# HG changeset patch
# User paulson <lp15@cam.ac.uk>
# Date 1524000948 -3600
# Node ID 73a5a33486ee2400ee0701d632dee57af52864c3
# Parent  ae76012879c600481edd49774089d5bcd9994643
Change of variables proof

diff -r ae76012879c6 -r 73a5a33486ee src/HOL/Analysis/Change_Of_Vars.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Analysis/Change_Of_Vars.thy	Tue Apr 17 22:35:48 2018 +0100
@@ -0,0 +1,3974 @@
+theory Change_Of_Vars
+  imports  "HOL-Analysis.Vitali_Covering_Theorem" "HOL-Analysis.Determinants"
+
+begin
+
+subsection\<open>Induction on matrix row operations\<close>
+
+lemma induct_matrix_row_operations:
+  fixes P :: "(real^'n, 'n::finite) vec \<Rightarrow> bool"
+  assumes zero_row: "\<And>A i. row i A = 0 \<Longrightarrow> P A"
+    and diagonal: "\<And>A. (\<And>i j. i \<noteq> j \<Longrightarrow> A$i$j = 0) \<Longrightarrow> P A"
+    and swap_cols: "\<And>A m n. \<lbrakk>P A; m \<noteq> n\<rbrakk> \<Longrightarrow> P(\<chi> i j. A $ i $ Fun.swap m n id j)"
+    and row_op: "\<And>A m n c. \<lbrakk>P A; m \<noteq> n\<rbrakk>
+                   \<Longrightarrow> P(\<chi> i. if i = m then row m A + c *\<^sub>R row n A else row i A)"
+  shows "P A"
+proof -
+  have "P A" if "(\<And>i j. \<lbrakk>j \<in> -K;  i \<noteq> j\<rbrakk> \<Longrightarrow> A$i$j = 0)" for A K
+  proof -
+    have "finite K"
+      by simp
+    then show ?thesis using that
+    proof (induction arbitrary: A rule: finite_induct)
+      case empty
+      with diagonal show ?case
+        by simp
+    next
+      case (insert k K)
+      note insertK = insert
+      have "P A" if kk: "A$k$k \<noteq> 0"
+        and 0: "\<And>i j. \<lbrakk>j \<in> - insert k K; i \<noteq> j\<rbrakk> \<Longrightarrow> A$i$j = 0"
+               "\<And>i. \<lbrakk>i \<in> -L; i \<noteq> k\<rbrakk> \<Longrightarrow> A$i$k = 0" for A L
+      proof -
+        have "finite L"
+          by simp
+        then show ?thesis using 0 kk
+        proof (induction arbitrary: A rule: finite_induct)
+          case (empty B)
+          show ?case
+          proof (rule insertK)
+            fix i j
+            assume "i \<in> - K" "j \<noteq> i"
+            show "B $ j $ i = 0"
+              using \<open>j \<noteq> i\<close> \<open>i \<in> - K\<close> empty
+              by (metis ComplD ComplI Compl_eq_Diff_UNIV Diff_empty UNIV_I insert_iff)
+          qed
+        next
+          case (insert l L B)
+          show ?case
+          proof (cases "k = l")
+            case True
+            with insert show ?thesis
+              by auto
+          next
+            case False
+            let ?C = "\<chi> i. if i = l then row l B - (B $ l $ k / B $ k $ k) *\<^sub>R row k B else row i B"
+            have 1: "\<lbrakk>j \<in> - insert k K; i \<noteq> j\<rbrakk> \<Longrightarrow> ?C $ i $ j = 0" for j i
+              by (auto simp: insert.prems(1) row_def)
+            have 2: "?C $ i $ k = 0"
+              if "i \<in> - L" "i \<noteq> k" for i
+            proof (cases "i=l")
+              case True
+              with that insert.prems show ?thesis
+                by (simp add: row_def)
+            next
+              case False
+              with that show ?thesis
+                by (simp add: insert.prems(2) row_def)
+            qed
+            have 3: "?C $ k $ k \<noteq> 0"
+              by (auto simp: insert.prems row_def \<open>k \<noteq> l\<close>)
+            have PC: "P ?C"
+              using insert.IH [OF 1 2 3] by auto
+            have eqB: "(\<chi> i. if i = l then row l ?C + (B $ l $ k / B $ k $ k) *\<^sub>R row k ?C else row i ?C) = B"
+              using \<open>k \<noteq> l\<close> by (simp add: vec_eq_iff row_def)
+            show ?thesis
+              using row_op [OF PC, of l k, where c = "B$l$k / B$k$k"] eqB \<open>k \<noteq> l\<close>
+              by (simp add: cong: if_cong)
+          qed
+        qed
+      qed
+      then have nonzero_hyp: "P A"
+        if kk: "A$k$k \<noteq> 0" and zeroes: "\<And>i j. j \<in> - insert k K \<and> i\<noteq>j \<Longrightarrow> A$i$j = 0" for A
+        by (auto simp: intro!: kk zeroes)
+      show ?case
+      proof (cases "row k A = 0")
+        case True
+        with zero_row show ?thesis by auto
+      next
+        case False
+        then obtain l where l: "A$k$l \<noteq> 0"
+          by (auto simp: row_def zero_vec_def vec_eq_iff)
+        show ?thesis
+        proof (cases "k = l")
+          case True
+          with l nonzero_hyp insert.prems show ?thesis
+            by blast
+        next
+          case False
+          have *: "A $ i $ Fun.swap k l id j = 0" if "j \<noteq> k" "j \<notin> K" "i \<noteq> j" for i j
+            using False l insert.prems that
+            by (auto simp: swap_def insert split: if_split_asm)
+          have "P (\<chi> i j. (\<chi> i j. A $ i $ Fun.swap k l id j) $ i $ Fun.swap k l id j)"
+            by (rule swap_cols [OF nonzero_hyp False]) (auto simp: l *)
+          moreover
+          have "(\<chi> i j. (\<chi> i j. A $ i $ Fun.swap k l id j) $ i $ Fun.swap k l id j) = A"
+            by (metis (no_types, lifting) id_apply o_apply swap_id_idempotent vec_lambda_unique vec_lambda_unique)
+          ultimately show ?thesis
+            by simp
+        qed
+      qed
+    qed
+  qed
+  then show ?thesis
+    by blast
+qed
+
+lemma induct_matrix_elementary:
+  fixes P :: "(real^'n, 'n::finite) vec \<Rightarrow> bool"
+  assumes mult: "\<And>A B. \<lbrakk>P A; P B\<rbrakk> \<Longrightarrow> P(A ** B)"
+    and zero_row: "\<And>A i. row i A = 0 \<Longrightarrow> P A"
+    and diagonal: "\<And>A. (\<And>i j. i \<noteq> j \<Longrightarrow> A$i$j = 0) \<Longrightarrow> P A"
+    and swap1: "\<And>m n. m \<noteq> n \<Longrightarrow> P(\<chi> i j. mat 1 $ i $ Fun.swap m n id j)"
+    and idplus: "\<And>m n c. m \<noteq> n \<Longrightarrow> P(\<chi> i j. if i = m \<and> j = n then c else of_bool (i = j))"
+  shows "P A"
+proof -
+  have swap: "P (\<chi> i j. A $ i $ Fun.swap m n id j)"  (is "P ?C")
+    if "P A" "m \<noteq> n" for A m n
+  proof -
+    have "A ** (\<chi> i j. mat 1 $ i $ Fun.swap m n id j) = ?C"
+      by (simp add: matrix_matrix_mult_def mat_def vec_eq_iff if_distrib sum.delta_remove)
+    then show ?thesis
+      using mult swap1 that by metis
+  qed
+  have row: "P (\<chi> i. if i = m then row m A + c *\<^sub>R row n A else row i A)"  (is "P ?C")
+    if "P A" "m \<noteq> n" for A m n c
+  proof -
+    let ?B = "\<chi> i j. if i = m \<and> j = n then c else of_bool (i = j)"
+    have "?B ** A = ?C"
+      using \<open>m \<noteq> n\<close> unfolding matrix_matrix_mult_def row_def of_bool_def
+      by (auto simp: vec_eq_iff if_distrib [of "\<lambda>x. x * y" for y] sum.remove cong: if_cong)
+    then show ?thesis
+      by (rule subst) (auto simp: that mult idplus)
+  qed
+  show ?thesis
+    by (rule induct_matrix_row_operations [OF zero_row diagonal swap row])
+qed
+
+lemma induct_matrix_elementary_alt:
+  fixes P :: "(real^'n, 'n::finite) vec \<Rightarrow> bool"
+  assumes mult: "\<And>A B. \<lbrakk>P A; P B\<rbrakk> \<Longrightarrow> P(A ** B)"
+    and zero_row: "\<And>A i. row i A = 0 \<Longrightarrow> P A"
+    and diagonal: "\<And>A. (\<And>i j. i \<noteq> j \<Longrightarrow> A$i$j = 0) \<Longrightarrow> P A"
+    and swap1: "\<And>m n. m \<noteq> n \<Longrightarrow> P(\<chi> i j. mat 1 $ i $ Fun.swap m n id j)"
+    and idplus: "\<And>m n. m \<noteq> n \<Longrightarrow> P(\<chi> i j. of_bool (i = m \<and> j = n \<or> i = j))"
+  shows "P A"
+proof -
+  have *: "P (\<chi> i j. if i = m \<and> j = n then c else of_bool (i = j))"
+    if "m \<noteq> n" for m n c
+  proof (cases "c = 0")
+    case True
+    with diagonal show ?thesis by auto
+  next
+    case False
+    then have eq: "(\<chi> i j. if i = m \<and> j = n then c else of_bool (i = j)) =
+                      (\<chi> i j. if i = j then (if j = n then inverse c else 1) else 0) **
+                      (\<chi> i j. of_bool (i = m \<and> j = n \<or> i = j)) **
+                      (\<chi> i j. if i = j then if j = n then c else 1 else 0)"
+      using \<open>m \<noteq> n\<close>
+      apply (simp add: matrix_matrix_mult_def vec_eq_iff of_bool_def if_distrib [of "\<lambda>x. y * x" for y] cong: if_cong)
+      apply (simp add: if_if_eq_conj sum.neutral conj_commute cong: conj_cong)
+      done
+    show ?thesis
+      apply (subst eq)
+      apply (intro mult idplus that)
+       apply (auto intro: diagonal)
+      done
+  qed
+  show ?thesis
+    by (rule induct_matrix_elementary) (auto intro: assms *)
+qed
+
+lemma induct_linear_elementary:
+  fixes f :: "real^'n \<Rightarrow> real^'n"
+  assumes "linear f"
+    and comp: "\<And>f g. \<lbrakk>linear f; linear g; P f; P g\<rbrakk> \<Longrightarrow> P(f \<circ> g)"
+    and zeroes: "\<And>f i. \<lbrakk>linear f; \<And>x. (f x) $ i = 0\<rbrakk> \<Longrightarrow> P f"
+    and const: "\<And>c. P(\<lambda>x. \<chi> i. c i * x$i)"
+    and swap: "\<And>m n::'n. m \<noteq> n \<Longrightarrow> P(\<lambda>x. \<chi> i. x $ Fun.swap m n id i)"
+    and idplus: "\<And>m n::'n. m \<noteq> n \<Longrightarrow> P(\<lambda>x. \<chi> i. if i = m then x$m + x$n else x$i)"
+  shows "P f"
+proof -
+  have "P (( *v) A)" for A
+  proof (rule induct_matrix_elementary_alt)
+    fix A B
+    assume "P (( *v) A)" and "P (( *v) B)"
+    then show "P (( *v) (A ** B))"
+      by (metis (no_types, lifting) comp linear_compose matrix_compose matrix_eq matrix_vector_mul matrix_vector_mul_linear)
+  next
+    fix A :: "((real, 'n) vec, 'n) vec" and i
+    assume "row i A = 0"
+    then show "P (( *v) A)"
+      by (metis inner_zero_left matrix_vector_mul_component matrix_vector_mul_linear row_def vec_eq_iff vec_lambda_beta zeroes)
+  next
+    fix A :: "((real, 'n) vec, 'n) vec"
+    assume 0: "\<And>i j. i \<noteq> j \<Longrightarrow> A $ i $ j = 0"
+    have "A $ i $ i * x $ i = (\<Sum>j\<in>UNIV. A $ i $ j * x $ j)" for x :: "(real, 'n) vec" and i :: "'n"
+      by (simp add: 0 comm_monoid_add_class.sum.remove [where x=i])
+    then have "(\<lambda>x. \<chi> i. A $ i $ i * x $ i) = (( *v) A)"
+      by (auto simp: 0 matrix_vector_mult_def)
+    then show "P (( *v) A)"
+      using const [of "\<lambda>i. A $ i $ i"] by simp
+  next
+    fix m n :: "'n"
+    assume "m \<noteq> n"
+    have eq: "(\<Sum>j\<in>UNIV. if i = Fun.swap m n id j then x $ j else 0) =
+              (\<Sum>j\<in>UNIV. if j = Fun.swap m n id i then x $ j else 0)"
+      for i and x :: "(real, 'n) vec"
+      unfolding swap_def by (rule sum.cong) auto
+    have "(\<lambda>x::real^'n. \<chi> i. x $ Fun.swap m n id i) = (( *v) (\<chi> i j. if i = Fun.swap m n id j then 1 else 0))"
+      by (auto simp: mat_def matrix_vector_mult_def eq if_distrib [of "\<lambda>x. x * y" for y] cong: if_cong)
+    with swap [OF \<open>m \<noteq> n\<close>] show "P (( *v) (\<chi> i j. mat 1 $ i $ Fun.swap m n id j))"
+      by (simp add: mat_def matrix_vector_mult_def)
+  next
+    fix m n :: "'n"
+    assume "m \<noteq> n"
+    then have "x $ m + x $ n = (\<Sum>j\<in>UNIV. of_bool (j = n \<or> m = j) * x $ j)" for x :: "(real, 'n) vec"
+      by (auto simp: of_bool_def if_distrib [of "\<lambda>x. x * y" for y] sum.remove cong: if_cong)
+    then have "(\<lambda>x::real^'n. \<chi> i. if i = m then x $ m + x $ n else x $ i) =
+               (( *v) (\<chi> i j. of_bool (i = m \<and> j = n \<or> i = j)))"
+      unfolding matrix_vector_mult_def of_bool_def
+      by (auto simp: vec_eq_iff if_distrib [of "\<lambda>x. x * y" for y] cong: if_cong)
+    then show "P (( *v) (\<chi> i j. of_bool (i = m \<and> j = n \<or> i = j)))"
+      using idplus [OF \<open>m \<noteq> n\<close>] by simp
+  qed
+  then show ?thesis
+    by (metis \<open>linear f\<close> matrix_vector_mul)
+qed
+
+
+proposition
+  fixes a :: "real^'n"
+  assumes "m \<noteq> n" and ab_ne: "cbox a b \<noteq> {}" and an: "0 \<le> a$n"
+  shows measurable_shear_interval: "(\<lambda>x. \<chi> i. if i = m then x$m + x$n else x$i) ` (cbox a b) \<in> lmeasurable"
+       (is  "?f ` _ \<in> _")
+   and measure_shear_interval: "measure lebesgue ((\<lambda>x. \<chi> i. if i = m then x$m + x$n else x$i) ` cbox a b)
+               = measure lebesgue (cbox a b)" (is "?Q")
+proof -
+  have lin: "linear ?f"
+    by (force simp: plus_vec_def scaleR_vec_def algebra_simps intro: linearI)
+  show fab: "?f ` cbox a b \<in> lmeasurable"
+    by (simp add: lin measurable_linear_image_interval)
+  let ?c = "\<chi> i. if i = m then b$m + b$n else b$i"
+  let ?mn = "axis m 1 - axis n (1::real)"
+  have eq1: "measure lebesgue (cbox a ?c)
+            = measure lebesgue (?f ` cbox a b)
+            + measure lebesgue (cbox a ?c \<inter> {x. ?mn \<bullet> x \<le> a$m})
+            + measure lebesgue (cbox a ?c \<inter> {x. ?mn \<bullet> x \<ge> b$m})"
+  proof (rule measure_Un3_negligible)
+    show "cbox a ?c \<inter> {x. ?mn \<bullet> x \<le> a$m} \<in> lmeasurable" "cbox a ?c \<inter> {x. ?mn \<bullet> x \<ge> b$m} \<in> lmeasurable"
+      by (auto simp: convex_Int convex_halfspace_le convex_halfspace_ge bounded_Int measurable_convex)
+    have "negligible {x. ?mn \<bullet> x = a$m}"
+      by (metis \<open>m \<noteq> n\<close> axis_index_axis eq_iff_diff_eq_0 negligible_hyperplane)
+    moreover have "?f ` cbox a b \<inter> (cbox a ?c \<inter> {x. ?mn \<bullet> x \<le> a $ m}) \<subseteq> {x. ?mn \<bullet> x = a$m}"
+      using \<open>m \<noteq> n\<close> antisym_conv by (fastforce simp: algebra_simps mem_box_cart inner_axis')
+    ultimately show "negligible ((?f ` cbox a b) \<inter> (cbox a ?c \<inter> {x. ?mn \<bullet> x \<le> a $ m}))"
+      by (rule negligible_subset)
+    have "negligible {x. ?mn \<bullet> x = b$m}"
+      by (metis \<open>m \<noteq> n\<close> axis_index_axis eq_iff_diff_eq_0 negligible_hyperplane)
+    moreover have "(?f ` cbox a b) \<inter> (cbox a ?c \<inter> {x. ?mn \<bullet> x \<ge> b$m}) \<subseteq> {x. ?mn \<bullet> x = b$m}"
+      using \<open>m \<noteq> n\<close> antisym_conv by (fastforce simp: algebra_simps mem_box_cart inner_axis')
+    ultimately show "negligible (?f ` cbox a b \<inter> (cbox a ?c \<inter> {x. ?mn \<bullet> x \<ge> b$m}))"
+      by (rule negligible_subset)
+    have "negligible {x. ?mn \<bullet> x = b$m}"
+      by (metis \<open>m \<noteq> n\<close> axis_index_axis eq_iff_diff_eq_0 negligible_hyperplane)
+    moreover have "(cbox a ?c \<inter> {x. ?mn \<bullet> x \<le> a $ m} \<inter> (cbox a ?c \<inter> {x. ?mn \<bullet> x \<ge> b$m})) \<subseteq> {x. ?mn \<bullet> x = b$m}"
+      using \<open>m \<noteq> n\<close> ab_ne
+      apply (auto simp: algebra_simps mem_box_cart inner_axis')
+      apply (drule_tac x=m in spec)+
+      apply simp
+      done
+    ultimately show "negligible (cbox a ?c \<inter> {x. ?mn \<bullet> x \<le> a $ m} \<inter> (cbox a ?c \<inter> {x. ?mn \<bullet> x \<ge> b$m}))"
+      by (rule negligible_subset)
+    show "?f ` cbox a b \<union> cbox a ?c \<inter> {x. ?mn \<bullet> x \<le> a $ m} \<union> cbox a ?c \<inter> {x. ?mn \<bullet> x \<ge> b$m} = cbox a ?c" (is "?lhs = _")
+    proof
+      show "?lhs \<subseteq> cbox a ?c"
+        by (auto simp: mem_box_cart add_mono) (meson add_increasing2 an order_trans)
+      show "cbox a ?c \<subseteq> ?lhs"
+        apply (auto simp: algebra_simps image_iff inner_axis' lambda_add_Galois [OF \<open>m \<noteq> n\<close>])
+        apply (auto simp: mem_box_cart split: if_split_asm)
+        done
+    qed
+  qed (fact fab)
+  let ?d = "\<chi> i. if i = m then a $ m - b $ m else 0"
+  have eq2: "measure lebesgue (cbox a ?c \<inter> {x. ?mn \<bullet> x \<le> a $ m}) + measure lebesgue (cbox a ?c \<inter> {x. ?mn \<bullet> x \<ge> b$m})
+           = measure lebesgue (cbox a (\<chi> i. if i = m then a $ m + b $ n else b $ i))"
+  proof (rule measure_translate_add[of "cbox a ?c \<inter> {x. ?mn \<bullet> x \<le> a$m}" "cbox a ?c \<inter> {x. ?mn \<bullet> x \<ge> b$m}"
+     "(\<chi> i. if i = m then a$m - b$m else 0)" "cbox a (\<chi> i. if i = m then a$m + b$n else b$i)"])
+    show "(cbox a ?c \<inter> {x. ?mn \<bullet> x \<le> a$m}) \<in> lmeasurable"
+      "cbox a ?c \<inter> {x. ?mn \<bullet> x \<ge> b$m} \<in> lmeasurable"
+      by (auto simp: convex_Int convex_halfspace_le convex_halfspace_ge bounded_Int measurable_convex)
+    have "\<And>x. \<lbrakk>x $ n + a $ m \<le> x $ m\<rbrakk>
+         \<Longrightarrow> x \<in> (+) (\<chi> i. if i = m then a $ m - b $ m else 0) ` {x. x $ n + b $ m \<le> x $ m}"
+      using \<open>m \<noteq> n\<close>
+      by (rule_tac x="x - (\<chi> i. if i = m then a$m - b$m else 0)" in image_eqI)
+         (simp_all add: mem_box_cart)
+    then have imeq: "(+) ?d ` {x. b $ m \<le> ?mn \<bullet> x} = {x. a $ m \<le> ?mn \<bullet> x}"
+      using \<open>m \<noteq> n\<close> by (auto simp: mem_box_cart inner_axis' algebra_simps)
+    have "\<And>x. \<lbrakk>0 \<le> a $ n; x $ n + a $ m \<le> x $ m;
+                \<forall>i. i \<noteq> m \<longrightarrow> a $ i \<le> x $ i \<and> x $ i \<le> b $ i\<rbrakk>
+         \<Longrightarrow> a $ m \<le> x $ m"
+      using \<open>m \<noteq> n\<close>  by force
+    then have "(+) ?d ` (cbox a ?c \<inter> {x. b $ m \<le> ?mn \<bullet> x})
+            = cbox a (\<chi> i. if i = m then a $ m + b $ n else b $ i) \<inter> {x. a $ m \<le> ?mn \<bullet> x}"
+      using an ab_ne
+      apply (simp add: cbox_translation [symmetric] translation_Int interval_ne_empty_cart imeq)
+      apply (auto simp: mem_box_cart inner_axis' algebra_simps if_distrib all_if_distrib)
+      by (metis (full_types) add_mono mult_2_right)
+    then show "cbox a ?c \<inter> {x. ?mn \<bullet> x \<le> a $ m} \<union>
+          (+) ?d ` (cbox a ?c \<inter> {x. b $ m \<le> ?mn \<bullet> x}) =
+          cbox a (\<chi> i. if i = m then a $ m + b $ n else b $ i)"  (is "?lhs = ?rhs")
+      using an \<open>m \<noteq> n\<close>
+      apply (auto simp: mem_box_cart inner_axis' algebra_simps if_distrib all_if_distrib, force)
+        apply (drule_tac x=n in spec)+
+      by (meson ab_ne add_mono_thms_linordered_semiring(3) dual_order.trans interval_ne_empty_cart(1))
+    have "negligible{x. ?mn \<bullet> x = a$m}"
+      by (metis \<open>m \<noteq> n\<close> axis_index_axis eq_iff_diff_eq_0 negligible_hyperplane)
+    moreover have "(cbox a ?c \<inter> {x. ?mn \<bullet> x \<le> a $ m} \<inter>
+                                 (+) ?d ` (cbox a ?c \<inter> {x. b $ m \<le> ?mn \<bullet> x})) \<subseteq> {x. ?mn \<bullet> x = a$m}"
+      using \<open>m \<noteq> n\<close> antisym_conv by (fastforce simp: algebra_simps mem_box_cart inner_axis')
+    ultimately show "negligible (cbox a ?c \<inter> {x. ?mn \<bullet> x \<le> a $ m} \<inter>
+                                 (+) ?d ` (cbox a ?c \<inter> {x. b $ m \<le> ?mn \<bullet> x}))"
+      by (rule negligible_subset)
+  qed
+  have ac_ne: "cbox a ?c \<noteq> {}"
+    using ab_ne an
+    by (clarsimp simp: interval_eq_empty_cart) (meson add_less_same_cancel1 le_less_linear less_le_trans)
+  have ax_ne: "cbox a (\<chi> i. if i = m then a $ m + b $ n else b $ i) \<noteq> {}"
+    using ab_ne an
+    by (clarsimp simp: interval_eq_empty_cart) (meson add_less_same_cancel1 le_less_linear less_le_trans)
+  have eq3: "measure lebesgue (cbox a ?c) = measure lebesgue (cbox a (\<chi> i. if i = m then a$m + b$n else b$i)) + measure lebesgue (cbox a b)"
+    by (simp add: content_cbox_if_cart ab_ne ac_ne ax_ne algebra_simps prod.delta_remove
+             if_distrib [of "\<lambda>u. u - z" for z] prod.remove)
+  show ?Q
+    using eq1 eq2 eq3
+    by (simp add: algebra_simps)
+qed
+
+
+
+proposition
+  fixes S :: "(real^'n) set"
+  assumes "S \<in> lmeasurable"
+  shows measurable_stretch: "((\<lambda>x. \<chi> k. m k * x$k) ` S) \<in> lmeasurable" (is  "?f ` S \<in> _")
+    and measure_stretch: "measure lebesgue ((\<lambda>x. \<chi> k. m k * x$k) ` S) = \<bar>prod m UNIV\<bar> * measure lebesgue S"
+    (is "?MEQ")
+proof -
+  have "(?f ` S) \<in> lmeasurable \<and> ?MEQ"
+  proof (cases "\<forall>k. m k \<noteq> 0")
+    case True
+    have m0: "0 < \<bar>prod m UNIV\<bar>"
+      using True by simp
+    have "(indicat_real (?f ` S) has_integral \<bar>prod m UNIV\<bar> * measure lebesgue S) UNIV"
+    proof (clarsimp simp add: has_integral_alt [where i=UNIV])
+      fix e :: "real"
+      assume "e > 0"
+      have "(indicat_real S has_integral (measure lebesgue S)) UNIV"
+        using assms lmeasurable_iff_has_integral by blast
+      then obtain B where "B>0"
+        and B: "\<And>a b. ball 0 B \<subseteq> cbox a b \<Longrightarrow>
+                        \<exists>z. (indicat_real S has_integral z) (cbox a b) \<and>
+                            \<bar>z - measure lebesgue S\<bar> < e / \<bar>prod m UNIV\<bar>"
+        by (simp add: has_integral_alt [where i=UNIV]) (metis (full_types) divide_pos_pos m0  m0 \<open>e > 0\<close>)
+      show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
+                  (\<exists>z. (indicat_real (?f ` S) has_integral z) (cbox a b) \<and>
+                       \<bar>z - \<bar>prod m UNIV\<bar> * measure lebesgue S\<bar> < e)"
+      proof (intro exI conjI allI)
+        let ?C = "Max (range (\<lambda>k. \<bar>m k\<bar>)) * B"
+        show "?C > 0"
+          using True \<open>B > 0\<close> by (simp add: Max_gr_iff)
+        show "ball 0 ?C \<subseteq> cbox u v \<longrightarrow>
+                  (\<exists>z. (indicat_real (?f ` S) has_integral z) (cbox u v) \<and>
+                       \<bar>z - \<bar>prod m UNIV\<bar> * measure lebesgue S\<bar> < e)" for u v
+        proof
+          assume uv: "ball 0 ?C \<subseteq> cbox u v"
+          with \<open>?C > 0\<close> have cbox_ne: "cbox u v \<noteq> {}"
+            using centre_in_ball by blast
+          let ?\<alpha> = "\<lambda>k. u$k / m k"
+          let ?\<beta> = "\<lambda>k. v$k / m k"
+          have invm0: "\<And>k. inverse (m k) \<noteq> 0"
+            using True by auto
+          have "ball 0 B \<subseteq> (\<lambda>x. \<chi> k. x $ k / m k) ` ball 0 ?C"
+          proof clarsimp
+            fix x :: "(real, 'n) vec"
+            assume x: "norm x < B"
+            have [simp]: "\<bar>Max (range (\<lambda>k. \<bar>m k\<bar>))\<bar> = Max (range (\<lambda>k. \<bar>m k\<bar>))"
+              by (meson Max_ge abs_ge_zero abs_of_nonneg finite finite_imageI order_trans rangeI)
+            have "norm (\<chi> k. m k * x $ k) \<le> norm (Max (range (\<lambda>k. \<bar>m k\<bar>)) *\<^sub>R x)"
+              by (rule norm_le_componentwise_cart) (auto simp: abs_mult intro: mult_right_mono)
+            also have "\<dots> < ?C"
+              using x by simp (metis \<open>B > 0\<close> \<open>?C > 0\<close> mult.commute real_mult_less_iff1 zero_less_mult_pos)
+            finally have "norm (\<chi> k. m k * x $ k) < ?C" .
+            then show "x \<in> (\<lambda>x. \<chi> k. x $ k / m k) ` ball 0 ?C"
+              using stretch_Galois [of "inverse \<circ> m"] True by (auto simp: image_iff field_simps)
+          qed
+          then have Bsub: "ball 0 B \<subseteq> cbox (\<chi> k. min (?\<alpha> k) (?\<beta> k)) (\<chi> k. max (?\<alpha> k) (?\<beta> k))"
+            using cbox_ne uv image_stretch_interval_cart [of "inverse \<circ> m" u v, symmetric]
+            by (force simp: field_simps)
+          obtain z where zint: "(indicat_real S has_integral z) (cbox (\<chi> k. min (?\<alpha> k) (?\<beta> k)) (\<chi> k. max (?\<alpha> k) (?\<beta> k)))"
+                   and zless: "\<bar>z - measure lebesgue S\<bar> < e / \<bar>prod m UNIV\<bar>"
+            using B [OF Bsub] by blast
+          have ind: "indicat_real (?f ` S) = (\<lambda>x. indicator S (\<chi> k. x$k / m k))"
+            using True stretch_Galois [of m] by (force simp: indicator_def)
+          show "\<exists>z. (indicat_real (?f ` S) has_integral z) (cbox u v) \<and>
+                       \<bar>z - \<bar>prod m UNIV\<bar> * measure lebesgue S\<bar> < e"
+          proof (simp add: ind, intro conjI exI)
+            have "((\<lambda>x. indicat_real S (\<chi> k. x $ k/ m k)) has_integral z *\<^sub>R \<bar>prod m UNIV\<bar>)
+                ((\<lambda>x. \<chi> k. x $ k * m k) ` cbox (\<chi> k. min (?\<alpha> k) (?\<beta> k)) (\<chi> k. max (?\<alpha> k) (?\<beta> k)))"
+              using True has_integral_stretch_cart [OF zint, of "inverse \<circ> m"]
+              by (simp add: field_simps prod_dividef)
+            moreover have "((\<lambda>x. \<chi> k. x $ k * m k) ` cbox (\<chi> k. min (?\<alpha> k) (?\<beta> k)) (\<chi> k. max (?\<alpha> k) (?\<beta> k))) = cbox u v"
+              using True image_stretch_interval_cart [of "inverse \<circ> m" u v, symmetric]
+                image_stretch_interval_cart [of "\<lambda>k. 1" u v, symmetric] \<open>cbox u v \<noteq> {}\<close>
+              by (simp add: field_simps image_comp o_def)
+            ultimately show "((\<lambda>x. indicat_real S (\<chi> k. x $ k/ m k)) has_integral z *\<^sub>R \<bar>prod m UNIV\<bar>) (cbox u v)"
+              by simp
+            have "\<bar>z *\<^sub>R \<bar>prod m UNIV\<bar> - \<bar>prod m UNIV\<bar> * measure lebesgue S\<bar>
+                 = \<bar>prod m UNIV\<bar> * \<bar>z - measure lebesgue S\<bar>"
+              by (metis (no_types, hide_lams) abs_abs abs_scaleR mult.commute real_scaleR_def right_diff_distrib')
+            also have "\<dots> < e"
+              using zless True by (simp add: field_simps)
+            finally show "\<bar>z *\<^sub>R \<bar>prod m UNIV\<bar> - \<bar>prod m UNIV\<bar> * measure lebesgue S\<bar> < e" .
+          qed
+        qed
+      qed
+    qed
+    then show ?thesis
+      by (auto simp: has_integral_integrable integral_unique lmeasure_integral_UNIV measurable_integrable)
+  next
+    case False
+    then obtain k where "m k = 0" and prm: "prod m UNIV = 0"
+      by auto
+    have nfS: "negligible (?f ` S)"
+      by (rule negligible_subset [OF negligible_standard_hyperplane_cart]) (use \<open>m k = 0\<close> in auto)
+    then have "(?f ` S) \<in> lmeasurable"
+      by (simp add: negligible_iff_measure)
+    with nfS show ?thesis
+      by (simp add: prm negligible_iff_measure0)
+  qed
+  then show "(?f ` S) \<in> lmeasurable" ?MEQ
+    by metis+
+qed
+
+
+
+
+
+proposition
+ fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
+  assumes "linear f" "S \<in> lmeasurable"
+  shows measurable_linear_image: "(f ` S) \<in> lmeasurable"
+    and measure_linear_image: "measure lebesgue (f ` S) = \<bar>det (matrix f)\<bar> * measure lebesgue S" (is "?Q f S")
+proof -
+  have "\<forall>S \<in> lmeasurable. (f ` S) \<in> lmeasurable \<and> ?Q f S"
+  proof (rule induct_linear_elementary [OF \<open>linear f\<close>]; intro ballI)
+    fix f g and S :: "(real,'n) vec set"
+    assume "linear f" and "linear g"
+      and f [rule_format]: "\<forall>S \<in> lmeasurable. f ` S \<in> lmeasurable \<and> ?Q f S"
+      and g [rule_format]: "\<forall>S \<in> lmeasurable. g ` S \<in> lmeasurable \<and> ?Q g S"
+      and S: "S \<in> lmeasurable"
+    then have gS: "g ` S \<in> lmeasurable"
+      by blast
+    show "(f \<circ> g) ` S \<in> lmeasurable \<and> ?Q (f \<circ> g) S"
+      using f [OF gS] g [OF S] matrix_compose [OF \<open>linear g\<close> \<open>linear f\<close>]
+      by (simp add: o_def image_comp abs_mult det_mul)
+  next
+    fix f :: "(real, 'n) vec \<Rightarrow> (real, 'n) vec" and i and S :: "(real, 'n) vec set"
+    assume "linear f" and 0: "\<And>x. f x $ i = 0" and "S \<in> lmeasurable"
+    then have "\<not> inj f"
+      by (metis (full_types) linear_injective_imp_surjective one_neq_zero surjE vec_component)
+    have detf: "det (matrix f) = 0"
+      by (metis "0" \<open>linear f\<close> invertible_det_nz invertible_right_inverse matrix_right_invertible_surjective matrix_vector_mul surjE vec_component)
+    show "f ` S \<in> lmeasurable \<and> ?Q f S"
+    proof
+      show "f ` S \<in> lmeasurable"
+        using lmeasurable_iff_indicator_has_integral \<open>linear f\<close> \<open>\<not> inj f\<close> negligible_UNIV negligible_linear_singular_image by blast
+      have "measure lebesgue (f ` S) = 0"
+        by (meson \<open>\<not> inj f\<close> \<open>linear f\<close> negligible_imp_measure0 negligible_linear_singular_image)
+      also have "\<dots> = \<bar>det (matrix f)\<bar> * measure lebesgue S"
+        by (simp add: detf)
+      finally show "?Q f S" .
+    qed
+  next
+    fix c and S :: "(real, 'n) vec set"
+    assume "S \<in> lmeasurable"
+    show "(\<lambda>a. \<chi> i. c i * a $ i) ` S \<in> lmeasurable \<and> ?Q (\<lambda>a. \<chi> i. c i * a $ i) S"
+    proof
+      show "(\<lambda>a. \<chi> i. c i * a $ i) ` S \<in> lmeasurable"
+        by (simp add: \<open>S \<in> lmeasurable\<close> measurable_stretch)
+      show "?Q (\<lambda>a. \<chi> i. c i * a $ i) S"
+        by (simp add: measure_stretch [OF \<open>S \<in> lmeasurable\<close>, of c] axis_def matrix_def det_diagonal)
+    qed
+  next
+    fix m :: "'n" and n :: "'n" and S :: "(real, 'n) vec set"
+    assume "m \<noteq> n" and "S \<in> lmeasurable"
+    let ?h = "\<lambda>v::(real, 'n) vec. \<chi> i. v $ Fun.swap m n id i"
+    have lin: "linear ?h"
+      by (simp add: plus_vec_def scaleR_vec_def linearI)
+    have meq: "measure lebesgue ((\<lambda>v::(real, 'n) vec. \<chi> i. v $ Fun.swap m n id i) ` cbox a b)
+             = measure lebesgue (cbox a b)" for a b
+    proof (cases "cbox a b = {}")
+      case True then show ?thesis
+        by simp
+    next
+      case False
+      then have him: "?h ` (cbox a b) \<noteq> {}"
+        by blast
+      have eq: "?h ` (cbox a b) = cbox (?h a) (?h b)"
+        by (auto simp: image_iff lambda_swap_Galois mem_box_cart) (metis swap_id_eq)+
+      show ?thesis
+        using him prod.permute [OF permutes_swap_id, where S=UNIV and g="\<lambda>i. (b - a)$i", symmetric]
+        by (simp add: eq content_cbox_cart False)
+    qed
+    have "(\<chi> i j. if Fun.swap m n id i = j then 1 else 0) = (\<chi> i j. if j = Fun.swap m n id i then 1 else (0::real))"
+      by (auto intro!: Cart_lambda_cong)
+    then have "matrix ?h = transpose(\<chi> i j. mat 1 $ i $ Fun.swap m n id j)"
+      by (auto simp: matrix_eq transpose_def axis_def mat_def matrix_def)
+    then have 1: "\<bar>det (matrix ?h)\<bar> = 1"
+      by (simp add: det_permute_columns permutes_swap_id sign_swap_id abs_mult)
+    show "?h ` S \<in> lmeasurable \<and> ?Q ?h S"
+    proof
+      show "?h ` S \<in> lmeasurable" "?Q ?h S"
+        using measure_linear_sufficient [OF lin \<open>S \<in> lmeasurable\<close>] meq 1 by force+
+    qed
+  next
+    fix m :: "'n" and n :: "'n" and S :: "(real, 'n) vec set"
+    assume "m \<noteq> n" and "S \<in> lmeasurable"
+    let ?h = "\<lambda>v::(real, 'n) vec. \<chi> i. if i = m then v $ m + v $ n else v $ i"
+    have lin: "linear ?h"
+      by (auto simp: algebra_simps plus_vec_def scaleR_vec_def vec_eq_iff intro: linearI)
+    consider "m < n" | " n < m"
+      using \<open>m \<noteq> n\<close> less_linear by blast
+    then have 1: "det(matrix ?h) = 1"
+    proof cases
+      assume "m < n"
+      have *: "matrix ?h $ i $ j = (0::real)" if "j < i" for i j :: 'n
+      proof -
+        have "axis j 1 = (\<chi> n. if n = j then 1 else (0::real))"
+          using axis_def by blast
+        then have "(\<chi> p q. if p = m then axis q 1 $ m + axis q 1 $ n else axis q 1 $ p) $ i $ j = (0::real)"
+          using \<open>j < i\<close> axis_def \<open>m < n\<close> by auto
+        with \<open>m < n\<close> show ?thesis
+          by (auto simp: matrix_def axis_def cong: if_cong)
+      qed
+      show ?thesis
+        using \<open>m \<noteq> n\<close> by (subst det_upperdiagonal [OF *]) (auto simp: matrix_def axis_def cong: if_cong)
+    next
+      assume "n < m"
+      have *: "matrix ?h $ i $ j = (0::real)" if "j > i" for i j :: 'n
+      proof -
+        have "axis j 1 = (\<chi> n. if n = j then 1 else (0::real))"
+          using axis_def by blast
+        then have "(\<chi> p q. if p = m then axis q 1 $ m + axis q 1 $ n else axis q 1 $ p) $ i $ j = (0::real)"
+          using \<open>j > i\<close> axis_def \<open>m > n\<close> by auto
+        with \<open>m > n\<close> show ?thesis
+          by (auto simp: matrix_def axis_def cong: if_cong)
+      qed
+      show ?thesis
+        using \<open>m \<noteq> n\<close>
+        by (subst det_lowerdiagonal [OF *]) (auto simp: matrix_def axis_def cong: if_cong)
+    qed
+    have meq: "measure lebesgue (?h ` (cbox a b)) = measure lebesgue (cbox a b)" for a b
+    proof (cases "cbox a b = {}")
+      case True then show ?thesis by simp
+    next
+      case False
+      then have ne: "(+) (\<chi> i. if i = n then - a $ n else 0) ` cbox a b \<noteq> {}"
+        by auto
+      let ?v = "\<chi> i. if i = n then - a $ n else 0"
+      have "?h ` cbox a b
+            = (+) (\<chi> i. if i = m \<or> i = n then a $ n else 0) ` ?h ` (+) ?v ` (cbox a b)"
+        using \<open>m \<noteq> n\<close> unfolding image_comp o_def by (force simp: vec_eq_iff)
+      then have "measure lebesgue (?h ` (cbox a b))
+               = measure lebesgue ((\<lambda>v. \<chi> i. if i = m then v $ m + v $ n else v $ i) `
+                                   (+) ?v ` cbox a b)"
+        by (rule ssubst) (rule measure_translation)
+      also have "\<dots> = measure lebesgue ((\<lambda>v. \<chi> i. if i = m then v $ m + v $ n else v $ i) ` cbox (?v +a) (?v + b))"
+        by (metis (no_types, lifting) cbox_translation)
+      also have "\<dots> = measure lebesgue ((+) (\<chi> i. if i = n then - a $ n else 0) ` cbox a b)"
+        apply (subst measure_shear_interval)
+        using \<open>m \<noteq> n\<close> ne apply auto
+        apply (simp add: cbox_translation)
+        by (metis cbox_borel cbox_translation measure_completion sets_lborel)
+      also have "\<dots> = measure lebesgue (cbox a b)"
+        by (rule measure_translation)
+        finally show ?thesis .
