"
+            using r by (simp add: ball_subset_ball_iff)
+          also have "\<dots> \<subseteq> ball z 1"
+            using ball_subset_ball_iff k by auto
+          finally show "ball x r \<subseteq> ball z 1" .
+          have "?\<epsilon> * ?\<mu> (ball x r) \<le> e * content (ball x r) / content (ball z 1)"
+            using r \<open>e > 0\<close> by (simp add: ord_class.min_def divide_simps)
+          with mU show "?\<mu> U < e * content (ball x r) / content (ball z 1)"
+            by auto
+        qed (use r U x in auto)
+      qed
+      have "\<exists>U. case p of (x,d) \<Rightarrow> S \<inter> ball x d \<subseteq> U \<and>
+                        U \<in> lmeasurable \<and> ?\<mu> U < e / ?\<mu> (ball z 1) * ?\<mu> (ball x d)"
+        if "p \<in> C" for p
+        using that Csub by auto
+      then obtain U where U:
+                "\<And>p. p \<in> C \<Longrightarrow>
+                       case p of (x,d) \<Rightarrow> S \<inter> ball x d \<subseteq> U p \<and>
+                        U p \<in> lmeasurable \<and> ?\<mu> (U p) < e / ?\<mu> (ball z 1) * ?\<mu> (ball x d)"
+        by (rule that [OF someI_ex])
+      let ?T = "((S \<inter> ball z 1) - (\<Union>(x,d)\<in>C. ball x d)) \<union> \<Union>(U ` C)"
+      show "\<exists>T. S \<inter> ball z 1 \<subseteq> T \<and> T \<in> lmeasurable \<and> ?\<mu> T \<le> e"
+      proof (intro exI conjI)
+        show "S \<inter> ball z 1 \<subseteq> ?T"
+          using U by fastforce
+        { have Um: "U i \<in> lmeasurable" if "i \<in> C" for i
+            using that U by blast
+          have lee: "?\<mu> (\<Union>i\<in>I. U i) \<le> e" if "I \<subseteq> C" "finite I" for I
+          proof -
+            have "?\<mu> (\<Union>(x,d)\<in>I. ball x d) \<le> ?\<mu> (ball z 1)"
+              apply (rule measure_mono_fmeasurable)
+              using \<open>I \<subseteq> C\<close> \<open>finite I\<close> Csub by (force simp: prod.case_eq_if sets.finite_UN)+
+            then have le1: "(?\<mu> (\<Union>(x,d)\<in>I. ball x d) / ?\<mu> (ball z 1)) \<le> 1"
+              by simp
+            have "?\<mu> (\<Union>i\<in>I. U i) \<le> (\<Sum>i\<in>I. ?\<mu> (U i))"
+              using that U by (blast intro: measure_UNION_le)
+            also have "\<dots> \<le> (\<Sum>(x,r)\<in>I. e / ?\<mu> (ball z 1) * ?\<mu> (ball x r))"
+              by (rule sum_mono) (use \<open>I \<subseteq> C\<close> U in force)
+            also have "\<dots> = (e / ?\<mu> (ball z 1)) * (\<Sum>(x,r)\<in>I. ?\<mu> (ball x r))"
+              by (simp add: case_prod_app prod.case_distrib sum_distrib_left)
+            also have "\<dots> = e * (?\<mu> (\<Union>(x,r)\<in>I. ball x r) / ?\<mu> (ball z 1))"
+              apply (subst measure_UNION')
+              using that pwC by (auto simp: case_prod_unfold elim: pairwise_mono)
+            also have "\<dots> \<le> e"
+              by (metis mult.commute mult.left_neutral real_mult_le_cancel_iff1 \<open>e > 0\<close> le1)
+            finally show ?thesis .
+          qed
+          have "UNION C U \<in> lmeasurable" "?\<mu> (\<Union>(U ` C)) \<le> e"
+            using \<open>e > 0\<close> Um lee
+            by(auto intro!: fmeasurable_UN_bound [OF \<open>countable C\<close>] measure_UN_bound [OF \<open>countable C\<close>])
+        }
+        moreover have "?\<mu> ?T = ?\<mu> (UNION C U)"
+        proof (rule measure_negligible_symdiff [OF \<open>UNION C U \<in> lmeasurable\<close>])
+          show "negligible((UNION C U - ?T) \<union> (?T - UNION C U))"
+            by (force intro!: negligible_subset [OF negC])
+        qed
+        ultimately show "?T \<in> lmeasurable"  "?\<mu> ?T \<le> 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 \<longleftrightarrow>
+    (\<forall>x\<in>S. \<forall>r>0. \<forall>e>0. \<exists>d. 0 < d \<and> d \<le> r \<and>
+                   (\<exists>U. S \<inter> ball x d \<subseteq> U \<and> U \<in> lmeasurable \<and> measure lebesgue U < e * measure lebesgue (ball x d)))"
+proof -
+  let ?Q = "\<lambda>x d e. \<exists>U. S \<inter> ball x d \<subseteq> U \<and> U \<in> lmeasurable \<and> measure lebesgue U < e * content (ball x d)"
+  have "(\<forall>e>0. \<exists>d>0. d \<le> e \<and> ?Q x d e) = (\<forall>r>0. \<forall>e>0. \<exists>d>0. d \<le> r \<and> ?Q x d e)"
+    if "x \<in> S" for x
+  proof (intro iffI allI impI)
+    fix r :: "real" and e :: "real"
+    assume L [rule_format]: "\<forall>e>0. \<exists>d>0. d \<le> e \<and> ?Q x d e" and "r > 0" "e > 0"
+    show "\<exists>d>0. d \<le> r \<and> ?Q x d e"
+      using L [of "min r e"] apply (rule ex_forward)
+      using \<open>r > 0\<close> \<open>e > 0\<close>  by (auto intro: less_le_trans elim!: ex_forward)
+  qed auto
+  then show ?thesis
+    by (force simp: negligible_eq_zero_density_alt)
+qed
+
+end```