src/HOL/Probability/Lebesgue_Measure.thy
 author huffman Fri Aug 19 14:17:28 2011 -0700 (2011-08-19) changeset 44311 42c5cbf68052 parent 43920 cedb5cb948fd child 44666 8670a39d4420 permissions -rw-r--r--
new isCont theorems;
simplify some proofs.
```     1 (*  Title:      HOL/Probability/Lebesgue_Measure.thy
```
```     2     Author:     Johannes Hölzl, TU München
```
```     3     Author:     Robert Himmelmann, TU München
```
```     4 *)
```
```     5
```
```     6 header {* Lebsegue measure *}
```
```     7
```
```     8 theory Lebesgue_Measure
```
```     9   imports Finite_Product_Measure
```
```    10 begin
```
```    11
```
```    12 subsection {* Standard Cubes *}
```
```    13
```
```    14 definition cube :: "nat \<Rightarrow> 'a::ordered_euclidean_space set" where
```
```    15   "cube n \<equiv> {\<chi>\<chi> i. - real n .. \<chi>\<chi> i. real n}"
```
```    16
```
```    17 lemma cube_closed[intro]: "closed (cube n)"
```
```    18   unfolding cube_def by auto
```
```    19
```
```    20 lemma cube_subset[intro]: "n \<le> N \<Longrightarrow> cube n \<subseteq> cube N"
```
```    21   by (fastsimp simp: eucl_le[where 'a='a] cube_def)
```
```    22
```
```    23 lemma cube_subset_iff:
```
```    24   "cube n \<subseteq> cube N \<longleftrightarrow> n \<le> N"
```
```    25 proof
```
```    26   assume subset: "cube n \<subseteq> (cube N::'a set)"
```
```    27   then have "((\<chi>\<chi> i. real n)::'a) \<in> cube N"
```
```    28     using DIM_positive[where 'a='a]
```
```    29     by (fastsimp simp: cube_def eucl_le[where 'a='a])
```
```    30   then show "n \<le> N"
```
```    31     by (fastsimp simp: cube_def eucl_le[where 'a='a])
```
```    32 next
```
```    33   assume "n \<le> N" then show "cube n \<subseteq> (cube N::'a set)" by (rule cube_subset)
```
```    34 qed
```
```    35
```
```    36 lemma ball_subset_cube:"ball (0::'a::ordered_euclidean_space) (real n) \<subseteq> cube n"
```
```    37   unfolding cube_def subset_eq mem_interval apply safe unfolding euclidean_lambda_beta'
```
```    38 proof- fix x::'a and i assume as:"x\<in>ball 0 (real n)" "i<DIM('a)"
```
```    39   thus "- real n \<le> x \$\$ i" "real n \<ge> x \$\$ i"
```
```    40     using component_le_norm[of x i] by(auto simp: dist_norm)
```
```    41 qed
```
```    42
```
```    43 lemma mem_big_cube: obtains n where "x \<in> cube n"
```
```    44 proof- from real_arch_lt[of "norm x"] guess n ..
```
```    45   thus ?thesis apply-apply(rule that[where n=n])
```
```    46     apply(rule ball_subset_cube[unfolded subset_eq,rule_format])
```
```    47     by (auto simp add:dist_norm)
```
```    48 qed
```
```    49
```
```    50 lemma cube_subset_Suc[intro]: "cube n \<subseteq> cube (Suc n)"
```
```    51   unfolding cube_def_raw subset_eq apply safe unfolding mem_interval by auto
```
```    52
```
```    53 subsection {* Lebesgue measure *}
```
```    54
```
```    55 definition lebesgue :: "'a::ordered_euclidean_space measure_space" where
```
```    56   "lebesgue = \<lparr> space = UNIV,
```
```    57     sets = {A. \<forall>n. (indicator A :: 'a \<Rightarrow> real) integrable_on cube n},
```
```    58     measure = \<lambda>A. SUP n. ereal (integral (cube n) (indicator A)) \<rparr>"
```
```    59
```
```    60 lemma space_lebesgue[simp]: "space lebesgue = UNIV"
```
```    61   unfolding lebesgue_def by simp
```
```    62
```
```    63 lemma lebesgueD: "A \<in> sets lebesgue \<Longrightarrow> (indicator A :: _ \<Rightarrow> real) integrable_on cube n"
```
```    64   unfolding lebesgue_def by simp
```
```    65
```
```    66 lemma lebesgueI: "(\<And>n. (indicator A :: _ \<Rightarrow> real) integrable_on cube n) \<Longrightarrow> A \<in> sets lebesgue"
```
```    67   unfolding lebesgue_def by simp
```
```    68
```
```    69 lemma absolutely_integrable_on_indicator[simp]:
```
```    70   fixes A :: "'a::ordered_euclidean_space set"
```
```    71   shows "((indicator A :: _ \<Rightarrow> real) absolutely_integrable_on X) \<longleftrightarrow>
```
```    72     (indicator A :: _ \<Rightarrow> real) integrable_on X"
```
```    73   unfolding absolutely_integrable_on_def by simp
```
```    74
```
```    75 lemma LIMSEQ_indicator_UN:
```
```    76   "(\<lambda>k. indicator (\<Union> i<k. A i) x) ----> (indicator (\<Union>i. A i) x :: real)"
```
```    77 proof cases
```
```    78   assume "\<exists>i. x \<in> A i" then guess i .. note i = this
```
```    79   then have *: "\<And>n. (indicator (\<Union> i<n + Suc i. A i) x :: real) = 1"
```
```    80     "(indicator (\<Union> i. A i) x :: real) = 1" by (auto simp: indicator_def)
```
```    81   show ?thesis
```
```    82     apply (rule LIMSEQ_offset[of _ "Suc i"]) unfolding * by auto
```
```    83 qed (auto simp: indicator_def)
```
```    84
```
```    85 lemma indicator_add:
```
```    86   "A \<inter> B = {} \<Longrightarrow> (indicator A x::_::monoid_add) + indicator B x = indicator (A \<union> B) x"
```
```    87   unfolding indicator_def by auto
```
```    88
```
```    89 interpretation lebesgue: sigma_algebra lebesgue
```
```    90 proof (intro sigma_algebra_iff2[THEN iffD2] conjI allI ballI impI lebesgueI)
```
```    91   fix A n assume A: "A \<in> sets lebesgue"
```
```    92   have "indicator (space lebesgue - A) = (\<lambda>x. 1 - indicator A x :: real)"
```
```    93     by (auto simp: fun_eq_iff indicator_def)
```
```    94   then show "(indicator (space lebesgue - A) :: _ \<Rightarrow> real) integrable_on cube n"
```
```    95     using A by (auto intro!: integrable_sub dest: lebesgueD simp: cube_def)
```
```    96 next
```
```    97   fix n show "(indicator {} :: _\<Rightarrow>real) integrable_on cube n"
```
```    98     by (auto simp: cube_def indicator_def_raw)
```
```    99 next
```
```   100   fix A :: "nat \<Rightarrow> 'a set" and n ::nat assume "range A \<subseteq> sets lebesgue"
```
```   101   then have A: "\<And>i. (indicator (A i) :: _ \<Rightarrow> real) integrable_on cube n"
```
```   102     by (auto dest: lebesgueD)
```
```   103   show "(indicator (\<Union>i. A i) :: _ \<Rightarrow> real) integrable_on cube n" (is "?g integrable_on _")
```
```   104   proof (intro dominated_convergence[where g="?g"] ballI)