+      qed
+    show "?h ` S \<in> lmeasurable \<and> ?Q ?h S"
+      using measure_linear_sufficient [OF lin \<open>S \<in> lmeasurable\<close>] meq 1 by force
+  qed
+  with assms show "(f ` S) \<in> lmeasurable" "?Q f S"
+    by metis+
+qed
+
+
+lemma
+ fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
+  assumes f: "orthogonal_transformation f" and S: "S \<in> lmeasurable"
+  shows measurable_orthogonal_image: "f ` S \<in> lmeasurable"
+    and measure_orthogonal_image: "measure lebesgue (f ` S) = measure lebesgue S"
+proof -
+  have "linear f"
+    by (simp add: f orthogonal_transformation_linear)
+  then show "f ` S \<in> lmeasurable"
+    by (metis S measurable_linear_image)
+  show "measure lebesgue (f ` S) = measure lebesgue S"
+    by (simp add: measure_linear_image \<open>linear f\<close> S f)
+qed
+
+lemma sets_lebesgue_inner_closed:
+  assumes "S \<in> sets lebesgue" "e > 0"
+  obtains T where "closed T" "T \<subseteq> S" "S-T \<in> lmeasurable" "measure lebesgue (S - T) < e"
+proof -
+  have "-S \<in> sets lebesgue"
+    using assms by (simp add: Compl_in_sets_lebesgue)
+  then obtain T where "open T" "-S \<subseteq> T"
+          and T: "(T - -S) \<in> lmeasurable" "measure lebesgue (T - -S) < e"
+    using lmeasurable_outer_open assms  by blast
+  show thesis
+  proof
+    show "closed (-T)"
+      using \<open>open T\<close> by blast
+    show "-T \<subseteq> S"
+      using \<open>- S \<subseteq> T\<close> by auto
+    show "S - ( -T) \<in> lmeasurable" "measure lebesgue (S - (- T)) < e"
+      using T by (auto simp: Int_commute)
+  qed
+qed
+
+subsection\<open>@{text F_sigma} and @{text G_delta} sets.\<close>
+
+(*https://en.wikipedia.org/wiki/F\<sigma>_set*)
+inductive fsigma :: "'a::topological_space set \<Rightarrow> bool" where
+  "(\<And>n::nat. closed (F n)) \<Longrightarrow> fsigma (UNION UNIV F)"
+
+inductive gdelta :: "'a::topological_space set \<Rightarrow> bool" where
+  "(\<And>n::nat. open (F n)) \<Longrightarrow> gdelta (INTER UNIV F)"
+
+
+lemma fsigma_Union_compact:
+  fixes S :: "'a::{real_normed_vector,heine_borel} set"
+  shows "fsigma S \<longleftrightarrow> (\<exists>F::nat \<Rightarrow> 'a set. range F \<subseteq> Collect compact \<and> S = UNION UNIV F)"
+proof safe
+  assume "fsigma S"
+  then obtain F :: "nat \<Rightarrow> 'a set" where F: "range F \<subseteq> Collect closed" "S = UNION UNIV F"
+    by (meson fsigma.cases image_subsetI mem_Collect_eq)
+  then have "\<exists>D::nat \<Rightarrow> 'a set. range D \<subseteq> Collect compact \<and> UNION UNIV D = F i" for i
+    using closed_Union_compact_subsets [of "F i"]
+    by (metis image_subsetI mem_Collect_eq range_subsetD)
+  then obtain D :: "nat \<Rightarrow> nat \<Rightarrow> 'a set"
+    where D: "\<And>i. range (D i) \<subseteq> Collect compact \<and> UNION UNIV (D i) = F i"
+    by metis
+  let ?DD = "\<lambda>n. (\<lambda>(i,j). D i j) (prod_decode n)"
+  show "\<exists>F::nat \<Rightarrow> 'a set. range F \<subseteq> Collect compact \<and> S = UNION UNIV F"
+  proof (intro exI conjI)
+    show "range ?DD \<subseteq> Collect compact"
+      using D by clarsimp (metis mem_Collect_eq rangeI split_conv subsetCE surj_pair)
+    show "S = UNION UNIV ?DD"
+    proof
+      show "S \<subseteq> UNION UNIV ?DD"
+        using D F
+        by clarsimp (metis UN_iff old.prod.case prod_decode_inverse prod_encode_eq)
+      show "UNION UNIV ?DD \<subseteq> S"
+        using D F  by fastforce
+    qed
+  qed
+next
+  fix F :: "nat \<Rightarrow> 'a set"
+  assume "range F \<subseteq> Collect compact" and "S = UNION UNIV F"
+  then show "fsigma (UNION UNIV F)"
+    by (simp add: compact_imp_closed fsigma.intros image_subset_iff)
+qed
+
+lemma gdelta_imp_fsigma: "gdelta S \<Longrightarrow> fsigma (- S)"
+proof (induction rule: gdelta.induct)
+  case (1 F)
+  have "- INTER UNIV F = (\<Union>i. -(F i))"
+    by auto
+  then show ?case
+    by (simp add: fsigma.intros closed_Compl 1)
+qed
+
+lemma fsigma_imp_gdelta: "fsigma S \<Longrightarrow> gdelta (- S)"
+proof (induction rule: fsigma.induct)
+  case (1 F)
+  have "- UNION UNIV F = (\<Inter>i. -(F i))"
+    by auto
+  then show ?case
+    by (simp add: 1 gdelta.intros open_closed)
+qed
+
+
+
+lemma gdelta_complement: "gdelta(- S) \<longleftrightarrow> fsigma S"
+  using fsigma_imp_gdelta gdelta_imp_fsigma by force
+
+
+text\<open>A Lebesgue set is almost an @{text F_sigma} or @{text G_delta}.\<close>
+lemma lebesgue_set_almost_fsigma:
+  assumes "S \<in> sets lebesgue"
+  obtains C T where "fsigma C" "negligible T" "C \<union> T = S" "disjnt C T"
+proof -
+  { fix n::nat
+    have "\<exists>T. closed T \<and> T \<subseteq> S \<and> S - T \<in> lmeasurable \<and> measure lebesgue (S-T) < 1 / Suc n"
+      using sets_lebesgue_inner_closed [OF assms]
+      by (metis divide_pos_pos less_numeral_extra(1) of_nat_0_less_iff zero_less_Suc)
+  }
+  then obtain F where F: "\<And>n::nat. closed (F n) \<and> F n \<subseteq> S \<and> S - F n \<in> lmeasurable \<and> measure lebesgue (S - F n) < 1 / Suc n"
+    by metis
+  let ?C = "UNION UNIV F"
+  show thesis
+  proof
+    show "fsigma ?C"
+      using F by (simp add: fsigma.intros)
+    show "negligible (S - ?C)"
+    proof (clarsimp simp add: negligible_outer_le)
+      fix e :: "real"
+      assume "0 < e"
+      then obtain n where n: "1 / Suc n < e"
+        using nat_approx_posE by metis
+      show "\<exists>T. S - (\<Union>x. F x) \<subseteq> T \<and> T \<in> lmeasurable \<and> measure lebesgue T \<le> e"
+      proof (intro exI conjI)
+        show "measure lebesgue (S - F n) \<le> e"
+          by (meson F n less_trans not_le order.asym)
+      qed (use F in auto)
+    qed
+    show "?C \<union> (S - ?C) = S"
+      using F by blast
+    show "disjnt ?C (S - ?C)"
+      by (auto simp: disjnt_def)
+  qed
+qed
+
+lemma lebesgue_set_almost_gdelta:
+  assumes "S \<in> sets lebesgue"
+  obtains C T where "gdelta C" "negligible T" "S \<union> T = C" "disjnt S T"
+proof -
+  have "-S \<in> sets lebesgue"
+    using assms Compl_in_sets_lebesgue by blast
+  then obtain C T where C: "fsigma C" "negligible T" "C \<union> T = -S" "disjnt C T"
+    using lebesgue_set_almost_fsigma by metis
+  show thesis
+  proof
+    show "gdelta (-C)"
+      by (simp add: \<open>fsigma C\<close> fsigma_imp_gdelta)
+    show "S \<union> T = -C" "disjnt S T"
+      using C by (auto simp: disjnt_def)
+  qed (use C in auto)
+qed
+
+
+proposition measure_semicontinuous_with_hausdist_explicit:
+  assumes "bounded S" and neg: "negligible(frontier S)" and "e > 0"
+  obtains d where "d > 0"
+                  "\<And>T. \<lbrakk>T \<in> lmeasurable; \<And>y. y \<in> T \<Longrightarrow> \<exists>x. x \<in> S \<and> dist x y < d\<rbrakk>
+                        \<Longrightarrow> measure lebesgue T < measure lebesgue S + e"
+proof (cases "S = {}")
+  case True
+  with that \<open>e > 0\<close> show ?thesis by force
+next
+  case False
+  then have frS: "frontier S \<noteq> {}"
+    using \<open>bounded S\<close> frontier_eq_empty not_bounded_UNIV by blast
+  have "S \<in> lmeasurable"
+    by (simp add: \<open>bounded S\<close> measurable_Jordan neg)
+  have null: "(frontier S) \<in> null_sets lebesgue"
+    by (metis neg negligible_iff_null_sets)
+  have "frontier S \<in> lmeasurable" and mS0: "measure lebesgue (frontier S) = 0"
+    using neg negligible_imp_measurable negligible_iff_measure by blast+
+  with \<open>e > 0\<close> lmeasurable_outer_open
+  obtain U where "open U"
+    and U: "frontier S \<subseteq> U" "U - frontier S \<in> lmeasurable" "measure lebesgue (U - frontier S) < e"
+    by (metis fmeasurableD)
+  with null have "U \<in> lmeasurable"
+    by (metis borel_open measurable_Diff_null_set sets_completionI_sets sets_lborel)
+  have "measure lebesgue (U - frontier S) = measure lebesgue U"
+    using mS0 by (simp add: \<open>U \<in> lmeasurable\<close> fmeasurableD measure_Diff_null_set null)
+  with U have mU: "measure lebesgue U < e"
+    by simp
+  show ?thesis
+  proof
+    have "U \<noteq> UNIV"
+      using \<open>U \<in> lmeasurable\<close> by auto
+    then have "- U \<noteq> {}"
+      by blast
+    with \<open>open U\<close> \<open>frontier S \<subseteq> U\<close> show "setdist (frontier S) (- U) > 0"
+      by (auto simp: \<open>bounded S\<close> open_closed compact_frontier_bounded setdist_gt_0_compact_closed frS)
+    fix T
+    assume "T \<in> lmeasurable"
+      and T: "\<And>t. t \<in> T \<Longrightarrow> \<exists>y. y \<in> S \<and> dist y t < setdist (frontier S) (- U)"
+    then have "measure lebesgue T - measure lebesgue S \<le> measure lebesgue (T - S)"
+      by (simp add: \<open>S \<in> lmeasurable\<close> measure_diff_le_measure_setdiff)
+    also have "\<dots>  \<le> measure lebesgue U"
+    proof -
+      have "T - S \<subseteq> U"
+      proof clarify
+        fix x
+        assume "x \<in> T" and "x \<notin> S"
+        then obtain y where "y \<in> S" and y: "dist y x < setdist (frontier S) (- U)"
+          using T by blast
+        have "closed_segment x y \<inter> frontier S \<noteq> {}"
+          using connected_Int_frontier \<open>x \<notin> S\<close> \<open>y \<in> S\<close> by blast
+        then obtain z where z: "z \<in> closed_segment x y" "z \<in> frontier S"
+          by auto
+        with y have "dist z x < setdist(frontier S) (- U)"
+          by (auto simp: dist_commute dest!: dist_in_closed_segment)
+        with z have False if "x \<in> -U"
+          using setdist_le_dist [OF \<open>z \<in> frontier S\<close> that] by auto
+        then show "x \<in> U"
+          by blast
+      qed
+      then show ?thesis
+        by (simp add: \<open>S \<in> lmeasurable\<close> \<open>T \<in> lmeasurable\<close> \<open>U \<in> lmeasurable\<close> fmeasurableD measure_mono_fmeasurable sets.Diff)
+    qed
+    finally have "measure lebesgue T - measure lebesgue S \<le> measure lebesgue U" .
+    with mU show "measure lebesgue T < measure lebesgue S + e"
+      by linarith
+  qed
+qed
+
+proposition lebesgue_regular_inner:
+ assumes "S \<in> sets lebesgue"
+ obtains K C where "negligible K" "\<And>n::nat. compact(C n)" "S = (\<Union>n. C n) \<union> K"
+proof -
+  have "\<exists>T. closed T \<and> T \<subseteq> S \<and> (S - T) \<in> lmeasurable \<and> measure lebesgue (S - T) < (1/2)^n" for n
+    using sets_lebesgue_inner_closed assms
+    by (metis sets_lebesgue_inner_closed zero_less_divide_1_iff zero_less_numeral zero_less_power)
+  then obtain C where clo: "\<And>n. closed (C n)" and subS: "\<And>n. C n \<subseteq> S"
+    and mea: "\<And>n. (S - C n) \<in> lmeasurable"
+    and less: "\<And>n. measure lebesgue (S - C n) < (1/2)^n"
+    by metis
+  have "\<exists>F. (\<forall>n::nat. compact(F n)) \<and> (\<Union>n. F n) = C m" for m::nat
+    by (metis clo closed_Union_compact_subsets)
+  then obtain D :: "[nat,nat] \<Rightarrow> 'a set" where D: "\<And>m n. compact(D m n)" "\<And>m. (\<Union>n. D m n) = C m"
+    by metis
+  let ?C = "from_nat_into (\<Union>m. range (D m))"
+  have "countable (\<Union>m. range (D m))"
+    by blast
+  have "range (from_nat_into (\<Union>m. range (D m))) = (\<Union>m. range (D m))"
+    using range_from_nat_into by simp
+  then have CD: "\<exists>m n. ?C k = D m n"  for k
+    by (metis (mono_tags, lifting) UN_iff rangeE range_eqI)
+  show thesis
+  proof
+    show "negligible (S - (\<Union>n. C n))"
+    proof (clarsimp simp: negligible_outer_le)
+      fix e :: "real"
+      assume "e > 0"
+      then obtain n where n: "(1/2)^n < e"
+        using real_arch_pow_inv [of e "1/2"] by auto
+      show "\<exists>T. S - (\<Union>n. C n) \<subseteq> T \<and> T \<in> lmeasurable \<and> measure lebesgue T \<le> e"
+      proof (intro exI conjI)
+        show "S - (\<Union>n. C n) \<subseteq> S - C n"
+          by blast
+        show "S - C n \<in> lmeasurable"
+          by (simp add: mea)
+        show "measure lebesgue (S - C n) \<le> e"
+          using less [of n] n by simp
+      qed
+    qed
+    show "compact (?C n)" for n
+      using CD D by metis
+    show "S = (\<Union>n. ?C n) \<union> (S - (\<Union>n. C n))" (is "_ = ?rhs")
+    proof
+      show "S \<subseteq> ?rhs"
+        using D by fastforce
+      show "?rhs \<subseteq> S"
+        using subS D CD by auto (metis Sup_upper range_eqI subsetCE)
+    qed
+  qed
+qed
+
+
+lemma sets_lebesgue_continuous_image:
+  assumes T: "T \<in> sets lebesgue" and contf: "continuous_on S f"
+    and negim: "\<And>T. \<lbrakk>negligible T; T \<subseteq> S\<rbrakk> \<Longrightarrow> negligible(f ` T)" and "T \<subseteq> S"
+ shows "f ` T \<in> sets lebesgue"
+proof -
+  obtain K C where "negligible K" and com: "\<And>n::nat. compact(C n)" and Teq: "T = (\<Union>n. C n) \<union> K"
+    using lebesgue_regular_inner [OF T] by metis
+  then have comf: "\<And>n::nat. compact(f ` C n)"
+    by (metis Un_subset_iff Union_upper \<open>T \<subseteq> S\<close> compact_continuous_image contf continuous_on_subset rangeI)
+  have "((\<Union>n. f ` C n) \<union> f ` K) \<in> sets lebesgue"
+  proof (rule sets.Un)
+    have "K \<subseteq> S"
+      using Teq \<open>T \<subseteq> S\<close> by blast
+    show "(\<Union>n. f ` C n) \<in> sets lebesgue"
+    proof (rule sets.countable_Union)
+      show "range (\<lambda>n. f ` C n) \<subseteq> sets lebesgue"
+        using borel_compact comf by (auto simp: borel_compact)
+    qed auto
+    show "f ` K \<in> sets lebesgue"
+      by (simp add: \<open>K \<subseteq> S\<close> \<open>negligible K\<close> negim negligible_imp_sets)
+  qed
+  then show ?thesis
+    by (simp add: Teq image_Un image_Union)
+qed
+
+lemma differentiable_image_in_sets_lebesgue:
+  fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
+  assumes S: "S \<in> sets lebesgue" and dim: "DIM('m) \<le> DIM('n)" and f: "f differentiable_on S"
+  shows "f`S \<in> sets lebesgue"
+proof (rule sets_lebesgue_continuous_image [OF S])
+  show "continuous_on S f"
+    by (meson differentiable_imp_continuous_on f)
+  show "\<And>T. \<lbrakk>negligible T; T \<subseteq> S\<rbrakk> \<Longrightarrow> negligible (f ` T)"
+    using differentiable_on_subset f
+    by (auto simp: intro!: negligible_differentiable_image_negligible [OF dim])
+qed auto
+
+lemma sets_lebesgue_on_continuous_image:
+  assumes S: "S \<in> sets lebesgue" and X: "X \<in> sets (lebesgue_on S)" and contf: "continuous_on S f"
+    and negim: "\<And>T. \<lbrakk>negligible T; T \<subseteq> S\<rbrakk> \<Longrightarrow> negligible(f ` T)"
+  shows "f ` X \<in> sets (lebesgue_on (f ` S))"
+proof -
+  have "X \<subseteq> S"
+    by (metis S X sets.Int_space_eq2 sets_restrict_space_iff)
+  moreover have "f ` S \<in> sets lebesgue"
+    using S contf negim sets_lebesgue_continuous_image by blast
+  moreover have "f ` X \<in> sets lebesgue"
+    by (metis S X contf negim sets_lebesgue_continuous_image sets_restrict_space_iff space_restrict_space space_restrict_space2)
+  ultimately show ?thesis
+    by (auto simp: sets_restrict_space_iff)
+qed
+
+lemma differentiable_image_in_sets_lebesgue_on:
+  fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
+  assumes S: "S \<in> sets lebesgue" and X: "X \<in> sets (lebesgue_on S)" and dim: "DIM('m) \<le> DIM('n)"
+       and f: "f differentiable_on S"
+     shows "f ` X \<in> sets (lebesgue_on (f`S))"
+proof (rule sets_lebesgue_on_continuous_image [OF S X])
+  show "continuous_on S f"
+    by (meson differentiable_imp_continuous_on f)
+  show "\<And>T. \<lbrakk>negligible T; T \<subseteq> S\<rbrakk> \<Longrightarrow> negligible (f ` T)"
+    using differentiable_on_subset f
+    by (auto simp: intro!: negligible_differentiable_image_negligible [OF dim])
+qed
+
+
+proposition
+ fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
+  assumes S: "S \<in> lmeasurable"
+  and deriv: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
+  and int: "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) integrable_on S"
+  and bounded: "\<And>x. x \<in> S \<Longrightarrow> \<bar>det (matrix (f' x))\<bar> \<le> B"
+  shows measurable_bounded_differentiable_image:
+       "f ` S \<in> lmeasurable"
+    and measure_bounded_differentiable_image:
+       "measure lebesgue (f ` S) \<le> B * measure lebesgue S" (is "?M")
+proof -
+  have "f ` S \<in> lmeasurable \<and> measure lebesgue (f ` S) \<le> B * measure lebesgue S"
+  proof (cases "B < 0")
+    case True
+    then have "S = {}"
+      by (meson abs_ge_zero bounded empty_iff equalityI less_le_trans linorder_not_less subsetI)
+    then show ?thesis
+      by auto
+  next
+    case False
+    then have "B \<ge> 0"
+      by arith
+    let ?\<mu> = "measure lebesgue"
+    have f_diff: "f differentiable_on S"
+      using deriv by (auto simp: differentiable_on_def differentiable_def)
+    have eps: "f ` S \<in> lmeasurable" "?\<mu> (f ` S) \<le> (B+e) * ?\<mu> S" (is "?ME")
+              if "e > 0" for e
+    proof -
+      have eps_d: "f ` S \<in> lmeasurable"  "?\<mu> (f ` S) \<le> (B+e) * (?\<mu> S + d)" (is "?MD")
+                  if "d > 0" for d
+      proof -
+        obtain T where "open T" "S \<subseteq> T" and TS: "(T-S) \<in> lmeasurable" and "?\<mu> (T-S) < d"
+          using S \<open>d > 0\<close> lmeasurable_outer_open by blast
+        with S have "T \<in> lmeasurable" and Tless: "?\<mu> T < ?\<mu> S + d"
+          by (auto simp: measurable_measure_Diff dest!: fmeasurable_Diff_D)
+        have "\<exists>r. 0 < r \<and> r < d \<and> ball x r \<subseteq> T \<and> f ` (S \<inter> ball x r) \<in> lmeasurable \<and>
+                  ?\<mu> (f ` (S \<inter> ball x r)) \<le> (B + e) * ?\<mu> (ball x r)"
+          if "x \<in> S" "d > 0" for x d
+        proof -
+          have lin: "linear (f' x)"
+            and lim0: "((\<lambda>y. (f y - (f x + f' x (y - x))) /\<^sub>R norm(y - x)) \<longlongrightarrow> 0) (at x within S)"
+            using deriv \<open>x \<in> S\<close> by (auto simp: has_derivative_within bounded_linear.linear field_simps)
+          have bo: "bounded (f' x ` ball 0 1)"
+            by (simp add: bounded_linear_image linear_linear lin)
+          have neg: "negligible (frontier (f' x ` ball 0 1))"
+            using deriv has_derivative_linear \<open>x \<in> S\<close>
+            by (auto intro!: negligible_convex_frontier [OF convex_linear_image])
+          have 0: "0 < e * unit_ball_vol (real CARD('n))"
+            using  \<open>e > 0\<close> by simp
+          obtain k where "k > 0" and k:
+                  "\<And>U. \<lbrakk>U \<in> lmeasurable; \<And>y. y \<in> U \<Longrightarrow> \<exists>z. z \<in> f' x ` ball 0 1 \<and> dist z y < k\<rbrakk>
+                        \<Longrightarrow> ?\<mu> U < ?\<mu> (f' x ` ball 0 1) + e * unit_ball_vol (CARD('n))"
+            using measure_semicontinuous_with_hausdist_explicit [OF bo neg 0] by blast
+          obtain l where "l > 0" and l: "ball x l \<subseteq> T"
+            using \<open>x \<in> S\<close> \<open>open T\<close> \<open>S \<subseteq> T\<close> openE by blast
+          obtain \<zeta> where "0 < \<zeta>"
+            and \<zeta>: "\<And>y. \<lbrakk>y \<in> S; y \<noteq> x; dist y x < \<zeta>\<rbrakk>
+                        \<Longrightarrow> norm (f y - (f x + f' x (y - x))) / norm (y - x) < k"
+            using lim0 \<open>k > 0\<close> by (force simp: Lim_within field_simps)
+          define r where "r \<equiv> min (min l (\<zeta>/2)) (min 1 (d/2))"
+          show ?thesis
+          proof (intro exI conjI)
+            show "r > 0" "r < d"
+              using \<open>l > 0\<close> \<open>\<zeta> > 0\<close> \<open>d > 0\<close> by (auto simp: r_def)
+            have "r \<le> l"
+              by (auto simp: r_def)
+            with l show "ball x r \<subseteq> T"
+              by auto
+            have ex_lessK: "\<exists>x' \<in> ball 0 1. dist (f' x x') ((f y - f x) /\<^sub>R r) < k"
+              if "y \<in> S" and "dist x y < r" for y
+            proof (cases "y = x")
+              case True
+              with lin linear_0 \<open>k > 0\<close> that show ?thesis
+                by (rule_tac x=0 in bexI) (auto simp: linear_0)
+            next
+              case False
+              then show ?thesis
+              proof (rule_tac x="(y - x) /\<^sub>R r" in bexI)
+                have "f' x ((y - x) /\<^sub>R r) = f' x (y - x) /\<^sub>R r"
+                  by (simp add: lin linear_cmul)
+                then have "dist (f' x ((y - x) /\<^sub>R r)) ((f y - f x) /\<^sub>R r) = norm (f' x (y - x) /\<^sub>R r - (f y - f x) /\<^sub>R r)"
+                  by (simp add: dist_norm)
+                also have "\<dots> = norm (f' x (y - x) - (f y - f x)) / r"
+                  using \<open>r > 0\<close> by (simp add: real_vector.scale_right_diff_distrib [symmetric] divide_simps)
+                also have "\<dots> \<le> norm (f y - (f x + f' x (y - x))) / norm (y - x)"
+                  using that \<open>r > 0\<close> False by (simp add: algebra_simps divide_simps dist_norm norm_minus_commute mult_right_mono)
+                also have "\<dots> < k"
+                  using that \<open>0 < \<zeta>\<close> by (simp add: dist_commute r_def  \<zeta> [OF \<open>y \<in> S\<close> False])
+                finally show "dist (f' x ((y - x) /\<^sub>R r)) ((f y - f x) /\<^sub>R r) < k" .
+                show "(y - x) /\<^sub>R r \<in> ball 0 1"
+                  using that \<open>r > 0\<close> by (simp add: dist_norm divide_simps norm_minus_commute)
+              qed
+            qed
+            let ?rfs = "(\<lambda>x. x /\<^sub>R r) ` (+) (- f x) ` f ` (S \<inter> ball x r)"
+            have rfs_mble: "?rfs \<in> lmeasurable"
+            proof (rule bounded_set_imp_lmeasurable)
+              have "f differentiable_on S \<inter> ball x r"
+                using f_diff by (auto simp: fmeasurableD differentiable_on_subset)
+              with S show "?rfs \<in> sets lebesgue"
+                by (auto simp: sets.Int intro!: lebesgue_sets_translation differentiable_image_in_sets_lebesgue)
+              let ?B = "(\<lambda>(x, y). x + y) ` (f' x ` ball 0 1 \<times> ball 0 k)"
+              have "bounded ?B"
+                by (simp add: bounded_plus [OF bo])
+              moreover have "?rfs \<subseteq> ?B"
+                apply (auto simp: dist_norm image_iff dest!: ex_lessK)
+                by (metis (no_types, hide_lams) add.commute diff_add_cancel dist_0_norm dist_commute dist_norm mem_ball)
+              ultimately show "bounded (?rfs)"
+                by (rule bounded_subset)
+            qed
+            then have "(\<lambda>x. r *\<^sub>R x) ` ?rfs \<in> lmeasurable"
+              by (simp add: measurable_linear_image)
+            with \<open>r > 0\<close> have "(+) (- f x) ` f ` (S \<inter> ball x r) \<in> lmeasurable"
+              by (simp add: image_comp o_def)
+            then have "(+) (f x) ` (+) (- f x) ` f ` (S \<inter> ball x r) \<in> lmeasurable"
+              using  measurable_translation by blast
+            then show fsb: "f ` (S \<inter> ball x r) \<in> lmeasurable"
+              by (simp add: image_comp o_def)
+            have "?\<mu> (f ` (S \<inter> ball x r)) = ?\<mu> (?rfs) * r ^ CARD('n)"
+              using \<open>r > 0\<close> by (simp add: measure_translation measure_linear_image measurable_translation fsb field_simps)
+            also have "\<dots> \<le> (\<bar>det (matrix (f' x))\<bar> * unit_ball_vol (CARD('n)) + e * unit_ball_vol (CARD('n))) * r ^ CARD('n)"
+            proof -
+              have "?\<mu> (?rfs) < ?\<mu> (f' x ` ball 0 1) + e * unit_ball_vol (CARD('n))"
+                using rfs_mble by (force intro: k dest!: ex_lessK)
+              then have "?\<mu> (?rfs) < \<bar>det (matrix (f' x))\<bar> * unit_ball_vol (CARD('n)) + e * unit_ball_vol (CARD('n))"
+                by (simp add: lin measure_linear_image [of "f' x"] content_ball)
+              with \<open>r > 0\<close> show ?thesis
+                by auto
+            qed
+            also have "\<dots> \<le> (B + e) * ?\<mu> (ball x r)"
+              using bounded [OF \<open>x \<in> S\<close>] \<open>r > 0\<close> by (simp add: content_ball algebra_simps)
+            finally show "?\<mu> (f ` (S \<inter> ball x r)) \<le> (B + e) * ?\<mu> (ball x r)" .
+          qed
+        qed
+        then obtain r where
+          r0d: "\<And>x d. \<lbrakk>x \<in> S; d > 0\<rbrakk> \<Longrightarrow> 0 < r x d \<and> r x d < d"
+          and rT: "\<And>x d. \<lbrakk>x \<in> S; d > 0\<rbrakk> \<Longrightarrow> ball x (r x d) \<subseteq> T"
+          and r: "\<And>x d. \<lbrakk>x \<in> S; d > 0\<rbrakk> \<Longrightarrow>
+                  (f ` (S \<inter> ball x (r x d))) \<in> lmeasurable \<and>
+                  ?\<mu> (f ` (S \<inter> ball x (r x d))) \<le> (B + e) * ?\<mu> (ball x (r x d))"
+          by metis
+        obtain C where "countable C" and Csub: "C \<subseteq> {(x,r x t) |x t. x \<in> S \<and> 0 < t}"
+          and pwC: "pairwise (\<lambda>i j. disjnt (ball (fst i) (snd i)) (ball (fst j) (snd j))) C"
+          and negC: "negligible(S - (\<Union>i \<in> C. ball (fst i) (snd i)))"
+          apply (rule Vitali_covering_theorem_balls [of S "{(x,r x t) |x t. x \<in> S \<and> 0 < t}" fst snd])
+           apply auto
+          by (metis dist_eq_0_iff r0d)
+        let ?UB = "(\<Union>(x,s) \<in> C. ball x s)"
+        have eq: "f ` (S \<inter> ?UB) = (\<Union>(x,s) \<in> C. f ` (S \<inter> ball x s))"
+          by auto
+        have mle: "?\<mu> (\<Union>(x,s) \<in> K. f ` (S \<inter> ball x s)) \<le> (B + e) * (?\<mu> S + d)"  (is "?l \<le> ?r")
+          if "K \<subseteq> C" and "finite K" for K
+        proof -
+          have gt0: "b > 0" if "(a, b) \<in> K" for a b
+            using Csub that \<open>K \<subseteq> C\<close> r0d by auto
+          have inj: "inj_on (\<lambda>(x, y). ball x y) K"
+            by (force simp: inj_on_def ball_eq_ball_iff dest: gt0)
+          have disjnt: "disjoint ((\<lambda>(x, y). ball x y) ` K)"
+            using pwC that
+            apply (clarsimp simp: pairwise_def case_prod_unfold ball_eq_ball_iff)
+            by (metis subsetD fst_conv snd_conv)
+          have "?l \<le> (\<Sum>i\<in>K. ?\<mu> (case i of (x, s) \<Rightarrow> f ` (S \<inter> ball x s)))"
+          proof (rule measure_UNION_le [OF \<open>finite K\<close>], clarify)
+            fix x r
+            assume "(x,r) \<in> K"
+            then have "x \<in> S"
+              using Csub \<open>K \<subseteq> C\<close> by auto
+            show "f ` (S \<inter> ball x r) \<in> sets lebesgue"
+              by (meson Int_lower1 S differentiable_on_subset f_diff fmeasurableD lmeasurable_ball order_refl sets.Int differentiable_image_in_sets_lebesgue)
+          qed
+          also have "\<dots> \<le> (\<Sum>(x,s) \<in> K. (B + e) * ?\<mu> (ball x s))"
+            apply (rule sum_mono)
+            using Csub r \<open>K \<subseteq> C\<close> by auto
+          also have "\<dots> = (B + e) * (\<Sum>(x,s) \<in> K. ?\<mu> (ball x s))"
+            by (simp add: prod.case_distrib sum_distrib_left)
+          also have "\<dots> = (B + e) * sum ?\<mu> ((\<lambda>(x, y). ball x y) ` K)"
+            using \<open>B \<ge> 0\<close> \<open>e > 0\<close> by (simp add: inj sum.reindex prod.case_distrib)
+          also have "\<dots> = (B + e) * ?\<mu> (\<Union>(x,s) \<in> K. ball x s)"
+            using \<open>B \<ge> 0\<close> \<open>e > 0\<close> that
+            by (subst measure_Union') (auto simp: disjnt measure_Union')
+          also have "\<dots> \<le> (B + e) * ?\<mu> T"
+            using \<open>B \<ge> 0\<close> \<open>e > 0\<close> that apply simp
+            apply (rule measure_mono_fmeasurable [OF _ _ \<open>T \<in> lmeasurable\<close>])
+            using Csub rT by force+
+          also have "\<dots> \<le> (B + e) * (?\<mu> S + d)"
+            using \<open>B \<ge> 0\<close> \<open>e > 0\<close> Tless by simp
+          finally show ?thesis .
+        qed
+        have fSUB_mble: "(f ` (S \<inter> ?UB)) \<in> lmeasurable"
+          unfolding eq using Csub r False \<open>e > 0\<close> that
+          by (auto simp: intro!: fmeasurable_UN_bound [OF \<open>countable C\<close> _ mle])
+        have fSUB_meas: "?\<mu> (f ` (S \<inter> ?UB)) \<le> (B + e) * (?\<mu> S + d)"  (is "?MUB")
+          unfolding eq using Csub r False \<open>e > 0\<close> that
+          by (auto simp: intro!: measure_UN_bound [OF \<open>countable C\<close> _ mle])
+        have neg: "negligible ((f ` (S \<inter> ?UB) - f ` S) \<union> (f ` S - f ` (S \<inter> ?UB)))"
+        proof (rule negligible_subset [OF negligible_differentiable_image_negligible [OF order_refl negC, where f=f]])
+          show "f differentiable_on S - (\<Union>i\<in>C. ball (fst i) (snd i))"
+            by (meson DiffE differentiable_on_subset subsetI f_diff)
+        qed force
+        show "f ` S \<in> lmeasurable"
+          by (rule lmeasurable_negligible_symdiff [OF fSUB_mble neg])
+        show ?MD
+          using fSUB_meas measure_negligible_symdiff [OF fSUB_mble neg] by simp
+      qed
+      show "f ` S \<in> lmeasurable"
+        using eps_d [of 1] by simp
+      show ?ME
+      proof (rule field_le_epsilon)
+        fix \<delta> :: real
+        assume "0 < \<delta>"
+        then show "?\<mu> (f ` S) \<le> (B + e) * ?\<mu> S + \<delta>"
+          using eps_d [of "\<delta> / (B+e)"] \<open>e > 0\<close> \<open>B \<ge> 0\<close> by (auto simp: divide_simps mult_ac)
+      qed
+    qed
+    show ?thesis
+    proof (cases "?\<mu> S = 0")
+      case True
+      with eps have "?\<mu> (f ` S) = 0"
+        by (metis mult_zero_right not_le zero_less_measure_iff)
+      then show ?thesis
+        using eps [of 1] by (simp add: True)
+    next
+      case False
+      have "?\<mu> (f ` S) \<le> B * ?\<mu> S"
+      proof (rule field_le_epsilon)
+        fix e :: real
+        assume "e > 0"
+        then show "?\<mu> (f ` S) \<le> B * ?\<mu> S + e"
+          using eps [of "e / ?\<mu> S"] False by (auto simp: algebra_simps zero_less_measure_iff)
+      qed
+      with eps [of 1] show ?thesis by auto
+    qed
+  qed
+  then show "f ` S \<in> lmeasurable" ?M by blast+
+qed
+
+lemma
+ fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
+  assumes S: "S \<in> lmeasurable"
+    and deriv: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
+    and int: "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) integrable_on S"
+  shows m_diff_image_weak: "f ` S \<in> lmeasurable \<and> measure lebesgue (f ` S) \<le> integral S (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)"
+proof -
+  let ?\<mu> = "measure lebesgue"
+  have aint_S: "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) absolutely_integrable_on S"
+    using int unfolding absolutely_integrable_on_def by auto
+  define m where "m \<equiv> integral S (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)"
+  have *: "f ` S \<in> lmeasurable" "?\<mu> (f ` S) \<le> m + e * ?\<mu> S"
+    if "e > 0" for e
+  proof -
+    define T where "T \<equiv> \<lambda>n. {x \<in> S. n * e \<le> \<bar>det (matrix (f' x))\<bar> \<and>
+                                     \<bar>det (matrix (f' x))\<bar> < (Suc n) * e}"
+    have meas_t: "T n \<in> lmeasurable" for n
+    proof -
+      have *: "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) \<in> borel_measurable (lebesgue_on S)"
+        using aint_S by (simp add: S borel_measurable_restrict_space_iff fmeasurableD set_integrable_def)
+      have [intro]: "x \<in> sets (lebesgue_on S) \<Longrightarrow> x \<in> sets lebesgue" for x
+        using S sets_restrict_space_subset by blast
+      have "{x \<in> S. real n * e \<le> \<bar>det (matrix (f' x))\<bar>} \<in> sets lebesgue"
+        using * by (auto simp: borel_measurable_iff_halfspace_ge space_restrict_space)
+      then have 1: "{x \<in> S. real n * e \<le> \<bar>det (matrix (f' x))\<bar>} \<in> lmeasurable"
+        using S by (simp add: fmeasurableI2)
+      have "{x \<in> S. \<bar>det (matrix (f' x))\<bar> < (1 + real n) * e} \<in> sets lebesgue"
+        using * by (auto simp: borel_measurable_iff_halfspace_less space_restrict_space)
+      then have 2: "{x \<in> S. \<bar>det (matrix (f' x))\<bar> < (1 + real n) * e} \<in> lmeasurable"
+        using S by (simp add: fmeasurableI2)
+      show ?thesis
+        using fmeasurable.Int [OF 1 2] by (simp add: T_def Int_def cong: conj_cong)
+    qed
+    have aint_T: "\<And>k. (\<lambda>x. \<bar>det (matrix (f' x))\<bar>) absolutely_integrable_on T k"
+      using set_integrable_subset [OF aint_S] meas_t T_def by blast
+    have Seq: "S = (\<Union>n. T n)"
+      apply (auto simp: T_def)
+      apply (rule_tac x="nat(floor(abs(det(matrix(f' x))) / e))" in exI)
+      using that apply auto
+      using of_int_floor_le pos_le_divide_eq apply blast
+      by (metis add.commute pos_divide_less_eq real_of_int_floor_add_one_gt)
+    have meas_ft: "f ` T n \<in> lmeasurable" for n
+    proof (rule measurable_bounded_differentiable_image)
+      show "T n \<in> lmeasurable"
+        by (simp add: meas_t)
+    next
+      fix x :: "(real,'n) vec"
+      assume "x \<in> T n"
+      show "(f has_derivative f' x) (at x within T n)"
+        by (metis (no_types, lifting) \<open>x \<in> T n\<close> deriv has_derivative_within_subset mem_Collect_eq subsetI T_def)
+      show "\<bar>det (matrix (f' x))\<bar> \<le> (Suc n) * e"
+        using \<open>x \<in> T n\<close> T_def by auto
+    next
+      show "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) integrable_on T n"
+        using aint_T absolutely_integrable_on_def by blast
+    qed
+    have disT: "disjoint (range T)"
+      unfolding disjoint_def
+    proof clarsimp
+      show "T m \<inter> T n = {}" if "T m \<noteq> T n" for m n
+        using that
+      proof (induction m n rule: linorder_less_wlog)
+        case (less m n)
+        with \<open>e > 0\<close> show ?case
+          unfolding T_def
+          proof (clarsimp simp add: Collect_conj_eq [symmetric])
+            fix x
+            assume "e > 0"  "m < n"  "n * e \<le> \<bar>det (matrix (f' x))\<bar>"  "\<bar>det (matrix (f' x))\<bar> < (1 + real m) * e"
+            then have "n < 1 + real m"
+              by (metis (no_types, hide_lams) less_le_trans mult.commute not_le real_mult_le_cancel_iff2)
+            then show "False"
+              using less.hyps by linarith
+          qed
+      qed auto
+    qed
+    have injT: "inj_on T ({n. T n \<noteq> {}})"
+      unfolding inj_on_def
+    proof clarsimp
+      show "m = n" if "T m = T n" "T n \<noteq> {}" for m n
+        using that
+      proof (induction m n rule: linorder_less_wlog)
+        case (less m n)
+        have False if "T n \<subseteq> T m" "x \<in> T n" for x
+          using \<open>e > 0\<close> \<open>m < n\<close> that
+          apply (auto simp: T_def  mult.commute intro: less_le_trans dest!: subsetD)
+          by (metis add.commute less_le_trans nat_less_real_le not_le real_mult_le_cancel_iff2)
+        then show ?case
+          using less.prems by blast
+      qed auto
+    qed
+    have sum_eq_Tim: "(\<Sum>k\<le>n. f (T k)) = sum f (T ` {..n})" if "f {} = 0" for f :: "_ \<Rightarrow> real" and n
+    proof (subst sum.reindex_nontrivial)
+      fix i j  assume "i \<in> {..n}" "j \<in> {..n}" "i \<noteq> j" "T i = T j"
+      with that  injT [unfolded inj_on_def] show "f (T i) = 0"
+        by simp metis
+    qed (use atMost_atLeast0 in auto)
+    let ?B = "m + e * ?\<mu> S"
+    have "(\<Sum>k\<le>n. ?\<mu> (f ` T k)) \<le> ?B" for n
+    proof -
+      have "(\<Sum>k\<le>n. ?\<mu> (f ` T k)) \<le> (\<Sum>k\<le>n. ((k+1) * e) * ?\<mu>(T k))"
+      proof (rule sum_mono [OF measure_bounded_differentiable_image])
+        show "(f has_derivative f' x) (at x within T k)" if "x \<in> T k" for k x
+          using that unfolding T_def by (blast intro: deriv has_derivative_within_subset)
+        show "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) integrable_on T k" for k
+          using absolutely_integrable_on_def aint_T by blast
+        show "\<bar>det (matrix (f' x))\<bar> \<le> real (k + 1) * e" if "x \<in> T k" for k x
+          using T_def that by auto
+      qed (use meas_t in auto)
+      also have "\<dots> \<le> (\<Sum>k\<le>n. (k * e) * ?\<mu>(T k)) + (\<Sum>k\<le>n. e * ?\<mu>(T k))"
+        by (simp add: algebra_simps sum.distrib)
+      also have "\<dots> \<le> ?B"
+      proof (rule add_mono)
+        have "(\<Sum>k\<le>n. real k * e * ?\<mu> (T k)) = (\<Sum>k\<le>n. integral (T k) (\<lambda>x. k * e))"
+          by (simp add: lmeasure_integral [OF meas_t]
+                        integral_mult_right [symmetric] integral_mult_left [symmetric]
+                   del: integral_mult_right integral_mult_left)
+        also have "\<dots> \<le> (\<Sum>k\<le>n. integral (T k) (\<lambda>x.  (abs (det (matrix (f' x))))))"
+        proof (rule sum_mono)
+          fix k
+          assume "k \<in> {..n}"
+          show "integral (T k) (\<lambda>x. k * e) \<le> integral (T k) (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)"
+          proof (rule integral_le [OF integrable_on_const [OF meas_t]])
+            show "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) integrable_on T k"
+              using absolutely_integrable_on_def aint_T by blast
+          next
+            fix x assume "x \<in> T k"
+            show "k * e \<le> \<bar>det (matrix (f' x))\<bar>"
+              using \<open>x \<in> T k\<close> T_def by blast
+          qed
+        qed
+        also have "\<dots> = sum (\<lambda>T. integral T (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)) (T ` {..n})"
+          by (auto intro: sum_eq_Tim)
+        also have "\<dots> = integral (\<Union>k\<le>n. T k) (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)"
+        proof (rule integral_unique [OF has_integral_Union, symmetric])
+          fix S  assume "S \<in> T ` {..n}"
+          then show "((\<lambda>x. \<bar>det (matrix (f' x))\<bar>) has_integral integral S (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)) S"
+          using absolutely_integrable_on_def aint_T by blast
+        next
+          show "pairwise (\<lambda>S S'. negligible (S \<inter> S')) (T ` {..n})"
+            using disT unfolding disjnt_iff by (auto simp: pairwise_def intro!: empty_imp_negligible)
+        qed auto
+        also have "\<dots> \<le> m"
+          unfolding m_def
+        proof (rule integral_subset_le)
+          have "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) absolutely_integrable_on (\<Union>k\<le>n. T k)"
+            apply (rule set_integrable_subset [OF aint_S])
+             apply (intro measurable meas_t fmeasurableD)
+            apply (force simp: Seq)
+            done
+          then show "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) integrable_on (\<Union>k\<le>n. T k)"
+            using absolutely_integrable_on_def by blast
+        qed (use Seq int in auto)
+        finally show "(\<Sum>k\<le>n. real k * e * ?\<mu> (T k)) \<le> m" .
+      next
+        have "(\<Sum>k\<le>n. ?\<mu> (T k)) = sum ?\<mu> (T ` {..n})"
+          by (auto intro: sum_eq_Tim)
+        also have "\<dots> = ?\<mu> (\<Union>k\<le>n. T k)"
+          using S disT by (auto simp: pairwise_def meas_t intro: measure_Union' [symmetric])
+        also have "\<dots> \<le> ?\<mu> S"
+          using S by (auto simp: Seq intro: meas_t fmeasurableD measure_mono_fmeasurable)
+        finally have "(\<Sum>k\<le>n. ?\<mu> (T k)) \<le> ?\<mu> S" .
+        then show "(\<Sum>k\<le>n. e * ?\<mu> (T k)) \<le> e * ?\<mu> S"
+          by (metis less_eq_real_def ordered_comm_semiring_class.comm_mult_left_mono sum_distrib_left that)
+      qed
+      finally show "(\<Sum>k\<le>n. ?\<mu> (f ` T k)) \<le> ?B" .
+    qed
+    moreover have "measure lebesgue (\<Union>k\<le>n. f ` T k) \<le> (\<Sum>k\<le>n. ?\<mu> (f ` T k))" for n
+      by (simp add: fmeasurableD meas_ft measure_UNION_le)
+    ultimately have B_ge_m: "?\<mu> (\<Union>k\<le>n. (f ` T k)) \<le> ?B" for n
+      by (meson order_trans)
+    have "(\<Union>n. f ` T n) \<in> lmeasurable"
+      by (rule fmeasurable_countable_Union [OF meas_ft B_ge_m])
+    moreover have "?\<mu> (\<Union>n. f ` T n) \<le> m + e * ?\<mu> S"
+      by (rule measure_countable_Union_le [OF meas_ft B_ge_m])
+    ultimately show "f ` S \<in> lmeasurable" "?\<mu> (f ` S) \<le> m + e * ?\<mu> S"
+      by (auto simp: Seq image_Union)
+  qed
+  show ?thesis
+  proof
+    show "f ` S \<in> lmeasurable"
+      using * linordered_field_no_ub by blast
+    let ?x = "m - ?\<mu> (f ` S)"
+    have False if "?\<mu> (f ` S) > integral S (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)"
+    proof -
+      have ml: "m < ?\<mu> (f ` S)"
+        using m_def that by blast
+      then have "?\<mu> S \<noteq> 0"
+        using "*"(2) bgauge_existence_lemma by fastforce
+      with ml have 0: "0 < - (m - ?\<mu> (f ` S))/2 / ?\<mu> S"
+        using that zero_less_measure_iff by force
+      then show ?thesis
+        using * [OF 0] that by (auto simp: divide_simps m_def split: if_split_asm)
+    qed
+    then show "?\<mu> (f ` S) \<le> integral S (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)"
+      by fastforce
+  qed
+qed
+
+
+theorem
+ fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
+  assumes S: "S \<in> sets lebesgue"
+    and deriv: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
+    and int: "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) integrable_on S"
+  shows measurable_differentiable_image: "f ` S \<in> lmeasurable"
+    and measure_differentiable_image:
+       "measure lebesgue (f ` S) \<le> integral S (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)" (is "?M")
+proof -
+  let ?I = "\<lambda>n::nat. cbox (vec (-n)) (vec n) \<inter> S"
+  let ?\<mu> = "measure lebesgue"
+  have "x \<in> cbox (vec (- real (nat \<lceil>norm x\<rceil>))) (vec (real (nat \<lceil>norm x\<rceil>)))" for x :: "real^'n::_"
+    apply (auto simp: mem_box_cart)
+    apply (metis abs_le_iff component_le_norm_cart minus_le_iff of_nat_ceiling order.trans)
+    by (meson abs_le_D1 norm_bound_component_le_cart real_nat_ceiling_ge)
+  then have Seq: "S = (\<Union>n. ?I n)"
+    by auto
+  have fIn: "f ` ?I n \<in> lmeasurable"
+       and mfIn: "?\<mu> (f ` ?I n) \<le> integral S (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)" (is ?MN) for n
+  proof -
+    have In: "?I n \<in> lmeasurable"
+      by (simp add: S bounded_Int bounded_set_imp_lmeasurable sets.Int)
+    moreover have "\<And>x. x \<in> ?I n \<Longrightarrow> (f has_derivative f' x) (at x within ?I n)"
+      by (meson Int_iff deriv has_derivative_within_subset subsetI)
+    moreover have int_In: "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) integrable_on ?I n"
+    proof -
+      have "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) absolutely_integrable_on S"
+        using int absolutely_integrable_integrable_bound by force
+      then have "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) absolutely_integrable_on ?I n"
+        by (metis (no_types) Int_lower1 In fmeasurableD inf_commute set_integrable_subset)
+      then show ?thesis
+        using absolutely_integrable_on_def by blast
+    qed
+    ultimately have "f ` ?I n \<in> lmeasurable" "?\<mu> (f ` ?I n) \<le> integral (?I n) (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)"
+      using m_diff_image_weak by metis+
+    moreover have "integral (?I n) (\<lambda>x. \<bar>det (matrix (f' x))\<bar>) \<le> integral S (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)"
+      by (simp add: int_In int integral_subset_le)
+    ultimately show "f ` ?I n \<in> lmeasurable" ?MN
+      by auto
+  qed
+  have "?I k \<subseteq> ?I n" if "k \<le> n" for k n
+    by (rule Int_mono) (use that in \<open>auto simp: subset_interval_imp_cart\<close>)
+  then have "(\<Union>k\<le>n. f ` ?I k) = f ` ?I n" for n
+    by (fastforce simp add:)
+  with mfIn have "?\<mu> (\<Union>k\<le>n. f ` ?I k) \<le> integral S (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)" for n
+    by simp
+  then have "(\<Union>n. f ` ?I n) \<in> lmeasurable" "?\<mu> (\<Union>n. f ` ?I n) \<le> integral S (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)"
+    by (rule fmeasurable_countable_Union [OF fIn] measure_countable_Union_le [OF fIn])+
+  then show "f ` S \<in> lmeasurable" ?M
+    by (metis Seq image_UN)+
+qed
+
+
+lemma borel_measurable_simple_function_limit_increasing:
+  fixes f :: "'a::euclidean_space \<Rightarrow> real"
+  shows "(f \<in> borel_measurable lebesgue \<and> (\<forall>x. 0 \<le> f x)) \<longleftrightarrow>
+         (\<exists>g. (\<forall>n x. 0 \<le> g n x \<and> g n x \<le> f x) \<and> (\<forall>n x. g n x \<le> (g(Suc n) x)) \<and>
+              (\<forall>n. g n \<in> borel_measurable lebesgue) \<and> (\<forall>n. finite(range (g n))) \<and>
+              (\<forall>x. (\<lambda>n. g n x) \<longlonglongrightarrow> f x))"
+         (is "?lhs = ?rhs")
+proof
+  assume f: ?lhs
+  have leb_f: "{x. a \<le> f x \<and> f x < b} \<in> sets lebesgue" for a b
+  proof -
+    have "{x. a \<le> f x \<and> f x < b} = {x. f x < b} - {x. f x < a}"
+      by auto
+    also have "\<dots> \<in> sets lebesgue"
+      using borel_measurable_vimage_halfspace_component_lt [of f UNIV] f by auto
+    finally show ?thesis .
+  qed
+  have "g n x \<le> f x"
+        if inc_g: "\<And>n x. 0 \<le> g n x \<and> g n x \<le> g (Suc n) x"
+           and meas_g: "\<And>n. g n \<in> borel_measurable lebesgue"
+           and fin: "\<And>n. finite(range (g n))" and lim: "\<And>x. (\<lambda>n. g n x) \<longlonglongrightarrow> f x" for g n x
+  proof -
+    have "\<exists>r>0. \<forall>N. \<exists>n\<ge>N. dist (g n x) (f x) \<ge> r" if "g n x > f x"
+    proof -
+      have g: "g n x \<le> g (N + n) x" for N
+        by (rule transitive_stepwise_le) (use inc_g in auto)
+      have "\<exists>na\<ge>N. g n x - f x \<le> dist (g na x) (f x)" for N
+        apply (rule_tac x="N+n" in exI)
+        using g [of N] by (auto simp: dist_norm)
+      with that show ?thesis
+        using diff_gt_0_iff_gt by blast
+    qed
+    with lim show ?thesis
+      apply (auto simp: lim_sequentially)
+      by (meson less_le_not_le not_le_imp_less)
+  qed
+  moreover
+  let ?\<Omega> = "\<lambda>n k. indicator {y. k/2^n \<le> f y \<and> f y < (k+1)/2^n}"
+  let ?g = "\<lambda>n x. (\<Sum>k::real | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n). k/2^n * ?\<Omega> n k x)"
+  have "\<exists>g. (\<forall>n x. 0 \<le> g n x \<and> g n x \<le> (g(Suc n) x)) \<and>
+             (\<forall>n. g n \<in> borel_measurable lebesgue) \<and> (\<forall>n. finite(range (g n))) \<and>(\<forall>x. (\<lambda>n. g n x) \<longlonglongrightarrow> f x)"
+  proof (intro exI allI conjI)
+    show "0 \<le> ?g n x" for n x
+    proof (clarify intro!: ordered_comm_monoid_add_class.sum_nonneg)
+      fix k::real
+      assume "k \<in> \<int>" and k: "\<bar>k\<bar> \<le> 2 ^ (2*n)"
+      show "0 \<le> k/2^n * ?\<Omega> n k x"
+        using f \<open>k \<in> \<int>\<close> apply (auto simp: indicator_def divide_simps Ints_def)
+        apply (drule spec [where x=x])
+        using zero_le_power [of "2::real" n] mult_nonneg_nonneg [of "f x" "2^n"]
+        by linarith
+    qed
+    show "?g n x \<le> ?g (Suc n) x" for n x
+    proof -
+      have "?g n x =
+            (\<Sum>k | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n).
+              k/2^n * (indicator {y. k/2^n \<le> f y \<and> f y < (k+1/2)/2^n} x +
+              indicator {y. (k+1/2)/2^n \<le> f y \<and> f y < (k+1)/2^n} x))"
+        by (rule sum.cong [OF refl]) (simp add: indicator_def divide_simps)
+      also have "\<dots> = (\<Sum>k | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n). k/2^n * indicator {y. k/2^n \<le> f y \<and> f y < (k+1/2)/2^n} x) +
+                       (\<Sum>k | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n). k/2^n * indicator {y. (k+1/2)/2^n \<le> f y \<and> f y < (k+1)/2^n} x)"
+        by (simp add:  comm_monoid_add_class.sum.distrib algebra_simps)
+      also have "\<dots> = (\<Sum>k | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n). (2 * k)/2 ^ Suc n * indicator {y. (2 * k)/2 ^ Suc n \<le> f y \<and> f y < (2 * k+1)/2 ^ Suc n} x) +
+                       (\<Sum>k | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n). (2 * k)/2 ^ Suc n * indicator {y. (2 * k+1)/2 ^ Suc n \<le> f y \<and> f y < ((2 * k+1) + 1)/2 ^ Suc n} x)"
+        by (force simp: field_simps indicator_def intro: sum.cong)
+      also have "\<dots> \<le> (\<Sum>k | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2 * Suc n). k/2 ^ Suc n * (indicator {y. k/2 ^ Suc n \<le> f y \<and> f y < (k+1)/2 ^ Suc n} x))"
+                (is "?a + _ \<le> ?b")
+      proof -
+        have *: "\<lbrakk>sum f I \<le> sum h I; a + sum h I \<le> b\<rbrakk> \<Longrightarrow> a + sum f I \<le> b" for I a b f and h :: "real\<Rightarrow>real"
+          by linarith
+        let ?h = "\<lambda>k. (2*k+1)/2 ^ Suc n *
+                      (indicator {y. (2 * k+1)/2 ^ Suc n \<le> f y \<and> f y < ((2*k+1) + 1)/2 ^ Suc n} x)"
+        show ?thesis
+        proof (rule *)
+          show "(\<Sum>k | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n).
+                  2 * k/2 ^ Suc n * indicator {y. (2 * k+1)/2 ^ Suc n \<le> f y \<and> f y < (2 * k+1 + 1)/2 ^ Suc n} x)
+                \<le> sum ?h {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)}"
+            by (rule sum_mono) (simp add: indicator_def divide_simps)
+        next
+          have \<alpha>: "?a = (\<Sum>k \<in> ( *)2 ` {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)}.
+                         k/2 ^ Suc n * indicator {y. k/2 ^ Suc n \<le> f y \<and> f y < (k+1)/2 ^ Suc n} x)"
+            by (auto simp: inj_on_def field_simps comm_monoid_add_class.sum.reindex)
+          have \<beta>: "sum ?h {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)}
+                   = (\<Sum>k \<in> (\<lambda>x. 2*x + 1) ` {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)}.
+                      k/2 ^ Suc n * indicator {y. k/2 ^ Suc n \<le> f y \<and> f y < (k+1)/2 ^ Suc n} x)"
+            by (auto simp: inj_on_def field_simps comm_monoid_add_class.sum.reindex)
+          have 0: "( *) 2 ` {k \<in> \<int>. P k} \<inter> (\<lambda>x. 2 * x + 1) ` {k \<in> \<int>. P k} = {}" for P :: "real \<Rightarrow> bool"
+          proof -
+            have "2 * i \<noteq> 2 * j + 1" for i j :: int by arith
+            thus ?thesis
+              unfolding Ints_def by auto (use of_int_eq_iff in fastforce)
+          qed
+          have "?a + sum ?h {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)}
+                = (\<Sum>k \<in> ( *)2 ` {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)} \<union> (\<lambda>x. 2*x + 1) ` {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)}.
+                  k/2 ^ Suc n * indicator {y. k/2 ^ Suc n \<le> f y \<and> f y < (k+1)/2 ^ Suc n} x)"
+            unfolding \<alpha> \<beta>
+            using finite_abs_int_segment [of "2 ^ (2*n)"]
+            by (subst sum_Un) (auto simp: 0)
+          also have "\<dots> \<le> ?b"
+          proof (rule sum_mono2)
+            show "finite {k::real. k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2 * Suc n)}"
+              by (rule finite_abs_int_segment)
+            show "( *) 2 ` {k::real. k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2^(2*n)} \<union> (\<lambda>x. 2*x + 1) ` {k \<in> \<int>. \<bar>k\<bar> \<le> 2^(2*n)} \<subseteq> {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2 * Suc n)}"
+              apply auto
+              using one_le_power [of "2::real" "2*n"]  by linarith
+            have *: "\<lbrakk>x \<in> (S \<union> T) - U; \<And>x. x \<in> S \<Longrightarrow> x \<in> U; \<And>x. x \<in> T \<Longrightarrow> x \<in> U\<rbrakk> \<Longrightarrow> P x" for S T U P
+              by blast
+            have "0 \<le> b" if "b \<in> \<int>" "f x * (2 * 2^n) < b + 1" for b
+            proof -
+              have "0 \<le> f x * (2 * 2^n)"
+                by (simp add: f)
+              also have "\<dots> < b+1"
+                by (simp add: that)
+              finally show "0 \<le> b"
+                using \<open>b \<in> \<int>\<close> by (auto simp: elim!: Ints_cases)
+            qed
+            then show "0 \<le> b/2 ^ Suc n * indicator {y. b/2 ^ Suc n \<le> f y \<and> f y < (b + 1)/2 ^ Suc n} x"
+                  if "b \<in> {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2 * Suc n)} -
+                          (( *) 2 ` {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)} \<union> (\<lambda>x. 2*x + 1) ` {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)})" for b
+              using that by (simp add: indicator_def divide_simps)
+          qed
+          finally show "?a + sum ?h {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)} \<le> ?b" .
+        qed
+      qed
+      finally show ?thesis .
+    qed
+    show "?g n \<in> borel_measurable lebesgue" for n
+      apply (intro borel_measurable_indicator borel_measurable_times borel_measurable_sum)
+      using leb_f sets_restrict_UNIV by auto
+    show "finite (range (?g n))" for n
+    proof -
+      have "(\<Sum>k | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n). k/2^n * ?\<Omega> n k x)
+              \<in> (\<lambda>k. k/2^n) ` {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)}" for x
+      proof (cases "\<exists>k. k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n) \<and> k/2^n \<le> f x \<and> f x < (k+1)/2^n")
+        case True
+        then show ?thesis
+          by (blast intro: indicator_sum_eq)
+      next
+        case False
+        then have "(\<Sum>k | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n). k/2^n * ?\<Omega> n k x) = 0"
+          by auto
+        then show ?thesis by force
+      qed
+      then have "range (?g n) \<subseteq> ((\<lambda>k. (k/2^n)) ` {k. k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n)})"
+        by auto
+      moreover have "finite ((\<lambda>k::real. (k/2^n)) ` {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)})"
+        by (intro finite_imageI finite_abs_int_segment)
+      ultimately show ?thesis
+        by (rule finite_subset)
+    qed
+    show "(\<lambda>n. ?g n x) \<longlonglongrightarrow> f x" for x
+    proof (clarsimp simp add: lim_sequentially)
+      fix e::real
+      assume "e > 0"
+      obtain N1 where N1: "2 ^ N1 > abs(f x)"
+        using real_arch_pow by fastforce
+      obtain N2 where N2: "(1/2) ^ N2 < e"
+        using real_arch_pow_inv \<open>e > 0\<close> by fastforce
+      have "dist (\<Sum>k | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n). k/2^n * ?\<Omega> n k x) (f x) < e" if "N1 + N2 \<le> n" for n
+      proof -
+        let ?m = "real_of_int \<lfloor>2^n * f x\<rfloor>"
+        have "\<bar>?m\<bar> \<le> 2^n * 2^N1"
+          using N1 apply (simp add: f)
+          by (meson floor_mono le_floor_iff less_le_not_le mult_le_cancel_left_pos zero_less_numeral zero_less_power)
+        also have "\<dots> \<le> 2 ^ (2*n)"
+          by (metis that add_leD1 add_le_cancel_left mult.commute mult_2_right one_less_numeral_iff
+                    power_add power_increasing_iff semiring_norm(76))
+        finally have m_le: "\<bar>?m\<bar> \<le> 2 ^ (2*n)" .
+        have "?m/2^n \<le> f x" "f x < (?m + 1)/2^n"
+          by (auto simp: mult.commute pos_divide_le_eq mult_imp_less_div_pos)
+        then have eq: "dist (\<Sum>k | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n). k/2^n * ?\<Omega> n k x) (f x)
+                     = dist (?m/2^n) (f x)"
+          by (subst indicator_sum_eq [of ?m]) (auto simp: m_le)
+        have "\<bar>2^n\<bar> * \<bar>?m/2^n - f x\<bar> = \<bar>2^n * (?m/2^n - f x)\<bar>"
+          by (simp add: abs_mult)
+        also have "\<dots> < 2 ^ N2 * e"
+          using N2 by (simp add: divide_simps mult.commute) linarith
+        also have "\<dots> \<le> \<bar>2^n\<bar> * e"
+          using that \<open>e > 0\<close> by auto
+        finally have "dist (?m/2^n) (f x) < e"
+          by (simp add: dist_norm)
+        then show ?thesis
+          using eq by linarith
+      qed
+      then show "\<exists>no. \<forall>n\<ge>no. dist (\<Sum>k | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n). k * ?\<Omega> n k x/2^n) (f x) < e"
+        by force
+    qed
+  qed
+  ultimately show ?rhs
+    by metis
+next
+  assume RHS: ?rhs
+  with borel_measurable_simple_function_limit [of f UNIV, unfolded borel_measurable_UNIV_eq]
+  show ?lhs
+    by (blast intro: order_trans)
+qed
+
+subsection\<open>Borel measurable Jacobian determinant\<close>
+
+lemma lemma_partial_derivatives0:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes "linear f" and lim0: "((\<lambda>x. f x /\<^sub>R norm x) \<longlongrightarrow> 0) (at 0 within S)"
+    and lb: "\<And>v. v \<noteq> 0 \<Longrightarrow> (\<exists>k>0. \<forall>e>0. \<exists>x. x \<in> S - {0} \<and> norm x < e \<and> k * norm x \<le> \<bar>v \<bullet> x\<bar>)"
+  shows "f x = 0"
+proof -
+  have "dim {x. f x = 0} \<le> DIM('a)"
+    using dim_subset_UNIV by blast
+  moreover have False if less: "dim {x. f x = 0} < DIM('a)"
+  proof -
+    obtain d where "d \<noteq> 0" and d: "\<And>y. f y = 0 \<Longrightarrow> d \<bullet> y = 0"
+      using orthogonal_to_subspace_exists [OF less] orthogonal_def
+      by (metis (mono_tags, lifting) mem_Collect_eq span_clauses(1))
+    then obtain k where "k > 0"
+      and k: "\<And>e. e > 0 \<Longrightarrow> \<exists>y. y \<in> S - {0} \<and> norm y < e \<and> k * norm y \<le> \<bar>d \<bullet> y\<bar>"
+      using lb by blast
+    have "\<exists>h. \<forall>n. ((h n \<in> S \<and> h n \<noteq> 0 \<and> k * norm (h n) \<le> \<bar>d \<bullet> h n\<bar>) \<and> norm (h n) < 1 / real (Suc n)) \<and>
+               norm (h (Suc n)) < norm (h n)"
+    proof (rule dependent_nat_choice)
+      show "\<exists>y. (y \<in> S \<and> y \<noteq> 0 \<and> k * norm y \<le> \<bar>d \<bullet> y\<bar>) \<and> norm y < 1 / real (Suc 0)"
+        by simp (metis DiffE insertCI k not_less not_one_le_zero)
+    qed (use k [of "min (norm x) (1/(Suc n + 1))" for x n] in auto)
+    then obtain \<alpha> where \<alpha>: "\<And>n. \<alpha> n \<in> S - {0}" and kd: "\<And>n. k * norm(\<alpha> n) \<le> \<bar>d \<bullet> \<alpha> n\<bar>"
+         and norm_lt: "\<And>n. norm(\<alpha> n) < 1/(Suc n)"
+      by force
+    let ?\<beta> = "\<lambda>n. \<alpha> n /\<^sub>R norm (\<alpha> n)"
+    have com: "\<And>g. (\<forall>n. g n \<in> sphere (0::'a) 1)
+              \<Longrightarrow> \<exists>l \<in> sphere 0 1. \<exists>\<rho>::nat\<Rightarrow>nat. strict_mono \<rho> \<and> (g \<circ> \<rho>) \<longlonglongrightarrow> l"
+      using compact_sphere compact_def by metis
+    moreover have "\<forall>n. ?\<beta> n \<in> sphere 0 1"
+      using \<alpha> by auto
+    ultimately obtain l::'a and \<rho>::"nat\<Rightarrow>nat"
+       where l: "l \<in> sphere 0 1" and "strict_mono \<rho>" and to_l: "(?\<beta> \<circ> \<rho>) \<longlonglongrightarrow> l"
+      by meson
+    moreover have "continuous (at l) (\<lambda>x. (\<bar>d \<bullet> x\<bar> - k))"
+      by (intro continuous_intros)
+    ultimately have lim_dl: "((\<lambda>x. (\<bar>d \<bullet> x\<bar> - k)) \<circ> (?\<beta> \<circ> \<rho>)) \<longlonglongrightarrow> (\<bar>d \<bullet> l\<bar> - k)"
+      by (meson continuous_imp_tendsto)
+    have "\<forall>\<^sub>F i in sequentially. 0 \<le> ((\<lambda>x. \<bar>d \<bullet> x\<bar> - k) \<circ> ((\<lambda>n. \<alpha> n /\<^sub>R norm (\<alpha> n)) \<circ> \<rho>)) i"
+      using \<alpha> kd by (auto simp: divide_simps)
+    then have "k \<le> \<bar>d \<bullet> l\<bar>"
+      using tendsto_lowerbound [OF lim_dl, of 0] by auto
+    moreover have "d \<bullet> l = 0"
+    proof (rule d)
+      show "f l = 0"
+      proof (rule LIMSEQ_unique [of "f \<circ> ?\<beta> \<circ> \<rho>"])
+        have "isCont f l"
+          using \<open>linear f\<close> linear_continuous_at linear_conv_bounded_linear by blast
+        then show "(f \<circ> (\<lambda>n. \<alpha> n /\<^sub>R norm (\<alpha> n)) \<circ> \<rho>) \<longlonglongrightarrow> f l"
+          unfolding comp_assoc
+          using to_l continuous_imp_tendsto by blast
+        have "\<alpha> \<longlonglongrightarrow> 0"
+          using norm_lt LIMSEQ_norm_0 by metis
+        with \<open>strict_mono \<rho>\<close> have "(\<alpha> \<circ> \<rho>) \<longlonglongrightarrow> 0"
+          by (metis LIMSEQ_subseq_LIMSEQ)
+        with lim0 \<alpha> have "((\<lambda>x. f x /\<^sub>R norm x) \<circ> (\<alpha> \<circ> \<rho>)) \<longlonglongrightarrow> 0"
+          by (force simp: tendsto_at_iff_sequentially)
+        then show "(f \<circ> (\<lambda>n. \<alpha> n /\<^sub>R norm (\<alpha> n)) \<circ> \<rho>) \<longlonglongrightarrow> 0"
+          by (simp add: o_def linear_cmul \<open>linear f\<close>)
+      qed
+    qed
+    ultimately show False
+      using \<open>k > 0\<close> by auto
+  qed
+  ultimately have dim: "dim {x. f x = 0} = DIM('a)"
+    by force
+  then show ?thesis
+    by (metis (mono_tags, lifting) UNIV_I assms(1) dim_eq_full linear_eq_0_span mem_Collect_eq)
+qed
+
+lemma lemma_partial_derivatives:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes "linear f" and lim: "((\<lambda>x. f (x - a) /\<^sub>R norm (x - a)) \<longlongrightarrow> 0) (at a within S)"
+    and lb: "\<And>v. v \<noteq> 0 \<Longrightarrow> (\<exists>k>0.  \<forall>e>0. \<exists>x \<in> S - {a}. norm(a - x) < e \<and> k * norm(a - x) \<le> \<bar>v \<bullet> (x - a)\<bar>)"
+  shows "f x = 0"
+proof -
+  have "((\<lambda>x. f x /\<^sub>R norm x) \<longlongrightarrow> 0) (at 0 within (\<lambda>x. x-a) ` S)"
+    using lim by (simp add: Lim_within dist_norm)
+  then show ?thesis
+  proof (rule lemma_partial_derivatives0 [OF \<open>linear f\<close>])
+    fix v :: "'a"
+    assume v: "v \<noteq> 0"
+    show "\<exists>k>0. \<forall>e>0. \<exists>x. x \<in> (\<lambda>x. x - a) ` S - {0} \<and> norm x < e \<and> k * norm x \<le> \<bar>v \<bullet> x\<bar>"
+      using lb [OF v] by (force simp:  norm_minus_commute)
+  qed
+qed
+
+
+proposition borel_measurable_partial_derivatives:
+  fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n"
+  assumes S: "S \<in> sets lebesgue"
+    and f: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
+  shows "(\<lambda>x. (matrix(f' x)$m$n)) \<in> borel_measurable (lebesgue_on S)"
+proof -
+  have contf: "continuous_on S f"
+    using continuous_on_eq_continuous_within f has_derivative_continuous by blast
+  have "{x \<in> S.  (matrix (f' x)$m$n) \<le> b} \<in> sets lebesgue" for b
+  proof (rule sets_negligible_symdiff)
+    let ?T = "{x \<in> S. \<forall>e>0. \<exists>d>0. \<exists>A. A$m$n < b \<and> (\<forall>i j. A$i$j \<in> \<rat>) \<and>
+                       (\<forall>y \<in> S. norm(y - x) < d \<longrightarrow> norm(f y - f x - A *v (y - x)) \<le> e * norm(y - x))}"
+    let ?U = "S \<inter>
+              (\<Inter>e \<in> {e \<in> \<rat>. e > 0}.
+                \<Union>A \<in> {A. A$m$n < b \<and> (\<forall>i j. A$i$j \<in> \<rat>)}.
+                  \<Union>d \<in> {d \<in> \<rat>. 0 < d}.
+                     S \<inter> (\<Inter>y \<in> S. {x \<in> S. norm(y - x) < d \<longrightarrow> norm(f y - f x - A *v (y - x)) \<le> e * norm(y - x)}))"
+    have "?T = ?U"
+    proof (intro set_eqI iffI)
+      fix x
+      assume xT: "x \<in> ?T"
+      then show "x \<in> ?U"
+      proof (clarsimp simp add:)
+        fix q :: real
+        assume "q \<in> \<rat>" "q > 0"
+        then obtain d A where "d > 0" and A: "A $ m $ n < b" "\<And>i j. A $ i $ j \<in> \<rat>"
+          "\<And>y. \<lbrakk>y\<in>S;  norm (y - x) < d\<rbrakk> \<Longrightarrow> norm (f y - f x - A *v (y - x)) \<le> q * norm (y - x)"
+          using xT by auto
+        then obtain \<delta> where "d > \<delta>" "\<delta> > 0" "\<delta> \<in> \<rat>"
+          using Rats_dense_in_real by blast
+        with A show "\<exists>A. A $ m $ n < b \<and> (\<forall>i j. A $ i $ j \<in> \<rat>) \<and>
+                         (\<exists>s. s \<in> \<rat> \<and> 0 < s \<and> (\<forall>y\<in>S. norm (y - x) < s \<longrightarrow> norm (f y - f x - A *v (y - x)) \<le> q * norm (y - x)))"
+          by force
+      qed
+    next
+      fix x
+      assume xU: "x \<in> ?U"
+      then show "x \<in> ?T"
+      proof clarsimp
+        fix e :: "real"
+        assume "e > 0"
+        then obtain \<epsilon> where \<epsilon>: "e > \<epsilon>" "\<epsilon> > 0" "\<epsilon> \<in> \<rat>"
+          using Rats_dense_in_real by blast
+        with xU obtain A r where "x \<in> S" and Ar: "A $ m $ n < b" "\<forall>i j. A $ i $ j \<in> \<rat>" "r \<in> \<rat>" "r > 0"
+          and "\<forall>y\<in>S. norm (y - x) < r \<longrightarrow> norm (f y - f x - A *v (y - x)) \<le> \<epsilon> * norm (y - x)"
+          by (auto simp: split: if_split_asm)
+        then have "\<forall>y\<in>S. norm (y - x) < r \<longrightarrow> norm (f y - f x - A *v (y - x)) \<le> e * norm (y - x)"
+          by (meson \<open>e > \<epsilon>\<close> less_eq_real_def mult_right_mono norm_ge_zero order_trans)
+        then show "\<exists>d>0. \<exists>A. A $ m $ n < b \<and> (\<forall>i j. A $ i $ j \<in> \<rat>) \<and> (\<forall>y\<in>S. norm (y - x) < d \<longrightarrow> norm (f y - f x - A *v (y - x)) \<le> e * norm (y - x))"
+          using \<open>x \<in> S\<close> Ar by blast
+      qed
+    qed
+    moreover have "?U \<in> sets lebesgue"
+    proof -
+      have coQ: "countable {e \<in> \<rat>. 0 < e}"
+        using countable_Collect countable_rat by blast
+      have ne: "{e \<in> \<rat>. (0::real) < e} \<noteq> {}"
+        using zero_less_one Rats_1 by blast
+      have coA: "countable {A. A $ m $ n < b \<and> (\<forall>i j. A $ i $ j \<in> \<rat>)}"
+      proof (rule countable_subset)
+        show "countable {A. \<forall>i j. A $ i $ j \<in> \<rat>}"
+          using countable_vector [OF countable_vector, of "\<lambda>i j. \<rat>"] by (simp add: countable_rat)
+      qed blast
+      have *: "\<lbrakk>U \<noteq> {} \<Longrightarrow> closedin (subtopology euclidean S) (S \<inter> \<Inter> U)\<rbrakk>
+               \<Longrightarrow> closedin (subtopology euclidean S) (S \<inter> \<Inter> U)" for U
+        by fastforce
+      have eq: "{x::(real,'m)vec. P x \<and> (Q x \<longrightarrow> R x)} = {x. P x \<and> \<not> Q x} \<union> {x. P x \<and> R x}" for P Q R
+        by auto
+      have sets: "S \<inter> (\<Inter>y\<in>S. {x \<in> S. norm (y - x) < d \<longrightarrow> norm (f y - f x - A *v (y - x)) \<le> e * norm (y - x)})
+                  \<in> sets lebesgue" for e A d
+      proof -
+        have clo: "closedin (subtopology euclidean S)
+                     {x \<in> S. norm (y - x) < d \<longrightarrow> norm (f y - f x - A *v (y - x)) \<le> e * norm (y - x)}"
+          for y
+        proof -
+          have cont1: "continuous_on S (\<lambda>x. norm (y - x))"
+          and  cont2: "continuous_on S (\<lambda>x. e * norm (y - x) - norm (f y - f x - (A *v y - A *v x)))"
+            by (force intro: contf continuous_intros)+
+          have clo1: "closedin (subtopology euclidean S) {x \<in> S. d \<le> norm (y - x)}"
+            using continuous_closedin_preimage [OF cont1, of "{d..}"] by (simp add: vimage_def Int_def)
+          have clo2: "closedin (subtopology euclidean S)
+                       {x \<in> S. norm (f y - f x - (A *v y - A *v x)) \<le> e * norm (y - x)}"
+            using continuous_closedin_preimage [OF cont2, of "{0..}"] by (simp add: vimage_def Int_def)
+          show ?thesis
+            by (auto simp: eq not_less matrix_vector_mult_diff_distrib intro: clo1 clo2)
+        qed
+        show ?thesis
+          by (rule lebesgue_closedin [of S]) (force intro: * S clo)+
+      qed
+      show ?thesis
+        by (intro sets sets.Int S sets.countable_UN'' sets.countable_INT'' coQ coA) auto
+    qed
+    ultimately show "?T \<in> sets lebesgue"
+      by simp
+    let ?M = "(?T - {x \<in> S. matrix (f' x) $ m $ n \<le> b} \<union> ({x \<in> S. matrix (f' x) $ m $ n \<le> b} - ?T))"
+    let ?\<Theta> = "\<lambda>x v. \<forall>\<xi>>0. \<exists>e>0. \<forall>y \<in> S-{x}. norm (x - y) < e \<longrightarrow> \<bar>v \<bullet> (y - x)\<bar> < \<xi> * norm (x - y)"
+    have nN: "negligible {x \<in> S. \<exists>v\<noteq>0. ?\<Theta> x v}"
+      unfolding negligible_eq_zero_density
+    proof clarsimp
+      fix x v and r e :: "real"
+      assume "x \<in> S" "v \<noteq> 0" "r > 0" "e > 0"
+      and Theta [rule_format]: "?\<Theta> x v"
+      moreover have "(norm v * e / 2) / CARD('m) ^ CARD('m) > 0"
+        by (simp add: \<open>v \<noteq> 0\<close> \<open>e > 0\<close>)
+      ultimately obtain d where "d > 0"
+         and dless: "\<And>y. \<lbrakk>y \<in> S - {x}; norm (x - y) < d\<rbrakk> \<Longrightarrow>
+                        \<bar>v \<bullet> (y - x)\<bar> < ((norm v * e / 2) / CARD('m) ^ CARD('m)) * norm (x - y)"
+        by metis
+      let ?W = "ball x (min d r) \<inter> {y. \<bar>v \<bullet> (y - x)\<bar> < (norm v * e/2 * min d r) / CARD('m) ^ CARD('m)}"
+      have "open {x. \<bar>v \<bullet> (x - a)\<bar> < b}" for a b
+        by (intro open_Collect_less continuous_intros)
+      show "\<exists>d>0. d \<le> r \<and>
+            (\<exists>U. {x' \<in> S. \<exists>v\<noteq>0. ?\<Theta> x' v} \<inter> ball x d \<subseteq> U \<and>
+                 U \<in> lmeasurable \<and> measure lebesgue U < e * content (ball x d))"
+      proof (intro exI conjI)
+        show "0 < min d r" "min d r \<le> r"
+          using \<open>r > 0\<close> \<open>d > 0\<close> by auto
+        show "{x' \<in> S. \<exists>v. v \<noteq> 0 \<and> (\<forall>\<xi>>0. \<exists>e>0. \<forall>z\<in>S - {x'}. norm (x' - z) < e \<longrightarrow> \<bar>v \<bullet> (z - x')\<bar> < \<xi> * norm (x' - z))} \<inter> ball x (min d r) \<subseteq> ?W"
+          proof (clarsimp simp: dist_norm norm_minus_commute)
+            fix y :: "(real, 'm) vec" and w :: "(real, 'm) vec"
+            assume "y \<in> S" "w \<noteq> 0"
+              and less [rule_format]:
+                    "\<forall>\<xi>>0. \<exists>e>0. \<forall>z\<in>S - {y}. norm (y - z) < e \<longrightarrow> \<bar>w \<bullet> (z - y)\<bar> < \<xi> * norm (y - z)"
+              and d: "norm (y - x) < d" and r: "norm (y - x) < r"
+            show "\<bar>v \<bullet> (y - x)\<bar> < norm v * e * min d r / (2 * real CARD('m) ^ CARD('m))"
+            proof (cases "y = x")
+              case True
+              with \<open>r > 0\<close> \<open>d > 0\<close> \<open>e > 0\<close> \<open>v \<noteq> 0\<close> show ?thesis
+                by simp
+            next
+              case False
+              have "\<bar>v \<bullet> (y - x)\<bar> < norm v * e / 2 / real (CARD('m) ^ CARD('m)) * norm (x - y)"
+                apply (rule dless)
+                using False \<open>y \<in> S\<close> d by (auto simp: norm_minus_commute)
+              also have "\<dots> \<le> norm v * e * min d r / (2 * real CARD('m) ^ CARD('m))"
+                using d r \<open>e > 0\<close> by (simp add: field_simps norm_minus_commute mult_left_mono)
+              finally show ?thesis .