```
```   105     fix k show "(indicator (\<Union>i<k. A i) :: _ \<Rightarrow> real) integrable_on cube n"
```
```   106     proof (induct k)
```
```   107       case (Suc k)
```
```   108       have *: "(\<Union> i<Suc k. A i) = (\<Union> i<k. A i) \<union> A k"
```
```   109         unfolding lessThan_Suc UN_insert by auto
```
```   110       have *: "(\<lambda>x. max (indicator (\<Union> i<k. A i) x) (indicator (A k) x) :: real) =
```
```   111           indicator (\<Union> i<Suc k. A i)" (is "(\<lambda>x. max (?f x) (?g x)) = _")
```
```   112         by (auto simp: fun_eq_iff * indicator_def)
```
```   113       show ?case
```
```   114         using absolutely_integrable_max[of ?f "cube n" ?g] A Suc by (simp add: *)
```
```   115     qed auto
```
```   116   qed (auto intro: LIMSEQ_indicator_UN simp: cube_def)
```
```   117 qed simp
```
```   118
```
```   119 interpretation lebesgue: measure_space lebesgue
```
```   120 proof
```
```   121   have *: "indicator {} = (\<lambda>x. 0 :: real)" by (simp add: fun_eq_iff)
```
```   122   show "positive lebesgue (measure lebesgue)"
```
```   123   proof (unfold positive_def, safe)
```
```   124     show "measure lebesgue {} = 0" by (simp add: integral_0 * lebesgue_def)
```
```   125     fix A assume "A \<in> sets lebesgue"
```
```   126     then show "0 \<le> measure lebesgue A"
```
```   127       unfolding lebesgue_def
```
```   128       by (auto intro!: le_SUPI2 integral_nonneg)
```
```   129   qed
```
```   130 next
```
```   131   show "countably_additive lebesgue (measure lebesgue)"
```
```   132   proof (intro countably_additive_def[THEN iffD2] allI impI)
```
```   133     fix A :: "nat \<Rightarrow> 'b set" assume rA: "range A \<subseteq> sets lebesgue" "disjoint_family A"
```
```   134     then have A[simp, intro]: "\<And>i n. (indicator (A i) :: _ \<Rightarrow> real) integrable_on cube n"
```
```   135       by (auto dest: lebesgueD)
```
```   136     let "?m n i" = "integral (cube n) (indicator (A i) :: _\<Rightarrow>real)"
```
```   137     let "?M n I" = "integral (cube n) (indicator (\<Union>i\<in>I. A i) :: _\<Rightarrow>real)"
```
```   138     have nn[simp, intro]: "\<And>i n. 0 \<le> ?m n i" by (auto intro!: integral_nonneg)
```
```   139     assume "(\<Union>i. A i) \<in> sets lebesgue"
```
```   140     then have UN_A[simp, intro]: "\<And>i n. (indicator (\<Union>i. A i) :: _ \<Rightarrow> real) integrable_on cube n"
```
```   141       by (auto dest: lebesgueD)
```
```   142     show "(\<Sum>n. measure lebesgue (A n)) = measure lebesgue (\<Union>i. A i)"
```
```   143     proof (simp add: lebesgue_def, subst suminf_SUP_eq, safe intro!: incseq_SucI)
```
```   144       fix i n show "ereal (?m n i) \<le> ereal (?m (Suc n) i)"
```
```   145         using cube_subset[of n "Suc n"] by (auto intro!: integral_subset_le incseq_SucI)
```
```   146     next
```
```   147       fix i n show "0 \<le> ereal (?m n i)"
```
```   148         using rA unfolding lebesgue_def
```
```   149         by (auto intro!: le_SUPI2 integral_nonneg)
```
```   150     next
```
```   151       show "(SUP n. \<Sum>i. ereal (?m n i)) = (SUP n. ereal (?M n UNIV))"
```
```   152       proof (intro arg_cong[where f="SUPR UNIV"] ext sums_unique[symmetric] sums_ereal[THEN iffD2] sums_def[THEN iffD2])
```
```   153         fix n
```
```   154         have "\<And>m. (UNION {..<m} A) \<in> sets lebesgue" using rA by auto
```
```   155         from lebesgueD[OF this]
```
```   156         have "(\<lambda>m. ?M n {..< m}) ----> ?M n UNIV"
```
```   157           (is "(\<lambda>m. integral _ (?A m)) ----> ?I")
```
```   158           by (intro dominated_convergence(2)[where f="?A" and h="\<lambda>x. 1::real"])
```
```   159              (auto intro: LIMSEQ_indicator_UN simp: cube_def)
```
```   160         moreover
```
```   161         { fix m have *: "(\<Sum>x<m. ?m n x) = ?M n {..< m}"
```
```   162           proof (induct m)
```
```   163             case (Suc m)
```
```   164             have "(\<Union>i<m. A i) \<in> sets lebesgue" using rA by auto
```
```   165             then have "(indicator (\<Union>i<m. A i) :: _\<Rightarrow>real) integrable_on (cube n)"
```
```   166               by (auto dest!: lebesgueD)
```
```   167             moreover
```
```   168             have "(\<Union>i<m. A i) \<inter> A m = {}"
```
```   169               using rA(2)[unfolded disjoint_family_on_def, THEN bspec, of m]
```
```   170               by auto
```
```   171             then have "\<And>x. indicator (\<Union>i<Suc m. A i) x =
```
```   172               indicator (\<Union>i<m. A i) x + (indicator (A m) x :: real)"
```
```   173               by (auto simp: indicator_add lessThan_Suc ac_simps)
```
```   174             ultimately show ?case
```
```   175               using Suc A by (simp add: integral_add[symmetric])
```
```   176           qed auto }
```
```   177         ultimately show "(\<lambda>m. \<Sum>x = 0..<m. ?m n x) ----> ?M n UNIV"
```
```   178           by (simp add: atLeast0LessThan)
```
```   179       qed
```
```   180     qed
```
```   181   qed
```
```   182 qed
```
```   183
```
```   184 lemma has_integral_interval_cube:
```
```   185   fixes a b :: "'a::ordered_euclidean_space"
```
```   186   shows "(indicator {a .. b} has_integral
```
```   187     content ({\<chi>\<chi> i. max (- real n) (a \$\$ i) .. \<chi>\<chi> i. min (real n) (b \$\$ i)} :: 'a set)) (cube n)"
```
```   188     (is "(?I has_integral content ?R) (cube n)")
```
```   189 proof -
```
```   190   let "{?N .. ?P}" = ?R
```
```   191   have [simp]: "(\<lambda>x. if x \<in> cube n then ?I x else 0) = indicator ?R"
```
```   192     by (auto simp: indicator_def cube_def fun_eq_iff eucl_le[where 'a='a])
```
```   193   have "(?I has_integral content ?R) (cube n) \<longleftrightarrow> (indicator ?R has_integral content ?R) UNIV"
```
```   194     unfolding has_integral_restrict_univ[where s="cube n", symmetric] by simp
```
```   195   also have "\<dots> \<longleftrightarrow> ((\<lambda>x. 1) has_integral content ?R) ?R"
```
```   196     unfolding indicator_def_raw has_integral_restrict_univ ..
```
```   197   finally show ?thesis
```
```   198     using has_integral_const[of "1::real" "?N" "?P"] by simp
```
```   199 qed
```
```   200
```
```   201 lemma lebesgueI_borel[intro, simp]:
```
```   202   fixes s::"'a::ordered_euclidean_space set"
```
```   203   assumes "s \<in> sets borel" shows "s \<in> sets lebesgue"
```
```   204 proof -
```
```   205   let ?S = "range (\<lambda>(a, b). {a .. (b :: 'a\<Colon>ordered_euclidean_space)})"
```
```   206   have *:"?S \<subseteq> sets lebesgue"
```
```   207   proof (safe intro!: lebesgueI)
```
```   208     fix n :: nat and a b :: 'a
```
```   209     let ?N = "\<chi>\<chi> i. max (- real n) (a \$\$ i)"
```
```   210     let ?P = "\<chi>\<chi> i. min (real n) (b \$\$ i)"
```
```   211     show "(indicator {a..b} :: 'a\<Rightarrow>real) integrable_on cube n"
```
```   212       unfolding integrable_on_def
```
```   213       using has_integral_interval_cube[of a b] by auto
```
```   214   qed
```
```   215   have "s \<in> sigma_sets UNIV ?S" using assms
```
```   216     unfolding borel_eq_atLeastAtMost by (simp add: sigma_def)
```
```   217   thus ?thesis
```
```   218     using lebesgue.sigma_subset[of "\<lparr> space = UNIV, sets = ?S\<rparr>", simplified, OF *]
```
```   219     by (auto simp: sigma_def)
```
```   220 qed
```
```   221
```
```   222 lemma lebesgueI_negligible[dest]: fixes s::"'a::ordered_euclidean_space set"
```
```   223   assumes "negligible s" shows "s \<in> sets lebesgue"
```
```   224   using assms by (force simp: cube_def integrable_on_def negligible_def intro!: lebesgueI)
```
```   225
```
```   226 lemma lmeasure_eq_0:
```
```   227   fixes S :: "'a::ordered_euclidean_space set" assumes "negligible S" shows "lebesgue.\<mu> S = 0"
```
```   228 proof -
```
```   229   have "\<And>n. integral (cube n) (indicator S :: 'a\<Rightarrow>real) = 0"
```
```   230     unfolding lebesgue_integral_def using assms
```
```   231     by (intro integral_unique some1_equality ex_ex1I)
```
```   232        (auto simp: cube_def negligible_def)
```
```   233   then show ?thesis by (auto simp: lebesgue_def)
```
```   234 qed
```
```   235
```
```   236 lemma lmeasure_iff_LIMSEQ:
```
```   237   assumes "A \<in> sets lebesgue" "0 \<le> m"
```
```   238   shows "lebesgue.\<mu> A = ereal m \<longleftrightarrow> (\<lambda>n. integral (cube n) (indicator A :: _ \<Rightarrow> real)) ----> m"
```
```   239 proof (simp add: lebesgue_def, intro SUP_eq_LIMSEQ)
```
```   240   show "mono (\<lambda>n. integral (cube n) (indicator A::_=>real))"
```
```   241     using cube_subset assms by (intro monoI integral_subset_le) (auto dest!: lebesgueD)
```
```   242 qed
```
```   243
```
```   244 lemma has_integral_indicator_UNIV:
```
```   245   fixes s A :: "'a::ordered_euclidean_space set" and x :: real
```
```   246   shows "((indicator (s \<inter> A) :: 'a\<Rightarrow>real) has_integral x) UNIV = ((indicator s :: _\<Rightarrow>real) has_integral x) A"
```
```   247 proof -
```
```   248   have "(\<lambda>x. if x \<in> A then indicator s x else 0) = (indicator (s \<inter> A) :: _\<Rightarrow>real)"
```
```   249     by (auto simp: fun_eq_iff indicator_def)
```
```   250   then show ?thesis
```
```   251     unfolding has_integral_restrict_univ[where s=A, symmetric] by simp
```
```   252 qed
```
```   253
```
```   254 lemma
```
```   255   fixes s a :: "'a::ordered_euclidean_space set"
```
```   256   shows integral_indicator_UNIV:
```
```   257     "integral UNIV (indicator (s \<inter> A) :: 'a\<Rightarrow>real) = integral A (indicator s :: _\<Rightarrow>real)"
```
```   258   and integrable_indicator_UNIV:
```
```   259     "(indicator (s \<inter> A) :: 'a\<Rightarrow>real) integrable_on UNIV \<longleftrightarrow> (indicator s :: 'a\<Rightarrow>real) integrable_on A"
```
```   260   unfolding integral_def integrable_on_def has_integral_indicator_UNIV by auto
```
```   261
```
```   262 lemma lmeasure_finite_has_integral:
```
```   263   fixes s :: "'a::ordered_euclidean_space set"
```
```   264   assumes "s \<in> sets lebesgue" "lebesgue.\<mu> s = ereal m" "0 \<le> m"
```
```   265   shows "(indicator s has_integral m) UNIV"
```
```   266 proof -
```
```   267   let ?I = "indicator :: 'a set \<Rightarrow> 'a \<Rightarrow> real"
```
```   268   have **: "(?I s) integrable_on UNIV \<and> (\<lambda>k. integral UNIV (?I (s \<inter> cube k))) ----> integral UNIV (?I s)"
```
```   269   proof (intro monotone_convergence_increasing allI ballI)
```
```   270     have LIMSEQ: "(\<lambda>n. integral (cube n) (?I s)) ----> m"
```
```   271       using assms(2) unfolding lmeasure_iff_LIMSEQ[OF assms(1, 3)] .
```
```   272     { fix n have "integral (cube n) (?I s) \<le> m"
```
```   273         using cube_subset assms
```
```   274         by (intro incseq_le[where L=m] LIMSEQ incseq_def[THEN iffD2] integral_subset_le allI impI)
```
```   275            (auto dest!: lebesgueD) }
```
```   276     moreover
```
```   277     { fix n have "0 \<le> integral (cube n) (?I s)"
```
```   278       using assms by (auto dest!: lebesgueD intro!: integral_nonneg) }
```
```   279     ultimately
```
```   280     show "bounded {integral UNIV (?I (s \<inter> cube k)) |k. True}"
```
```   281       unfolding bounded_def
```
```   282       apply (rule_tac exI[of _ 0])
```
```   283       apply (rule_tac exI[of _ m])
```
```   284       by (auto simp: dist_real_def integral_indicator_UNIV)
```
```   285     fix k show "?I (s \<inter> cube k) integrable_on UNIV"
```
```   286       unfolding integrable_indicator_UNIV using assms by (auto dest!: lebesgueD)
```
```   287     fix x show "?I (s \<inter> cube k) x \<le> ?I (s \<inter> cube (Suc k)) x"
```
```   288       using cube_subset[of k "Suc k"] by (auto simp: indicator_def)
```
```   289   next
```
```   290     fix x :: 'a
```
```   291     from mem_big_cube obtain k where k: "x \<in> cube k" .
```
```   292     { fix n have "?I (s \<inter> cube (n + k)) x = ?I s x"
```
```   293       using k cube_subset[of k "n + k"] by (auto simp: indicator_def) }
```
```   294     note * = this
```
```   295     show "(\<lambda>k. ?I (s \<inter> cube k) x) ----> ?I s x"
```
```   296       by (rule LIMSEQ_offset[where k=k]) (auto simp: *)
```
```   297   qed
```
```   298   note ** = conjunctD2[OF this]
```
```   299   have m: "m = integral UNIV (?I s)"
```
```   300     apply (intro LIMSEQ_unique[OF _ **(2)])
```
```   301     using assms(2) unfolding lmeasure_iff_LIMSEQ[OF assms(1,3)] integral_indicator_UNIV .