+            qed
+          qed
+          show "?W \<in> lmeasurable"
+            by (simp add: fmeasurable_Int_fmeasurable borel_open)
+          obtain k::'m where True
+            by metis
+          obtain T where T: "orthogonal_transformation T" and v: "v = T(norm v *\<^sub>R axis k (1::real))"
+            using rotation_rightward_line by metis
+          define b where "b \<equiv> norm v"
+          have "b > 0"
+            using \<open>v \<noteq> 0\<close> by (auto simp: b_def)
+          obtain eqb: "inv T v = b *\<^sub>R axis k (1::real)" and "inj T" "bij T" and invT: "orthogonal_transformation (inv T)"
+            by (metis UNIV_I b_def  T v bij_betw_inv_into_left orthogonal_transformation_inj orthogonal_transformation_bij orthogonal_transformation_inv)
+          let ?v = "\<chi> i. min d r / CARD('m)"
+          let ?v' = "\<chi> i. if i = k then (e/2 * min d r) / CARD('m) ^ CARD('m) else min d r"
+          let ?x' = "inv T x"
+          let ?W' = "(ball ?x' (min d r) \<inter> {y. \<bar>(y - ?x')$k\<bar> < e * min d r / (2 * CARD('m) ^ CARD('m))})"
+          have abs: "x - e \<le> y \<and> y \<le> x + e \<longleftrightarrow> abs(y - x) \<le> e" for x y e::real
+            by auto
+          have "?W = T ` ?W'"
+          proof -
+            have 1: "T ` (ball (inv T x) (min d r)) = ball x (min d r)"
+              by (simp add: T image_orthogonal_transformation_ball orthogonal_transformation_surj surj_f_inv_f)
+            have 2: "{y. \<bar>v \<bullet> (y - x)\<bar> < b * e * min d r / (2 * real CARD('m) ^ CARD('m))} =
+                      T ` {y. \<bar>y $ k - ?x' $ k\<bar> < e * min d r / (2 * real CARD('m) ^ CARD('m))}"
+            proof -
+              have *: "\<bar>T (b *\<^sub>R axis k 1) \<bullet> (y - x)\<bar> = b * \<bar>inv T y $ k - ?x' $ k\<bar>" for y
+              proof -
+                have "\<bar>T (b *\<^sub>R axis k 1) \<bullet> (y - x)\<bar> = \<bar>(b *\<^sub>R axis k 1) \<bullet> inv T (y - x)\<bar>"
+                  by (metis (no_types, hide_lams) b_def eqb invT orthogonal_transformation_def v)
+                also have "\<dots> = b * \<bar>(axis k 1) \<bullet> inv T (y - x)\<bar>"
+                  using \<open>b > 0\<close> by (simp add: abs_mult)
+                also have "\<dots> = b * \<bar>inv T y $ k - ?x' $ k\<bar>"
+                  using orthogonal_transformation_linear [OF invT]
+                  by (simp add: inner_axis' linear_diff)
+                finally show ?thesis
+                  by simp
+              qed
+              show ?thesis
+                using v b_def [symmetric]
+                using \<open>b > 0\<close> by (simp add: * bij_image_Collect_eq [OF \<open>bij T\<close>] mult_less_cancel_left_pos times_divide_eq_right [symmetric] del: times_divide_eq_right)
+            qed
+            show ?thesis
+              using \<open>b > 0\<close> by (simp add: image_Int \<open>inj T\<close> 1 2 b_def [symmetric])
+          qed
+          moreover have "?W' \<in> lmeasurable"
+            by (auto intro: fmeasurable_Int_fmeasurable)
+          ultimately have "measure lebesgue ?W = measure lebesgue ?W'"
+            by (metis measure_orthogonal_image T)
+          also have "\<dots> \<le> measure lebesgue (cbox (?x' - ?v') (?x' + ?v'))"
+          proof (rule measure_mono_fmeasurable)
+            show "?W' \<subseteq> cbox (?x' - ?v') (?x' + ?v')"
+              apply (clarsimp simp add: mem_box_cart abs dist_norm norm_minus_commute simp del: min_less_iff_conj min.bounded_iff)
+              by (metis component_le_norm_cart less_eq_real_def le_less_trans vector_minus_component)
+          qed auto
+          also have "\<dots> \<le> e/2 * measure lebesgue (cbox (?x' - ?v) (?x' + ?v))"
+          proof -
+            have "cbox (?x' - ?v) (?x' + ?v) \<noteq> {}"
+              using \<open>r > 0\<close> \<open>d > 0\<close> by (auto simp: interval_eq_empty_cart divide_less_0_iff)
+            with \<open>r > 0\<close> \<open>d > 0\<close> \<open>e > 0\<close> show ?thesis
+              apply (simp add: content_cbox_if_cart mem_box_cart)
+              apply (auto simp: prod_nonneg)
+              apply (simp add: abs if_distrib prod.delta_remove prod_constant field_simps power_diff split: if_split_asm)
+              done
+          qed
+          also have "\<dots> \<le> e/2 * measure lebesgue (cball ?x' (min d r))"
+          proof (rule mult_left_mono [OF measure_mono_fmeasurable])
+            have *: "norm (?x' - y) \<le> min d r"
+              if y: "\<And>i. \<bar>?x' $ i - y $ i\<bar> \<le> min d r / real CARD('m)" for y
+            proof -
+              have "norm (?x' - y) \<le> (\<Sum>i\<in>UNIV. \<bar>(?x' - y) $ i\<bar>)"
+                by (rule norm_le_l1_cart)
+              also have "\<dots> \<le> real CARD('m) * (min d r / real CARD('m))"
+                by (rule sum_bounded_above) (use y in auto)
+              finally show ?thesis
+                by simp
+            qed
+            show "cbox (?x' - ?v) (?x' + ?v) \<subseteq> cball ?x' (min d r)"
+              apply (clarsimp simp only: mem_box_cart dist_norm mem_cball intro!: *)
+              by (simp add: abs_diff_le_iff abs_minus_commute)
+          qed (use \<open>e > 0\<close> in auto)
+          also have "\<dots> < e * content (cball ?x' (min d r))"
+            using \<open>r > 0\<close> \<open>d > 0\<close> \<open>e > 0\<close> by auto
+          also have "\<dots> = e * content (ball x (min d r))"
+            using \<open>r > 0\<close> \<open>d > 0\<close> by (simp add: content_cball content_ball)
+          finally show "measure lebesgue ?W < e * content (ball x (min d r))" .
+      qed
+    qed
+    have *: "(\<And>x. (x \<notin> S) \<Longrightarrow> (x \<in> T \<longleftrightarrow> x \<in> U)) \<Longrightarrow> (T - U) \<union> (U - T) \<subseteq> S" for S T U :: "(real,'m) vec set"
+      by blast
+    have MN: "?M \<subseteq> {x \<in> S. \<exists>v\<noteq>0. ?\<Theta> x v}"
+    proof (rule *)
+      fix x
+      assume x: "x \<notin> {x \<in> S. \<exists>v\<noteq>0. ?\<Theta> x v}"
+      show "(x \<in> ?T) \<longleftrightarrow> (x \<in> {x \<in> S. matrix (f' x) $ m $ n \<le> b})"
+      proof (cases "x \<in> S")
+        case True
+        then have x: "\<not> ?\<Theta> x v" if "v \<noteq> 0" for v
+          using x that by force
+        show ?thesis
+        proof (rule iffI; clarsimp)
+          assume b: "\<forall>e>0. \<exists>d>0. \<exists>A. A $ m $ n < b \<and> (\<forall>i j. A $ i $ j \<in> \<rat>) \<and>
+                                    (\<forall>y\<in>S. norm (y - x) < d \<longrightarrow> norm (f y - f x - A *v (y - x)) \<le> e * norm (y - x))"
+                     (is "\<forall>e>0. \<exists>d>0. \<exists>A. ?\<Phi> e d A")
+          then have "\<forall>k. \<exists>d>0. \<exists>A. ?\<Phi> (1 / Suc k) d A"
+            by (metis (no_types, hide_lams) less_Suc_eq_0_disj of_nat_0_less_iff zero_less_divide_1_iff)
+          then obtain \<delta> A where \<delta>: "\<And>k. \<delta> k > 0"
+                           and Ab: "\<And>k. A k $ m $ n < b"
+                           and A: "\<And>k y. \<lbrakk>y \<in> S; norm (y - x) < \<delta> k\<rbrakk> \<Longrightarrow>
+                                          norm (f y - f x - A k *v (y - x)) \<le> 1/(Suc k) * norm (y - x)"
+            by metis
+          have "\<forall>i j. \<exists>a. (\<lambda>n. A n $ i $ j) \<longlonglongrightarrow> a"
+          proof (intro allI)
+            fix i j
+            have vax: "(A n *v axis j 1) $ i = A n $ i $ j" for n
+              by (metis cart_eq_inner_axis matrix_vector_mul_component)
+            let ?CA = "{x. Cauchy (\<lambda>n. (A n) *v x)}"
+            have "subspace ?CA"
+              unfolding subspace_def convergent_eq_Cauchy [symmetric]
+                by (force simp: algebra_simps intro: tendsto_intros)
+            then have CA_eq: "?CA = span ?CA"
+              by (metis span_eq)
+            also have "\<dots> = UNIV"
+            proof -
+              have "dim ?CA \<le> CARD('m)"
+                by (rule dim_subset_UNIV_cart)
+              moreover have "False" if less: "dim ?CA < CARD('m)"
+              proof -
+                obtain d where "d \<noteq> 0" and d: "\<And>y. y \<in> span ?CA \<Longrightarrow> orthogonal d y"
+                  using less by (force intro: orthogonal_to_subspace_exists [of ?CA])
+                with x [OF \<open>d \<noteq> 0\<close>] obtain \<xi> where "\<xi> > 0"
+                  and \<xi>: "\<And>e. e > 0 \<Longrightarrow> \<exists>y \<in> S - {x}. norm (x - y) < e \<and> \<xi> * norm (x - y) \<le> \<bar>d \<bullet> (y - x)\<bar>"
+                  by (fastforce simp: not_le Bex_def)
+                obtain \<gamma> z where \<gamma>Sx: "\<And>i. \<gamma> i \<in> S - {x}"
+                           and \<gamma>le:   "\<And>i. \<xi> * norm(\<gamma> i - x) \<le> \<bar>d \<bullet> (\<gamma> i - x)\<bar>"
+                           and \<gamma>x:    "\<gamma> \<longlonglongrightarrow> x"
+                           and z:     "(\<lambda>n. (\<gamma> n - x) /\<^sub>R norm (\<gamma> n - x)) \<longlonglongrightarrow> z"
+                proof -
+                  have "\<exists>\<gamma>. (\<forall>i. (\<gamma> i \<in> S - {x} \<and>
+                                  \<xi> * norm(\<gamma> i - x) \<le> \<bar>d \<bullet> (\<gamma> i - x)\<bar> \<and> norm(\<gamma> i - x) < 1/Suc i) \<and>
+                                 norm(\<gamma>(Suc i) - x) < norm(\<gamma> i - x))"
+                  proof (rule dependent_nat_choice)
+                    show "\<exists>y. y \<in> S - {x} \<and> \<xi> * norm (y - x) \<le> \<bar>d \<bullet> (y - x)\<bar> \<and> norm (y - x) < 1 / Suc 0"
+                      using \<xi> [of 1] by (auto simp: dist_norm norm_minus_commute)
+                  next
+                    fix y i
+                    assume "y \<in> S - {x} \<and> \<xi> * norm (y - x) \<le> \<bar>d \<bullet> (y - x)\<bar> \<and> norm (y - x) < 1/Suc i"
+                    then have "min (norm(y - x)) (1/((Suc i) + 1)) > 0"
+                      by auto
+                    then obtain y' where "y' \<in> S - {x}" and y': "norm (x - y') < min (norm (y - x)) (1/((Suc i) + 1))"
+                                         "\<xi> * norm (x - y') \<le> \<bar>d \<bullet> (y' - x)\<bar>"
+                      using \<xi> by metis
+                    with \<xi> show "\<exists>y'. (y' \<in> S - {x} \<and> \<xi> * norm (y' - x) \<le> \<bar>d \<bullet> (y' - x)\<bar> \<and>
+                              norm (y' - x) < 1/(Suc (Suc i))) \<and> norm (y' - x) < norm (y - x)"
+                      by (auto simp: dist_norm norm_minus_commute)
+                  qed
+                  then obtain \<gamma> where
+                        \<gamma>Sx: "\<And>i. \<gamma> i \<in> S - {x}"
+                        and \<gamma>le: "\<And>i. \<xi> * norm(\<gamma> i - x) \<le> \<bar>d \<bullet> (\<gamma> i - x)\<bar>"
+                        and \<gamma>conv: "\<And>i. norm(\<gamma> i - x) < 1/(Suc i)"
+                    by blast
+                  let ?f = "\<lambda>i. (\<gamma> i - x) /\<^sub>R norm (\<gamma> i - x)"
+                  have "?f i \<in> sphere 0 1" for i
+                    using \<gamma>Sx by auto
+                  then obtain l \<rho> where "l \<in> sphere 0 1" "strict_mono \<rho>" and l: "(?f \<circ> \<rho>) \<longlonglongrightarrow> l"
+                    using compact_sphere [of "0::(real,'m) vec" 1]  unfolding compact_def by meson
+                  show thesis
+                  proof
+                    show "(\<gamma> \<circ> \<rho>) i \<in> S - {x}" "\<xi> * norm ((\<gamma> \<circ> \<rho>) i - x) \<le> \<bar>d \<bullet> ((\<gamma> \<circ> \<rho>) i - x)\<bar>" for i
+                      using \<gamma>Sx \<gamma>le by auto
+                    have "\<gamma> \<longlonglongrightarrow> x"
+                    proof (clarsimp simp add: LIMSEQ_def dist_norm)
+                      fix r :: "real"
+                      assume "r > 0"
+                      with real_arch_invD obtain no where "no \<noteq> 0" "real no > 1/r"
+                        by (metis divide_less_0_1_iff not_less_iff_gr_or_eq of_nat_0_eq_iff reals_Archimedean2)
+                      with \<gamma>conv show "\<exists>no. \<forall>n\<ge>no. norm (\<gamma> n - x) < r"
+                        by (metis \<open>r > 0\<close> add.commute divide_inverse inverse_inverse_eq inverse_less_imp_less less_trans mult.left_neutral nat_le_real_less of_nat_Suc)
+                    qed
+                    with \<open>strict_mono \<rho>\<close> show "(\<gamma> \<circ> \<rho>) \<longlonglongrightarrow> x"
+                      by (metis LIMSEQ_subseq_LIMSEQ)
+                    show "(\<lambda>n. ((\<gamma> \<circ> \<rho>) n - x) /\<^sub>R norm ((\<gamma> \<circ> \<rho>) n - x)) \<longlonglongrightarrow> l"
+                      using l by (auto simp: o_def)
+                  qed
+                qed
+                have "isCont (\<lambda>x. (\<bar>d \<bullet> x\<bar> - \<xi>)) z"
+                  by (intro continuous_intros)
+                from isCont_tendsto_compose [OF this z]
+                have lim: "(\<lambda>y. \<bar>d \<bullet> ((\<gamma> y - x) /\<^sub>R norm (\<gamma> y - x))\<bar> - \<xi>) \<longlonglongrightarrow> \<bar>d \<bullet> z\<bar> - \<xi>"
+                  by auto
+                moreover have "\<forall>\<^sub>F i in sequentially. 0 \<le> \<bar>d \<bullet> ((\<gamma> i - x) /\<^sub>R norm (\<gamma> i - x))\<bar> - \<xi>"
+                proof (rule eventuallyI)
+                  fix n
+                  show "0 \<le> \<bar>d \<bullet> ((\<gamma> n - x) /\<^sub>R norm (\<gamma> n - x))\<bar> - \<xi>"
+                  using \<gamma>le [of n] \<gamma>Sx by (auto simp: abs_mult divide_simps)
+                qed
+                ultimately have "\<xi> \<le> \<bar>d \<bullet> z\<bar>"
+                  using tendsto_lowerbound [where a=0] by fastforce
+                have "Cauchy (\<lambda>n. (A n) *v z)"
+                proof (clarsimp simp add: Cauchy_def)
+                  fix \<epsilon> :: "real"
+                  assume "0 < \<epsilon>"
+                  then obtain N::nat where "N > 0" and N: "\<epsilon>/2 > 1/N"
+                    by (metis half_gt_zero inverse_eq_divide neq0_conv real_arch_inverse)
+                  show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (A m *v z) (A n *v z) < \<epsilon>"
+                  proof (intro exI allI impI)
+                    fix i j
+                    assume ij: "N \<le> i" "N \<le> j"
+                    let ?V = "\<lambda>i k. A i *v ((\<gamma> k - x) /\<^sub>R norm (\<gamma> k - x))"
+                    have "\<forall>\<^sub>F k in sequentially. dist (\<gamma> k) x < min (\<delta> i) (\<delta> j)"
+                      using \<gamma>x [unfolded tendsto_iff] by (meson min_less_iff_conj \<delta>)
+                    then have even: "\<forall>\<^sub>F k in sequentially. norm (?V i k - ?V j k) - 2 / N \<le> 0"
+                    proof (rule eventually_mono, clarsimp)
+                      fix p
+                      assume p: "dist (\<gamma> p) x < \<delta> i" "dist (\<gamma> p) x < \<delta> j"
+                      let ?C = "\<lambda>k. f (\<gamma> p) - f x - A k *v (\<gamma> p - x)"
+                      have "norm ((A i - A j) *v (\<gamma> p - x)) = norm (?C j - ?C i)"
+                        by (simp add: algebra_simps)
+                      also have "\<dots> \<le> norm (?C j) + norm (?C i)"
+                        using norm_triangle_ineq4 by blast
+                      also have "\<dots> \<le> 1/(Suc j) * norm (\<gamma> p - x) + 1/(Suc i) * norm (\<gamma> p - x)"
+                        by (metis A Diff_iff \<gamma>Sx dist_norm p add_mono)
+                      also have "\<dots> \<le> 1/N * norm (\<gamma> p - x) + 1/N * norm (\<gamma> p - x)"
+                        apply (intro add_mono mult_right_mono)
+                        using ij \<open>N > 0\<close> by (auto simp: field_simps)
+                      also have "\<dots> = 2 / N * norm (\<gamma> p - x)"
+                        by simp
+                      finally have no_le: "norm ((A i - A j) *v (\<gamma> p - x)) \<le> 2 / N * norm (\<gamma> p - x)" .
+                      have "norm (?V i p - ?V j p) =
+                            norm ((A i - A j) *v ((\<gamma> p - x) /\<^sub>R norm (\<gamma> p - x)))"
+                        by (simp add: algebra_simps)
+                      also have "\<dots> = norm ((A i - A j) *v (\<gamma> p - x)) / norm (\<gamma> p - x)"
+                        by (simp add: divide_inverse matrix_vector_mult_scaleR)
+                      also have "\<dots> \<le> 2 / N"
+                        using no_le by (auto simp: divide_simps)
+                      finally show "norm (?V i p - ?V j p) \<le> 2 / N" .
+                    qed
+                    have "isCont (\<lambda>w. (norm(A i *v w - A j *v w) - 2 / N)) z"
+                      by (intro continuous_intros)
+                    from isCont_tendsto_compose [OF this z]
+                    have lim: "(\<lambda>w. norm (A i *v ((\<gamma> w - x) /\<^sub>R norm (\<gamma> w - x)) -
+                                    A j *v ((\<gamma> w - x) /\<^sub>R norm (\<gamma> w - x))) - 2 / N)
+                               \<longlonglongrightarrow> norm (A i *v z - A j *v z) - 2 / N"
+                      by auto
+                    have "dist (A i *v z) (A j *v z) \<le> 2 / N"
+                      using tendsto_upperbound [OF lim even] by (auto simp: dist_norm)
+                    with N show "dist (A i *v z) (A j *v z) < \<epsilon>"
+                      by linarith
+                  qed
+                qed
+                then have "d \<bullet> z = 0"
+                  using CA_eq d orthogonal_def by auto
+                then show False
+                  using \<open>0 < \<xi>\<close> \<open>\<xi> \<le> \<bar>d \<bullet> z\<bar>\<close> by auto
+              qed
+              ultimately show ?thesis
+                using dim_eq_full by fastforce
+            qed
+            finally have "?CA = UNIV" .
+            then have "Cauchy (\<lambda>n. (A n) *v axis j 1)"
+              by auto
+            then obtain L where "(\<lambda>n. A n *v axis j 1) \<longlonglongrightarrow> L"
+              by (auto simp: Cauchy_convergent_iff convergent_def)
+            then have "(\<lambda>x. (A x *v axis j 1) $ i) \<longlonglongrightarrow> L $ i"
+              by (rule tendsto_vec_nth)
+            then show "\<exists>a. (\<lambda>n. A n $ i $ j) \<longlonglongrightarrow> a"
+              by (force simp: vax)
+          qed
+          then obtain B where B: "\<And>i j. (\<lambda>n. A n $ i $ j) \<longlonglongrightarrow> B $ i $ j"
+            by (auto simp: lambda_skolem)
+          have lin_df: "linear (f' x)"
+               and lim_df: "((\<lambda>y. (1 / norm (y - x)) *\<^sub>R (f y - (f x + f' x (y - x)))) \<longlongrightarrow> 0) (at x within S)"
+            using \<open>x \<in> S\<close> assms by (auto simp: has_derivative_within linear_linear)
+          moreover have "(matrix (f' x) - B) *v w = 0" for w
+          proof (rule lemma_partial_derivatives [of "( *v) (matrix (f' x) - B)"])
+            show "linear (( *v) (matrix (f' x) - B))"
+              by (rule matrix_vector_mul_linear)
+            have "((\<lambda>y. ((f x + f' x (y - x)) - f y) /\<^sub>R norm (y - x)) \<longlongrightarrow> 0) (at x within S)"
+              using tendsto_minus [OF lim_df] by (simp add: algebra_simps divide_simps)
+            then show "((\<lambda>y. (matrix (f' x) - B) *v (y - x) /\<^sub>R norm (y - x)) \<longlongrightarrow> 0) (at x within S)"
+            proof (rule Lim_transform)
+              have "((\<lambda>y. ((f y + B *v x - (f x + B *v y)) /\<^sub>R norm (y - x))) \<longlongrightarrow> 0) (at x within S)"
+              proof (clarsimp simp add: Lim_within dist_norm)
+                fix e :: "real"
+                assume "e > 0"
+                then obtain q::nat where "q \<noteq> 0" and qe2: "1/q < e/2"
+                  by (metis divide_pos_pos inverse_eq_divide real_arch_inverse zero_less_numeral)
+                let ?g = "\<lambda>p. sum  (\<lambda>i. sum (\<lambda>j. abs((A p - B)$i$j)) UNIV) UNIV"
+                have "(\<lambda>k. onorm (\<lambda>y. (A k - B) *v y)) \<longlonglongrightarrow> 0"
+                proof (rule Lim_null_comparison)
+                  show "\<forall>\<^sub>F k in sequentially. norm (onorm (\<lambda>y. (A k - B) *v y)) \<le> ?g k"
+                  proof (rule eventually_sequentiallyI)
+                    fix k :: "nat"
+                    assume "0 \<le> k"
+                    have "0 \<le> onorm (( *v) (A k - B))"
+                      by (simp add: linear_linear onorm_pos_le matrix_vector_mul_linear)
+                    then show "norm (onorm (( *v) (A k - B))) \<le> (\<Sum>i\<in>UNIV. \<Sum>j\<in>UNIV. \<bar>(A k - B) $ i $ j\<bar>)"
+                      by (simp add: onorm_le_matrix_component_sum del: vector_minus_component)
+                  qed
+                next
+                  show "?g \<longlonglongrightarrow> 0"
+                    using B Lim_null tendsto_rabs_zero_iff by (fastforce intro!: tendsto_null_sum)
+                qed
+                with \<open>e > 0\<close> obtain p where "\<And>n. n \<ge> p \<Longrightarrow> \<bar>onorm (( *v) (A n - B))\<bar> < e/2"
+                  unfolding lim_sequentially by (metis diff_zero dist_real_def divide_pos_pos zero_less_numeral)
+                then have pqe2: "\<bar>onorm (( *v) (A (p + q) - B))\<bar> < e/2" (*17 [`abs (onorm (\y. A (p + q) ** y - B ** y)) < e / &2`]*)
+                  using le_add1 by blast
+                show "\<exists>d>0. \<forall>y\<in>S. y \<noteq> x \<and> norm (y - x) < d \<longrightarrow>
+                           inverse (norm (y - x)) * norm (f y + B *v x - (f x + B *v y)) < e"
+                proof (intro exI, safe)
+                  show "0 < \<delta>(p + q)"
+                    by (simp add: \<delta>)
+                next
+                  fix y
+                  assume y: "y \<in> S" "norm (y - x) < \<delta>(p + q)" and "y \<noteq> x"
+                  have *: "\<lbrakk>norm(b - c) < e - d; norm(y - x - b) \<le> d\<rbrakk> \<Longrightarrow> norm(y - x - c) < e"
+                    for b c d e x and y:: "real^'n"
+                    using norm_triangle_ineq2 [of "y - x - c" "y - x - b"] by simp
+                  have "norm (f y - f x - B *v (y - x)) < e * norm (y - x)"
+                  proof (rule *)
+                    show "norm (f y - f x - A (p + q) *v (y - x)) \<le> norm (y - x) / (Suc (p + q))"
+                      using A [OF y] by simp
+                    have "norm (A (p + q) *v (y - x) - B *v (y - x)) \<le> onorm(\<lambda>x. (A(p + q) - B) *v x) * norm(y - x)"
+                      by (metis linear_linear matrix_vector_mul_linear matrix_vector_mult_diff_rdistrib onorm)
+                    also have "\<dots> < (e/2) * norm (y - x)"
+                      using \<open>y \<noteq> x\<close> pqe2 by auto
+                    also have "\<dots> \<le> (e - 1 / (Suc (p + q))) * norm (y - x)"
+                    proof (rule mult_right_mono)
+                      have "1 / Suc (p + q) \<le> 1 / q"
+                        using \<open>q \<noteq> 0\<close> by (auto simp: divide_simps)
+                      also have "\<dots> < e/2"
+                        using qe2 by auto
+                      finally show "e / 2 \<le> e - 1 / real (Suc (p + q))"
+                        by linarith
+                    qed auto
+                    finally show "norm (A (p + q) *v (y - x) - B *v (y - x)) < e * norm (y - x) - norm (y - x) / real (Suc (p + q))"
+                      by (simp add: algebra_simps)
+                  qed
+                  then show "inverse (norm (y - x)) * norm (f y + B *v x - (f x + B *v y)) < e"
+                    using \<open>y \<noteq> x\<close> by (simp add: divide_simps algebra_simps)
+                qed
+              qed
+              then show "((\<lambda>y. (matrix (f' x) - B) *v (y - x) /\<^sub>R
+                           norm (y - x) - (f x + f' x (y - x) - f y) /\<^sub>R norm (y - x)) \<longlongrightarrow> 0)
+                          (at x within S)"
+                by (simp add: algebra_simps lin_df linear_diff matrix_vector_mul_linear)
+            qed
+          qed (use x in \<open>simp; auto simp: not_less\<close>)
+          ultimately have "f' x = ( *v) B"
+            by (force simp: algebra_simps)
+          show "matrix (f' x) $ m $ n \<le> b"
+          proof (rule tendsto_upperbound [of "\<lambda>i. (A i $ m $ n)" _ sequentially])
+            show "(\<lambda>i. A i $ m $ n) \<longlonglongrightarrow> matrix (f' x) $ m $ n"
+              by (simp add: B \<open>f' x = ( *v) B\<close>)
+            show "\<forall>\<^sub>F i in sequentially. A i $ m $ n \<le> b"
+              by (simp add: Ab less_eq_real_def)
+          qed auto
+        next
+          fix e :: "real"
+          assume "x \<in> S" and b: "matrix (f' x) $ m $ n \<le> b" and "e > 0"
+          then obtain d where "d>0"
+            and d: "\<And>y. y\<in>S \<Longrightarrow> 0 < dist y x \<and> dist y x < d \<longrightarrow> norm (f y - f x - f' x (y - x)) / (norm (y - x))
+                  < e/2"
+            using f [OF \<open>x \<in> S\<close>] unfolding Deriv.has_derivative_at_within Lim_within
+            by (auto simp: field_simps dest: spec [of _ "e/2"])
+          let ?A = "matrix(f' x) - (\<chi> i j. if i = m \<and> j = n then e / 4 else 0)"
+          obtain B where BRats: "\<And>i j. B$i$j \<in> \<rat>" and Bo_e6: "onorm(( *v) (?A - B)) < e/6"
+            using matrix_rational_approximation \<open>e > 0\<close>
+            by (metis zero_less_divide_iff zero_less_numeral)
+          show "\<exists>d>0. \<exists>A. A $ m $ n < b \<and> (\<forall>i j. A $ i $ j \<in> \<rat>) \<and>
+                (\<forall>y\<in>S. norm (y - x) < d \<longrightarrow> norm (f y - f x - A *v (y - x)) \<le> e * norm (y - x))"
+          proof (intro exI conjI ballI allI impI)
+            show "d>0"
+              by (rule \<open>d>0\<close>)
+            show "B $ m $ n < b"
+            proof -
+              have "\<bar>matrix (( *v) (?A - B)) $ m $ n\<bar> \<le> onorm (( *v) (?A - B))"
+                using component_le_onorm [OF matrix_vector_mul_linear, of _ m n] by metis
+              then show ?thesis
+                using b Bo_e6 by simp
+            qed
+            show "B $ i $ j \<in> \<rat>" for i j
+              using BRats by auto
+            show "norm (f y - f x - B *v (y - x)) \<le> e * norm (y - x)"
+              if "y \<in> S" and y: "norm (y - x) < d" for y
+            proof (cases "y = x")
+              case True then show ?thesis
+                by simp
+            next
+              case False
+              have *: "norm(d' - d) \<le> e/2 \<Longrightarrow> norm(y - (x + d')) < e/2 \<Longrightarrow> norm(y - x - d) \<le> e" for d d' e and x y::"real^'n"
+                using norm_triangle_le [of "d' - d" "y - (x + d')"] by simp
+              show ?thesis
+              proof (rule *)
+                have split246: "\<lbrakk>norm y \<le> e / 6; norm(x - y) \<le> e / 4\<rbrakk> \<Longrightarrow> norm x \<le> e/2" if "e > 0" for e and x y :: "real^'n"
+                  using norm_triangle_le [of y "x-y" "e/2"] \<open>e > 0\<close> by simp
+                have "linear (f' x)"
+                  using True f has_derivative_linear by blast
+                then have "norm (f' x (y - x) - B *v (y - x)) = norm ((matrix (f' x) - B) *v (y - x))"
+                  by (metis matrix_vector_mul matrix_vector_mult_diff_rdistrib)
+                also have "\<dots> \<le> (e * norm (y - x)) / 2"
+                proof (rule split246)
+                  have "norm ((?A - B) *v (y - x)) / norm (y - x) \<le> onorm(\<lambda>x. (?A - B) *v x)"
+                    by (simp add: le_onorm linear_linear matrix_vector_mul_linear)
+                  also have  "\<dots> < e/6"
+                    by (rule Bo_e6)
+                  finally have "norm ((?A - B) *v (y - x)) / norm (y - x) < e / 6" .
+                  then show "norm ((?A - B) *v (y - x)) \<le> e * norm (y - x) / 6"
+                    by (simp add: divide_simps False)
+                  have "norm ((matrix (f' x) - B) *v (y - x) - ((?A - B) *v (y - x))) = norm ((\<chi> i j. if i = m \<and> j = n then e / 4 else 0) *v (y - x))"
+                    by (simp add: algebra_simps)
+                  also have "\<dots> = norm((e/4) *\<^sub>R (y - x)$n *\<^sub>R axis m (1::real))"
+                  proof -
+                    have "(\<Sum>j\<in>UNIV. (if i = m \<and> j = n then e / 4 else 0) * (y $ j - x $ j)) * 4 = e * (y $ n - x $ n) * axis m 1 $ i" for i
+                    proof (cases "i=m")
+                      case True then show ?thesis
+                        by (auto simp: if_distrib [of "\<lambda>z. z * _"] cong: if_cong)
+                    next
+                      case False then show ?thesis
+                        by (simp add: axis_def)
+                    qed
+                    then have "(\<chi> i j. if i = m \<and> j = n then e / 4 else 0) *v (y - x) = (e/4) *\<^sub>R (y - x)$n *\<^sub>R axis m (1::real)"
+                      by (auto simp: vec_eq_iff matrix_vector_mult_def)
+                    then show ?thesis
+                      by metis
+                  qed
+                  also have "\<dots> \<le> e * norm (y - x) / 4"
+                    using \<open>e > 0\<close> apply (simp add: norm_mult abs_mult)
+                    by (metis component_le_norm_cart vector_minus_component)
+                  finally show "norm ((matrix (f' x) - B) *v (y - x) - ((?A - B) *v (y - x))) \<le> e * norm (y - x) / 4" .
+                  show "0 < e * norm (y - x)"
+                    by (simp add: False \<open>e > 0\<close>)
+                qed
+                finally show "norm (f' x (y - x) - B *v (y - x)) \<le> (e * norm (y - x)) / 2" .
+                show "norm (f y - (f x + f' x (y - x))) < (e * norm (y - x)) / 2"
+                  using False d [OF \<open>y \<in> S\<close>] y by (simp add: dist_norm field_simps)
+              qed
+            qed
+          qed
+        qed
+      qed auto
+    qed
+    show "negligible ?M"
+      using negligible_subset [OF nN MN] .
+  qed
+  then show ?thesis
+    by (simp add: borel_measurable_vimage_halfspace_component_le sets_restrict_space_iff assms)
+qed
+
+
+theorem borel_measurable_det_Jacobian:
+ fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
+  assumes S: "S \<in> sets lebesgue" and f: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
+  shows "(\<lambda>x. det(matrix(f' x))) \<in> borel_measurable (lebesgue_on S)"
+  unfolding det_def
+  by (intro measurable) (auto intro: f borel_measurable_partial_derivatives [OF S])
+
+text\<open>The localisation wrt S uses the same argument for many similar results.
+See HOL Light's MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_BOREL, etc.\<close>
+lemma borel_measurable_lebesgue_on_preimage_borel:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes "S \<in> sets lebesgue"
+  shows "f \<in> borel_measurable (lebesgue_on S) \<longleftrightarrow>
+         (\<forall>T. T \<in> sets borel \<longrightarrow> {x \<in> S. f x \<in> T} \<in> sets lebesgue)"
+proof -
+  have "{x. (if x \<in> S then f x else 0) \<in> T} \<in> sets lebesgue \<longleftrightarrow> {x \<in> S. f x \<in> T} \<in> sets lebesgue"
+         if "T \<in> sets borel" for T
+    proof (cases "0 \<in> T")
+      case True
+      then have "{x \<in> S. f x \<in> T} = {x. (if x \<in> S then f x else 0) \<in> T} \<inter> S"
+                "{x. (if x \<in> S then f x else 0) \<in> T} = {x \<in> S. f x \<in> T} \<union> -S"
+        by auto
+      then show ?thesis
+        by (metis (no_types, lifting) Compl_in_sets_lebesgue assms sets.Int sets.Un)
+    next
+      case False
+      then have "{x. (if x \<in> S then f x else 0) \<in> T} = {x \<in> S. f x \<in> T}"
+        by auto
+      then show ?thesis
+        by auto
+    qed
+    then show ?thesis
+      unfolding borel_measurable_lebesgue_preimage_borel borel_measurable_UNIV [OF assms, symmetric]
+      by blast
+qed
+
+lemma sets_lebesgue_almost_borel:
+  assumes "S \<in> sets lebesgue"
+  obtains B N where "B \<in> sets borel" "negligible N" "B \<union> N = S"
+proof -
+  obtain T N N' where "S = T \<union> N" "N \<subseteq> N'" "N' \<in> null_sets lborel" "T \<in> sets borel"
+    using sets_completionE [OF assms] by auto
+  then show thesis
+    by (metis negligible_iff_null_sets negligible_subset null_sets_completionI that)
+qed
+
+lemma double_lebesgue_sets:
+ assumes S: "S \<in> sets lebesgue" and T: "T \<in> sets lebesgue" and fim: "f ` S \<subseteq> T"
+ shows "(\<forall>U. U \<in> sets lebesgue \<and> U \<subseteq> T \<longrightarrow> {x \<in> S. f x \<in> U} \<in> sets lebesgue) \<longleftrightarrow>
+          f \<in> borel_measurable (lebesgue_on S) \<and>
+          (\<forall>U. negligible U \<and> U \<subseteq> T \<longrightarrow> {x \<in> S. f x \<in> U} \<in> sets lebesgue)"
+         (is "?lhs \<longleftrightarrow> _ \<and> ?rhs")
+  unfolding borel_measurable_lebesgue_on_preimage_borel [OF S]
+proof (intro iffI allI conjI impI, safe)
+  fix V :: "'b set"
+  assume *: "\<forall>U. U \<in> sets lebesgue \<and> U \<subseteq> T \<longrightarrow> {x \<in> S. f x \<in> U} \<in> sets lebesgue"
+    and "V \<in> sets borel"
+  then have V: "V \<in> sets lebesgue"
+    by simp
+  have "{x \<in> S. f x \<in> V} = {x \<in> S. f x \<in> T \<inter> V}"
+    using fim by blast
+  also have "{x \<in> S. f x \<in> T \<inter> V} \<in> sets lebesgue"
+    using T V * le_inf_iff by blast
+  finally show "{x \<in> S. f x \<in> V} \<in> sets lebesgue" .
+next
+  fix U :: "'b set"
+  assume "\<forall>U. U \<in> sets lebesgue \<and> U \<subseteq> T \<longrightarrow> {x \<in> S. f x \<in> U} \<in> sets lebesgue"
+         "negligible U" "U \<subseteq> T"
+  then show "{x \<in> S. f x \<in> U} \<in> sets lebesgue"
+    using negligible_imp_sets by blast
+next
+  fix U :: "'b set"
+  assume 1 [rule_format]: "(\<forall>T. T \<in> sets borel \<longrightarrow> {x \<in> S. f x \<in> T} \<in> sets lebesgue)"
+     and 2 [rule_format]: "\<forall>U. negligible U \<and> U \<subseteq> T \<longrightarrow> {x \<in> S. f x \<in> U} \<in> sets lebesgue"
+     and "U \<in> sets lebesgue" "U \<subseteq> T"
+  then obtain C N where C: "C \<in> sets borel \<and> negligible N \<and> C \<union> N = U"
+    using sets_lebesgue_almost_borel
+    by metis
+  then have "{x \<in> S. f x \<in> C} \<in> sets lebesgue"
+    by (blast intro: 1)
+  moreover have "{x \<in> S. f x \<in> N} \<in> sets lebesgue"
+    using C \<open>U \<subseteq> T\<close> by (blast intro: 2)
+  moreover have "{x \<in> S. f x \<in> C \<union> N} = {x \<in> S. f x \<in> C} \<union> {x \<in> S. f x \<in> N}"
+    by auto
+  ultimately show "{x \<in> S. f x \<in> U} \<in> sets lebesgue"
+    using C by auto
+qed
+
+
+
+thm integrable_on_subcbox
+
+proposition measurable_bounded_by_integrable_imp_integrable:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes f: "f \<in> borel_measurable (lebesgue_on S)" and g: "g integrable_on S"
+    and normf: "\<And>x. x \<in> S \<Longrightarrow> norm(f x) \<le> g x" and S: "S \<in> sets lebesgue"
+  shows "f integrable_on S"
+proof (rule integrable_on_all_intervals_integrable_bound [OF _ normf g])
+  show "(\<lambda>x. if x \<in> S then f x else 0) integrable_on cbox a b" for a b
+  proof (rule measurable_bounded_lemma)
+    show "(\<lambda>x. if x \<in> S then f x else 0) \<in> borel_measurable lebesgue"
+      by (simp add: S borel_measurable_UNIV f)
+    show "(\<lambda>x. if x \<in> S then g x else 0) integrable_on cbox a b"
+      by (simp add: g integrable_altD(1))
+    show "norm (if x \<in> S then f x else 0) \<le> (if x \<in> S then g x else 0)" for x
+      using normf by simp
+  qed
+qed
+
+
+subsection\<open>Simplest case of Sard's theorem (we don't need continuity of derivative)\<close>
+
+lemma Sard_lemma00:
+  fixes P :: "'b::euclidean_space set"
+  assumes "a \<ge> 0" and a: "a *\<^sub>R i \<noteq> 0" and i: "i \<in> Basis"
+    and P: "P \<subseteq> {x. a *\<^sub>R i \<bullet> x = 0}"
+    and "0 \<le> m" "0 \<le> e"
+ obtains S where "S \<in> lmeasurable"
+            and "{z. norm z \<le> m \<and> (\<exists>t \<in> P. norm(z - t) \<le> e)} \<subseteq> S"
+            and "measure lebesgue S \<le> (2 * e) * (2 * m) ^ (DIM('b) - 1)"
+proof -
+  have "a > 0"
+    using assms by simp
+  let ?v = "(\<Sum>j\<in>Basis. (if j = i then e else m) *\<^sub>R j)"
+  show thesis
+  proof
+    have "- e \<le> x \<bullet> i" "x \<bullet> i \<le> e"
+      if "t \<in> P" "norm (x - t) \<le> e" for x t
+      using \<open>a > 0\<close> that Basis_le_norm [of i "x-t"] P i
+      by (auto simp: inner_commute algebra_simps)
+    moreover have "- m \<le> x \<bullet> j" "x \<bullet> j \<le> m"
+      if "norm x \<le> m" "t \<in> P" "norm (x - t) \<le> e" "j \<in> Basis" and "j \<noteq> i"
+      for x t j
+      using that Basis_le_norm [of j x] by auto
+    ultimately
+    show "{z. norm z \<le> m \<and> (\<exists>t\<in>P. norm (z - t) \<le> e)} \<subseteq> cbox (-?v) ?v"
+      by (auto simp: mem_box)
+    have *: "\<forall>k\<in>Basis. - ?v \<bullet> k \<le> ?v \<bullet> k"
+      using \<open>0 \<le> m\<close> \<open>0 \<le> e\<close> by (auto simp: inner_Basis)
+    have 2: "2 ^ DIM('b) = 2 * 2 ^ (DIM('b) - Suc 0)"
+      by (metis DIM_positive Suc_pred power_Suc)
+    show "measure lebesgue (cbox (-?v) ?v) \<le> 2 * e * (2 * m) ^ (DIM('b) - 1)"
+      using \<open>i \<in> Basis\<close>
+      by (simp add: content_cbox [OF *] prod.distrib prod.If_cases Diff_eq [symmetric] 2)
+  qed blast
+qed
+
+text\<open>As above, but reorienting the vector (HOL Light's @text{GEOM_BASIS_MULTIPLE_TAC})\<close>
+lemma Sard_lemma0:
+  fixes P :: "(real^'n::{finite,wellorder}) set"
+  assumes "a \<noteq> 0"
+    and P: "P \<subseteq> {x. a \<bullet> x = 0}" and "0 \<le> m" "0 \<le> e"
+  obtains S where "S \<in> lmeasurable"
+    and "{z. norm z \<le> m \<and> (\<exists>t \<in> P. norm(z - t) \<le> e)} \<subseteq> S"
+    and "measure lebesgue S \<le> (2 * e) * (2 * m) ^ (CARD('n) - 1)"
+proof -
+  obtain T and k::'n where T: "orthogonal_transformation T" and a: "a = T (norm a *\<^sub>R axis k (1::real))"
+    using rotation_rightward_line by metis
+  have Tinv [simp]: "T (inv T x) = x" for x
+    by (simp add: T orthogonal_transformation_surj surj_f_inv_f)
+  obtain S where S: "S \<in> lmeasurable"
+    and subS: "{z. norm z \<le> m \<and> (\<exists>t \<in> T-`P. norm(z - t) \<le> e)} \<subseteq> S"
+    and mS: "measure lebesgue S \<le> (2 * e) * (2 * m) ^ (CARD('n) - 1)"
+  proof (rule Sard_lemma00 [of "norm a" "axis k (1::real)" "T-`P" m e])
+    have "norm a *\<^sub>R axis k 1 \<bullet> x = 0" if "T x \<in> P" for x
+    proof -
+      have "a \<bullet> T x = 0"
+        using P that by blast
+      then show ?thesis
+        by (metis (no_types, lifting) T a orthogonal_orthogonal_transformation orthogonal_def)
+    qed
+    then show "T -` P \<subseteq> {x. norm a *\<^sub>R axis k 1 \<bullet> x = 0}"
+      by auto
+  qed (use assms T in auto)
+  show thesis
+  proof
+    show "T ` S \<in> lmeasurable"
+      using S measurable_orthogonal_image T by blast
+    have "{z. norm z \<le> m \<and> (\<exists>t\<in>P. norm (z - t) \<le> e)} \<subseteq> T ` {z. norm z \<le> m \<and> (\<exists>t\<in>T -` P. norm (z - t) \<le> e)}"
+    proof clarsimp
+      fix x t
+      assume "norm x \<le> m" "t \<in> P" "norm (x - t) \<le> e"
+      then have "norm (inv T x) \<le> m"
+        using orthogonal_transformation_inv [OF T] by (simp add: orthogonal_transformation_norm)
+      moreover have "\<exists>t\<in>T -` P. norm (inv T x - t) \<le> e"
+      proof
+        have "T (inv T x - inv T t) = x - t"
+          using T linear_diff orthogonal_transformation_def by fastforce
+        then have "norm (inv T x - inv T t) = norm (x - t)"
+          by (metis T orthogonal_transformation_norm)
+        then show "norm (inv T x - inv T t) \<le> e"
+          using \<open>norm (x - t) \<le> e\<close> by linarith
+       next
+         show "inv T t \<in> T -` P"
+           using \<open>t \<in> P\<close> by force
+      qed
+      ultimately show "x \<in> T ` {z. norm z \<le> m \<and> (\<exists>t\<in>T -` P. norm (z - t) \<le> e)}"
+        by force
+    qed
+    then show "{z. norm z \<le> m \<and> (\<exists>t\<in>P. norm (z - t) \<le> e)} \<subseteq> T ` S"
+      using image_mono [OF subS] by (rule order_trans)
+    show "measure lebesgue (T ` S) \<le> 2 * e * (2 * m) ^ (CARD('n) - 1)"
+      using mS T by (simp add: S measure_orthogonal_image)
+  qed
+qed
+
+(*As above, but translating the sets (HOL Light's GEN_GEOM_ORIGIN_TAC)*)
+lemma Sard_lemma1:
+  fixes P :: "(real^'n::{finite,wellorder}) set"
+   assumes P: "dim P < CARD('n)" and "0 \<le> m" "0 \<le> e"
+ obtains S where "S \<in> lmeasurable"
+            and "{z. norm(z - w) \<le> m \<and> (\<exists>t \<in> P. norm(z - w - t) \<le> e)} \<subseteq> S"
+            and "measure lebesgue S \<le> (2 * e) * (2 * m) ^ (CARD('n) - 1)"
+proof -
+  obtain a where "a \<noteq> 0" "P \<subseteq> {x. a \<bullet> x = 0}"
+    using lowdim_subset_hyperplane [of P] P span_inc by auto
+  then obtain S where S: "S \<in> lmeasurable"
+    and subS: "{z. norm z \<le> m \<and> (\<exists>t \<in> P. norm(z - t) \<le> e)} \<subseteq> S"
+    and mS: "measure lebesgue S \<le> (2 * e) * (2 * m) ^ (CARD('n) - 1)"
+    by (rule Sard_lemma0 [OF _ _ \<open>0 \<le> m\<close> \<open>0 \<le> e\<close>])
+  show thesis
+  proof
+    show "(+)w ` S \<in> lmeasurable"
+      by (metis measurable_translation S)
+    show "{z. norm (z - w) \<le> m \<and> (\<exists>t\<in>P. norm (z - w - t) \<le> e)} \<subseteq> (+)w ` S"
+      using subS by force
+    show "measure lebesgue ((+)w ` S) \<le> 2 * e * (2 * m) ^ (CARD('n) - 1)"
+      by (metis measure_translation mS)
+  qed
+qed
+
+lemma Sard_lemma2:
+  fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n::{finite,wellorder}"
+  assumes mlen: "CARD('m) \<le> CARD('n)" (is "?m \<le> ?n")
+    and "B > 0" "bounded S"
+    and derS: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
+    and rank: "\<And>x. x \<in> S \<Longrightarrow> rank(matrix(f' x)) < CARD('n)"
+    and B: "\<And>x. x \<in> S \<Longrightarrow> onorm(f' x) \<le> B"
+  shows "negligible(f ` S)"
+proof -
+  have lin_f': "\<And>x. x \<in> S \<Longrightarrow> linear(f' x)"
+    using derS has_derivative_linear by blast
+  show ?thesis
+  proof (clarsimp simp add: negligible_outer_le)
+    fix e :: "real"
+    assume "e > 0"
+    obtain c where csub: "S \<subseteq> cbox (- (vec c)) (vec c)" and "c > 0"
+    proof -
+      obtain b where b: "\<And>x. x \<in> S \<Longrightarrow> norm x \<le> b"
+        using \<open>bounded S\<close> by (auto simp: bounded_iff)
+      show thesis
+      proof
+        have "- \<bar>b\<bar> - 1 \<le> x $ i \<and> x $ i \<le> \<bar>b\<bar> + 1" if "x \<in> S" for x i
+          using component_le_norm_cart [of x i] b [OF that] by auto
+        then show "S \<subseteq> cbox (- vec (\<bar>b\<bar> + 1)) (vec (\<bar>b\<bar> + 1))"
+          by (auto simp: mem_box_cart)
+      qed auto
+    qed
+    then have box_cc: "box (- (vec c)) (vec c) \<noteq> {}" and cbox_cc: "cbox (- (vec c)) (vec c) \<noteq> {}"
+      by (auto simp: interval_eq_empty_cart)
+    obtain d where "d > 0" "d \<le> B"
+             and d: "(d * 2) * (4 * B) ^ (?n - 1) \<le> e / (2*c) ^ ?m / ?m ^ ?m"
+      apply (rule that [of "min B (e / (2*c) ^ ?m / ?m ^ ?m / (4 * B) ^ (?n - 1) / 2)"])
+      using \<open>B > 0\<close> \<open>c > 0\<close> \<open>e > 0\<close>
+      by (simp_all add: divide_simps min_mult_distrib_right)
+    have "\<exists>r. 0 < r \<and> r \<le> 1/2 \<and>
+              (x \<in> S
+               \<longrightarrow> (\<forall>y. y \<in> S \<and> norm(y - x) < r
+                       \<longrightarrow> norm(f y - f x - f' x (y - x)) \<le> d * norm(y - x)))" for x
+    proof (cases "x \<in> S")
+      case True
+      then obtain r where "r > 0"
+              and "\<And>y. \<lbrakk>y \<in> S; norm (y - x) < r\<rbrakk>
+                       \<Longrightarrow> norm (f y - f x - f' x (y - x)) \<le> d * norm (y - x)"
+        using derS \<open>d > 0\<close> by (force simp: has_derivative_within_alt)
+      then show ?thesis
+        by (rule_tac x="min r (1/2)" in exI) simp
+    next
+      case False
+      then show ?thesis
+        by (rule_tac x="1/2" in exI) simp
+    qed
+    then obtain r where r12: "\<And>x. 0 < r x \<and> r x \<le> 1/2"
+            and r: "\<And>x y. \<lbrakk>x \<in> S; y \<in> S; norm(y - x) < r x\<rbrakk>
+                          \<Longrightarrow> norm(f y - f x - f' x (y - x)) \<le> d * norm(y - x)"
+      by metis
+    then have ga: "gauge (\<lambda>x. ball x (r x))"
+      by (auto simp: gauge_def)
+    obtain \<D> where \<D>: "countable \<D>" and sub_cc: "\<Union>\<D> \<subseteq> cbox (- vec c) (vec c)"
+      and cbox: "\<And>K. K \<in> \<D> \<Longrightarrow> interior K \<noteq> {} \<and> (\<exists>u v. K = cbox u v)"
+      and djointish: "pairwise (\<lambda>A B. interior A \<inter> interior B = {}) \<D>"
+      and covered: "\<And>K. K \<in> \<D> \<Longrightarrow> \<exists>x \<in> S \<inter> K. K \<subseteq> ball x (r x)"
+      and close: "\<And>u v. cbox u v \<in> \<D> \<Longrightarrow> \<exists>n. \<forall>i::'m. v $ i - u $ i = 2*c / 2^n"
+      and covers: "S \<subseteq> \<Union>\<D>"
+      apply (rule covering_lemma [OF csub box_cc ga])
+      apply (auto simp: Basis_vec_def cart_eq_inner_axis [symmetric])
+      done
+    let ?\<mu> = "measure lebesgue"
+    have "\<exists>T. T \<in> lmeasurable \<and> f ` (K \<inter> S) \<subseteq> T \<and> ?\<mu> T \<le> e / (2*c) ^ ?m * ?\<mu> K"
+      if "K \<in> \<D>" for K
+    proof -
+      obtain u v where uv: "K = cbox u v"
+        using cbox \<open>K \<in> \<D>\<close> by blast
+      then have uv_ne: "cbox u v \<noteq> {}"
+        using cbox that by fastforce
+      obtain x where x: "x \<in> S \<inter> cbox u v" "cbox u v \<subseteq> ball x (r x)"
+        using \<open>K \<in> \<D>\<close> covered uv by blast
+      then have "dim (range (f' x)) < ?n"
+        using rank_dim_range [of "matrix (f' x)"] lin_f' rank by fastforce
+      then obtain T where T: "T \<in> lmeasurable"
+            and subT: "{z. norm(z - f x) \<le> (2 * B) * norm(v - u) \<and> (\<exists>t \<in> range (f' x). norm(z - f x - t) \<le> d * norm(v - u))} \<subseteq> T"
+            and measT: "?\<mu> T \<le> (2 * (d * norm(v - u))) * (2 * ((2 * B) * norm(v - u))) ^ (?n - 1)"
+                        (is "_ \<le> ?DVU")
+        apply (rule Sard_lemma1 [of "range (f' x)" "(2 * B) * norm(v - u)" "d * norm(v - u)" "f x"])
+        using \<open>B > 0\<close> \<open>d > 0\<close> by simp_all
+      show ?thesis
+      proof (intro exI conjI)
+        have "f ` (K \<inter> S) \<subseteq> {z. norm(z - f x) \<le> (2 * B) * norm(v - u) \<and> (\<exists>t \<in> range (f' x). norm(z - f x - t) \<le> d * norm(v - u))}"
+          unfolding uv
+        proof (clarsimp simp: mult.assoc, intro conjI)
+          fix y
+          assume y: "y \<in> cbox u v" and "y \<in> S"
+          then have "norm (y - x) < r x"
+            by (metis dist_norm mem_ball norm_minus_commute subsetCE x(2))
+          then have le_dyx: "norm (f y - f x - f' x (y - x)) \<le> d * norm (y - x)"
+            using r [of x y] x \<open>y \<in> S\<close> by blast
+          have yx_le: "norm (y - x) \<le> norm (v - u)"
+          proof (rule norm_le_componentwise_cart)
+            show "\<bar>(y - x) $ i\<bar> \<le> \<bar>(v - u) $ i\<bar>" for i
+            using x y by (force simp: mem_box_cart dest!: spec [where x=i])
+          qed
+          have *: "\<lbrakk>norm(y - x - z) \<le> d; norm z \<le> B; d \<le> B\<rbrakk> \<Longrightarrow> norm(y - x) \<le> 2 * B"
+            for x y z :: "real^'n::_" and d B
+            using norm_triangle_ineq2 [of "y - x" z] by auto
+          show "norm (f y - f x) \<le> 2 * (B * norm (v - u))"
+          proof (rule * [OF le_dyx])
+            have "norm (f' x (y - x)) \<le> onorm (f' x) * norm (y - x)"
+              using onorm [of "f' x" "y-x"] by (meson IntE lin_f' linear_linear x(1))
+            also have "\<dots> \<le> B * norm (v - u)"
+            proof (rule mult_mono)
+              show "onorm (f' x) \<le> B"
+                using B x by blast
+            qed (use \<open>B > 0\<close> yx_le in auto)
+            finally show "norm (f' x (y - x)) \<le> B * norm (v - u)" .
+            show "d * norm (y - x) \<le> B * norm (v - u)"
+              using \<open>B > 0\<close> by (auto intro: mult_mono [OF \<open>d \<le> B\<close> yx_le])
+          qed
+          show "\<exists>t. norm (f y - f x - f' x t) \<le> d * norm (v - u)"
+            apply (rule_tac x="y-x" in exI)
+            using \<open>d > 0\<close> yx_le le_dyx mult_left_mono [where c=d]
+            by (meson order_trans real_mult_le_cancel_iff2)
+        qed
+        with subT show "f ` (K \<inter> S) \<subseteq> T" by blast
+        show "?\<mu> T \<le> e / (2*c) ^ ?m * ?\<mu> K"
+        proof (rule order_trans [OF measT])
+          have "?DVU = (d * 2 * (4 * B) ^ (?n - 1)) * norm (v - u)^?n"
+            using \<open>c > 0\<close>
+            apply (simp add: algebra_simps power_mult_distrib)
+            by (metis Suc_pred power_Suc zero_less_card_finite)
+          also have "\<dots> \<le> (e / (2*c) ^ ?m / (?m ^ ?m)) * norm(v - u) ^ ?n"
+            by (rule mult_right_mono [OF d]) auto
+          also have "\<dots> \<le> e / (2*c) ^ ?m * ?\<mu> K"
+          proof -
+            have "u \<in> ball (x) (r x)" "v \<in> ball x (r x)"
+              using box_ne_empty(1) contra_subsetD [OF x(2)] mem_box(2) uv_ne by fastforce+
+            moreover have "r x \<le> 1/2"
+              using r12 by auto
+            ultimately have "norm (v - u) \<le> 1"
+              using norm_triangle_half_r [of x u 1 v]
+              by (metis (no_types, hide_lams) dist_commute dist_norm less_eq_real_def less_le_trans mem_ball)
+            then have "norm (v - u) ^ ?n \<le> norm (v - u) ^ ?m"
+              by (simp add: power_decreasing [OF mlen])
+            also have "\<dots> \<le> ?\<mu> K * real (?m ^ ?m)"
+            proof -
+              obtain n where n: "\<And>i. v$i - u$i = 2 * c / 2^n"
+                using close [of u v] \<open>K \<in> \<D>\<close> uv by blast
+              have "norm (v - u) ^ ?m \<le> (\<Sum>i\<in>UNIV. \<bar>(v - u) $ i\<bar>) ^ ?m"
+                by (intro norm_le_l1_cart power_mono) auto
+              also have "\<dots> \<le> (\<Prod>i\<in>UNIV. v $ i - u $ i) * real CARD('m) ^ CARD('m)"
+                by (simp add: n field_simps \<open>c > 0\<close> less_eq_real_def)
+              also have "\<dots> = ?\<mu> K * real (?m ^ ?m)"
+                by (simp add: uv uv_ne content_cbox_cart)
+              finally show ?thesis .
+            qed
+            finally have *: "1 / real (?m ^ ?m) * norm (v - u) ^ ?n \<le> ?\<mu> K"
+              by (simp add: divide_simps)
+            show ?thesis
+              using mult_left_mono [OF *, of "e / (2*c) ^ ?m"] \<open>c > 0\<close> \<open>e > 0\<close> by auto
+          qed
+          finally show "?DVU \<le> e / (2*c) ^ ?m * ?\<mu> K" .
+        qed
+      qed (use T in auto)
+    qed
+    then obtain g where meas_g: "\<And>K. K \<in> \<D> \<Longrightarrow> g K \<in> lmeasurable"
+                    and sub_g: "\<And>K. K \<in> \<D> \<Longrightarrow> f ` (K \<inter> S) \<subseteq> g K"
+                    and le_g: "\<And>K. K \<in> \<D> \<Longrightarrow> ?\<mu> (g K) \<le> e / (2*c)^?m * ?\<mu> K"
+      by metis
+    have le_e: "?\<mu> (\<Union>i\<in>\<F>. g i) \<le> e"
+      if "\<F> \<subseteq> \<D>" "finite \<F>" for \<F>
+    proof -
+      have "?\<mu> (\<Union>i\<in>\<F>. g i) \<le> (\<Sum>i\<in>\<F>. ?\<mu> (g i))"
+        using meas_g \<open>\<F> \<subseteq> \<D>\<close> by (auto intro: measure_UNION_le [OF \<open>finite \<F>\<close>])
+      also have "\<dots> \<le> (\<Sum>K\<in>\<F>. e / (2*c) ^ ?m * ?\<mu> K)"
+        using \<open>\<F> \<subseteq> \<D>\<close> sum_mono [OF le_g] by (meson le_g subsetCE sum_mono)
+      also have "\<dots> = e / (2*c) ^ ?m * (\<Sum>K\<in>\<F>. ?\<mu> K)"
+        by (simp add: sum_distrib_left)
+      also have "\<dots> \<le> e"
+      proof -
+        have "\<F> division_of \<Union>\<F>"
+        proof (rule division_ofI)
+          show "K \<subseteq> \<Union>\<F>"  "K \<noteq> {}" "\<exists>a b. K = cbox a b" if "K \<in> \<F>" for K
+            using \<open>K \<in> \<F>\<close> covered cbox \<open>\<F> \<subseteq> \<D>\<close> by (auto simp: Union_upper)
+          show "interior K \<inter> interior L = {}" if "K \<in> \<F>" and "L \<in> \<F>" and "K \<noteq> L" for K L
+            by (metis (mono_tags, lifting) \<open>\<F> \<subseteq> \<D>\<close> pairwiseD djointish pairwise_subset that)
+        qed (use that in auto)
+        then have "sum ?\<mu> \<F> \<le> ?\<mu> (\<Union>\<F>)"
+          by (simp add: content_division)
+        also have "\<dots> \<le> ?\<mu> (cbox (- vec c) (vec c) :: (real, 'm) vec set)"
+        proof (rule measure_mono_fmeasurable)
+          show "\<Union>\<F> \<subseteq> cbox (- vec c) (vec c)"
+            by (meson Sup_subset_mono sub_cc order_trans \<open>\<F> \<subseteq> \<D>\<close>)
+        qed (use \<open>\<F> division_of \<Union>\<F>\<close> lmeasurable_division in auto)
+        also have "\<dots> = content (cbox (- vec c) (vec c) :: (real, 'm) vec set)"
+          by simp
+        also have "\<dots> \<le> (2 ^ ?m * c ^ ?m)"
+          using \<open>c > 0\<close> by (simp add: content_cbox_if_cart)
+        finally have "sum ?\<mu> \<F> \<le> (2 ^ ?m * c ^ ?m)" .
+        then show ?thesis
+          using \<open>e > 0\<close> \<open>c > 0\<close> by (auto simp: divide_simps mult_less_0_iff)
+      qed
+      finally show ?thesis .
+    qed
+    show "\<exists>T. f ` S \<subseteq> T \<and> T \<in> lmeasurable \<and> ?\<mu> T \<le> e"
+    proof (intro exI conjI)
+      show "f ` S \<subseteq> UNION \<D> g"
+        using covers sub_g by force
+      show "UNION \<D> g \<in> lmeasurable"
+        by (rule fmeasurable_UN_bound [OF \<open>countable \<D>\<close> meas_g le_e])
+      show "?\<mu> (UNION \<D> g) \<le> e"
+        by (rule measure_UN_bound [OF \<open>countable \<D>\<close> meas_g le_e])
+    qed
+  qed
+qed
+
+
+theorem baby_Sard:
+  fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n::{finite,wellorder}"
+  assumes mlen: "CARD('m) \<le> CARD('n)"
+    and der: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
+    and rank: "\<And>x. x \<in> S \<Longrightarrow> rank(matrix(f' x)) < CARD('n)"
+  shows "negligible(f ` S)"
+proof -
+  let ?U = "\<lambda>n. {x \<in> S. norm(x) \<le> n \<and> onorm(f' x) \<le> real n}"
+  have "\<And>x. x \<in> S \<Longrightarrow> \<exists>n. norm x \<le> real n \<and> onorm (f' x) \<le> real n"
+    by (meson linear order_trans real_arch_simple)
+  then have eq: "S = (\<Union>n. ?U n)"
+    by auto
+  have "negligible (f ` ?U n)" for n
+  proof (rule Sard_lemma2 [OF mlen])
+    show "0 < real n + 1"
+      by auto
+    show "bounded (?U n)"
+      using bounded_iff by blast
+    show "(f has_derivative f' x) (at x within ?U n)" if "x \<in> ?U n" for x
+      using der that by (force intro: has_derivative_subset)
+  qed (use rank in auto)
+  then show ?thesis
+    by (subst eq) (simp add: image_Union negligible_Union_nat)
+qed
+
+
+subsection\<open>A one-way version of change-of-variables not assuming injectivity. \<close>
+
+lemma integral_on_image_ubound_weak:
+  fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real"
+  assumes S: "S \<in> sets lebesgue"
+      and f: "f \<in> borel_measurable (lebesgue_on (g ` S))"
+      and nonneg_fg:  "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f(g x)"
+      and der_g:   "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
+      and det_int_fg: "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x)) integrable_on S"
+      and meas_gim: "\<And>T. \<lbrakk>T \<subseteq> g ` S; T \<in> sets lebesgue\<rbrakk> \<Longrightarrow> {x \<in> S. g x \<in> T} \<in> sets lebesgue"
+    shows "f integrable_on (g ` S) \<and>
+           integral (g ` S) f \<le> integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x))"
+         (is "_ \<and> _ \<le> ?b")
+proof -
+  let ?D = "\<lambda>x. \<bar>det (matrix (g' x))\<bar>"
+  have cont_g: "continuous_on S g"
+    using der_g has_derivative_continuous_on by blast
+  have [simp]: "space (lebesgue_on S) = S"
+    by (simp add: S)
+  have gS_in_sets_leb: "g ` S \<in> sets lebesgue"
+    apply (rule differentiable_image_in_sets_lebesgue)
+    using der_g by (auto simp: S differentiable_def differentiable_on_def)
+  obtain h where nonneg_h: "\<And>n x. 0 \<le> h n x"
+    and h_le_f: "\<And>n x. x \<in> S \<Longrightarrow> h n (g x) \<le> f (g x)"
+    and h_inc: "\<And>n x. h n x \<le> h (Suc n) x"
+    and h_meas: "\<And>n. h n \<in> borel_measurable lebesgue"
+    and fin_R: "\<And>n. finite(range (h n))"
+    and lim: "\<And>x. x \<in> g ` S \<Longrightarrow> (\<lambda>n. h n x) \<longlonglongrightarrow> f x"
+  proof -
+    let ?f = "\<lambda>x. if x \<in> g ` S then f x else 0"
+    have "?f \<in> borel_measurable lebesgue \<and> (\<forall>x. 0 \<le> ?f x)"
+      by (auto simp: gS_in_sets_leb f nonneg_fg measurable_restrict_space_iff [symmetric])
+    then show ?thesis
+      apply (clarsimp simp add: borel_measurable_simple_function_limit_increasing)
+      apply (rename_tac h)
+      by (rule_tac h=h in that) (auto split: if_split_asm)
+  qed
+  have h_lmeas: "{t. h n (g t) = y} \<inter> S \<in> sets lebesgue" for y n
+  proof -
+    have "space (lebesgue_on (UNIV::(real,'n) vec set)) = UNIV"
+      by simp
+    then have "((h n) -`{y} \<inter> g ` S) \<in> sets (lebesgue_on (g ` S))"
+      by (metis Int_commute borel_measurable_vimage h_meas image_eqI inf_top.right_neutral sets_restrict_space space_borel space_completion space_lborel)
+    then have "({u. h n u = y} \<inter> g ` S) \<in> sets lebesgue"
+      using gS_in_sets_leb
+      by (simp add: integral_indicator fmeasurableI2 sets_restrict_space_iff vimage_def)
+    then have "{x \<in> S. g x \<in> ({u. h n u = y} \<inter> g ` S)} \<in> sets lebesgue"
+      using meas_gim[of "({u. h n u = y} \<inter> g ` S)"] by force
+    moreover have "{t. h n (g t) = y} \<inter> S = {x \<in> S. g x \<in> ({u. h n u = y} \<inter> g ` S)}"
+      by blast
+    ultimately show ?thesis
+      by auto
+  qed
+  have hint: "h n integrable_on g ` S \<and> integral (g ` S) (h n) \<le> integral S (\<lambda>x. ?D x * h n (g x))"
+          (is "?INT \<and> ?lhs \<le> ?rhs")  for n
+  proof -
+    let ?R = "range (h n)"
+    have hn_eq: "h n = (\<lambda>x. \<Sum>y\<in>?R. y * indicat_real {x. h n x = y} x)"
+      by (simp add: indicator_def if_distrib fin_R cong: if_cong)
+    have yind: "(\<lambda>t. y * indicator{x. h n x = y} t) integrable_on (g ` S) \<and>
+                (integral (g ` S) (\<lambda>t. y * indicator {x. h n x = y} t))
+                 \<le> integral S (\<lambda>t. \<bar>det (matrix (g' t))\<bar> * y * indicator {x. h n x = y} (g t))"
+       if y: "y \<in> ?R" for y::real
+    proof (cases "y=0")
+      case True
+      then show ?thesis using gS_in_sets_leb integrable_0 by force
+    next
+      case False
+      with that have "y > 0"
+        using less_eq_real_def nonneg_h by fastforce
+      have "(\<lambda>x. if x \<in> {t. h n (g t) = y} then ?D x else 0) integrable_on S"
+      proof (rule measurable_bounded_by_integrable_imp_integrable)
+        have "(\<lambda>x. ?D x) \<in> borel_measurable (lebesgue_on ({t. h n (g t) = y} \<inter> S))"
+          apply (intro borel_measurable_abs borel_measurable_det_Jacobian [OF h_lmeas, where f=g])
+          by (meson der_g IntD2 has_derivative_within_subset inf_le2)
+        then have "(\<lambda>x. if x \<in> {t. h n (g t) = y} \<inter> S then ?D x else 0) \<in> borel_measurable lebesgue"
+          by (rule borel_measurable_If_I [OF _ h_lmeas])
+        then show "(\<lambda>x. if x \<in> {t. h n (g t) = y} then ?D x else 0) \<in> borel_measurable (lebesgue_on S)"
+          by (simp add: if_if_eq_conj Int_commute borel_measurable_UNIV [OF S, symmetric])
+        show "(\<lambda>x. ?D x *\<^sub>R f (g x) /\<^sub>R y) integrable_on S"
+          by (rule integrable_cmul) (use det_int_fg in auto)
+        show "norm (if x \<in> {t. h n (g t) = y} then ?D x else 0) \<le> ?D x *\<^sub>R f (g x) /\<^sub>R y"
+          if "x \<in> S" for x
+          using nonneg_h [of n x] \<open>y > 0\<close> nonneg_fg [of x] h_le_f [of x n] that
+          by (auto simp: divide_simps ordered_semiring_class.mult_left_mono)
+      qed (use S in auto)
+      then have int_det: "(\<lambda>t. \<bar>det (matrix (g' t))\<bar>) integrable_on ({t. h n (g t) = y} \<inter> S)"
+        using integrable_restrict_Int by force
+      have "(g ` ({t. h n (g t) = y} \<inter> S)) \<in> lmeasurable"
+        apply (rule measurable_differentiable_image [OF h_lmeas])
+         apply (blast intro: has_derivative_within_subset [OF der_g])
+        apply (rule int_det)
+        done
+      moreover have "g ` ({t. h n (g t) = y} \<inter> S) = {x. h n x = y} \<inter> g ` S"
+        by blast
+      moreover have "measure lebesgue (g ` ({t. h n (g t) = y} \<inter> S))
+                     \<le> integral ({t. h n (g t) = y} \<inter> S) (\<lambda>t. \<bar>det (matrix (g' t))\<bar>)"
+        apply (rule measure_differentiable_image [OF h_lmeas _ int_det])
+        apply (blast intro: has_derivative_within_subset [OF der_g])
+        done
+      ultimately show ?thesis
+        using \<open>y > 0\<close> integral_restrict_Int [of S "{t. h n (g t) = y}" "\<lambda>t. \<bar>det (matrix (g' t))\<bar> * y"]
+        apply (simp add: integrable_on_indicator integrable_on_cmult_iff integral_indicator)
+        apply (simp add: indicator_def if_distrib cong: if_cong)
+        done
+    qed
+    have hn_int: "h n integrable_on g ` S"
+      apply (subst hn_eq)
+      using yind by (force intro: integrable_sum [OF fin_R])
+    then show ?thesis
+    proof
+      have "?lhs = integral (g ` S) (\<lambda>x. \<Sum>y\<in>range (h n). y * indicat_real {x. h n x = y} x)"
+        by (metis hn_eq)
+      also have "\<dots> = (\<Sum>y\<in>range (h n). integral (g ` S) (\<lambda>x. y * indicat_real {x. h n x = y} x))"
+        by (rule integral_sum [OF fin_R]) (use yind in blast)
+      also have "\<dots> \<le> (\<Sum>y\<in>range (h n). integral S (\<lambda>u. \<bar>det (matrix (g' u))\<bar> * y * indicat_real {x. h n x = y} (g u)))"
+        using yind by (force intro: sum_mono)
+      also have "\<dots> = integral S (\<lambda>u. \<Sum>y\<in>range (h n). \<bar>det (matrix (g' u))\<bar> * y * indicat_real {x. h n x = y} (g u))"
+      proof (rule integral_sum [OF fin_R, symmetric])
+        fix y assume y: "y \<in> ?R"
+        with nonneg_h have "y \<ge> 0"
+          by auto
+        show "(\<lambda>u. \<bar>det (matrix (g' u))\<bar> * y * indicat_real {x. h n x = y} (g u)) integrable_on S"
+        proof (rule measurable_bounded_by_integrable_imp_integrable)
+          have "(\<lambda>x. indicat_real {x. h n x = y} (g x)) \<in> borel_measurable (lebesgue_on S)"
+            using h_lmeas S
+            by (auto simp: indicator_vimage [symmetric] borel_measurable_indicator_iff sets_restrict_space_iff)
+          then show "(\<lambda>u. \<bar>det (matrix (g' u))\<bar> * y * indicat_real {x. h n x = y} (g u)) \<in> borel_measurable (lebesgue_on S)"
+            by (intro borel_measurable_times borel_measurable_abs borel_measurable_const borel_measurable_det_Jacobian [OF S der_g])
+        next
+          fix x
+          assume "x \<in> S"
+          have "y * indicat_real {x. h n x = y} (g x) \<le> f (g x)"
+            by (metis (full_types) \<open>x \<in> S\<close> h_le_f indicator_def mem_Collect_eq mult.right_neutral mult_zero_right nonneg_fg)
+          with \<open>y \<ge> 0\<close> show "norm (?D x * y * indicat_real {x. h n x = y} (g x)) \<le> ?D x * f(g x)"
+            by (simp add: abs_mult mult.assoc mult_left_mono)
+        qed (use S det_int_fg in auto)
+      qed
+      also have "\<dots> = integral S (\<lambda>T. \<bar>det (matrix (g' T))\<bar> *
+                                        (\<Sum>y\<in>range (h n). y * indicat_real {x. h n x = y} (g T)))"
+        by (simp add: sum_distrib_left mult.assoc)
+      also have "\<dots> = ?rhs"
+        by (metis hn_eq)
+      finally show "integral (g ` S) (h n) \<le> ?rhs" .
+    qed
+  qed
+  have le: "integral S (\<lambda>T. \<bar>det (matrix (g' T))\<bar> * h n (g T)) \<le> ?b" for n
+  proof (rule integral_le)
+    show "(\<lambda>T. \<bar>det (matrix (g' T))\<bar> * h n (g T)) integrable_on S"
+    proof (rule measurable_bounded_by_integrable_imp_integrable)
+      have "(\<lambda>T. \<bar>det (matrix (g' T))\<bar> *\<^sub>R h n (g T)) \<in> borel_measurable (lebesgue_on S)"
+      proof (intro borel_measurable_scaleR borel_measurable_abs borel_measurable_det_Jacobian \<open>S \<in> sets lebesgue\<close>)
+        have eq: "{x \<in> S. f x \<le> a} = (\<Union>b \<in> (f ` S) \<inter> atMost a. {x. f x = b} \<inter> S)" for f and a::real
+          by auto
+        have "finite ((\<lambda>x. h n (g x)) ` S \<inter> {..a})" for a
+          by (force intro: finite_subset [OF _ fin_R])
+        with h_lmeas [of n] show "(\<lambda>x. h n (g x)) \<in> borel_measurable (lebesgue_on S)"
+          apply (simp add: borel_measurable_vimage_halfspace_component_le \<open>S \<in> sets lebesgue\<close> sets_restrict_space_iff eq)
+          by (metis (mono_tags) SUP_inf sets.finite_UN)
+      qed (use der_g in blast)
+      then show "(\<lambda>T. \<bar>det (matrix (g' T))\<bar> * h n (g T)) \<in> borel_measurable (lebesgue_on S)"
+        by simp
+      show "norm (?D x * h n (g x)) \<le> ?D x *\<^sub>R f (g x)"
+        if "x \<in> S" for x
+        by (simp add: h_le_f mult_left_mono nonneg_h that)
+    qed (use S det_int_fg in auto)
+    show "?D x * h n (g x) \<le> ?D x * f (g x)" if "x \<in> S" for x
+      by (simp add: \<open>x \<in> S\<close> h_le_f mult_left_mono)
+    show "(\<lambda>x. ?D x * f (g x)) integrable_on S"
+      using det_int_fg by blast
+  qed
+  have "f integrable_on g ` S \<and> (\<lambda>k. integral (g ` S) (h k)) \<longlonglongrightarrow> integral (g ` S) f"
+  proof (rule monotone_convergence_increasing)
+    have "\<bar>integral (g ` S) (h n)\<bar> \<le> integral S (\<lambda>x. ?D x * f (g x))" for n
+    proof -
+      have "\<bar>integral (g ` S) (h n)\<bar> = integral (g ` S) (h n)"
+        using hint by (simp add: integral_nonneg nonneg_h)
+      also have "\<dots> \<le> integral S (\<lambda>x. ?D x * f (g x))"
+        using hint le by (meson order_trans)
+      finally show ?thesis .