```
```   302   show ?thesis
```
```   303     unfolding m by (intro integrable_integral **)
```
```   304 qed
```
```   305
```
```   306 lemma lmeasure_finite_integrable: assumes s: "s \<in> sets lebesgue" and "lebesgue.\<mu> s \<noteq> \<infinity>"
```
```   307   shows "(indicator s :: _ \<Rightarrow> real) integrable_on UNIV"
```
```   308 proof (cases "lebesgue.\<mu> s")
```
```   309   case (real m)
```
```   310   with lmeasure_finite_has_integral[OF `s \<in> sets lebesgue` this]
```
```   311     lebesgue.positive_measure[OF s]
```
```   312   show ?thesis unfolding integrable_on_def by auto
```
```   313 qed (insert assms lebesgue.positive_measure[OF s], auto)
```
```   314
```
```   315 lemma has_integral_lebesgue: assumes "((indicator s :: _\<Rightarrow>real) has_integral m) UNIV"
```
```   316   shows "s \<in> sets lebesgue"
```
```   317 proof (intro lebesgueI)
```
```   318   let ?I = "indicator :: 'a set \<Rightarrow> 'a \<Rightarrow> real"
```
```   319   fix n show "(?I s) integrable_on cube n" unfolding cube_def
```
```   320   proof (intro integrable_on_subinterval)
```
```   321     show "(?I s) integrable_on UNIV"
```
```   322       unfolding integrable_on_def using assms by auto
```
```   323   qed auto
```
```   324 qed
```
```   325
```
```   326 lemma has_integral_lmeasure: assumes "((indicator s :: _\<Rightarrow>real) has_integral m) UNIV"
```
```   327   shows "lebesgue.\<mu> s = ereal m"
```
```   328 proof (intro lmeasure_iff_LIMSEQ[THEN iffD2])
```
```   329   let ?I = "indicator :: 'a set \<Rightarrow> 'a \<Rightarrow> real"
```
```   330   show "s \<in> sets lebesgue" using has_integral_lebesgue[OF assms] .
```
```   331   show "0 \<le> m" using assms by (rule has_integral_nonneg) auto
```
```   332   have "(\<lambda>n. integral UNIV (?I (s \<inter> cube n))) ----> integral UNIV (?I s)"
```
```   333   proof (intro dominated_convergence(2) ballI)
```
```   334     show "(?I s) integrable_on UNIV" unfolding integrable_on_def using assms by auto
```
```   335     fix n show "?I (s \<inter> cube n) integrable_on UNIV"
```
```   336       unfolding integrable_indicator_UNIV using `s \<in> sets lebesgue` by (auto dest: lebesgueD)
```
```   337     fix x show "norm (?I (s \<inter> cube n) x) \<le> ?I s x" by (auto simp: indicator_def)
```
```   338   next
```
```   339     fix x :: 'a
```
```   340     from mem_big_cube obtain k where k: "x \<in> cube k" .
```
```   341     { fix n have "?I (s \<inter> cube (n + k)) x = ?I s x"
```
```   342       using k cube_subset[of k "n + k"] by (auto simp: indicator_def) }
```
```   343     note * = this
```
```   344     show "(\<lambda>k. ?I (s \<inter> cube k) x) ----> ?I s x"
```
```   345       by (rule LIMSEQ_offset[where k=k]) (auto simp: *)
```
```   346   qed
```
```   347   then show "(\<lambda>n. integral (cube n) (?I s)) ----> m"
```
```   348     unfolding integral_unique[OF assms] integral_indicator_UNIV by simp
```
```   349 qed
```
```   350
```
```   351 lemma has_integral_iff_lmeasure:
```
```   352   "(indicator A has_integral m) UNIV \<longleftrightarrow> (A \<in> sets lebesgue \<and> 0 \<le> m \<and> lebesgue.\<mu> A = ereal m)"
```
```   353 proof
```
```   354   assume "(indicator A has_integral m) UNIV"
```
```   355   with has_integral_lmeasure[OF this] has_integral_lebesgue[OF this]
```
```   356   show "A \<in> sets lebesgue \<and> 0 \<le> m \<and> lebesgue.\<mu> A = ereal m"
```
```   357     by (auto intro: has_integral_nonneg)
```
```   358 next
```
```   359   assume "A \<in> sets lebesgue \<and> 0 \<le> m \<and> lebesgue.\<mu> A = ereal m"
```
```   360   then show "(indicator A has_integral m) UNIV" by (intro lmeasure_finite_has_integral) auto
```
```   361 qed
```
```   362
```
```   363 lemma lmeasure_eq_integral: assumes "(indicator s::_\<Rightarrow>real) integrable_on UNIV"
```
```   364   shows "lebesgue.\<mu> s = ereal (integral UNIV (indicator s))"
```
```   365   using assms unfolding integrable_on_def
```
```   366 proof safe
```
```   367   fix y :: real assume "(indicator s has_integral y) UNIV"
```
```   368   from this[unfolded has_integral_iff_lmeasure] integral_unique[OF this]
```
```   369   show "lebesgue.\<mu> s = ereal (integral UNIV (indicator s))" by simp
```
```   370 qed
```
```   371
```
```   372 lemma lebesgue_simple_function_indicator:
```
```   373   fixes f::"'a::ordered_euclidean_space \<Rightarrow> ereal"
```
```   374   assumes f:"simple_function lebesgue f"
```
```   375   shows "f = (\<lambda>x. (\<Sum>y \<in> f ` UNIV. y * indicator (f -` {y}) x))"
```
```   376   by (rule, subst lebesgue.simple_function_indicator_representation[OF f]) auto
```
```   377
```
```   378 lemma integral_eq_lmeasure:
```
```   379   "(indicator s::_\<Rightarrow>real) integrable_on UNIV \<Longrightarrow> integral UNIV (indicator s) = real (lebesgue.\<mu> s)"
```
```   380   by (subst lmeasure_eq_integral) (auto intro!: integral_nonneg)
```
```   381
```
```   382 lemma lmeasure_finite: assumes "(indicator s::_\<Rightarrow>real) integrable_on UNIV" shows "lebesgue.\<mu> s \<noteq> \<infinity>"
```
```   383   using lmeasure_eq_integral[OF assms] by auto
```
```   384
```
```   385 lemma negligible_iff_lebesgue_null_sets:
```
```   386   "negligible A \<longleftrightarrow> A \<in> lebesgue.null_sets"
```
```   387 proof
```
```   388   assume "negligible A"
```
```   389   from this[THEN lebesgueI_negligible] this[THEN lmeasure_eq_0]
```
```   390   show "A \<in> lebesgue.null_sets" by auto
```
```   391 next
```
```   392   assume A: "A \<in> lebesgue.null_sets"
```
```   393   then have *:"((indicator A) has_integral (0::real)) UNIV" using lmeasure_finite_has_integral[of A] by auto
```
```   394   show "negligible A" unfolding negligible_def
```
```   395   proof (intro allI)
```
```   396     fix a b :: 'a
```
```   397     have integrable: "(indicator A :: _\<Rightarrow>real) integrable_on {a..b}"
```
```   398       by (intro integrable_on_subinterval has_integral_integrable) (auto intro: *)
```
```   399     then have "integral {a..b} (indicator A) \<le> (integral UNIV (indicator A) :: real)"
```
```   400       using * by (auto intro!: integral_subset_le has_integral_integrable)
```
```   401     moreover have "(0::real) \<le> integral {a..b} (indicator A)"
```
```   402       using integrable by (auto intro!: integral_nonneg)
```
```   403     ultimately have "integral {a..b} (indicator A) = (0::real)"
```
```   404       using integral_unique[OF *] by auto
```
```   405     then show "(indicator A has_integral (0::real)) {a..b}"
```
```   406       using integrable_integral[OF integrable] by simp
```
```   407   qed
```
```   408 qed
```
```   409
```
```   410 lemma integral_const[simp]:
```
```   411   fixes a b :: "'a::ordered_euclidean_space"
```
```   412   shows "integral {a .. b} (\<lambda>x. c) = content {a .. b} *\<^sub>R c"
```
```   413   by (rule integral_unique) (rule has_integral_const)
```
```   414
```
```   415 lemma lmeasure_UNIV[intro]: "lebesgue.\<mu> (UNIV::'a::ordered_euclidean_space set) = \<infinity>"
```
```   416 proof (simp add: lebesgue_def, intro SUP_PInfty bexI)
```
```   417   fix n :: nat
```
```   418   have "indicator UNIV = (\<lambda>x::'a. 1 :: real)" by auto
```
```   419   moreover
```
```   420   { have "real n \<le> (2 * real n) ^ DIM('a)"
```
```   421     proof (cases n)
```
```   422       case 0 then show ?thesis by auto
```
```   423     next
```
```   424       case (Suc n')
```
```   425       have "real n \<le> (2 * real n)^1" by auto
```
```   426       also have "(2 * real n)^1 \<le> (2 * real n) ^ DIM('a)"
```
```   427         using Suc DIM_positive[where 'a='a] by (intro power_increasing) (auto simp: real_of_nat_Suc)
```
```   428       finally show ?thesis .
```
```   429     qed }
```
```   430   ultimately show "ereal (real n) \<le> ereal (integral (cube n) (indicator UNIV::'a\<Rightarrow>real))"
```
```   431     using integral_const DIM_positive[where 'a='a]
```
```   432     by (auto simp: cube_def content_closed_interval_cases setprod_constant)
```
```   433 qed simp
```
```   434
```
```   435 lemma
```
```   436   fixes a b ::"'a::ordered_euclidean_space"
```
```   437   shows lmeasure_atLeastAtMost[simp]: "lebesgue.\<mu> {a..b} = ereal (content {a..b})"
```
```   438 proof -
```
```   439   have "(indicator (UNIV \<inter> {a..b})::_\<Rightarrow>real) integrable_on UNIV"
```
```   440     unfolding integrable_indicator_UNIV by (simp add: integrable_const indicator_def_raw)
```
```   441   from lmeasure_eq_integral[OF this] show ?thesis unfolding integral_indicator_UNIV
```
```   442     by (simp add: indicator_def_raw)
```
```   443 qed
```
```   444
```
```   445 lemma atLeastAtMost_singleton_euclidean[simp]:
```
```   446   fixes a :: "'a::ordered_euclidean_space" shows "{a .. a} = {a}"
```
```   447   by (force simp: eucl_le[where 'a='a] euclidean_eq[where 'a='a])
```
```   448
```
```   449 lemma content_singleton[simp]: "content {a} = 0"
```
```   450 proof -
```
```   451   have "content {a .. a} = 0"
```
```   452     by (subst content_closed_interval) auto
```
```   453   then show ?thesis by simp
```
```   454 qed
```
```   455
```
```   456 lemma lmeasure_singleton[simp]:
```
```   457   fixes a :: "'a::ordered_euclidean_space" shows "lebesgue.\<mu> {a} = 0"
```
```   458   using lmeasure_atLeastAtMost[of a a] by simp
```
```   459
```
```   460 declare content_real[simp]
```
```   461
```
```   462 lemma
```
```   463   fixes a b :: real
```
```   464   shows lmeasure_real_greaterThanAtMost[simp]:
```
```   465     "lebesgue.\<mu> {a <.. b} = ereal (if a \<le> b then b - a else 0)"
```
```   466 proof cases
```
```   467   assume "a < b"
```
```   468   then have "lebesgue.\<mu> {a <.. b} = lebesgue.\<mu> {a .. b} - lebesgue.\<mu> {a}"
```
```   469     by (subst lebesgue.measure_Diff[symmetric])
```
```   470        (auto intro!: arg_cong[where f=lebesgue.\<mu>])
```
```   471   then show ?thesis by auto
```
```   472 qed auto
```
```   473
```
```   474 lemma
```
```   475   fixes a b :: real
```
```   476   shows lmeasure_real_atLeastLessThan[simp]:
```
```   477     "lebesgue.\<mu> {a ..< b} = ereal (if a \<le> b then b - a else 0)"
```
```   478 proof cases
```
```   479   assume "a < b"
```
```   480   then have "lebesgue.\<mu> {a ..< b} = lebesgue.\<mu> {a .. b} - lebesgue.\<mu> {b}"
```
```   481     by (subst lebesgue.measure_Diff[symmetric])
```
```   482        (auto intro!: arg_cong[where f=lebesgue.\<mu>])
```
```   483   then show ?thesis by auto
```
```   484 qed auto
```
```   485
```
```   486 lemma
```
```   487   fixes a b :: real
```
```   488   shows lmeasure_real_greaterThanLessThan[simp]:
```
```   489     "lebesgue.\<mu> {a <..< b} = ereal (if a \<le> b then b - a else 0)"
```
```   490 proof cases
```
```   491   assume "a < b"
```
```   492   then have "lebesgue.\<mu> {a <..< b} = lebesgue.\<mu> {a <.. b} - lebesgue.\<mu> {b}"
```
```   493     by (subst lebesgue.measure_Diff[symmetric])
```
```   494        (auto intro!: arg_cong[where f=lebesgue.\<mu>])
```
```   495   then show ?thesis by auto
```
```   496 qed auto
```
```   497
```
```   498 subsection {* Lebesgue-Borel measure *}
```
```   499
```
```   500 definition "lborel = lebesgue \<lparr> sets := sets borel \<rparr>"
```
```   501
```
```   502 lemma
```
```   503   shows space_lborel[simp]: "space lborel = UNIV"
```
```   504   and sets_lborel[simp]: "sets lborel = sets borel"
```
```   505   and measure_lborel[simp]: "measure lborel = lebesgue.\<mu>"
```
```   506   and measurable_lborel[simp]: "measurable lborel = measurable borel"
```
```   507   by (simp_all add: measurable_def_raw lborel_def)
```
```   508
```
```   509 interpretation lborel: measure_space "lborel :: ('a::ordered_euclidean_space) measure_space"
```
```   510   where "space lborel = UNIV"
```
```   511   and "sets lborel = sets borel"
```
```   512   and "measure lborel = lebesgue.\<mu>"
```
```   513   and "measurable lborel = measurable borel"
```
```   514 proof (rule lebesgue.measure_space_subalgebra)
```
```   515   have "sigma_algebra (lborel::'a measure_space) \<longleftrightarrow> sigma_algebra (borel::'a algebra)"
```
```   516     unfolding sigma_algebra_iff2 lborel_def by simp
```
```   517   then show "sigma_algebra (lborel::'a measure_space)" by simp default
```
```   518 qed auto
```
```   519
```
```   520 interpretation lborel: sigma_finite_measure lborel
```
```   521   where "space lborel = UNIV"
```
```   522   and "sets lborel = sets borel"
```
```   523   and "measure lborel = lebesgue.\<mu>"
```
```   524   and "measurable lborel = measurable borel"
```
```   525 proof -
```
```   526   show "sigma_finite_measure lborel"
```
```   527   proof (default, intro conjI exI[of _ "\<lambda>n. cube n"])
```
```   528     show "range cube \<subseteq> sets lborel" by (auto intro: borel_closed)
```
```   529     { fix x have "\<exists>n. x\<in>cube n" using mem_big_cube by auto }
```
```   530     thus "(\<Union>i. cube i) = space lborel" by auto
```
```   531     show "\<forall>i. measure lborel (cube i) \<noteq> \<infinity>" by (simp add: cube_def)
```
```   532   qed
```
```   533 qed simp_all
```
```   534
```
```   535 interpretation lebesgue: sigma_finite_measure lebesgue
```
```   536 proof
```
```   537   from lborel.sigma_finite guess A ..