+    qed
+    then show "bounded (range (\<lambda>k. integral (g ` S) (h k)))"
+      by (force simp: bounded_iff)
+  qed (use h_inc lim hint in auto)
+  moreover have "integral (g ` S) (h n) \<le> integral S (\<lambda>x. ?D x * f (g x))" for n
+    using hint by (blast intro: le order_trans)
+  ultimately show ?thesis
+    by (auto intro: Lim_bounded_ereal)
+qed
+
+
+lemma integral_on_image_ubound_nonneg:
+  fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real"
+  assumes nonneg_fg: "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f(g x)"
+      and der_g:   "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
+      and intS: "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x)) integrable_on S"
+  shows "f integrable_on (g ` S) \<and> integral (g ` S) f \<le> integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x))"
+         (is "_ \<and> _ \<le> ?b")
+proof -
+  let ?D = "\<lambda>x. det (matrix (g' x))"
+  define S' where "S' \<equiv> {x \<in> S. ?D x * f(g x) \<noteq> 0}"
+  then have der_gS': "\<And>x. x \<in> S' \<Longrightarrow> (g has_derivative g' x) (at x within S')"
+    by (metis (mono_tags, lifting) der_g has_derivative_within_subset mem_Collect_eq subset_iff)
+  have "(\<lambda>x. if x \<in> S then \<bar>?D x\<bar> * f (g x) else 0) integrable_on UNIV"
+    by (simp add: integrable_restrict_UNIV intS)
+  then have Df_borel: "(\<lambda>x. if x \<in> S then \<bar>?D x\<bar> * f (g x) else 0) \<in> borel_measurable lebesgue"
+    using integrable_imp_measurable borel_measurable_UNIV_eq by blast
+  have S': "S' \<in> sets lebesgue"
+  proof -
+    from Df_borel borel_measurable_vimage_open [of _ UNIV]
+    have "{x. (if x \<in> S then \<bar>?D x\<bar> * f (g x) else 0) \<in> T} \<in> sets lebesgue"
+      if "open T" for T
+      using that unfolding borel_measurable_UNIV_eq
+      by (fastforce simp add: dest!: spec)
+    then have "{x. (if x \<in> S then \<bar>?D x\<bar> * f (g x) else 0) \<in> -{0}} \<in> sets lebesgue"
+      using open_Compl by blast
+    then show ?thesis
+      by (simp add: S'_def conj_ac split: if_split_asm cong: conj_cong)
+  qed
+  then have gS': "g ` S' \<in> sets lebesgue"
+  proof (rule differentiable_image_in_sets_lebesgue)
+    show "g differentiable_on S'"
+      using der_g unfolding S'_def differentiable_def differentiable_on_def
+      by (blast intro: has_derivative_within_subset)
+  qed auto
+  have f: "f \<in> borel_measurable (lebesgue_on (g ` S'))"
+  proof (clarsimp simp add: borel_measurable_vimage_open)
+    fix T :: "real set"
+    assume "open T"
+    have "{x \<in> g ` S'. f x \<in> T} = g ` {x \<in> S'. f(g x) \<in> T}"
+      by blast
+    moreover have "g ` {x \<in> S'. f(g x) \<in> T} \<in> sets lebesgue"
+    proof (rule differentiable_image_in_sets_lebesgue)
+      let ?h = "\<lambda>x. \<bar>?D x\<bar> * f (g x) /\<^sub>R \<bar>?D x\<bar>"
+      have "(\<lambda>x. if x \<in> S' then \<bar>?D x\<bar> * f (g x) else 0) = (\<lambda>x. if x \<in> S then \<bar>?D x\<bar> * f (g x) else 0)"
+        by (auto simp: S'_def)
+      also have "\<dots> \<in> borel_measurable lebesgue"
+        by (rule Df_borel)
+      finally have *: "(\<lambda>x. \<bar>?D x\<bar> * f (g x)) \<in> borel_measurable (lebesgue_on S')"
+        by (simp add: borel_measurable_If_D)
+      have "?h \<in> borel_measurable (lebesgue_on S')"
+        by (intro * S' der_gS' borel_measurable_det_Jacobian measurable) (blast intro: der_gS')
+      moreover have "?h x = f(g x)" if "x \<in> S'" for x
+        using that by (auto simp: S'_def)
+      ultimately have "(\<lambda>x. f(g x)) \<in> borel_measurable (lebesgue_on S')"
+        by (metis (no_types, lifting) measurable_lebesgue_cong)
+      then show "{x \<in> S'. f (g x) \<in> T} \<in> sets lebesgue"
+        by (simp add: \<open>S' \<in> sets lebesgue\<close> \<open>open T\<close> borel_measurable_vimage_open sets_restrict_space_iff)
+      show "g differentiable_on {x \<in> S'. f (g x) \<in> T}"
+        using der_g unfolding S'_def differentiable_def differentiable_on_def
+        by (blast intro: has_derivative_within_subset)
+    qed auto
+    ultimately have "{x \<in> g ` S'. f x \<in> T} \<in> sets lebesgue"
+      by metis
+    then show "{x \<in> g ` S'. f x \<in> T} \<in> sets (lebesgue_on (g ` S'))"
+      by (simp add: \<open>g ` S' \<in> sets lebesgue\<close> sets_restrict_space_iff)
+  qed
+  have intS': "(\<lambda>x. \<bar>?D x\<bar> * f (g x)) integrable_on S'"
+    using intS
+    by (rule integrable_spike_set) (auto simp: S'_def intro: empty_imp_negligible)
+  have lebS': "{x \<in> S'. g x \<in> T} \<in> sets lebesgue" if "T \<subseteq> g ` S'" "T \<in> sets lebesgue" for T
+  proof -
+    have "g \<in> borel_measurable (lebesgue_on S')"
+      using der_gS' has_derivative_continuous_on S'
+      by (blast intro: continuous_imp_measurable_on_sets_lebesgue)
+    moreover have "{x \<in> S'. g x \<in> U} \<in> sets lebesgue" if "negligible U" "U \<subseteq> g ` S'" for U
+    proof (intro negligible_imp_sets negligible_differentiable_vimage that)
+      fix x
+      assume x: "x \<in> S'"
+      then have "linear (g' x)"
+        using der_gS' has_derivative_linear by blast
+      with x show "inj (g' x)"
+        by (auto simp: S'_def det_nz_iff_inj)
+    qed (use der_gS' in auto)
+    ultimately show ?thesis
+      using double_lebesgue_sets [OF S' gS' order_refl] that by blast
+  qed
+  have int_gS': "f integrable_on g ` S' \<and> integral (g ` S') f \<le> integral S' (\<lambda>x. \<bar>?D x\<bar> * f(g x))"
+    using integral_on_image_ubound_weak [OF S' f nonneg_fg der_gS' intS' lebS'] S'_def by blast
+  have "negligible (g ` {x \<in> S. det(matrix(g' x)) = 0})"
+  proof (rule baby_Sard, simp_all)
+    fix x
+    assume x: "x \<in> S \<and> det (matrix (g' x)) = 0"
+    then show "(g has_derivative g' x) (at x within {x \<in> S. det (matrix (g' x)) = 0})"
+      by (metis (no_types, lifting) der_g has_derivative_within_subset mem_Collect_eq subsetI)
+    then show "rank (matrix (g' x)) < CARD('n)"
+      using det_nz_iff_inj matrix_vector_mul_linear x
+      by (fastforce simp add: less_rank_noninjective)
+  qed
+  then have negg: "negligible (g ` S - g ` {x \<in> S. ?D x \<noteq> 0})"
+    by (rule negligible_subset) (auto simp: S'_def)
+  have null: "g ` {x \<in> S. ?D x \<noteq> 0} - g ` S = {}"
+    by (auto simp: S'_def)
+  let ?F = "{x \<in> S. f (g x) \<noteq> 0}"
+  have eq: "g ` S' = g ` ?F \<inter> g ` {x \<in> S. ?D x \<noteq> 0}"
+    by (auto simp: S'_def image_iff)
+  show ?thesis
+  proof
+    have "((\<lambda>x. if x \<in> g ` ?F then f x else 0) integrable_on g ` {x \<in> S. ?D x \<noteq> 0})"
+      using int_gS' eq integrable_restrict_Int [where f=f]
+      by simp
+    then have "f integrable_on g ` {x \<in> S. ?D x \<noteq> 0}"
+      by (auto simp: image_iff elim!: integrable_eq)
+    then show "f integrable_on g ` S"
+      apply (rule integrable_spike_set [OF _ empty_imp_negligible negligible_subset])
+      using negg null by auto
+    have "integral (g ` S) f = integral (g ` {x \<in> S. ?D x \<noteq> 0}) f"
+      using negg by (auto intro: negligible_subset integral_spike_set)
+    also have "\<dots> = integral (g ` {x \<in> S. ?D x \<noteq> 0}) (\<lambda>x. if x \<in> g ` ?F then f x else 0)"
+      by (auto simp: image_iff intro!: integral_cong)
+    also have "\<dots> = integral (g ` S') f"
+      using  eq integral_restrict_Int by simp
+    also have "\<dots> \<le> integral S' (\<lambda>x. \<bar>?D x\<bar> * f(g x))"
+      by (metis int_gS')
+    also have "\<dots> \<le> ?b"
+      by (rule integral_subset_le [OF _ intS' intS]) (use nonneg_fg S'_def in auto)
+    finally show "integral (g ` S) f \<le> ?b" .
+  qed
+qed
+
+
+
+lemma absolutely_integrable_on_image_real:
+  fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real" and g :: "real^'n::_ \<Rightarrow> real^'n::_"
+  assumes der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
+    and intS: "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x)) absolutely_integrable_on S"
+  shows "f absolutely_integrable_on (g ` S)"
+proof -
+  let ?D = "\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f (g x)"
+  let ?N = "{x \<in> S. f (g x) < 0}" and ?P = "{x \<in> S. f (g x) > 0}"
+  have eq: "{x. (if x \<in> S then ?D x else 0) > 0} = {x \<in> S. ?D x > 0}"
+           "{x. (if x \<in> S then ?D x else 0) < 0} = {x \<in> S. ?D x < 0}"
+    by auto
+  have "?D integrable_on S"
+    using intS absolutely_integrable_on_def by blast
+  then have "(\<lambda>x. if x \<in> S then ?D x else 0) integrable_on UNIV"
+    by (simp add: integrable_restrict_UNIV)
+  then have D_borel: "(\<lambda>x. if x \<in> S then ?D x else 0) \<in> borel_measurable (lebesgue_on UNIV)"
+    using integrable_imp_measurable borel_measurable_UNIV_eq by blast
+  then have Dlt: "{x \<in> S. ?D x < 0} \<in> sets lebesgue"
+    unfolding borel_measurable_vimage_halfspace_component_lt
+    by (drule_tac x=0 in spec) (auto simp: eq)
+  from D_borel have Dgt: "{x \<in> S. ?D x > 0} \<in> sets lebesgue"
+    unfolding borel_measurable_vimage_halfspace_component_gt
+    by (drule_tac x=0 in spec) (auto simp: eq)
+
+  have dfgbm: "?D \<in> borel_measurable (lebesgue_on S)"
+    using intS absolutely_integrable_on_def integrable_imp_measurable by blast
+  have der_gN: "(g has_derivative g' x) (at x within ?N)" if "x \<in> ?N" for x
+      using der_g has_derivative_within_subset that by force
+  have "(\<lambda>x. - f x) integrable_on g ` ?N \<and>
+         integral (g ` ?N) (\<lambda>x. - f x) \<le> integral ?N (\<lambda>x. \<bar>det (matrix (g' x))\<bar> * - f (g x))"
+  proof (rule integral_on_image_ubound_nonneg [OF _ der_gN])
+    have 1: "?D integrable_on {x \<in> S. ?D x < 0}"
+      using Dlt
+      by (auto intro: set_lebesgue_integral_eq_integral [OF set_integrable_subset] intS)
+    have "uminus \<circ> (\<lambda>x. \<bar>det (matrix (g' x))\<bar> * - f (g x)) integrable_on ?N"
+      by (simp add: o_def mult_less_0_iff empty_imp_negligible integrable_spike_set [OF 1])
+    then show "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> * - f (g x)) integrable_on ?N"
+      by (simp add: integrable_neg_iff o_def)
+  qed auto
+  then have "f integrable_on g ` ?N"
+    by (simp add: integrable_neg_iff)
+  moreover have "g ` ?N = {y \<in> g ` S. f y < 0}"
+    by auto
+  ultimately have "f integrable_on {y \<in> g ` S. f y < 0}"
+    by simp
+  then have N: "f absolutely_integrable_on {y \<in> g ` S. f y < 0}"
+    by (rule absolutely_integrable_absolutely_integrable_ubound) auto
+
+  have der_gP: "(g has_derivative g' x) (at x within ?P)" if "x \<in> ?P" for x
+      using der_g has_derivative_within_subset that by force
+  have "f integrable_on g ` ?P \<and> integral (g ` ?P) f \<le> integral ?P ?D"
+  proof (rule integral_on_image_ubound_nonneg [OF _ der_gP])
+    have "?D integrable_on {x \<in> S. 0 < ?D x}"
+      using Dgt
+      by (auto intro: set_lebesgue_integral_eq_integral [OF set_integrable_subset] intS)
+    then show "?D integrable_on ?P"
+      apply (rule integrable_spike_set)
+      by (auto simp: zero_less_mult_iff empty_imp_negligible)
+  qed auto
+  then have "f integrable_on g ` ?P"
+    by metis
+  moreover have "g ` ?P = {y \<in> g ` S. f y > 0}"
+    by auto
+  ultimately have "f integrable_on {y \<in> g ` S. f y > 0}"
+    by simp
+  then have P: "f absolutely_integrable_on {y \<in> g ` S. f y > 0}"
+    by (rule absolutely_integrable_absolutely_integrable_lbound) auto
+  have "(\<lambda>x. if x \<in> g ` S \<and> f x < 0 \<or> x \<in> g ` S \<and> 0 < f x then f x else 0) = (\<lambda>x. if x \<in> g ` S then f x else 0)"
+    by auto
+  then show ?thesis
+    using absolutely_integrable_Un [OF N P] absolutely_integrable_restrict_UNIV [symmetric, where f=f]
+    by simp
+qed
+
+
+proposition absolutely_integrable_on_image:
+  fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
+  assumes der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
+    and intS: "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S"
+  shows "f absolutely_integrable_on (g ` S)"
+  apply (rule absolutely_integrable_componentwise [OF absolutely_integrable_on_image_real [OF der_g]])
+  using absolutely_integrable_component [OF intS]  by auto
+
+proposition integral_on_image_ubound:
+  fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real" and g :: "real^'n::_ \<Rightarrow> real^'n::_"
+  assumes"\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f(g x)"
+    and "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
+    and "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x)) integrable_on S"
+  shows "integral (g ` S) f \<le> integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x))"
+  using integral_on_image_ubound_nonneg [OF assms] by simp
+
+
+
+subsection\<open>Change-of-variables theorem\<close>
+
+text\<open>The classic change-of-variables theorem. We have two versions with quite general hypotheses,
+the first that the transforming function has a continuous inverse, the second that the base set is
+Lebesgue measurable.\<close>
+lemma cov_invertible_nonneg_le:
+  fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real" and g :: "real^'n::_ \<Rightarrow> real^'n::_"
+  assumes der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
+    and der_h: "\<And>y. y \<in> T \<Longrightarrow> (h has_derivative h' y) (at y within T)"
+    and f0: "\<And>y. y \<in> T \<Longrightarrow> 0 \<le> f y"
+    and hg: "\<And>x. x \<in> S \<Longrightarrow> g x \<in> T \<and> h(g x) = x"
+    and gh: "\<And>y. y \<in> T \<Longrightarrow> h y \<in> S \<and> g(h y) = y"
+    and id: "\<And>y. y \<in> T \<Longrightarrow> h' y \<circ> g'(h y) = id"
+  shows "f integrable_on T \<and> (integral T f) \<le> b \<longleftrightarrow>
+             (\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x)) integrable_on S \<and>
+             integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x)) \<le> b"
+        (is "?lhs = ?rhs")
+proof -
+  have Teq: "T = g`S" and Seq: "S = h`T"
+    using hg gh image_iff by fastforce+
+  have gS: "g differentiable_on S"
+    by (meson der_g differentiable_def differentiable_on_def)
+  let ?D = "\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f (g x)"
+  show ?thesis
+  proof
+    assume ?lhs
+    then have fT: "f integrable_on T" and intf: "integral T f \<le> b"
+      by blast+
+    show ?rhs
+    proof
+      let ?fgh = "\<lambda>x. \<bar>det (matrix (h' x))\<bar> * (\<bar>det (matrix (g' (h x)))\<bar> * f (g (h x)))"
+      have ddf: "?fgh x = f x"
+        if "x \<in> T" for x
+      proof -
+        have "matrix (h' x) ** matrix (g' (h x)) = mat 1"
+          using that id matrix_compose
+          by (metis der_g gh has_derivative_linear left_inverse_linear matrix_id_mat_1)
+        then have "\<bar>det (matrix (h' x))\<bar> * \<bar>det (matrix (g' (h x)))\<bar> = 1"
+          by (metis abs_1 abs_mult det_I det_mul)
+        then show ?thesis
+          by (simp add: gh that)
+      qed
+      have "?D integrable_on (h ` T)"
+      proof (intro set_lebesgue_integral_eq_integral absolutely_integrable_on_image_real)
+        show "(\<lambda>x. ?fgh x) absolutely_integrable_on T"
+        proof (subst absolutely_integrable_on_iff_nonneg)
+          show "(\<lambda>x. ?fgh x) integrable_on T"
+            using ddf fT integrable_eq by force
+        qed (simp add: zero_le_mult_iff f0 gh)
+      qed (use der_h in auto)
+      with Seq show "(\<lambda>x. ?D x) integrable_on S"
+        by simp
+      have "integral S (\<lambda>x. ?D x) \<le> integral T (\<lambda>x. ?fgh x)"
+        unfolding Seq
+      proof (rule integral_on_image_ubound)
+        show "(\<lambda>x. ?fgh x) integrable_on T"
+        using ddf fT integrable_eq by force
+      qed (use f0 gh der_h in auto)
+      also have "\<dots> = integral T f"
+        by (force simp: ddf intro: integral_cong)
+      also have "\<dots> \<le> b"
+        by (rule intf)
+      finally show "integral S (\<lambda>x. ?D x) \<le> b" .
+    qed
+  next
+    assume R: ?rhs
+    then have "f integrable_on g ` S"
+      using der_g f0 hg integral_on_image_ubound_nonneg by blast
+    moreover have "integral (g ` S) f \<le> integral S (\<lambda>x. ?D x)"
+      by (rule integral_on_image_ubound [OF f0 der_g]) (use R Teq in auto)
+    ultimately show ?lhs
+      using R by (simp add: Teq)
+  qed
+qed
+
+
+lemma cov_invertible_nonneg_eq:
+  fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real" and g :: "real^'n::_ \<Rightarrow> real^'n::_"
+  assumes "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
+      and "\<And>y. y \<in> T \<Longrightarrow> (h has_derivative h' y) (at y within T)"
+      and "\<And>y. y \<in> T \<Longrightarrow> 0 \<le> f y"
+      and "\<And>x. x \<in> S \<Longrightarrow> g x \<in> T \<and> h(g x) = x"
+      and "\<And>y. y \<in> T \<Longrightarrow> h y \<in> S \<and> g(h y) = y"
+      and "\<And>y. y \<in> T \<Longrightarrow> h' y \<circ> g'(h y) = id"
+  shows "((\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x)) has_integral b) S \<longleftrightarrow> (f has_integral b) T"
+  using cov_invertible_nonneg_le [OF assms]
+  by (simp add: has_integral_iff) (meson eq_iff)
+
+
+lemma cov_invertible_real:
+  fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real" and g :: "real^'n::_ \<Rightarrow> real^'n::_"
+  assumes der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
+      and der_h: "\<And>y. y \<in> T \<Longrightarrow> (h has_derivative h' y) (at y within T)"
+      and hg: "\<And>x. x \<in> S \<Longrightarrow> g x \<in> T \<and> h(g x) = x"
+      and gh: "\<And>y. y \<in> T \<Longrightarrow> h y \<in> S \<and> g(h y) = y"
+      and id: "\<And>y. y \<in> T \<Longrightarrow> h' y \<circ> g'(h y) = id"
+  shows "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x)) absolutely_integrable_on S \<and>
+           integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x)) = b \<longleftrightarrow>
+         f absolutely_integrable_on T \<and> integral T f = b"
+         (is "?lhs = ?rhs")
+proof -
+  have Teq: "T = g`S" and Seq: "S = h`T"
+    using hg gh image_iff by fastforce+
+  let ?DP = "\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x)" and ?DN = "\<lambda>x. \<bar>det (matrix (g' x))\<bar> * -f(g x)"
+  have "+": "(?DP has_integral b) {x \<in> S. f (g x) > 0} \<longleftrightarrow> (f has_integral b) {y \<in> T. f y > 0}" for b
+  proof (rule cov_invertible_nonneg_eq)
+    have *: "(\<lambda>x. f (g x)) -` Y \<inter> {x \<in> S. f (g x) > 0}
+          = ((\<lambda>x. f (g x)) -` Y \<inter> S) \<inter> {x \<in> S. f (g x) > 0}" for Y
+      by auto
+    show "(g has_derivative g' x) (at x within {x \<in> S. f (g x) > 0})" if "x \<in> {x \<in> S. f (g x) > 0}" for x
+      using that der_g has_derivative_within_subset by fastforce
+    show "(h has_derivative h' y) (at y within {y \<in> T. f y > 0})" if "y \<in> {y \<in> T. f y > 0}" for y
+      using that der_h has_derivative_within_subset by fastforce
+  qed (use gh hg id in auto)
+  have "-": "(?DN has_integral b) {x \<in> S. f (g x) < 0} \<longleftrightarrow> ((\<lambda>x. - f x) has_integral b) {y \<in> T. f y < 0}" for b
+  proof (rule cov_invertible_nonneg_eq)
+    have *: "(\<lambda>x. - f (g x)) -` y \<inter> {x \<in> S. f (g x) < 0}
+          = ((\<lambda>x. f (g x)) -` uminus ` y \<inter> S) \<inter> {x \<in> S. f (g x) < 0}" for y
+      using image_iff by fastforce
+    show "(g has_derivative g' x) (at x within {x \<in> S. f (g x) < 0})" if "x \<in> {x \<in> S. f (g x) < 0}" for x
+      using that der_g has_derivative_within_subset by fastforce
+    show "(h has_derivative h' y) (at y within {y \<in> T. f y < 0})" if "y \<in> {y \<in> T. f y < 0}" for y
+      using that der_h has_derivative_within_subset by fastforce
+  qed (use gh hg id in auto)
+  show ?thesis
+  proof
+    assume LHS: ?lhs
+    have eq: "{x. (if x \<in> S then ?DP x else 0) > 0} = {x \<in> S. ?DP x > 0}"
+      "{x. (if x \<in> S then ?DP x else 0) < 0} = {x \<in> S. ?DP x < 0}"
+      by auto
+    have "?DP integrable_on S"
+      using LHS absolutely_integrable_on_def by blast
+    then have "(\<lambda>x. if x \<in> S then ?DP x else 0) integrable_on UNIV"
+      by (simp add: integrable_restrict_UNIV)
+    then have D_borel: "(\<lambda>x. if x \<in> S then ?DP x else 0) \<in> borel_measurable (lebesgue_on UNIV)"
+      using integrable_imp_measurable borel_measurable_UNIV_eq by blast
+    then have SN: "{x \<in> S. ?DP x < 0} \<in> sets lebesgue"
+      unfolding borel_measurable_vimage_halfspace_component_lt
+      by (drule_tac x=0 in spec) (auto simp: eq)
+    from D_borel have SP: "{x \<in> S. ?DP x > 0} \<in> sets lebesgue"
+      unfolding borel_measurable_vimage_halfspace_component_gt
+      by (drule_tac x=0 in spec) (auto simp: eq)
+    have "?DP absolutely_integrable_on {x \<in> S. ?DP x > 0}"
+      using LHS by (fast intro!: set_integrable_subset [OF _, of _ S] SP)
+    then have aP: "?DP absolutely_integrable_on {x \<in> S. f (g x) > 0}"
+      by (rule absolutely_integrable_spike_set) (auto simp: zero_less_mult_iff empty_imp_negligible)
+    have "?DP absolutely_integrable_on {x \<in> S. ?DP x < 0}"
+      using LHS by (fast intro!: set_integrable_subset [OF _, of _ S] SN)
+    then have aN: "?DP absolutely_integrable_on {x \<in> S. f (g x) < 0}"
+      by (rule absolutely_integrable_spike_set) (auto simp: mult_less_0_iff empty_imp_negligible)
+    have fN: "f integrable_on {y \<in> T. f y < 0}"
+             "integral {y \<in> T. f y < 0} f = integral {x \<in> S. f (g x) < 0} ?DP"
+      using "-" [of "integral {x \<in> S. f(g x) < 0} ?DN"] aN
+      by (auto simp: set_lebesgue_integral_eq_integral has_integral_iff integrable_neg_iff)
+    have faN: "f absolutely_integrable_on {y \<in> T. f y < 0}"
+      apply (rule absolutely_integrable_integrable_bound [where g = "\<lambda>x. - f x"])
+      using fN by (auto simp: integrable_neg_iff)
+    have fP: "f integrable_on {y \<in> T. f y > 0}"
+             "integral {y \<in> T. f y > 0} f = integral {x \<in> S. f (g x) > 0} ?DP"
+      using "+" [of "integral {x \<in> S. f(g x) > 0} ?DP"] aP
+      by (auto simp: set_lebesgue_integral_eq_integral has_integral_iff integrable_neg_iff)
+    have faP: "f absolutely_integrable_on {y \<in> T. f y > 0}"
+      apply (rule absolutely_integrable_integrable_bound [where g = f])
+      using fP by auto
+    have fa: "f absolutely_integrable_on ({y \<in> T. f y < 0} \<union> {y \<in> T. f y > 0})"
+      by (rule absolutely_integrable_Un [OF faN faP])
+    show ?rhs
+    proof
+      have eq: "((if x \<in> T \<and> f x < 0 \<or> x \<in> T \<and> 0 < f x then 1 else 0) * f x)
+              = (if x \<in> T then 1 else 0) * f x" for x
+        by auto
+      show "f absolutely_integrable_on T"
+        using fa by (simp add: indicator_def set_integrable_def eq)
+      have [simp]: "{y \<in> T. f y < 0} \<inter> {y \<in> T. 0 < f y} = {}" for T and f :: "(real^'n::_) \<Rightarrow> real"
+        by auto
+      have "integral T f = integral ({y \<in> T. f y < 0} \<union> {y \<in> T. f y > 0}) f"
+        by (intro empty_imp_negligible integral_spike_set) (auto simp: eq)
+      also have "\<dots> = integral {y \<in> T. f y < 0} f + integral {y \<in> T. f y > 0} f"
+        using fN fP by simp
+      also have "\<dots> = integral {x \<in> S. f (g x) < 0} ?DP + integral {x \<in> S. 0 < f (g x)} ?DP"
+        by (simp add: fN fP)
+      also have "\<dots> = integral ({x \<in> S. f (g x) < 0} \<union> {x \<in> S. 0 < f (g x)}) ?DP"
+        using aP aN by (simp add: set_lebesgue_integral_eq_integral)
+      also have "\<dots> = integral S ?DP"
+        by (intro empty_imp_negligible integral_spike_set) auto
+      also have "\<dots> = b"
+        using LHS by simp
+      finally show "integral T f = b" .
+    qed
+  next
+    assume RHS: ?rhs
+    have eq: "{x. (if x \<in> T then f x else 0) > 0} = {x \<in> T. f x > 0}"
+             "{x. (if x \<in> T then f x else 0) < 0} = {x \<in> T. f x < 0}"
+      by auto
+    have "f integrable_on T"
+      using RHS absolutely_integrable_on_def by blast
+    then have "(\<lambda>x. if x \<in> T then f x else 0) integrable_on UNIV"
+      by (simp add: integrable_restrict_UNIV)
+    then have D_borel: "(\<lambda>x. if x \<in> T then f x else 0) \<in> borel_measurable (lebesgue_on UNIV)"
+      using integrable_imp_measurable borel_measurable_UNIV_eq by blast
+    then have TN: "{x \<in> T. f x < 0} \<in> sets lebesgue"
+      unfolding borel_measurable_vimage_halfspace_component_lt
+      by (drule_tac x=0 in spec) (auto simp: eq)
+    from D_borel have TP: "{x \<in> T. f x > 0} \<in> sets lebesgue"
+      unfolding borel_measurable_vimage_halfspace_component_gt
+      by (drule_tac x=0 in spec) (auto simp: eq)
+    have aint: "f absolutely_integrable_on {y. y \<in> T \<and> 0 < (f y)}"
+               "f absolutely_integrable_on {y. y \<in> T \<and> (f y) < 0}"
+         and intT: "integral T f = b"
+      using set_integrable_subset [of _ T] TP TN RHS
+      by blast+
+    show ?lhs
+    proof
+      have fN: "f integrable_on {v \<in> T. f v < 0}"
+        using absolutely_integrable_on_def aint by blast
+      then have DN: "(?DN has_integral integral {y \<in> T. f y < 0} (\<lambda>x. - f x)) {x \<in> S. f (g x) < 0}"
+        using "-" [of "integral {y \<in> T. f y < 0} (\<lambda>x. - f x)"]
+        by (simp add: has_integral_neg_iff integrable_integral)
+      have aDN: "?DP absolutely_integrable_on {x \<in> S. f (g x) < 0}"
+        apply (rule absolutely_integrable_integrable_bound [where g = ?DN])
+        using DN hg by (fastforce simp: abs_mult integrable_neg_iff)+
+      have fP: "f integrable_on {v \<in> T. f v > 0}"
+        using absolutely_integrable_on_def aint by blast
+      then have DP: "(?DP has_integral integral {y \<in> T. f y > 0} f) {x \<in> S. f (g x) > 0}"
+        using "+" [of "integral {y \<in> T. f y > 0} f"]
+        by (simp add: has_integral_neg_iff integrable_integral)
+      have aDP: "?DP absolutely_integrable_on {x \<in> S. f (g x) > 0}"
+        apply (rule absolutely_integrable_integrable_bound [where g = ?DP])
+        using DP hg by (fastforce simp: integrable_neg_iff)+
+      have eq: "(if x \<in> S then 1 else 0) * ?DP x = (if x \<in> S \<and> f (g x) < 0 \<or> x \<in> S \<and> f (g x) > 0 then 1 else 0) * ?DP x" for x
+        by force
+      have "?DP absolutely_integrable_on ({x \<in> S. f (g x) < 0} \<union> {x \<in> S. f (g x) > 0})"
+        by (rule absolutely_integrable_Un [OF aDN aDP])
+      then show I: "?DP absolutely_integrable_on S"
+        by (simp add: indicator_def eq set_integrable_def)
+      have [simp]: "{y \<in> S. f y < 0} \<inter> {y \<in> S. 0 < f y} = {}" for S and f :: "(real^'n::_) \<Rightarrow> real"
+        by auto
+      have "integral S ?DP = integral ({x \<in> S. f (g x) < 0} \<union> {x \<in> S. f (g x) > 0}) ?DP"
+        by (intro empty_imp_negligible integral_spike_set) auto
+      also have "\<dots> = integral {x \<in> S. f (g x) < 0} ?DP + integral {x \<in> S. 0 < f (g x)} ?DP"
+        using aDN aDP by (simp add: set_lebesgue_integral_eq_integral)
+      also have "\<dots> = - integral {y \<in> T. f y < 0} (\<lambda>x. - f x) + integral {y \<in> T. f y > 0} f"
+        using DN DP by (auto simp: has_integral_iff)
+      also have "\<dots> = integral ({x \<in> T. f x < 0} \<union> {x \<in> T. 0 < f x}) f"
+        by (simp add: fN fP)
+      also have "\<dots> = integral T f"
+        by (intro empty_imp_negligible integral_spike_set) auto
+      also have "\<dots> = b"
+        using intT by simp
+      finally show "integral S ?DP = b" .
+    qed
+  qed
+qed
+
+
+lemma cv_inv_version3:
+  fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
+  assumes der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
+    and der_h: "\<And>y. y \<in> T \<Longrightarrow> (h has_derivative h' y) (at y within T)"
+    and hg: "\<And>x. x \<in> S \<Longrightarrow> g x \<in> T \<and> h(g x) = x"
+    and gh: "\<And>y. y \<in> T \<Longrightarrow> h y \<in> S \<and> g(h y) = y"
+    and id: "\<And>y. y \<in> T \<Longrightarrow> h' y \<circ> g'(h y) = id"
+  shows "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S \<and>
+             integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) = b
+         \<longleftrightarrow> f absolutely_integrable_on T \<and> integral T f = b"
+proof -
+  let ?D = "\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)"
+  have "((\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x) $ i) absolutely_integrable_on S \<and> integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> * (f(g x) $ i)) = b $ i) \<longleftrightarrow>
+        ((\<lambda>x. f x $ i) absolutely_integrable_on T \<and> integral T (\<lambda>x. f x $ i) = b $ i)" for i
+    by (rule cov_invertible_real [OF der_g der_h hg gh id])
+  then have "?D absolutely_integrable_on S \<and> (?D has_integral b) S \<longleftrightarrow>
+        f absolutely_integrable_on T \<and> (f has_integral b) T"
+    unfolding absolutely_integrable_componentwise_iff [where f=f] has_integral_componentwise_iff [of f]
+              absolutely_integrable_componentwise_iff [where f="?D"] has_integral_componentwise_iff [of ?D]
+    by (auto simp: all_conj_distrib Basis_vec_def cart_eq_inner_axis [symmetric]
+           has_integral_iff set_lebesgue_integral_eq_integral)
+  then show ?thesis
+    using absolutely_integrable_on_def by blast
+qed
+
+
+lemma cv_inv_version4:
+  fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
+  assumes der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S) \<and> invertible(matrix(g' x))"
+    and hg: "\<And>x. x \<in> S \<Longrightarrow> continuous_on (g ` S) h \<and> h(g x) = x"
+  shows "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S \<and>
+             integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) = b
+         \<longleftrightarrow> f absolutely_integrable_on (g ` S) \<and> integral (g ` S) f = b"
+proof -
+  have "\<forall>x. \<exists>h'. x \<in> S
+           \<longrightarrow> (g has_derivative g' x) (at x within S) \<and> linear h' \<and> g' x \<circ> h' = id \<and> h' \<circ> g' x = id"
+    using der_g matrix_invertible has_derivative_linear by blast
+  then obtain h' where h':
+    "\<And>x. x \<in> S
+           \<Longrightarrow> (g has_derivative g' x) (at x within S) \<and>
+               linear (h' x) \<and> g' x \<circ> (h' x) = id \<and> (h' x) \<circ> g' x = id"
+    by metis
+  show ?thesis
+  proof (rule cv_inv_version3)
+    show "\<And>y. y \<in> g ` S \<Longrightarrow> (h has_derivative h' (h y)) (at y within g ` S)"
+      using h' hg
+      by (force simp: continuous_on_eq_continuous_within intro!: has_derivative_inverse_within)
+  qed (use h' hg in auto)
+qed
+
+
+proposition has_absolute_integral_change_of_variables_invertible:
+  fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
+  assumes der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
+      and hg: "\<And>x. x \<in> S \<Longrightarrow> h(g x) = x"
+      and conth: "continuous_on (g ` S) h"
+  shows "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S \<and> integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) = b \<longleftrightarrow>
+         f absolutely_integrable_on (g ` S) \<and> integral (g ` S) f = b"
+    (is "?lhs = ?rhs")
+proof -
+  let ?S = "{x \<in> S. invertible (matrix (g' x))}" and ?D = "\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)"
+  have *: "?D absolutely_integrable_on ?S \<and> integral ?S ?D = b
+           \<longleftrightarrow> f absolutely_integrable_on (g ` ?S) \<and> integral (g ` ?S) f = b"
+  proof (rule cv_inv_version4)
+    show "(g has_derivative g' x) (at x within ?S) \<and> invertible (matrix (g' x))"
+      if "x \<in> ?S" for x
+      using der_g that has_derivative_within_subset that by fastforce
+    show "continuous_on (g ` ?S) h \<and> h (g x) = x"
+      if "x \<in> ?S" for x
+      using that continuous_on_subset [OF conth]  by (simp add: hg image_mono)
+  qed
+  have "(g has_derivative g' x) (at x within {x \<in> S. rank (matrix (g' x)) < CARD('m)})" if "x \<in> S" for x
+    by (metis (no_types, lifting) der_g has_derivative_within_subset mem_Collect_eq subsetI that)
+  then have "negligible (g ` {x \<in> S. \<not> invertible (matrix (g' x))})"
+    by (auto simp: invertible_det_nz det_eq_0_rank intro: baby_Sard)
+  then have neg: "negligible {x \<in> g ` S. x \<notin> g ` ?S \<and> f x \<noteq> 0}"
+    by (auto intro: negligible_subset)
+  have [simp]: "{x \<in> g ` ?S. x \<notin> g ` S \<and> f x \<noteq> 0} = {}"
+    by auto
+  have "?D absolutely_integrable_on ?S \<and> integral ?S ?D = b
+    \<longleftrightarrow> ?D absolutely_integrable_on S \<and> integral S ?D = b"
+    apply (intro conj_cong absolutely_integrable_spike_set_eq)
+      apply(auto simp: integral_spike_set invertible_det_nz empty_imp_negligible neg)
+    done
+  moreover
+  have "f absolutely_integrable_on (g ` ?S) \<and> integral (g ` ?S) f = b
+    \<longleftrightarrow> f absolutely_integrable_on (g ` S) \<and> integral (g ` S) f = b"
+    by (auto intro!: conj_cong absolutely_integrable_spike_set_eq integral_spike_set neg)
+  ultimately
+  show ?thesis
+    using "*" by blast
+qed
+
+
+
+lemma has_absolute_integral_change_of_variables_compact:
+  fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
+  assumes "compact S"
+      and der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
+      and inj: "inj_on g S"
+  shows "((\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S \<and>
+             integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) = b
+      \<longleftrightarrow> f absolutely_integrable_on (g ` S) \<and> integral (g ` S) f = b)"
+proof -
+  obtain h where hg: "\<And>x. x \<in> S \<Longrightarrow> h(g x) = x"
+    using inj by (metis the_inv_into_f_f)
+  have conth: "continuous_on (g ` S) h"
+    by (metis \<open>compact S\<close> continuous_on_inv der_g has_derivative_continuous_on hg)
+  show ?thesis
+    by (rule has_absolute_integral_change_of_variables_invertible [OF der_g hg conth])
+qed
+
+
+lemma has_absolute_integral_change_of_variables_compact_family:
+  fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
+  assumes compact: "\<And>n::nat. compact (F n)"
+      and der_g: "\<And>x. x \<in> (\<Union>n. F n) \<Longrightarrow> (g has_derivative g' x) (at x within (\<Union>n. F n))"
+      and inj: "inj_on g (\<Union>n. F n)"
+  shows "((\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) absolutely_integrable_on (\<Union>n. F n) \<and>
+             integral (\<Union>n. F n) (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) = b
+      \<longleftrightarrow> f absolutely_integrable_on (g ` (\<Union>n. F n)) \<and> integral (g ` (\<Union>n. F n)) f = b)"
+proof -
+  let ?D = "\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f (g x)"
+  let ?U = "\<lambda>n. \<Union>m\<le>n. F m"
+  let ?lift = "vec::real\<Rightarrow>real^1"
+  have F_leb: "F m \<in> sets lebesgue" for m
+    by (simp add: compact borel_compact)
+  have iff: "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f (g x)) absolutely_integrable_on (?U n) \<and>
+             integral (?U n) (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f (g x)) = b
+         \<longleftrightarrow> f absolutely_integrable_on (g ` (?U n)) \<and> integral (g ` (?U n)) f = b" for n b and f :: "real^'m::_ \<Rightarrow> real^'k"
+  proof (rule has_absolute_integral_change_of_variables_compact)
+    show "compact (?U n)"
+      by (simp add: compact compact_UN)
+    show "(g has_derivative g' x) (at x within (?U n))"
+      if "x \<in> ?U n" for x
+      using that by (blast intro!: has_derivative_within_subset [OF der_g])
+    show "inj_on g (?U n)"
+      using inj by (auto simp: inj_on_def)
+  qed
+  show ?thesis
+    unfolding image_UN
+  proof safe
+    assume DS: "?D absolutely_integrable_on (\<Union>n. F n)"
+      and b: "b = integral (\<Union>n. F n) ?D"
+    have DU: "\<And>n. ?D absolutely_integrable_on (?U n)"
+             "(\<lambda>n. integral (?U n) ?D) \<longlonglongrightarrow> integral (\<Union>n. F n) ?D"
+      using integral_countable_UN [OF DS F_leb] by auto
+    with iff have fag: "f absolutely_integrable_on g ` (?U n)"
+      and fg_int: "integral (\<Union>m\<le>n. g ` F m) f = integral (?U n) ?D" for n
+      by (auto simp: image_UN)
+    let ?h = "\<lambda>x. if x \<in> (\<Union>m. g ` F m) then norm(f x) else 0"
+    have "(\<lambda>x. if x \<in> (\<Union>m. g ` F m) then f x else 0) absolutely_integrable_on UNIV"
+    proof (rule dominated_convergence_absolutely_integrable)
+      show "(\<lambda>x. if x \<in> (\<Union>m\<le>k. g ` F m) then f x else 0) absolutely_integrable_on UNIV" for k
+        unfolding absolutely_integrable_restrict_UNIV
+        using fag by (simp add: image_UN)
+      let ?nf = "\<lambda>n x. if x \<in> (\<Union>m\<le>n. g ` F m) then norm(f x) else 0"
+      show "?h integrable_on UNIV"
+      proof (rule monotone_convergence_increasing [THEN conjunct1])
+        show "?nf k integrable_on UNIV" for k
+          using fag
+          unfolding integrable_restrict_UNIV absolutely_integrable_on_def by (simp add: image_UN)
+        { fix n
+          have "(norm \<circ> ?D) absolutely_integrable_on ?U n"
+            by (intro absolutely_integrable_norm DU)
+          then have "integral (g ` ?U n) (norm \<circ> f) = integral (?U n) (norm \<circ> ?D)"
+            using iff [of n "vec \<circ> norm \<circ> f" "integral (?U n) (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R (?lift \<circ> norm \<circ> f) (g x))"]
+            unfolding absolutely_integrable_on_1_iff integral_on_1_eq by (auto simp: o_def)
+        }
+        moreover have "bounded (range (\<lambda>k. integral (?U k) (norm \<circ> ?D)))"
+          unfolding bounded_iff
+        proof (rule exI, clarify)
+          fix k
+          show "norm (integral (?U k) (norm \<circ> ?D)) \<le> integral (\<Union>n. F n) (norm \<circ> ?D)"
+            unfolding integral_restrict_UNIV [of _ "norm \<circ> ?D", symmetric]
+          proof (rule integral_norm_bound_integral)
+            show "(\<lambda>x. if x \<in> UNION {..k} F then (norm \<circ> ?D) x else 0) integrable_on UNIV"
+              "(\<lambda>x. if x \<in> (\<Union>n. F n) then (norm \<circ> ?D) x else 0) integrable_on UNIV"
+              using DU(1) DS
+              unfolding absolutely_integrable_on_def o_def integrable_restrict_UNIV by auto
+          qed auto
+        qed
+        ultimately show "bounded (range (\<lambda>k. integral UNIV (?nf k)))"
+          by (simp add: integral_restrict_UNIV image_UN [symmetric] o_def)
+      next
+        show "(\<lambda>k. if x \<in> (\<Union>m\<le>k. g ` F m) then norm (f x) else 0)
+              \<longlonglongrightarrow> (if x \<in> (\<Union>m. g ` F m) then norm (f x) else 0)" for x
+          by (force intro: Lim_eventually eventually_sequentiallyI)
+      qed auto
+    next
+      show "(\<lambda>k. if x \<in> (\<Union>m\<le>k. g ` F m) then f x else 0)
+            \<longlonglongrightarrow> (if x \<in> (\<Union>m. g ` F m) then f x else 0)" for x
+      proof clarsimp
+        fix m y
+        assume "y \<in> F m"
+        show "(\<lambda>k. if \<exists>x\<in>{..k}. g y \<in> g ` F x then f (g y) else 0) \<longlonglongrightarrow> f (g y)"
+          using \<open>y \<in> F m\<close> by (force intro: Lim_eventually eventually_sequentiallyI [of m])
+      qed
+    qed auto
+    then show fai: "f absolutely_integrable_on (\<Union>m. g ` F m)"
+      using absolutely_integrable_restrict_UNIV by blast
+    show "integral ((\<Union>x. g ` F x)) f = integral (\<Union>n. F n) ?D"
+    proof (rule LIMSEQ_unique)
+      show "(\<lambda>n. integral (?U n) ?D) \<longlonglongrightarrow> integral (\<Union>x. g ` F x) f"
+        unfolding fg_int [symmetric]
+      proof (rule integral_countable_UN [OF fai])
+        show "g ` F m \<in> sets lebesgue" for m
+        proof (rule differentiable_image_in_sets_lebesgue [OF F_leb])
+          show "g differentiable_on F m"
+            by (meson der_g differentiableI UnionI differentiable_on_def differentiable_on_subset rangeI subsetI)
+        qed auto
+      qed
+    next
+      show "(\<lambda>n. integral (?U n) ?D) \<longlonglongrightarrow> integral (\<Union>n. F n) ?D"
+        by (rule DU)
+    qed
+  next
+    assume fs: "f absolutely_integrable_on (\<Union>x. g ` F x)"
+      and b: "b = integral ((\<Union>x. g ` F x)) f"
+    have gF_leb: "g ` F m \<in> sets lebesgue" for m
+    proof (rule differentiable_image_in_sets_lebesgue [OF F_leb])
+      show "g differentiable_on F m"
+        using der_g unfolding differentiable_def differentiable_on_def
+        by (meson Sup_upper UNIV_I UnionI has_derivative_within_subset image_eqI)
+    qed auto
+    have fgU: "\<And>n. f absolutely_integrable_on (\<Union>m\<le>n. g ` F m)"
+      "(\<lambda>n. integral (\<Union>m\<le>n. g ` F m) f) \<longlonglongrightarrow> integral (\<Union>m. g ` F m) f"
+      using integral_countable_UN [OF fs gF_leb] by auto
+    with iff have DUn: "?D absolutely_integrable_on ?U n"
+      and D_int: "integral (?U n) ?D = integral (\<Union>m\<le>n. g ` F m) f" for n
+      by (auto simp: image_UN)
+    let ?h = "\<lambda>x. if x \<in> (\<Union>n. F n) then norm(?D x) else 0"
+    have "(\<lambda>x. if x \<in> (\<Union>n. F n) then ?D x else 0) absolutely_integrable_on UNIV"
+    proof (rule dominated_convergence_absolutely_integrable)
+      show "(\<lambda>x. if x \<in> ?U k then ?D x else 0) absolutely_integrable_on UNIV" for k
+        unfolding absolutely_integrable_restrict_UNIV using DUn by simp
+      let ?nD = "\<lambda>n x. if x \<in> ?U n then norm(?D x) else 0"
+      show "?h integrable_on UNIV"
+      proof (rule monotone_convergence_increasing [THEN conjunct1])
+        show "?nD k integrable_on UNIV" for k
+          using DUn
+          unfolding integrable_restrict_UNIV absolutely_integrable_on_def by (simp add: image_UN)
+        { fix n::nat
+          have "(norm \<circ> f) absolutely_integrable_on (\<Union>m\<le>n. g ` F m)"
+            apply (rule absolutely_integrable_norm)
+            using fgU by blast
+          then have "integral (?U n) (norm \<circ> ?D) = integral (g ` ?U n) (norm \<circ> f)"
+            using iff [of n "?lift \<circ> norm \<circ> f" "integral (g ` ?U n) (?lift \<circ> norm \<circ> f)"]
+            unfolding absolutely_integrable_on_1_iff integral_on_1_eq image_UN by (auto simp: o_def)
+        }
+        moreover have "bounded (range (\<lambda>k. integral (g ` ?U k) (norm \<circ> f)))"
+          unfolding bounded_iff
+        proof (rule exI, clarify)
+          fix k
+          show "norm (integral (g ` ?U k) (norm \<circ> f)) \<le> integral (g ` (\<Union>n. F n)) (norm \<circ> f)"
+            unfolding integral_restrict_UNIV [of _ "norm \<circ> f", symmetric]
+          proof (rule integral_norm_bound_integral)
+            show "(\<lambda>x. if x \<in> g ` ?U k then (norm \<circ> f) x else 0) integrable_on UNIV"
+                 "(\<lambda>x. if x \<in> g ` (\<Union>n. F n) then (norm \<circ> f) x else 0) integrable_on UNIV"
+              using fgU fs
+              unfolding absolutely_integrable_on_def o_def integrable_restrict_UNIV
+              by (auto simp: image_UN)
+          qed auto
+        qed
+        ultimately show "bounded (range (\<lambda>k. integral UNIV (?nD k)))"
+          unfolding integral_restrict_UNIV image_UN [symmetric] o_def by simp
+      next
+        show "(\<lambda>k. if x \<in> ?U k then norm (?D x) else 0) \<longlonglongrightarrow> (if x \<in> (\<Union>n. F n) then norm (?D x) else 0)" for x
+          by (force intro: Lim_eventually eventually_sequentiallyI)
+      qed auto
+    next
+      show "(\<lambda>k. if x \<in> ?U k then ?D x else 0) \<longlonglongrightarrow> (if x \<in> (\<Union>n. F n) then ?D x else 0)" for x
+      proof clarsimp
+        fix n
+        assume "x \<in> F n"
+        show "(\<lambda>m. if \<exists>j\<in>{..m}. x \<in> F j then ?D x else 0) \<longlonglongrightarrow> ?D x"
+          using \<open>x \<in> F n\<close> by (auto intro!: Lim_eventually eventually_sequentiallyI [of n])
+      qed
+    qed auto
+    then show Dai: "?D absolutely_integrable_on (\<Union>n. F n)"
+      unfolding absolutely_integrable_restrict_UNIV by simp
+    show "integral (\<Union>n. F n) ?D = integral ((\<Union>x. g ` F x)) f"
+    proof (rule LIMSEQ_unique)
+      show "(\<lambda>n. integral (\<Union>m\<le>n. g ` F m) f) \<longlonglongrightarrow> integral (\<Union>x. g ` F x) f"
+        by (rule fgU)
+      show "(\<lambda>n. integral (\<Union>m\<le>n. g ` F m) f) \<longlonglongrightarrow> integral (\<Union>n. F n) ?D"
+        unfolding D_int [symmetric] by (rule integral_countable_UN [OF Dai F_leb])
+    qed
+  qed
+qed
+
+
+proposition has_absolute_integral_change_of_variables:
+  fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
+  assumes S: "S \<in> sets lebesgue"
+    and der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
+    and inj: "inj_on g S"
+  shows "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S \<and>
+           integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) = b
+     \<longleftrightarrow> f absolutely_integrable_on (g ` S) \<and> integral (g ` S) f = b"
+proof -
+  obtain C N where "fsigma C" "negligible N" and CNS: "C \<union> N = S" and "disjnt C N"
+    using lebesgue_set_almost_fsigma [OF S] .