```
```   538   moreover then have "range A \<subseteq> sets lebesgue" using lebesgueI_borel by blast
```
```   539   ultimately show "\<exists>A::nat \<Rightarrow> 'b set. range A \<subseteq> sets lebesgue \<and> (\<Union>i. A i) = space lebesgue \<and> (\<forall>i. lebesgue.\<mu> (A i) \<noteq> \<infinity>)"
```
```   540     by auto
```
```   541 qed
```
```   542
```
```   543 subsection {* Lebesgue integrable implies Gauge integrable *}
```
```   544
```
```   545 lemma has_integral_cmult_real:
```
```   546   fixes c :: real
```
```   547   assumes "c \<noteq> 0 \<Longrightarrow> (f has_integral x) A"
```
```   548   shows "((\<lambda>x. c * f x) has_integral c * x) A"
```
```   549 proof cases
```
```   550   assume "c \<noteq> 0"
```
```   551   from has_integral_cmul[OF assms[OF this], of c] show ?thesis
```
```   552     unfolding real_scaleR_def .
```
```   553 qed simp
```
```   554
```
```   555 lemma simple_function_has_integral:
```
```   556   fixes f::"'a::ordered_euclidean_space \<Rightarrow> ereal"
```
```   557   assumes f:"simple_function lebesgue f"
```
```   558   and f':"range f \<subseteq> {0..<\<infinity>}"
```
```   559   and om:"\<And>x. x \<in> range f \<Longrightarrow> lebesgue.\<mu> (f -` {x} \<inter> UNIV) = \<infinity> \<Longrightarrow> x = 0"
```
```   560   shows "((\<lambda>x. real (f x)) has_integral (real (integral\<^isup>S lebesgue f))) UNIV"
```
```   561   unfolding simple_integral_def space_lebesgue
```
```   562 proof (subst lebesgue_simple_function_indicator)
```
```   563   let "?M x" = "lebesgue.\<mu> (f -` {x} \<inter> UNIV)"
```
```   564   let "?F x" = "indicator (f -` {x})"
```
```   565   { fix x y assume "y \<in> range f"
```
```   566     from subsetD[OF f' this] have "y * ?F y x = ereal (real y * ?F y x)"
```
```   567       by (cases rule: ereal2_cases[of y "?F y x"])
```
```   568          (auto simp: indicator_def one_ereal_def split: split_if_asm) }
```
```   569   moreover
```
```   570   { fix x assume x: "x\<in>range f"
```
```   571     have "x * ?M x = real x * real (?M x)"
```
```   572     proof cases
```
```   573       assume "x \<noteq> 0" with om[OF x] have "?M x \<noteq> \<infinity>" by auto
```
```   574       with subsetD[OF f' x] f[THEN lebesgue.simple_functionD(2)] show ?thesis
```
```   575         by (cases rule: ereal2_cases[of x "?M x"]) auto
```
```   576     qed simp }
```
```   577   ultimately
```
```   578   have "((\<lambda>x. real (\<Sum>y\<in>range f. y * ?F y x)) has_integral real (\<Sum>x\<in>range f. x * ?M x)) UNIV \<longleftrightarrow>
```
```   579     ((\<lambda>x. \<Sum>y\<in>range f. real y * ?F y x) has_integral (\<Sum>x\<in>range f. real x * real (?M x))) UNIV"
```
```   580     by simp
```
```   581   also have \<dots>
```
```   582   proof (intro has_integral_setsum has_integral_cmult_real lmeasure_finite_has_integral
```
```   583                real_of_ereal_pos lebesgue.positive_measure ballI)
```
```   584     show *: "finite (range f)" "\<And>y. f -` {y} \<in> sets lebesgue" "\<And>y. f -` {y} \<inter> UNIV \<in> sets lebesgue"
```
```   585       using lebesgue.simple_functionD[OF f] by auto
```
```   586     fix y assume "real y \<noteq> 0" "y \<in> range f"
```
```   587     with * om[OF this(2)] show "lebesgue.\<mu> (f -` {y}) = ereal (real (?M y))"
```
```   588       by (auto simp: ereal_real)
```
```   589   qed
```
```   590   finally show "((\<lambda>x. real (\<Sum>y\<in>range f. y * ?F y x)) has_integral real (\<Sum>x\<in>range f. x * ?M x)) UNIV" .