+  then obtain F :: "nat \<Rightarrow> (real^'m::_) set"
+    where F: "range F \<subseteq> Collect compact" and Ceq: "C = Union(range F)"
+    using fsigma_Union_compact by metis
+  let ?D = "\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f (g x)"
+  have "?D absolutely_integrable_on C \<and> integral C ?D = b
+    \<longleftrightarrow> f absolutely_integrable_on (g ` C) \<and> integral (g ` C) f = b"
+    unfolding Ceq
+  proof (rule has_absolute_integral_change_of_variables_compact_family)
+    fix n x
+    assume "x \<in> UNION UNIV F"
+    then show "(g has_derivative g' x) (at x within UNION UNIV F)"
+      using Ceq \<open>C \<union> N = S\<close> der_g has_derivative_within_subset by blast
+  next
+    have "UNION UNIV F \<subseteq> S"
+      using Ceq \<open>C \<union> N = S\<close> by blast
+    then show "inj_on g (UNION UNIV F)"
+      using inj by (meson inj_on_subset)
+  qed (use F in auto)
+  moreover
+  have "?D absolutely_integrable_on C \<and> integral C ?D = b
+    \<longleftrightarrow> ?D absolutely_integrable_on S \<and> integral S ?D = b"
+  proof (rule conj_cong)
+    have neg: "negligible {x \<in> C - S. ?D x \<noteq> 0}" "negligible {x \<in> S - C. ?D x \<noteq> 0}"
+      using CNS by (blast intro: negligible_subset [OF \<open>negligible N\<close>])+
+    then show "(?D absolutely_integrable_on C) = (?D absolutely_integrable_on S)"
+      by (rule absolutely_integrable_spike_set_eq)
+    show "(integral C ?D = b) \<longleftrightarrow> (integral S ?D = b)"
+      using integral_spike_set [OF neg] by simp
+  qed
+  moreover
+  have "f absolutely_integrable_on (g ` C) \<and> integral (g ` C) f = b
+    \<longleftrightarrow> f absolutely_integrable_on (g ` S) \<and> integral (g ` S) f = b"
+  proof (rule conj_cong)
+    have "g differentiable_on N"
+      by (metis CNS der_g differentiable_def differentiable_on_def differentiable_on_subset sup.cobounded2)
+    with \<open>negligible N\<close>
+    have neg_gN: "negligible (g ` N)"
+      by (blast intro: negligible_differentiable_image_negligible)
+    have neg: "negligible {x \<in> g ` C - g ` S. f x \<noteq> 0}"
+              "negligible {x \<in> g ` S - g ` C. f x \<noteq> 0}"
+      using CNS by (blast intro: negligible_subset [OF neg_gN])+
+    then show "(f absolutely_integrable_on g ` C) = (f absolutely_integrable_on g ` S)"
+      by (rule absolutely_integrable_spike_set_eq)
+    show "(integral (g ` C) f = b) \<longleftrightarrow> (integral (g ` S) f = b)"
+      using integral_spike_set [OF neg] by simp
+  qed
+  ultimately show ?thesis
+    by simp
+qed
+
+
+corollary absolutely_integrable_change_of_variables:
+  fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
+  assumes "S \<in> sets lebesgue"
+    and "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
+    and "inj_on g S"
+  shows "f absolutely_integrable_on (g ` S)
+     \<longleftrightarrow> (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S"
+  using assms has_absolute_integral_change_of_variables by blast
+
+corollary integral_change_of_variables:
+  fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
+  assumes S: "S \<in> sets lebesgue"
+    and der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
+    and inj: "inj_on g S"
+    and disj: "(f absolutely_integrable_on (g ` S) \<or>
+        (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S)"
+  shows "integral (g ` S) f = integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x))"
+  using has_absolute_integral_change_of_variables [OF S der_g inj] disj
+  by blast
+
+lemma has_absolute_integral_change_of_variables_1:
+  fixes f :: "real \<Rightarrow> real^'n::{finite,wellorder}" and g :: "real \<Rightarrow> real"
+  assumes S: "S \<in> sets lebesgue"
+    and der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_vector_derivative g' x) (at x within S)"
+    and inj: "inj_on g S"
+  shows "(\<lambda>x. \<bar>g' x\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S \<and>
+           integral S (\<lambda>x. \<bar>g' x\<bar> *\<^sub>R f(g x)) = b
+     \<longleftrightarrow> f absolutely_integrable_on (g ` S) \<and> integral (g ` S) f = b"
+proof -
+  let ?lift = "vec :: real \<Rightarrow> real^1"
+  let ?drop = "(\<lambda>x::real^1. x $ 1)"
+  have S': "?lift ` S \<in> sets lebesgue"
+    by (auto intro: differentiable_image_in_sets_lebesgue [OF S] differentiable_vec)
+  have "((\<lambda>x. vec (g (x $ 1))) has_derivative ( *\<^sub>R) (g' z)) (at (vec z) within ?lift ` S)"
+    if "z \<in> S" for z
+    using der_g [OF that]
+    by (simp add: has_vector_derivative_def has_derivative_vector_1)
+  then have der': "\<And>x. x \<in> ?lift ` S \<Longrightarrow>
+        (?lift \<circ> g \<circ> ?drop has_derivative ( *\<^sub>R) (g' (?drop x))) (at x within ?lift ` S)"
+    by (auto simp: o_def)
+  have inj': "inj_on (vec \<circ> g \<circ> ?drop) (vec ` S)"
+    using inj by (simp add: inj_on_def)
+  let ?fg = "\<lambda>x. \<bar>g' x\<bar> *\<^sub>R f(g x)"
+  have "((\<lambda>x. ?fg x $ i) absolutely_integrable_on S \<and> ((\<lambda>x. ?fg x $ i) has_integral b $ i) S
+    \<longleftrightarrow> (\<lambda>x. f x $ i) absolutely_integrable_on g ` S \<and> ((\<lambda>x. f x $ i) has_integral b $ i) (g ` S))" for i
+    using has_absolute_integral_change_of_variables [OF S' der' inj', of "\<lambda>x. ?lift(f (?drop x) $ i)" "?lift (b$i)"]
+    unfolding integrable_on_1_iff integral_on_1_eq absolutely_integrable_on_1_iff absolutely_integrable_drop absolutely_integrable_on_def
+    by (auto simp: image_comp o_def integral_vec1_eq has_integral_iff)
+  then have "?fg absolutely_integrable_on S \<and> (?fg has_integral b) S
+         \<longleftrightarrow> f absolutely_integrable_on (g ` S) \<and> (f has_integral b) (g ` S)"
+    unfolding has_integral_componentwise_iff [where y=b]
+           absolutely_integrable_componentwise_iff [where f=f]
+           absolutely_integrable_componentwise_iff [where f = ?fg]
+    by (force simp: Basis_vec_def cart_eq_inner_axis)
+  then show ?thesis
+    using absolutely_integrable_on_def by blast
+qed
+
+
+corollary absolutely_integrable_change_of_variables_1:
+  fixes f :: "real \<Rightarrow> real^'n::{finite,wellorder}" and g :: "real \<Rightarrow> real"
+  assumes S: "S \<in> sets lebesgue"
+    and der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_vector_derivative g' x) (at x within S)"
+    and inj: "inj_on g S"
+  shows "(f absolutely_integrable_on g ` S \<longleftrightarrow>
+             (\<lambda>x. \<bar>g' x\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S)"
+  using has_absolute_integral_change_of_variables_1 [OF assms] by auto
+
+
+subsection\<open>Change of variables for integrals: special case of linear function\<close>
+
+lemma has_absolute_integral_change_of_variables_linear:
+  fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
+  assumes "linear g"
+  shows "(\<lambda>x. \<bar>det (matrix g)\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S \<and>
+           integral S (\<lambda>x. \<bar>det (matrix g)\<bar> *\<^sub>R f(g x)) = b
+     \<longleftrightarrow> f absolutely_integrable_on (g ` S) \<and> integral (g ` S) f = b"
+proof (cases "det(matrix g) = 0")
+  case True
+  then have "negligible(g ` S)"
+    using assms det_nz_iff_inj negligible_linear_singular_image by blast
+  with True show ?thesis
+    by (auto simp: absolutely_integrable_on_def integrable_negligible integral_negligible)
+next
+  case False
+  then obtain h where h: "\<And>x. x \<in> S \<Longrightarrow> h (g x) = x" "linear h"
+    using assms det_nz_iff_inj linear_injective_isomorphism by blast
+  show ?thesis
+  proof (rule has_absolute_integral_change_of_variables_invertible)
+    show "(g has_derivative g) (at x within S)" for x
+      by (simp add: assms linear_imp_has_derivative)
+    show "continuous_on (g ` S) h"
+      using continuous_on_eq_continuous_within has_derivative_continuous linear_imp_has_derivative h by blast
+  qed (use h in auto)
+qed
+
+lemma absolutely_integrable_change_of_variables_linear:
+  fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
+  assumes "linear g"
+  shows "(\<lambda>x. \<bar>det (matrix g)\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S
+     \<longleftrightarrow> f absolutely_integrable_on (g ` S)"
+  using assms has_absolute_integral_change_of_variables_linear by blast
+
+lemma absolutely_integrable_on_linear_image:
+  fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
+  assumes "linear g"
+  shows "f absolutely_integrable_on (g ` S)
+     \<longleftrightarrow> (f \<circ> g) absolutely_integrable_on S \<or> det(matrix g) = 0"
+  unfolding assms absolutely_integrable_change_of_variables_linear [OF assms, symmetric] absolutely_integrable_on_scaleR_iff
+  by (auto simp: set_integrable_def)
+
+lemma integral_change_of_variables_linear:
+  fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
+  assumes "linear g"
+      and "f absolutely_integrable_on (g ` S) \<or> (f \<circ> g) absolutely_integrable_on S"
+    shows "integral (g ` S) f = \<bar>det (matrix g)\<bar> *\<^sub>R integral S (f \<circ> g)"
+proof -
+  have "((\<lambda>x. \<bar>det (matrix g)\<bar> *\<^sub>R f (g x)) absolutely_integrable_on S) \<or> (f absolutely_integrable_on g ` S)"
+    using absolutely_integrable_on_linear_image assms by blast
+  moreover
+  have ?thesis if "((\<lambda>x. \<bar>det (matrix g)\<bar> *\<^sub>R f (g x)) absolutely_integrable_on S)" "(f absolutely_integrable_on g ` S)"
+    using has_absolute_integral_change_of_variables_linear [OF \<open>linear g\<close>] that
+    by (auto simp: o_def)
+  ultimately show ?thesis
+    using absolutely_integrable_change_of_variables_linear [OF \<open>linear g\<close>]
+    by blast
+qed
+
+subsection\<open>Change of variable for measure\<close>
+
+lemma has_measure_differentiable_image:
+  fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
+  assumes "S \<in> sets lebesgue"
+      and "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
+      and "inj_on f S"
+  shows "f ` S \<in> lmeasurable \<and> measure lebesgue (f ` S) = m
+     \<longleftrightarrow> ((\<lambda>x. \<bar>det (matrix (f' x))\<bar>) has_integral m) S"
+  using has_absolute_integral_change_of_variables [OF assms, of "\<lambda>x. (1::real^1)" "vec m"]
+  unfolding absolutely_integrable_on_1_iff integral_on_1_eq integrable_on_1_iff absolutely_integrable_on_def
+  by (auto simp: has_integral_iff lmeasurable_iff_integrable_on lmeasure_integral)
+
+lemma measurable_differentiable_image_eq:
+  fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
+  assumes "S \<in> sets lebesgue"
+      and "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
+      and "inj_on f S"
+  shows "f ` S \<in> lmeasurable \<longleftrightarrow> (\<lambda>x. \<bar>det (matrix (f' x))\<bar>) integrable_on S"
+  using has_measure_differentiable_image [OF assms]
+  by blast
+
+lemma measurable_differentiable_image_alt:
+  fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
+  assumes "S \<in> sets lebesgue"
+    and "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
+    and "inj_on f S"
+  shows "f ` S \<in> lmeasurable \<longleftrightarrow> (\<lambda>x. \<bar>det (matrix (f' x))\<bar>) absolutely_integrable_on S"
+  using measurable_differentiable_image_eq [OF assms]
+  by (simp only: absolutely_integrable_on_iff_nonneg)
+
+lemma measure_differentiable_image_eq:
+  fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
+  assumes S: "S \<in> sets lebesgue"
+    and der_f: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
+    and inj: "inj_on f S"
+    and intS: "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) integrable_on S"
+  shows "measure lebesgue (f ` S) = integral S (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)"
+  using measurable_differentiable_image_eq [OF S der_f inj]
+        assms has_measure_differentiable_image by blast
+
+end
diff -r ae76012879c6 -r 73a5a33486ee src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy
--- a/src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy	Tue Apr 17 18:04:49 2018 +0100
+++ b/src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy	Tue Apr 17 22:35:48 2018 +0100
@@ -1550,14 +1550,15 @@
   assume "0 < e"
   have "S \<in> lmeasurable"
     using \<open>negligible S\<close> by (simp add: negligible_iff_null_sets fmeasurableI_null_sets)
+  then have "S \<in> sets lebesgue"
+    by blast
   have e22: "0 < e/2 / (2 * B * real DIM('M)) ^ DIM('N)"
     using \<open>0 < e\<close> \<open>0 < B\<close> by (simp add: divide_simps)
-  obtain T
-    where "open T" "S \<subseteq> T" "T \<in> lmeasurable"
-      and "measure lebesgue T \<le> measure lebesgue S + e/2 / (2 * B * DIM('M)) ^ DIM('N)"
-    by (rule lmeasurable_outer_open [OF \<open>S \<in> lmeasurable\<close> e22])
+  obtain T where "open T" "S \<subseteq> T" "(T - S) \<in> lmeasurable" 
+                 "measure lebesgue (T - S) < e/2 / (2 * B * DIM('M)) ^ DIM('N)"
+    by (rule lmeasurable_outer_open [OF \<open>S \<in> sets lebesgue\<close> e22])
   then have T: "measure lebesgue T \<le> e/2 / (2 * B * DIM('M)) ^ DIM('N)"
-    using \<open>negligible S\<close> by (simp add: negligible_iff_null_sets measure_eq_0_null_sets)
+    using \<open>negligible S\<close> by (simp add: measure_Diff_null_set negligible_iff_null_sets)
   have "\<exists>r. 0 < r \<and> r \<le> 1/2 \<and>
             (x \<in> S \<longrightarrow> (\<forall>y. norm(y - x) < r
                        \<longrightarrow> y \<in> T \<and> (y \<in> S \<longrightarrow> norm(f y - f x) \<le> B * norm(y - x))))"
@@ -1690,7 +1691,7 @@
         using pairwise_subset [OF pw \<open>\<D>' \<subseteq> \<D>\<close>] unfolding pairwise_def apply force+
         done
       have le_meaT: "measure lebesgue (\<Union>\<D>') \<le> measure lebesgue T"
-      proof (rule measure_mono_fmeasurable [OF _ _ \<open>T \<in> lmeasurable\<close>])
+      proof (rule measure_mono_fmeasurable)
         show "(\<Union>\<D>') \<in> sets lebesgue"
           using div lmeasurable_division by auto
         have "\<Union>\<D>' \<subseteq> \<Union>\<D>"
@@ -1704,7 +1705,9 @@
             by (metis \<open>x \<in> D\<close> Int_iff dist_norm mem_ball norm_minus_commute subsetD RT)
         qed
         finally show "\<Union>\<D>' \<subseteq> T" .
-      qed
+        show "T \<in> lmeasurable"
+          using \<open>S \<in> lmeasurable\<close> \<open>S \<subseteq> T\<close> \<open>T - S \<in> lmeasurable\<close> fmeasurable_Diff_D by blast
+      qed 
       have "sum (measure lebesgue) \<D>' = sum content \<D>'"
         using  \<open>\<D>' \<subseteq> \<D>\<close> cbox by (force intro: sum.cong)
       then have "(2 * B * DIM('M)) ^ DIM('N) * sum (measure lebesgue) \<D>' =
@@ -3991,4 +3994,705 @@
       (auto simp: emeasure_lborel_box_eq emeasure_lborel_cbox_eq algebra_simps mem_box)
 qed
 
+subsection\<open>Various common equivalent forms of function measurability\<close>
+
+lemma indicator_sum_eq:
+  fixes m::real and f :: "'a \<Rightarrow> real"
+  assumes "\<bar>m\<bar> \<le> 2 ^ (2*n)" "m/2^n \<le> f x" "f x < (m+1)/2^n" "m \<in> \<int>"
+  shows   "(\<Sum>k::real | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n).
+            k/2^n * indicator {y. k/2^n \<le> f y \<and> f y < (k+1)/2^n} x)  = m/2^n"
+          (is "sum ?f ?S = _")
+proof -
+  have "sum ?f ?S = sum (\<lambda>k. k/2^n * indicator {y. k/2^n \<le> f y \<and> f y < (k+1)/2^n} x) {m}"
+  proof (rule comm_monoid_add_class.sum.mono_neutral_right)
+    show "finite ?S"
+      by (rule finite_abs_int_segment)
+    show "{m} \<subseteq> {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)}"
+      using assms by auto
+    show "\<forall>i\<in>{k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)} - {m}. ?f i = 0"
+      using assms by (auto simp: indicator_def Ints_def abs_le_iff divide_simps)
+  qed
+  also have "\<dots> = m/2^n"
+    using assms by (auto simp: indicator_def not_less)
+  finally show ?thesis .
+qed
+
+lemma measurable_on_sf_limit_lemma1:
+  fixes f :: "'a::euclidean_space \<Rightarrow> real"
+  assumes "\<And>a b. {x \<in> S. a \<le> f x \<and> f x < b} \<in> sets (lebesgue_on S)"
+  obtains g where "\<And>n. g n \<in> borel_measurable (lebesgue_on S)"
+                  "\<And>n. finite(range (g n))"
+                  "\<And>x. (\<lambda>n. g n x) \<longlonglongrightarrow> f x"
+proof
+  show "(\<lambda>x. sum (\<lambda>k::real. k/2^n * indicator {y. k/2^n \<le> f y \<and> f y < (k+1)/2^n} x)
+                 {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)}) \<in> borel_measurable (lebesgue_on S)"
+        (is "?g \<in> _")  for n
+  proof -
+    have "\<And>k. \<lbrakk>k \<in> \<int>; \<bar>k\<bar> \<le> 2 ^ (2*n)\<rbrakk>
+         \<Longrightarrow> Measurable.pred (lebesgue_on S) (\<lambda>x. k / (2^n) \<le> f x \<and> f x < (k+1) / (2^n))"
+      using assms by (force simp: pred_def space_restrict_space)
+    then show ?thesis
+      by (simp add: field_class.field_divide_inverse)
+  qed
+  show "finite (range (?g n))" for n
+  proof -
+    have "range (?g n) \<subseteq> (\<lambda>k. k/2^n) ` {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)}"
+    proof clarify
+      fix x
+      show "?g n x  \<in> (\<lambda>k. k/2^n) ` {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)}"
+      proof (cases "\<exists>k::real. k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n) \<and> k/2^n \<le> (f x) \<and> (f x) < (k+1)/2^n")
+        case True
+        then show ?thesis
+          apply clarify
+          by (subst indicator_sum_eq) auto
+      next
+        case False
+        then have "?g n x = 0" by auto
+        then show ?thesis by force
+      qed
+    qed
+    moreover have "finite ((\<lambda>k::real. (k/2^n)) ` {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)})"
+      by (simp add: finite_abs_int_segment)
+    ultimately show ?thesis
+      using finite_subset by blast
+  qed
+  show "(\<lambda>n. ?g n x) \<longlonglongrightarrow> f x" for x
+  proof (rule LIMSEQ_I)
+    fix e::real
+    assume "e > 0"
+    obtain N1 where N1: "\<bar>f x\<bar> < 2 ^ N1"
+      using real_arch_pow by fastforce
+    obtain N2 where N2: "(1/2) ^ N2 < e"
+      using real_arch_pow_inv \<open>e > 0\<close> by force
+    have "norm (?g n x - f x) < e" if n: "n \<ge> max N1 N2" for n
+    proof -
+      define m where "m \<equiv> floor(2^n * (f x))"
+      have "1 \<le> \<bar>2^n\<bar> * e"
+        using n N2 \<open>e > 0\<close> less_eq_real_def less_le_trans by (fastforce simp add: divide_simps)
+      then have *: "\<lbrakk>x \<le> y; y < x + 1\<rbrakk> \<Longrightarrow> abs(x - y) < \<bar>2^n\<bar> * e" for x y::real
+        by linarith
+      have "\<bar>2^n\<bar> * \<bar>m/2^n - f x\<bar> = \<bar>2^n * (m/2^n - f x)\<bar>"
+        by (simp add: abs_mult)
+      also have "\<dots> = \<bar>real_of_int \<lfloor>2^n * f x\<rfloor> - f x * 2^n\<bar>"
+        by (simp add: algebra_simps m_def)
+      also have "\<dots> < \<bar>2^n\<bar> * e"
+        by (rule *; simp add: mult.commute)
+      finally have "\<bar>2^n\<bar> * \<bar>m/2^n - f x\<bar> < \<bar>2^n\<bar> * e" .
+      then have me: "\<bar>m/2^n - f x\<bar> < e"
+        by simp
+      have "\<bar>real_of_int m\<bar> \<le> 2 ^ (2*n)"
+      proof (cases "f x < 0")
+        case True
+        then have "-\<lfloor>f x\<rfloor> \<le> \<lfloor>(2::real) ^ N1\<rfloor>"
+          using N1 le_floor_iff minus_le_iff by fastforce
+        with n True have "\<bar>real_of_int\<lfloor>f x\<rfloor>\<bar> \<le> 2 ^ N1"
+          by linarith
+        also have "\<dots> \<le> 2^n"
+          using n by (simp add: m_def)
+        finally have "\<bar>real_of_int \<lfloor>f x\<rfloor>\<bar> * 2^n \<le> 2^n * 2^n"
+          by simp
+        moreover
+        have "\<bar>real_of_int \<lfloor>2^n * f x\<rfloor>\<bar> \<le> \<bar>real_of_int \<lfloor>f x\<rfloor>\<bar> * 2^n"
+        proof -
+          have "\<bar>real_of_int \<lfloor>2^n * f x\<rfloor>\<bar> = - (real_of_int \<lfloor>2^n * f x\<rfloor>)"
+            using True by (simp add: abs_if mult_less_0_iff)
+          also have "\<dots> \<le> - (real_of_int (\<lfloor>(2::real) ^ n\<rfloor> * \<lfloor>f x\<rfloor>))"
+            using le_mult_floor_Ints [of "(2::real)^n"] by simp
+          also have "\<dots> \<le> \<bar>real_of_int \<lfloor>f x\<rfloor>\<bar> * 2^n"
+            using True
+            by simp
+          finally show ?thesis .
+        qed
+        ultimately show ?thesis
+          by (metis (no_types, hide_lams) m_def order_trans power2_eq_square power_even_eq)
+      next
+        case False
+        with n N1 have "f x \<le> 2^n"
+          by (simp add: not_less) (meson less_eq_real_def one_le_numeral order_trans power_increasing)
+        moreover have "0 \<le> m"
+          using False m_def by force
+        ultimately show ?thesis
+          by (metis abs_of_nonneg floor_mono le_floor_iff m_def of_int_0_le_iff power2_eq_square power_mult real_mult_le_cancel_iff1 zero_less_numeral mult.commute zero_less_power)
+      qed
+      then have "?g n x = m/2^n"
+        by (rule indicator_sum_eq) (auto simp: m_def mult.commute divide_simps)
+      then have "norm (?g n x - f x) = norm (m/2^n - f x)"
+        by simp
+      also have "\<dots> < e"
+        by (simp add: me)
+      finally show ?thesis .
+    qed
+    then show "\<exists>no. \<forall>n\<ge>no. norm (?g n x - f x) < e"
+      by blast
+  qed
+qed
+
+
+lemma borel_measurable_vimage_halfspace_component_lt:
+     "f \<in> borel_measurable (lebesgue_on S) \<longleftrightarrow>
+      (\<forall>a i. i \<in> Basis \<longrightarrow> {x \<in> S. f x \<bullet> i < a} \<in> sets (lebesgue_on S))"
+  apply (rule trans [OF borel_measurable_iff_halfspace_less])
+  apply (fastforce simp add: space_restrict_space)
+  done
+
+lemma borel_measurable_simple_function_limit:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  shows "f \<in> borel_measurable (lebesgue_on S) \<longleftrightarrow>
+         (\<exists>g. (\<forall>n. (g n) \<in> borel_measurable (lebesgue_on S)) \<and>
+              (\<forall>n. finite (range (g n))) \<and> (\<forall>x. (\<lambda>n. g n x) \<longlonglongrightarrow> f x))"
+proof -
+  have "\<exists>g. (\<forall>n. (g n) \<in> borel_measurable (lebesgue_on S)) \<and>
+            (\<forall>n. finite (range (g n))) \<and> (\<forall>x. (\<lambda>n. g n x) \<longlonglongrightarrow> f x)"
+       if f: "\<And>a i. i \<in> Basis \<Longrightarrow> {x \<in> S. f x \<bullet> i < a} \<in> sets (lebesgue_on S)"
+  proof -
+    have "\<exists>g. (\<forall>n. (g n) \<in> borel_measurable (lebesgue_on S)) \<and>
+                  (\<forall>n. finite(image (g n) UNIV)) \<and>
+                  (\<forall>x. ((\<lambda>n. g n x) \<longlonglongrightarrow> f x \<bullet> i))" if "i \<in> Basis" for i
+    proof (rule measurable_on_sf_limit_lemma1 [of S "\<lambda>x. f x \<bullet> i"])
+      show "{x \<in> S. a \<le> f x \<bullet> i \<and> f x \<bullet> i < b} \<in> sets (lebesgue_on S)" for a b
+      proof -
+        have "{x \<in> S. a \<le> f x \<bullet> i \<and> f x \<bullet> i < b} = {x \<in> S. f x \<bullet> i < b} - {x \<in> S. a > f x \<bullet> i}"
+          by auto
+        also have "\<dots> \<in> sets (lebesgue_on S)"
+          using f that by blast
+        finally show ?thesis .
+      qed
+    qed blast
+    then obtain g where g:
+          "\<And>i n. i \<in> Basis \<Longrightarrow> g i n \<in> borel_measurable (lebesgue_on S)"
+          "\<And>i n. i \<in> Basis \<Longrightarrow> finite(range (g i n))"
+          "\<And>i x. i \<in> Basis \<Longrightarrow> ((\<lambda>n. g i n x) \<longlonglongrightarrow> f x \<bullet> i)"
+      by metis
+    show ?thesis
+    proof (intro conjI allI exI)
+      show "(\<lambda>x. \<Sum>i\<in>Basis. g i n x *\<^sub>R i) \<in> borel_measurable (lebesgue_on S)" for n
+        by (intro borel_measurable_sum borel_measurable_scaleR) (auto intro: g)
+      show "finite (range (\<lambda>x. \<Sum>i\<in>Basis. g i n x *\<^sub>R i))" for n
+      proof -
+        have "range (\<lambda>x. \<Sum>i\<in>Basis. g i n x *\<^sub>R i) \<subseteq> (\<lambda>h. \<Sum>i\<in>Basis. h i *\<^sub>R i) ` PiE Basis (\<lambda>i. range (g i n))"
+        proof clarify
+          fix x
+          show "(\<Sum>i\<in>Basis. g i n x *\<^sub>R i) \<in> (\<lambda>h. \<Sum>i\<in>Basis. h i *\<^sub>R i) ` (\<Pi>\<^sub>E i\<in>Basis. range (g i n))"
+            by (rule_tac x="\<lambda>i\<in>Basis. g i n x" in image_eqI) auto
+        qed
+        moreover have "finite(PiE Basis (\<lambda>i. range (g i n)))"
+          by (simp add: g finite_PiE)
+        ultimately show ?thesis
+          by (metis (mono_tags, lifting) finite_surj)
+      qed
+      show "(\<lambda>n. \<Sum>i\<in>Basis. g i n x *\<^sub>R i) \<longlonglongrightarrow> f x" for x
+      proof -
+        have "(\<lambda>n. \<Sum>i\<in>Basis. g i n x *\<^sub>R i) \<longlonglongrightarrow> (\<Sum>i\<in>Basis. (f x \<bullet> i) *\<^sub>R i)"
+          by (auto intro!: tendsto_sum tendsto_scaleR g)
+        moreover have "(\<Sum>i\<in>Basis. (f x \<bullet> i) *\<^sub>R i) = f x"
+          using euclidean_representation by blast
+        ultimately show ?thesis
+          by metis
+      qed
+    qed
+  qed
+  moreover have "f \<in> borel_measurable (lebesgue_on S)"
+              if meas_g: "\<And>n. g n \<in> borel_measurable (lebesgue_on S)"
+                 and fin: "\<And>n. finite (range (g n))"
+                 and to_f: "\<And>x. (\<lambda>n. g n x) \<longlonglongrightarrow> f x" for  g
+    by (rule borel_measurable_LIMSEQ_metric [OF meas_g to_f])
+  ultimately show ?thesis
+    using borel_measurable_vimage_halfspace_component_lt by blast
+qed
+
+lemma borel_measurable_vimage_halfspace_component_ge:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  shows "f \<in> borel_measurable (lebesgue_on S) \<longleftrightarrow>
+         (\<forall>a i. i \<in> Basis \<longrightarrow> {x \<in> S. f x \<bullet> i \<ge> a} \<in> sets (lebesgue_on S))"
+  apply (rule trans [OF borel_measurable_iff_halfspace_ge])
+  apply (fastforce simp add: space_restrict_space)
+  done
+
+lemma borel_measurable_vimage_halfspace_component_gt:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  shows "f \<in> borel_measurable (lebesgue_on S) \<longleftrightarrow>
+         (\<forall>a i. i \<in> Basis \<longrightarrow> {x \<in> S. f x \<bullet> i > a} \<in> sets (lebesgue_on S))"
+  apply (rule trans [OF borel_measurable_iff_halfspace_greater])
+  apply (fastforce simp add: space_restrict_space)
+  done
+
+lemma borel_measurable_vimage_halfspace_component_le:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  shows "f \<in> borel_measurable (lebesgue_on S) \<longleftrightarrow>
+         (\<forall>a i. i \<in> Basis \<longrightarrow> {x \<in> S. f x \<bullet> i \<le> a} \<in> sets (lebesgue_on S))"
+  apply (rule trans [OF borel_measurable_iff_halfspace_le])
+  apply (fastforce simp add: space_restrict_space)
+  done
+
+lemma
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  shows borel_measurable_vimage_open_interval:
+         "f \<in> borel_measurable (lebesgue_on S) \<longleftrightarrow>
+         (\<forall>a b. {x \<in> S. f x \<in> box a b} \<in> sets (lebesgue_on S))" (is ?thesis1)
+   and borel_measurable_vimage_open:
+         "f \<in> borel_measurable (lebesgue_on S) \<longleftrightarrow>
+         (\<forall>T. open T \<longrightarrow> {x \<in> S. f x \<in> T} \<in> sets (lebesgue_on S))" (is ?thesis2)
+proof -
+  have "{x \<in> S. f x \<in> box a b} \<in> sets (lebesgue_on S)" if "f \<in> borel_measurable (lebesgue_on S)" for a b
+  proof -
+    have "S = S \<inter> space lebesgue"
+      by simp
+    then have "S \<inter> (f -` box a b) \<in> sets (lebesgue_on S)"
+      by (metis (no_types) box_borel in_borel_measurable_borel inf_sup_aci(1) space_restrict_space that)
+    then show ?thesis
+      by (simp add: Collect_conj_eq vimage_def)
+  qed
+  moreover
+  have "{x \<in> S. f x \<in> T} \<in> sets (lebesgue_on S)"
+       if T: "\<And>a b. {x \<in> S. f x \<in> box a b} \<in> sets (lebesgue_on S)" "open T" for T
+  proof -
+    obtain \<D> where "countable \<D>" and \<D>: "\<And>X. X \<in> \<D> \<Longrightarrow> \<exists>a b. X = box a b" "\<Union>\<D> = T"
+      using open_countable_Union_open_box that \<open>open T\<close> by metis
+    then have eq: "{x \<in> S. f x \<in> T} = (\<Union>U \<in> \<D>. {x \<in> S. f x \<in> U})"
+      by blast
+    have "{x \<in> S. f x \<in> U} \<in> sets (lebesgue_on S)" if "U \<in> \<D>" for U
+      using that T \<D> by blast
+    then show ?thesis
+      by (auto simp: eq intro: Sigma_Algebra.sets.countable_UN' [OF \<open>countable \<D>\<close>])
+  qed
+  moreover
+  have eq: "{x \<in> S. f x \<bullet> i < a} = {x \<in> S. f x \<in> {y. y \<bullet> i < a}}" for i a
+    by auto
+  have "f \<in> borel_measurable (lebesgue_on S)"
+    if "\<And>T. open T \<Longrightarrow> {x \<in> S. f x \<in> T} \<in> sets (lebesgue_on S)"
+    by (metis (no_types) eq borel_measurable_vimage_halfspace_component_lt open_halfspace_component_lt that)
+  ultimately show "?thesis1" "?thesis2"
+    by blast+
+qed
+
+
+lemma borel_measurable_vimage_closed:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  shows "f \<in> borel_measurable (lebesgue_on S) \<longleftrightarrow>
+         (\<forall>T. closed T \<longrightarrow> {x \<in> S. f x \<in> T} \<in> sets (lebesgue_on S))"
+        (is "?lhs = ?rhs")
+proof -
+  have eq: "{x \<in> S. f x \<in> T} = S - {x \<in> S. f x \<in> (- T)}" for T
+    by auto
+  show ?thesis
+    apply (simp add: borel_measurable_vimage_open, safe)
+     apply (simp_all (no_asm) add: eq)
+     apply (intro sets.Diff sets_lebesgue_on_refl, force simp: closed_open)
+    apply (intro sets.Diff sets_lebesgue_on_refl, force simp: open_closed)
+    done
+qed
+
+lemma borel_measurable_vimage_closed_interval:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  shows "f \<in> borel_measurable (lebesgue_on S) \<longleftrightarrow>
+         (\<forall>a b. {x \<in> S. f x \<in> cbox a b} \<in> sets (lebesgue_on S))"
+        (is "?lhs = ?rhs")
+proof
+  assume ?lhs then show ?rhs
+    using borel_measurable_vimage_closed by blast
+next
+  assume RHS: ?rhs
+  have "{x \<in> S. f x \<in> T} \<in> sets (lebesgue_on S)" if "open T" for T
+  proof -
+    obtain \<D> where "countable \<D>" and \<D>: "\<D> \<subseteq> Pow T" "\<And>X. X \<in> \<D> \<Longrightarrow> \<exists>a b. X = cbox a b" "\<Union>\<D> = T"
+      using open_countable_Union_open_cbox that \<open>open T\<close> by metis
+    then have eq: "{x \<in> S. f x \<in> T} = (\<Union>U \<in> \<D>. {x \<in> S. f x \<in> U})"
+      by blast
+    have "{x \<in> S. f x \<in> U} \<in> sets (lebesgue_on S)" if "U \<in> \<D>" for U
+      using that \<D> by (metis RHS)
+    then show ?thesis
+      by (auto simp: eq intro: Sigma_Algebra.sets.countable_UN' [OF \<open>countable \<D>\<close>])
+  qed
+  then show ?lhs
+    by (simp add: borel_measurable_vimage_open)
+qed
+
+lemma borel_measurable_UNIV_eq: "borel_measurable (lebesgue_on UNIV) = borel_measurable lebesgue"
+  by auto
+
+lemma borel_measurable_vimage_borel:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  shows "f \<in> borel_measurable (lebesgue_on S) \<longleftrightarrow>
+         (\<forall>T. T \<in> sets borel \<longrightarrow> {x \<in> S. f x \<in> T} \<in> sets (lebesgue_on S))"
+        (is "?lhs = ?rhs")
+proof
+  assume f: ?lhs
+  then show ?rhs
+    using measurable_sets [OF f]
+      by (simp add: Collect_conj_eq inf_sup_aci(1) space_restrict_space vimage_def)
+qed (simp add: borel_measurable_vimage_open_interval)
+
+lemma lebesgue_measurable_vimage_borel:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes "f \<in> borel_measurable lebesgue" "T \<in> sets borel"
+  shows "{x. f x \<in> T} \<in> sets lebesgue"
+  using assms borel_measurable_vimage_borel [of f UNIV] by auto
+
+lemma borel_measurable_If_I:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes f: "f \<in> borel_measurable (lebesgue_on S)" and S: "S \<in> sets lebesgue"
+  shows "(\<lambda>x. if x \<in> S then f x else 0) \<in> borel_measurable lebesgue"
+proof -
+  have eq: "{x. x \<notin> S} \<union> {x. f x \<in> Y} = {x. x \<notin> S} \<union> {x. f x \<in> Y} \<inter> S" for Y
+    by blast
+  show ?thesis
+  using f S
+  apply (simp add: vimage_def in_borel_measurable_borel Ball_def)
+  apply (elim all_forward imp_forward asm_rl)
+  apply (simp only: Collect_conj_eq Collect_disj_eq imp_conv_disj eq)
+  apply (auto simp: Compl_eq [symmetric] Compl_in_sets_lebesgue sets_restrict_space_iff)
+  done
+qed
+
+lemma borel_measurable_If_D:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes "(\<lambda>x. if x \<in> S then f x else 0) \<in> borel_measurable lebesgue"
+  shows "f \<in> borel_measurable (lebesgue_on S)"
+  using assms
+  apply (simp add: in_borel_measurable_borel Ball_def)
+  apply (elim all_forward imp_forward asm_rl)
+  apply (force simp: space_restrict_space sets_restrict_space image_iff intro: rev_bexI)
+  done
+
+lemma borel_measurable_UNIV:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes "S \<in> sets lebesgue"
+  shows "(\<lambda>x. if x \<in> S then f x else 0) \<in> borel_measurable lebesgue \<longleftrightarrow> f \<in> borel_measurable (lebesgue_on S)"
+  using assms borel_measurable_If_D borel_measurable_If_I by blast
+
+lemma borel_measurable_lebesgue_preimage_borel:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  shows "f \<in> borel_measurable lebesgue \<longleftrightarrow>
+         (\<forall>T. T \<in> sets borel \<longrightarrow> {x. f x \<in> T} \<in> sets lebesgue)"
+  apply (intro iffI allI impI lebesgue_measurable_vimage_borel)
+    apply (auto simp: in_borel_measurable_borel vimage_def)
+  done
+
+subsection\<open>More results on integrability\<close>
+
+lemma integrable_on_all_intervals_UNIV:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
+  assumes intf: "\<And>a b. f integrable_on cbox a b"
+    and normf: "\<And>x. norm(f x) \<le> g x" and g: "g integrable_on UNIV"
+  shows "f integrable_on UNIV"
+proof -
+have intg: "(\<forall>a b. g integrable_on cbox a b)"
+    and gle_e: "\<forall>e>0. \<exists>B>0. \<forall>a b c d.