```
```   591 qed fact
```
```   592
```
```   593 lemma bounded_realI: assumes "\<forall>x\<in>s. abs (x::real) \<le> B" shows "bounded s"
```
```   594   unfolding bounded_def dist_real_def apply(rule_tac x=0 in exI)
```
```   595   using assms by auto
```
```   596
```
```   597 lemma simple_function_has_integral':
```
```   598   fixes f::"'a::ordered_euclidean_space \<Rightarrow> ereal"
```
```   599   assumes f: "simple_function lebesgue f" "\<And>x. 0 \<le> f x"
```
```   600   and i: "integral\<^isup>S lebesgue f \<noteq> \<infinity>"
```
```   601   shows "((\<lambda>x. real (f x)) has_integral (real (integral\<^isup>S lebesgue f))) UNIV"
```
```   602 proof -
```
```   603   let ?f = "\<lambda>x. if x \<in> f -` {\<infinity>} then 0 else f x"
```
```   604   note f(1)[THEN lebesgue.simple_functionD(2)]
```
```   605   then have [simp, intro]: "\<And>X. f -` X \<in> sets lebesgue" by auto
```
```   606   have f': "simple_function lebesgue ?f"
```
```   607     using f by (intro lebesgue.simple_function_If_set) auto
```
```   608   have rng: "range ?f \<subseteq> {0..<\<infinity>}" using f by auto
```
```   609   have "AE x in lebesgue. f x = ?f x"
```
```   610     using lebesgue.simple_integral_PInf[OF f i]
```
```   611     by (intro lebesgue.AE_I[where N="f -` {\<infinity>} \<inter> space lebesgue"]) auto
```
```   612   from f(1) f' this have eq: "integral\<^isup>S lebesgue f = integral\<^isup>S lebesgue ?f"
```
```   613     by (rule lebesgue.simple_integral_cong_AE)
```
```   614   have real_eq: "\<And>x. real (f x) = real (?f x)" by auto
```
```   615
```
```   616   show ?thesis
```
```   617     unfolding eq real_eq
```
```   618   proof (rule simple_function_has_integral[OF f' rng])
```
```   619     fix x assume x: "x \<in> range ?f" and inf: "lebesgue.\<mu> (?f -` {x} \<inter> UNIV) = \<infinity>"
```
```   620     have "x * lebesgue.\<mu> (?f -` {x} \<inter> UNIV) = (\<integral>\<^isup>S y. x * indicator (?f -` {x}) y \<partial>lebesgue)"
```
```   621       using f'[THEN lebesgue.simple_functionD(2)]
```
```   622       by (simp add: lebesgue.simple_integral_cmult_indicator)
```
```   623     also have "\<dots> \<le> integral\<^isup>S lebesgue f"
```
```   624       using f'[THEN lebesgue.simple_functionD(2)] f
```
```   625       by (intro lebesgue.simple_integral_mono lebesgue.simple_function_mult lebesgue.simple_function_indicator)
```
```   626          (auto split: split_indicator)
```
```   627     finally show "x = 0" unfolding inf using i subsetD[OF rng x] by (auto split: split_if_asm)
```
```   628   qed
```
```   629 qed
```
```   630
```
```   631 lemma positive_integral_has_integral:
```
```   632   fixes f :: "'a::ordered_euclidean_space \<Rightarrow> ereal"
```
```   633   assumes f: "f \<in> borel_measurable lebesgue" "range f \<subseteq> {0..<\<infinity>}" "integral\<^isup>P lebesgue f \<noteq> \<infinity>"
```
```   634   shows "((\<lambda>x. real (f x)) has_integral (real (integral\<^isup>P lebesgue f))) UNIV"
```
```   635 proof -
```
```   636   from lebesgue.borel_measurable_implies_simple_function_sequence'[OF f(1)]
```
```   637   guess u . note u = this
```
```   638   have SUP_eq: "\<And>x. (SUP i. u i x) = f x"
```
```   639     using u(4) f(2)[THEN subsetD] by (auto split: split_max)
```
```   640   let "?u i x" = "real (u i x)"
```
```   641   note u_eq = lebesgue.positive_integral_eq_simple_integral[OF u(1,5), symmetric]
```
```   642   { fix i
```
```   643     note u_eq
```
```   644     also have "integral\<^isup>P lebesgue (u i) \<le> (\<integral>\<^isup>+x. max 0 (f x) \<partial>lebesgue)"
```
```   645       by (intro lebesgue.positive_integral_mono) (auto intro: le_SUPI simp: u(4)[symmetric])
```
```   646     finally have "integral\<^isup>S lebesgue (u i) \<noteq> \<infinity>"
```
```   647       unfolding positive_integral_max_0 using f by auto }
```
```   648   note u_fin = this
```
```   649   then have u_int: "\<And>i. (?u i has_integral real (integral\<^isup>S lebesgue (u i))) UNIV"
```
```   650     by (rule simple_function_has_integral'[OF u(1,5)])
```
```   651   have "\<forall>x. \<exists>r\<ge>0. f x = ereal r"
```
```   652   proof
```
```   653     fix x from f(2) have "0 \<le> f x" "f x \<noteq> \<infinity>" by (auto simp: subset_eq)
```
```   654     then show "\<exists>r\<ge>0. f x = ereal r" by (cases "f x") auto
```
```   655   qed
```
```   656   from choice[OF this] obtain f' where f': "f = (\<lambda>x. ereal (f' x))" "\<And>x. 0 \<le> f' x" by auto
```
```   657
```
```   658   have "\<forall>i. \<exists>r. \<forall>x. 0 \<le> r x \<and> u i x = ereal (r x)"
```
```   659   proof
```
```   660     fix i show "\<exists>r. \<forall>x. 0 \<le> r x \<and> u i x = ereal (r x)"
```
```   661     proof (intro choice allI)
```
```   662       fix i x have "u i x \<noteq> \<infinity>" using u(3)[of i] by (auto simp: image_iff) metis
```
```   663       then show "\<exists>r\<ge>0. u i x = ereal r" using u(5)[of i x] by (cases "u i x") auto
```
```   664     qed
```
```   665   qed
```
```   666   from choice[OF this] obtain u' where
```
```   667       u': "u = (\<lambda>i x. ereal (u' i x))" "\<And>i x. 0 \<le> u' i x" by (auto simp: fun_eq_iff)
```
```   668
```
```   669   have convergent: "f' integrable_on UNIV \<and>
```
```   670     (\<lambda>k. integral UNIV (u' k)) ----> integral UNIV f'"
```
```   671   proof (intro monotone_convergence_increasing allI ballI)
```
```   672     show int: "\<And>k. (u' k) integrable_on UNIV"
```
```   673       using u_int unfolding integrable_on_def u' by auto
```
```   674     show "\<And>k x. u' k x \<le> u' (Suc k) x" using u(2,3,5)
```
```   675       by (auto simp: incseq_Suc_iff le_fun_def image_iff u' intro!: real_of_ereal_positive_mono)
```
```   676     show "\<And>x. (\<lambda>k. u' k x) ----> f' x"
```
```   677       using SUP_eq u(2)
```
```   678       by (intro SUP_eq_LIMSEQ[THEN iffD1]) (auto simp: u' f' incseq_mono incseq_Suc_iff le_fun_def)
```
```   679     show "bounded {integral UNIV (u' k)|k. True}"
```
```   680     proof (safe intro!: bounded_realI)
```
```   681       fix k
```
```   682       have "\<bar>integral UNIV (u' k)\<bar> = integral UNIV (u' k)"
```
```   683         by (intro abs_of_nonneg integral_nonneg int ballI u')
```
```   684       also have "\<dots> = real (integral\<^isup>S lebesgue (u k))"
```
```   685         using u_int[THEN integral_unique] by (simp add: u')
```
```   686       also have "\<dots> = real (integral\<^isup>P lebesgue (u k))"
```
```   687         using lebesgue.positive_integral_eq_simple_integral[OF u(1,5)] by simp
```
```   688       also have "\<dots> \<le> real (integral\<^isup>P lebesgue f)" using f
```
```   689         by (auto intro!: real_of_ereal_positive_mono lebesgue.positive_integral_positive
```
```   690              lebesgue.positive_integral_mono le_SUPI simp: SUP_eq[symmetric])
```
```   691       finally show "\<bar>integral UNIV (u' k)\<bar> \<le> real (integral\<^isup>P lebesgue f)" .
```
```   692     qed
```
```   693   qed
```
```   694
```
```   695   have "integral\<^isup>P lebesgue f = ereal (integral UNIV f')"
```
```   696   proof (rule tendsto_unique[OF trivial_limit_sequentially])
```
```   697     have "(\<lambda>i. integral\<^isup>S lebesgue (u i)) ----> (SUP i. integral\<^isup>P lebesgue (u i))"
```
```   698       unfolding u_eq by (intro LIMSEQ_ereal_SUPR lebesgue.incseq_positive_integral u)
```
```   699     also note lebesgue.positive_integral_monotone_convergence_SUP
```
```   700       [OF u(2)  lebesgue.borel_measurable_simple_function[OF u(1)] u(5), symmetric]
```
```   701     finally show "(\<lambda>k. integral\<^isup>S lebesgue (u k)) ----> integral\<^isup>P lebesgue f"
```
```   702       unfolding SUP_eq .