+                    ball 0 B \<subseteq> cbox a b \<and> cbox a b \<subseteq> cbox c d \<longrightarrow>
+                    \<bar>integral (cbox a b) g - integral (cbox c d) g\<bar>
+                    < e"
+    using g
+    by (auto simp: integrable_alt_subset [of _ UNIV] intf)
+  have le: "norm (integral (cbox a b) f - integral (cbox c d) f) \<le> \<bar>integral (cbox a b) g - integral (cbox c d) g\<bar>"
+    if "cbox a b \<subseteq> cbox c d" for a b c d
+  proof -
+    have "norm (integral (cbox a b) f - integral (cbox c d) f) = norm (integral (cbox c d - cbox a b) f)"
+      using intf that by (simp add: norm_minus_commute integral_setdiff)
+    also have "\<dots> \<le> integral (cbox c d - cbox a b) g"
+    proof (rule integral_norm_bound_integral [OF _ _ normf])
+      show "f integrable_on cbox c d - cbox a b" "g integrable_on cbox c d - cbox a b"
+        by (meson integrable_integral integrable_setdiff intf intg negligible_setdiff that)+
+    qed
+    also have "\<dots> = integral (cbox c d) g - integral (cbox a b) g"
+      using intg that by (simp add: integral_setdiff)
+    also have "\<dots> \<le> \<bar>integral (cbox a b) g - integral (cbox c d) g\<bar>"
+      by simp
+    finally show ?thesis .
+  qed
+  show ?thesis
+    using gle_e
+    apply (simp add: integrable_alt_subset [of _ UNIV] intf)
+    apply (erule imp_forward all_forward ex_forward asm_rl)+
+    by (meson not_less order_trans le)
+qed
+
+lemma integrable_on_all_intervals_integrable_bound:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
+  assumes intf: "\<And>a b. (\<lambda>x. if x \<in> S then f x else 0) integrable_on cbox a b"
+    and normf: "\<And>x. x \<in> S \<Longrightarrow> norm(f x) \<le> g x" and g: "g integrable_on S"
+  shows "f integrable_on S"
+  using integrable_on_all_intervals_UNIV [OF intf, of "(\<lambda>x. if x \<in> S then g x else 0)"]
+  by (simp add: g integrable_restrict_UNIV normf)
+
+lemma measurable_bounded_lemma:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes f: "f \<in> borel_measurable lebesgue" and g: "g integrable_on cbox a b"
+    and normf: "\<And>x. x \<in> cbox a b \<Longrightarrow> norm(f x) \<le> g x"
+  shows "f integrable_on cbox a b"
+proof -
+  have "g absolutely_integrable_on cbox a b"
+    by (metis (full_types) add_increasing g le_add_same_cancel1 nonnegative_absolutely_integrable_1 norm_ge_zero normf)
+  then have "integrable (lebesgue_on (cbox a b)) g"
+    by (simp add: integrable_restrict_space set_integrable_def)
+  then have "integrable (lebesgue_on (cbox a b)) f"
+  proof (rule Bochner_Integration.integrable_bound)
+    show "AE x in lebesgue_on (cbox a b). norm (f x) \<le> norm (g x)"
+      by (rule AE_I2) (auto intro: normf order_trans)
+  qed (simp add: f measurable_restrict_space1)
+  then show ?thesis
+    by (simp add: integrable_on_lebesgue_on)
+qed
+
+proposition measurable_bounded_by_integrable_imp_integrable:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes f: "f \<in> borel_measurable (lebesgue_on S)" and g: "g integrable_on S"
+    and normf: "\<And>x. x \<in> S \<Longrightarrow> norm(f x) \<le> g x" and S: "S \<in> sets lebesgue"
+  shows "f integrable_on S"
+proof (rule integrable_on_all_intervals_integrable_bound [OF _ normf g])
+  show "(\<lambda>x. if x \<in> S then f x else 0) integrable_on cbox a b" for a b
+  proof (rule measurable_bounded_lemma)
+    show "(\<lambda>x. if x \<in> S then f x else 0) \<in> borel_measurable lebesgue"
+      by (simp add: S borel_measurable_UNIV f)
+    show "(\<lambda>x. if x \<in> S then g x else 0) integrable_on cbox a b"
+      by (simp add: g integrable_altD(1))
+    show "norm (if x \<in> S then f x else 0) \<le> (if x \<in> S then g x else 0)" for x
+      using normf by simp
+  qed
+qed
+
+subsection\<open> Relation between Borel measurability and integrability.\<close>
+
+lemma integrable_imp_measurable_weak:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes "S \<in> sets lebesgue" "f integrable_on S"
+  shows "f \<in> borel_measurable (lebesgue_on S)"
+  by (metis (mono_tags, lifting) assms has_integral_implies_lebesgue_measurable borel_measurable_restrict_space_iff integrable_on_def sets.Int_space_eq2)
+
+lemma integrable_imp_measurable:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes "f integrable_on S"
+  shows "f \<in> borel_measurable (lebesgue_on S)"
+proof -
+  have "(UNIV::'a set) \<in> sets lborel"
+    by simp
+  then show ?thesis
+    using assms borel_measurable_If_D borel_measurable_UNIV_eq integrable_imp_measurable_weak integrable_restrict_UNIV by blast
+qed
+
+proposition negligible_differentiable_vimage:
+  fixes f :: "'a \<Rightarrow> 'a::euclidean_space"
+  assumes "negligible T"
+    and f': "\<And>x. x \<in> S \<Longrightarrow> inj(f' x)"
+    and derf: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
+  shows "negligible {x \<in> S. f x \<in> T}"
+proof -
+  define U where
+    "U \<equiv> \<lambda>n::nat. {x \<in> S. \<forall>y. y \<in> S \<and> norm(y - x) < 1/n
+                     \<longrightarrow> norm(y - x) \<le> n * norm(f y - f x)}"
+  have "negligible {x \<in> U n. f x \<in> T}" if "n > 0" for n
+  proof (subst locally_negligible_alt, clarify)
+    fix a
+    assume a: "a \<in> U n" and fa: "f a \<in> T"
+    define V where "V \<equiv> {x. x \<in> U n \<and> f x \<in> T} \<inter> ball a (1 / n / 2)"
+    show "\<exists>V. openin (subtopology euclidean {x \<in> U n. f x \<in> T}) V \<and> a \<in> V \<and> negligible V"
+    proof (intro exI conjI)
+      have noxy: "norm(x - y) \<le> n * norm(f x - f y)" if "x \<in> V" "y \<in> V" for x y
+        using that unfolding U_def V_def mem_Collect_eq Int_iff mem_ball dist_norm
+        by (meson norm_triangle_half_r)
+      then have "inj_on f V"
+        by (force simp: inj_on_def)
+      then obtain g where g: "\<And>x. x \<in> V \<Longrightarrow> g(f x) = x"
+        by (metis inv_into_f_f)
+      have "\<exists>T' B. open T' \<and> f x \<in> T' \<and>
+                   (\<forall>y\<in>f ` V \<inter> T \<inter> T'. norm (g y - g (f x)) \<le> B * norm (y - f x))"
+        if "f x \<in> T" "x \<in> V" for x
+        apply (rule_tac x = "ball (f x) 1" in exI)
+        using that noxy by (force simp: g)
+      then have "negligible (g ` (f ` V \<inter> T))"
+        by (force simp: \<open>negligible T\<close> negligible_Int intro!: negligible_locally_Lipschitz_image)
+      moreover have "V \<subseteq> g ` (f ` V \<inter> T)"
+        by (force simp: g image_iff V_def)
+      ultimately show "negligible V"
+        by (rule negligible_subset)
+    qed (use a fa V_def that in auto)
+  qed
+  with negligible_countable_Union have "negligible (\<Union>n \<in> {0<..}. {x. x \<in> U n \<and> f x \<in> T})"
+    by auto
+  moreover have "{x \<in> S. f x \<in> T} \<subseteq> (\<Union>n \<in> {0<..}. {x. x \<in> U n \<and> f x \<in> T})"
+  proof clarsimp
+    fix x
+    assume "x \<in> S" and "f x \<in> T"
+    then obtain inj: "inj(f' x)" and der: "(f has_derivative f' x) (at x within S)"
+      using assms by metis
+    moreover have "linear(f' x)"
+      and eps: "\<And>\<epsilon>. \<epsilon> > 0 \<Longrightarrow> \<exists>\<delta>>0. \<forall>y\<in>S. norm (y - x) < \<delta> \<longrightarrow>
+                      norm (f y - f x - f' x (y - x)) \<le> \<epsilon> * norm (y - x)"
+      using der by (auto simp: has_derivative_within_alt linear_linear)
+    ultimately obtain g where "linear g" and g: "g \<circ> f' x = id"
+      using linear_injective_left_inverse by metis
+    then obtain B where "B > 0" and B: "\<And>z. B * norm z \<le> norm(f' x z)"
+      using linear_invertible_bounded_below_pos \<open>linear (f' x)\<close> by blast
+    then obtain i where "i \<noteq> 0" and i: "1 / real i < B"
+      by (metis (full_types) inverse_eq_divide real_arch_invD)
+    then obtain \<delta> where "\<delta> > 0"
+         and \<delta>: "\<And>y. \<lbrakk>y\<in>S; norm (y - x) < \<delta>\<rbrakk> \<Longrightarrow>
+                  norm (f y - f x - f' x (y - x)) \<le> (B - 1 / real i) * norm (y - x)"
+      using eps [of "B - 1/i"] by auto
+    then obtain j where "j \<noteq> 0" and j: "inverse (real j) < \<delta>"
+      using real_arch_inverse by blast
+    have "norm (y - x)/(max i j) \<le> norm (f y - f x)"
+      if "y \<in> S" and less: "norm (y - x) < 1 / (max i j)" for y
+    proof -
+      have "1 / real (max i j) < \<delta>"
+        using j \<open>j \<noteq> 0\<close> \<open>0 < \<delta>\<close>
+        by (auto simp: divide_simps max_mult_distrib_left of_nat_max)
+    then have "norm (y - x) < \<delta>"
+      using less by linarith
+    with \<delta> \<open>y \<in> S\<close> have le: "norm (f y - f x - f' x (y - x)) \<le> B * norm (y - x) - norm (y - x)/i"
+      by (auto simp: algebra_simps)
+    have *: "\<lbrakk>norm(f - f' - y) \<le> b - c; b \<le> norm y; d \<le> c\<rbrakk> \<Longrightarrow> d \<le> norm(f - f')"
+      for b c d and y f f'::'a
+      using norm_triangle_ineq3 [of "f - f'" y] by simp
+    show ?thesis
+      apply (rule * [OF le B])
+      using \<open>i \<noteq> 0\<close> \<open>j \<noteq> 0\<close> by (simp add: divide_simps max_mult_distrib_left of_nat_max less_max_iff_disj)
+  qed
+  with \<open>x \<in> S\<close> \<open>i \<noteq> 0\<close> \<open>j \<noteq> 0\<close> show "\<exists>n\<in>{0<..}. x \<in> U n"
+    by (rule_tac x="max i j" in bexI) (auto simp: field_simps U_def less_max_iff_disj)
+qed
+  ultimately show ?thesis
+    by (rule negligible_subset)
+qed
+
+lemma absolutely_integrable_Un:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes S: "f absolutely_integrable_on S" and T: "f absolutely_integrable_on T"
+  shows "f absolutely_integrable_on (S \<union> T)"
+proof -
+  have [simp]: "{x. (if x \<in> A then f x else 0) \<noteq> 0} = {x \<in> A. f x \<noteq> 0}" for A
+    by auto
+  let ?ST = "{x \<in> S. f x \<noteq> 0} \<inter> {x \<in> T. f x \<noteq> 0}"
+  have "?ST \<in> sets lebesgue"
+  proof (rule Sigma_Algebra.sets.Int)
+    have "f integrable_on S"
+      using S absolutely_integrable_on_def by blast
+    then have "(\<lambda>x. if x \<in> S then f x else 0) integrable_on UNIV"
+      by (simp add: integrable_restrict_UNIV)
+    then have borel: "(\<lambda>x. if x \<in> S then f x else 0) \<in> borel_measurable (lebesgue_on UNIV)"
+      using integrable_imp_measurable borel_measurable_UNIV_eq by blast
+    then show "{x \<in> S. f x \<noteq> 0} \<in> sets lebesgue"
+      unfolding borel_measurable_vimage_open
+      by (rule allE [where x = "-{0}"]) auto
+  next
+    have "f integrable_on T"
+      using T absolutely_integrable_on_def by blast
+    then have "(\<lambda>x. if x \<in> T then f x else 0) integrable_on UNIV"
+      by (simp add: integrable_restrict_UNIV)
+    then have borel: "(\<lambda>x. if x \<in> T then f x else 0) \<in> borel_measurable (lebesgue_on UNIV)"
+      using integrable_imp_measurable borel_measurable_UNIV_eq by blast
+    then show "{x \<in> T. f x \<noteq> 0} \<in> sets lebesgue"
+      unfolding borel_measurable_vimage_open
+      by (rule allE [where x = "-{0}"]) auto
+  qed
+  then have "f absolutely_integrable_on ?ST"
+    by (rule set_integrable_subset [OF S]) auto
+  then have Int: "(\<lambda>x. if x \<in> ?ST then f x else 0) absolutely_integrable_on UNIV"
+    using absolutely_integrable_restrict_UNIV by blast
+  have "(\<lambda>x. if x \<in> S then f x else 0) absolutely_integrable_on UNIV"
+       "(\<lambda>x. if x \<in> T then f x else 0) absolutely_integrable_on UNIV"
+    using S T absolutely_integrable_restrict_UNIV by blast+
+  then have "(\<lambda>x. (if x \<in> S then f x else 0) + (if x \<in> T then f x else 0)) absolutely_integrable_on UNIV"
+    by (rule absolutely_integrable_add)
+  then have "(\<lambda>x. ((if x \<in> S then f x else 0) + (if x \<in> T then f x else 0)) - (if x \<in> ?ST then f x else 0)) absolutely_integrable_on UNIV"
+    using Int by (rule absolutely_integrable_diff)
+  then have "(\<lambda>x. if x \<in> S \<union> T then f x else 0) absolutely_integrable_on UNIV"
+    by (rule absolutely_integrable_spike) (auto intro: empty_imp_negligible)
+  then show ?thesis
+    unfolding absolutely_integrable_restrict_UNIV .
+qed
+
+
+
+
+subsubsection\<open>Differentiability of inverse function (most basic form)\<close>
+
+proposition has_derivative_inverse_within:
+  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::euclidean_space"
+  assumes der_f: "(f has_derivative f') (at a within S)"
+    and cont_g: "continuous (at (f a) within f ` S) g"
+    and "a \<in> S" "linear g'" and id: "g' \<circ> f' = id"
+    and gf: "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x"
+  shows "(g has_derivative g') (at (f a) within f ` S)"
+proof -
+  have [simp]: "g' (f' x) = x" for x
+    by (simp add: local.id pointfree_idE)
+  have "bounded_linear f'"
+    and f': "\<And>e. e>0 \<Longrightarrow> \<exists>d>0. \<forall>y\<in>S. norm (y - a) < d \<longrightarrow>
+                        norm (f y - f a - f' (y - a)) \<le> e * norm (y - a)"
+    using der_f by (auto simp: has_derivative_within_alt)
+  obtain C where "C > 0" and C: "\<And>x. norm (g' x) \<le> C * norm x"
+    using linear_bounded_pos [OF \<open>linear g'\<close>] by metis
+  obtain B k where "B > 0" "k > 0"
+    and Bk: "\<And>x. \<lbrakk>x \<in> S; norm(f x - f a) < k\<rbrakk> \<Longrightarrow> norm(x - a) \<le> B * norm(f x - f a)"
+  proof -
+    obtain B where "B > 0" and B: "\<And>x. B * norm x \<le> norm (f' x)"
+      using linear_inj_bounded_below_pos [of f'] \<open>linear g'\<close> id der_f has_derivative_linear
+        linear_invertible_bounded_below_pos by blast
+    then obtain d where "d>0"
+      and d: "\<And>y. \<lbrakk>y \<in> S; norm (y - a) < d\<rbrakk> \<Longrightarrow>
+                    norm (f y - f a - f' (y - a)) \<le> B / 2 * norm (y - a)"
+      using f' [of "B/2"] by auto
+    then obtain e where "e > 0"
+      and e: "\<And>x. \<lbrakk>x \<in> S; norm (f x - f a) < e\<rbrakk> \<Longrightarrow> norm (g (f x) - g (f a)) < d"
+      using cont_g by (auto simp: continuous_within_eps_delta dist_norm)
+    show thesis
+    proof
+      show "2/B > 0"
+        using \<open>B > 0\<close> by simp
+      show "norm (x - a) \<le> 2 / B * norm (f x - f a)"
+        if "x \<in> S" "norm (f x - f a) < e" for x
+      proof -
+        have xa: "norm (x - a) < d"
+          using e [OF that] gf by (simp add: \<open>a \<in> S\<close> that)
+        have *: "\<lbrakk>norm(y - f') \<le> B / 2 * norm x; B * norm x \<le> norm f'\<rbrakk>
+                 \<Longrightarrow> norm y \<ge> B / 2 * norm x" for y f'::'b and x::'a
+          using norm_triangle_ineq3 [of y f'] by linarith
+        show ?thesis
+          using * [OF d [OF \<open>x \<in> S\<close> xa] B] \<open>B > 0\<close> by (simp add: field_simps)
+      qed
+    qed (use \<open>e > 0\<close> in auto)
+  qed
+  show ?thesis
+    unfolding has_derivative_within_alt
+  proof (intro conjI impI allI)
+    show "bounded_linear g'"
+      using \<open>linear g'\<close> by (simp add: linear_linear)
+  next
+    fix e :: "real"
+    assume "e > 0"
+    then obtain d where "d>0"
+      and d: "\<And>y. \<lbrakk>y \<in> S; norm (y - a) < d\<rbrakk> \<Longrightarrow>
+                    norm (f y - f a - f' (y - a)) \<le> e / (B * C) * norm (y - a)"
+      using f' [of "e / (B * C)"] \<open>B > 0\<close> \<open>C > 0\<close> by auto
+    have "norm (x - a - g' (f x - f a)) \<le> e * norm (f x - f a)"
+      if "x \<in> S" and lt_k: "norm (f x - f a) < k" and lt_dB: "norm (f x - f a) < d/B" for x
+    proof -
+      have "norm (x - a) \<le> B * norm(f x - f a)"
+        using Bk lt_k \<open>x \<in> S\<close> by blast
+      also have "\<dots> < d"
+        by (metis \<open>0 < B\<close> lt_dB mult.commute pos_less_divide_eq)
+      finally have lt_d: "norm (x - a) < d" .
+      have "norm (x - a - g' (f x - f a)) \<le> norm(g'(f x - f a - (f' (x - a))))"
+        by (simp add: linear_diff [OF \<open>linear g'\<close>] norm_minus_commute)
+      also have "\<dots> \<le> C * norm (f x - f a - f' (x - a))"
+        using C by blast
+      also have "\<dots> \<le> e * norm (f x - f a)"
+      proof -
+        have "norm (f x - f a - f' (x - a)) \<le> e / (B * C) * norm (x - a)"
+          using d [OF \<open>x \<in> S\<close> lt_d] .
+        also have "\<dots> \<le> (norm (f x - f a) * e) / C"
+          using \<open>B > 0\<close> \<open>C > 0\<close> \<open>e > 0\<close> by (simp add: field_simps Bk lt_k \<open>x \<in> S\<close>)
+        finally show ?thesis
+          using \<open>C > 0\<close> by (simp add: field_simps)
+      qed
+    finally show ?thesis .
+    qed
+    then show "\<exists>d>0. \<forall>y\<in>f ` S.
+               norm (y - f a) < d \<longrightarrow>
+               norm (g y - g (f a) - g' (y - f a)) \<le> e * norm (y - f a)"
+      apply (rule_tac x="min k (d / B)" in exI)
+      using \<open>k > 0\<close> \<open>B > 0\<close> \<open>d > 0\<close> \<open>a \<in> S\<close> by (auto simp: gf)
+  qed
+qed
+
 end
diff -r ae76012879c6 -r 73a5a33486ee src/HOL/Analysis/Henstock_Kurzweil_Integration.thy
--- a/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy	Tue Apr 17 18:04:49 2018 +0100
+++ b/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy	Tue Apr 17 22:35:48 2018 +0100
@@ -5069,6 +5069,56 @@
       norm (integral (cbox a b) (\<lambda>x. if x \<in> s then f x else 0) - integral (cbox c d)  (\<lambda>x. if x \<in> s then f x else 0)) < e"
   using assms[unfolded integrable_alt[of f]] by auto
 
+lemma integrable_alt_subset:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
+  shows
+     "f integrable_on S \<longleftrightarrow>
+      (\<forall>a b. (\<lambda>x. if x \<in> S then f x else 0) integrable_on cbox a b) \<and>
+      (\<forall>e>0. \<exists>B>0. \<forall>a b c d.
+                      ball 0 B \<subseteq> cbox a b \<and> cbox a b \<subseteq> cbox c d
+                      \<longrightarrow> norm(integral (cbox a b) (\<lambda>x. if x \<in> S then f x else 0) -
+                               integral (cbox c d) (\<lambda>x. if x \<in> S then f x else 0)) < e)"
+      (is "_ = ?rhs")
+proof -
+  let ?g = "\<lambda>x. if x \<in> S then f x else 0"
+  have "f integrable_on S \<longleftrightarrow>
+        (\<forall>a b. ?g integrable_on cbox a b) \<and>
+        (\<forall>e>0. \<exists>B>0. \<forall>a b c d. ball 0 B \<subseteq> cbox a b \<and> ball 0 B \<subseteq> cbox c d \<longrightarrow>
+           norm (integral (cbox a b) ?g - integral (cbox c d)  ?g) < e)"
+    by (rule integrable_alt)
+  also have "\<dots> = ?rhs"
+  proof -
+    { fix e :: "real"
+      assume e: "\<And>e. e>0 \<Longrightarrow> \<exists>B>0. \<forall>a b c d. ball 0 B \<subseteq> cbox a b \<and> cbox a b \<subseteq> cbox c d \<longrightarrow>
+                                   norm (integral (cbox a b) ?g - integral (cbox c d) ?g) < e"
+        and "e > 0"
+      obtain B where "B > 0"
+        and B: "\<And>a b c d. \<lbrakk>ball 0 B \<subseteq> cbox a b; cbox a b \<subseteq> cbox c d\<rbrakk> \<Longrightarrow>
+                           norm (integral (cbox a b) ?g - integral (cbox c d) ?g) < e/2"
+        using \<open>e > 0\<close> e [of "e/2"] by force
+      have "\<exists>B>0. \<forall>a b c d.
+               ball 0 B \<subseteq> cbox a b \<and> ball 0 B \<subseteq> cbox c d \<longrightarrow>
+               norm (integral (cbox a b) ?g - integral (cbox c d) ?g) < e"
+      proof (intro exI allI conjI impI)
+        fix a b c d :: "'a"
+        let ?\<alpha> = "\<Sum>i\<in>Basis. max (a \<bullet> i) (c \<bullet> i) *\<^sub>R i"
+        let ?\<beta> = "\<Sum>i\<in>Basis. min (b \<bullet> i) (d \<bullet> i) *\<^sub>R i"
+        show "norm (integral (cbox a b) ?g - integral (cbox c d) ?g) < e"
+          if ball: "ball 0 B \<subseteq> cbox a b \<and> ball 0 B \<subseteq> cbox c d"
+        proof -
+          have B': "norm (integral (cbox a b \<inter> cbox c d) ?g - integral (cbox x y) ?g) < e/2"
+            if "cbox a b \<inter> cbox c d \<subseteq> cbox x y" for x y
+            using B [of ?\<alpha> ?\<beta> x y] ball that by (simp add: Int_interval [symmetric])
+          show ?thesis
+            using B' [of a b] B' [of c d] norm_triangle_half_r by blast
+        qed
+      qed (use \<open>B > 0\<close> in auto)}
+  then show ?thesis
+    by force
+qed
+  finally show ?thesis .
+qed
+
 lemma integrable_on_subcbox:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes intf: "f integrable_on S"
diff -r ae76012879c6 -r 73a5a33486ee src/HOL/Analysis/Lebesgue_Measure.thy
--- a/src/HOL/Analysis/Lebesgue_Measure.thy	Tue Apr 17 18:04:49 2018 +0100
+++ b/src/HOL/Analysis/Lebesgue_Measure.thy	Tue Apr 17 22:35:48 2018 +0100
@@ -1168,29 +1168,27 @@
   "compact S \<Longrightarrow> (\<And>c x. \<lbrakk>(c *\<^sub>R x) \<in> S; 0 \<le> c; x \<in> S\<rbrakk> \<Longrightarrow> c = 1) \<Longrightarrow> S \<in> null_sets lebesgue"
   using starlike_negligible_bounded_gmeasurable[of S] by (auto simp: compact_eq_bounded_closed)
 
-lemma outer_regular_lborel:
-  assumes B: "B \<in> fmeasurable lborel" "0 < (e::real)"
-  shows "\<exists>U. open U \<and> B \<subseteq> U \<and> emeasure lborel U \<le> emeasure lborel B + e"
+proposition outer_regular_lborel_le:
+  assumes B[measurable]: "B \<in> sets borel" and "0 < (e::real)"
+  obtains U where "open U" "B \<subseteq> U" and "emeasure lborel (U - B) \<le> e"
 proof -
   let ?\<mu> = "emeasure lborel"
   let ?B = "\<lambda>n::nat. ball 0 n :: 'a set"
-  have B[measurable]: "B \<in> sets borel"
-    using B by auto
   let ?e = "\<lambda>n. e*((1/2)^Suc n)"
   have "\<forall>n. \<exists>U. open U \<and> ?B n \<inter> B \<subseteq> U \<and> ?\<mu> (U - B) < ?e n"
   proof
     fix n :: nat
     let ?A = "density lborel (indicator (?B n))"
     have emeasure_A: "X \<in> sets borel \<Longrightarrow> emeasure ?A X = ?\<mu> (?B n \<inter> X)" for X
-      by (auto simp add: emeasure_density borel_measurable_indicator indicator_inter_arith[symmetric])
+      by (auto simp: emeasure_density borel_measurable_indicator indicator_inter_arith[symmetric])
 
     have finite_A: "emeasure ?A (space ?A) \<noteq> \<infinity>"
-      using emeasure_bounded_finite[of "?B n"] by (auto simp add: emeasure_A)
+      using emeasure_bounded_finite[of "?B n"] by (auto simp: emeasure_A)
     interpret A: finite_measure ?A
       by rule fact
     have "emeasure ?A B + ?e n > (INF U:{U. B \<subseteq> U \<and> open U}. emeasure ?A U)"
       using \<open>0<e\<close> by (auto simp: outer_regular[OF _ finite_A B, symmetric])
-    then obtain U where U: "B \<subseteq> U" "open U" "?\<mu> (?B n \<inter> B) + ?e n > ?\<mu> (?B n \<inter> U)"
+    then obtain U where U: "B \<subseteq> U" "open U" and muU: "?\<mu> (?B n \<inter> B) + ?e n > ?\<mu> (?B n \<inter> U)"
       unfolding INF_less_iff by (auto simp: emeasure_A)
     moreover
     { have "?\<mu> ((?B n \<inter> U) - B) = ?\<mu> ((?B n \<inter> U) - (?B n \<inter> B))"
@@ -1199,7 +1197,7 @@
         using U A.emeasure_finite[of B]
         by (intro emeasure_Diff) (auto simp del: A.emeasure_finite simp: emeasure_A)
       also have "\<dots> < ?e n"
-        using U(1,2,3) A.emeasure_finite[of B]
+        using U muU A.emeasure_finite[of B]
         by (subst minus_less_iff_ennreal)
           (auto simp del: A.emeasure_finite simp: emeasure_A less_top ac_simps intro!: emeasure_mono)
       finally have "?\<mu> ((?B n \<inter> U) - B) < ?e n" . }
@@ -1209,50 +1207,93 @@
   then obtain U
     where U: "\<And>n. open (U n)" "\<And>n. ?B n \<inter> B \<subseteq> U n" "\<And>n. ?\<mu> (U n - B) < ?e n"
     by metis
-  then show ?thesis
-  proof (intro exI conjI)
+  show ?thesis
+  proof
     { fix x assume "x \<in> B"
       moreover
-      have "\<exists>n. norm x < real n"
-        by (simp add: reals_Archimedean2)
-      then guess n ..
+      obtain n where "norm x < real n"
+        using reals_Archimedean2 by blast
       ultimately have "x \<in> (\<Union>n. U n)"
         using U(2)[of n] by auto }
     note * = this
     then show "open (\<Union>n. U n)" "B \<subseteq> (\<Union>n. U n)"
-      using U(1,2) by auto
-    have "?\<mu> (\<Union>n. U n) = ?\<mu> (B \<union> (\<Union>n. U n - B))"
-      using * U(2) by (intro arg_cong[where ?f="?\<mu>"]) auto
-    also have "\<dots> = ?\<mu> B + ?\<mu> (\<Union>n. U n - B)"
-      using U(1) by (intro plus_emeasure[symmetric]) auto
-    also have "\<dots> \<le> ?\<mu> B + (\<Sum>n. ?\<mu> (U n - B))"
-      using U(1) by (intro add_mono emeasure_subadditive_countably) auto
-    also have "\<dots> \<le> ?\<mu> B + (\<Sum>n. ennreal (?e n))"
-      using U(3) by (intro add_mono suminf_le) (auto intro: less_imp_le)
-    also have "(\<Sum>n. ennreal (?e n)) = ennreal (e * 1)"
+      using U by auto
+    have "?\<mu> (\<Union>n. U n - B) \<le> (\<Sum>n. ?\<mu> (U n - B))"
+      using U(1) by (intro emeasure_subadditive_countably) auto
+    also have "\<dots> \<le> (\<Sum>n. ennreal (?e n))"
+      using U(3) by (intro suminf_le) (auto intro: less_imp_le)
+    also have "\<dots> = ennreal (e * 1)"
       using \<open>0<e\<close> by (intro suminf_ennreal_eq sums_mult power_half_series) auto
-    finally show "emeasure lborel (\<Union>n. U n) \<le> emeasure lborel B + ennreal e"
+    finally show "emeasure lborel ((\<Union>n. U n) - B) \<le> ennreal e"
       by simp
   qed
 qed
 
-lemma lmeasurable_outer_open:
-  assumes S: "S \<in> lmeasurable" and "0 < e"
-  obtains T where "open T" "S \<subseteq> T" "T \<in> lmeasurable" "measure lebesgue T \<le> measure lebesgue S + e"
+lemma outer_regular_lborel:
+  assumes B: "B \<in> sets borel" and "0 < (e::real)"
+  obtains U where "open U" "B \<subseteq> U" "emeasure lborel (U - B) < e"
+proof -
+  obtain U where U: "open U" "B \<subseteq> U" and "emeasure lborel (U-B) \<le> e/2"
+    using outer_regular_lborel_le [OF B, of "e/2"] \<open>e > 0\<close>
+    by force
+  moreover have "ennreal (e/2) < ennreal e"
+    using \<open>e > 0\<close> by (simp add: ennreal_lessI)
+  ultimately have "emeasure lborel (U-B) < e"
+    by auto
+  with U show ?thesis
+    using that by auto
+qed
+
+lemma completion_upper:
+  assumes A: "A \<in> sets (completion M)"
+  obtains A' where "A \<subseteq> A'" "A' \<in> sets M" "A' - A \<in> null_sets (completion M)"
+                   "emeasure (completion M) A = emeasure M A'"
 proof -
-  obtain S' where S': "S \<subseteq> S'" "S' \<in> sets borel" "emeasure lborel S' = emeasure lebesgue S"
+  from AE_notin_null_part[OF A] obtain N where N: "N \<in> null_sets M" "null_part M A \<subseteq> N"
+    unfolding eventually_ae_filter using null_part_null_sets[OF A, THEN null_setsD2, THEN sets.sets_into_space] by auto
+  let ?A' = "main_part M A \<union> N"
+  show ?thesis
+  proof
+    show "A \<subseteq> ?A'"
+      using \<open>null_part M A \<subseteq> N\<close> by (subst main_part_null_part_Un[symmetric, OF A]) auto
+    have "main_part M A \<subseteq> A"
+      using assms main_part_null_part_Un by auto
+    then have "?A' - A \<subseteq> N"
+      by blast
+    with N show "?A' - A \<in> null_sets (completion M)"
+      by (blast intro: null_sets_completionI completion.complete_measure_axioms complete_measure.complete2)
+    show "emeasure (completion M) A = emeasure M (main_part M A \<union> N)"
+      using A \<open>N \<in> null_sets M\<close> by (simp add: emeasure_Un_null_set)
+  qed (use A N in auto)
+qed
+
+lemma lmeasurable_outer_open:
+  assumes S: "S \<in> sets lebesgue" and "e > 0"
+  obtains T where "open T" "S \<subseteq> T" "(T - S) \<in> lmeasurable" "measure lebesgue (T - S) < e"
+proof -
+  obtain S' where S': "S \<subseteq> S'" "S' \<in> sets borel"
+              and null: "S' - S \<in> null_sets lebesgue"
+              and em: "emeasure lebesgue S = emeasure lborel S'"
     using completion_upper[of S lborel] S by auto
-  then have f_S': "S' \<in> fmeasurable lborel"
+  then have f_S': "S' \<in> sets borel"
     using S by (auto simp: fmeasurable_def)
-  from outer_regular_lborel[OF this \<open>0<e\<close>] guess U .. note U = this
+  with outer_regular_lborel[OF _ \<open>0<e\<close>]
+  obtain U where U: "open U" "S' \<subseteq> U" "emeasure lborel (U - S') < e"
+    by blast
   show thesis
-  proof (rule that)
-    show "open U" "S \<subseteq> U" "U \<in> lmeasurable"
-      using f_S' U S' by (auto simp: fmeasurable_def less_top[symmetric] top_unique)
-    then have "U \<in> fmeasurable lborel"
-      by (auto simp: fmeasurable_def)
-    with S U \<open>0<e\<close> show "measure lebesgue U \<le> measure lebesgue S + e"
-      unfolding S'(3) by (simp add: emeasure_eq_measure2 ennreal_plus[symmetric] del: ennreal_plus)
+  proof
+    show "open U" "S \<subseteq> U"
+      using f_S' U S' by auto
+  have "(U - S) = (U - S') \<union> (S' - S)"
+    using S' U by auto
+  then have eq: "emeasure lebesgue (U - S) = emeasure lborel (U - S')"
+    using null  by (simp add: U(1) emeasure_Un_null_set f_S' sets.Diff)
+  have "(U - S) \<in> sets lebesgue"
+    by (simp add: S U(1) sets.Diff)
+  then show "(U - S) \<in> lmeasurable"
+    unfolding fmeasurable_def using U(3) eq less_le_trans by fastforce
+  with eq U show "measure lebesgue (U - S) < e"
+    by (metis \<open>U - S \<in> lmeasurable\<close> emeasure_eq_measure2 ennreal_leI not_le)
   qed
 qed
 
diff -r ae76012879c6 -r 73a5a33486ee src/HOL/Analysis/Vitali_Covering_Theorem.thy
--- a/src/HOL/Analysis/Vitali_Covering_Theorem.thy	Tue Apr 17 18:04:49 2018 +0100
+++ b/src/HOL/Analysis/Vitali_Covering_Theorem.thy	Tue Apr 17 22:35:48 2018 +0100
@@ -629,7 +629,6 @@
     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>