```
```   703
```
```   704     { fix k
```
```   705       have "0 \<le> integral\<^isup>S lebesgue (u k)"
```
```   706         using u by (auto intro!: lebesgue.simple_integral_positive)
```
```   707       then have "integral\<^isup>S lebesgue (u k) = ereal (real (integral\<^isup>S lebesgue (u k)))"
```
```   708         using u_fin by (auto simp: ereal_real) }
```
```   709     note * = this
```
```   710     show "(\<lambda>k. integral\<^isup>S lebesgue (u k)) ----> ereal (integral UNIV f')"
```
```   711       using convergent using u_int[THEN integral_unique, symmetric]
```
```   712       by (subst *) (simp add: lim_ereal u')
```
```   713   qed
```
```   714   then show ?thesis using convergent by (simp add: f' integrable_integral)
```
```   715 qed
```
```   716
```
```   717 lemma lebesgue_integral_has_integral:
```
```   718   fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real"
```
```   719   assumes f: "integrable lebesgue f"
```
```   720   shows "(f has_integral (integral\<^isup>L lebesgue f)) UNIV"
```
```   721 proof -
```
```   722   let ?n = "\<lambda>x. real (ereal (max 0 (- f x)))" and ?p = "\<lambda>x. real (ereal (max 0 (f x)))"
```
```   723   have *: "f = (\<lambda>x. ?p x - ?n x)" by (auto simp del: ereal_max)
```
```   724   { fix f have "(\<integral>\<^isup>+ x. ereal (f x) \<partial>lebesgue) = (\<integral>\<^isup>+ x. ereal (max 0 (f x)) \<partial>lebesgue)"
```
```   725       by (intro lebesgue.positive_integral_cong_pos) (auto split: split_max) }
```
```   726   note eq = this
```
```   727   show ?thesis
```
```   728     unfolding lebesgue_integral_def
```
```   729     apply (subst *)
```
```   730     apply (rule has_integral_sub)
```
```   731     unfolding eq[of f] eq[of "\<lambda>x. - f x"]
```
```   732     apply (safe intro!: positive_integral_has_integral)
```
```   733     using integrableD[OF f]
```
```   734     by (auto simp: zero_ereal_def[symmetric] positive_integral_max_0  split: split_max
```
```   735              intro!: lebesgue.measurable_If lebesgue.borel_measurable_ereal)
```
```   736 qed
```
```   737
```
```   738 lemma lebesgue_positive_integral_eq_borel:
```
```   739   assumes f: "f \<in> borel_measurable borel"
```
```   740   shows "integral\<^isup>P lebesgue f = integral\<^isup>P lborel f"
```
```   741 proof -
```
```   742   from f have "integral\<^isup>P lebesgue (\<lambda>x. max 0 (f x)) = integral\<^isup>P lborel (\<lambda>x. max 0 (f x))"
```
```   743     by (auto intro!: lebesgue.positive_integral_subalgebra[symmetric]) default
```
```   744   then show ?thesis unfolding positive_integral_max_0 .
```
```   745 qed
```
```   746
```
```   747 lemma lebesgue_integral_eq_borel:
```
```   748   assumes "f \<in> borel_measurable borel"
```
```   749   shows "integrable lebesgue f \<longleftrightarrow> integrable lborel f" (is ?P)
```
```   750     and "integral\<^isup>L lebesgue f = integral\<^isup>L lborel f" (is ?I)
```
```   751 proof -
```
```   752   have *: "sigma_algebra lborel" by default
```
```   753   have "sets lborel \<subseteq> sets lebesgue" by auto
```
```   754   from lebesgue.integral_subalgebra[of f lborel, OF _ this _ _ *] assms
```
```   755   show ?P ?I by auto
```
```   756 qed
```
```   757
```
```   758 lemma borel_integral_has_integral:
```
```   759   fixes f::"'a::ordered_euclidean_space => real"
```
```   760   assumes f:"integrable lborel f"
```
```   761   shows "(f has_integral (integral\<^isup>L lborel f)) UNIV"
```
```   762 proof -
```
```   763   have borel: "f \<in> borel_measurable borel"
```
```   764     using f unfolding integrable_def by auto
```
```   765   from f show ?thesis
```
```   766     using lebesgue_integral_has_integral[of f]
```
```   767     unfolding lebesgue_integral_eq_borel[OF borel] by simp
```
```   768 qed
```
```   769
```
```   770 subsection {* Equivalence between product spaces and euclidean spaces *}
```
```   771
```
```   772 definition e2p :: "'a::ordered_euclidean_space \<Rightarrow> (nat \<Rightarrow> real)" where
```
```   773   "e2p x = (\<lambda>i\<in>{..<DIM('a)}. x\$\$i)"
```
```   774
```
```   775 definition p2e :: "(nat \<Rightarrow> real) \<Rightarrow> 'a::ordered_euclidean_space" where
```
```   776   "p2e x = (\<chi>\<chi> i. x i)"
```
```   777
```
```   778 lemma e2p_p2e[simp]:
```
```   779   "x \<in> extensional {..<DIM('a)} \<Longrightarrow> e2p (p2e x::'a::ordered_euclidean_space) = x"
```
```   780   by (auto simp: fun_eq_iff extensional_def p2e_def e2p_def)
```
```   781
```
```   782 lemma p2e_e2p[simp]:
```
```   783   "p2e (e2p x) = (x::'a::ordered_euclidean_space)"
```
```   784   by (auto simp: euclidean_eq[where 'a='a] p2e_def e2p_def)
```
```   785
```
```   786 interpretation lborel_product: product_sigma_finite "\<lambda>x. lborel::real measure_space"
```
```   787   by default
```
```   788
```
```   789 interpretation lborel_space: finite_product_sigma_finite "\<lambda>x. lborel::real measure_space" "{..<n}" for n :: nat
```
```   790   where "space lborel = UNIV"
```
```   791   and "sets lborel = sets borel"
```
```   792   and "measure lborel = lebesgue.\<mu>"
```
```   793   and "measurable lborel = measurable borel"
```
```   794 proof -
```
```   795   show "finite_product_sigma_finite (\<lambda>x. lborel::real measure_space) {..<n}"
```
```   796     by default simp
```
```   797 qed simp_all
```
```   798
```
```   799 lemma sets_product_borel:
```
```   800   assumes [intro]: "finite I"
```
```   801   shows "sets (\<Pi>\<^isub>M i\<in>I.
```
```   802      \<lparr> space = UNIV::real set, sets = range lessThan, measure = lebesgue.\<mu> \<rparr>) =
```
```   803    sets (\<Pi>\<^isub>M i\<in>I. lborel)" (is "sets ?G = _")
```
```   804 proof -
```
```   805   have "sets ?G = sets (\<Pi>\<^isub>M i\<in>I.
```
```   806        sigma \<lparr> space = UNIV::real set, sets = range lessThan, measure = lebesgue.\<mu> \<rparr>)"
```
```   807     by (subst sigma_product_algebra_sigma_eq[of I "\<lambda>_ i. {..<real i}" ])
```
```   808        (auto intro!: measurable_sigma_sigma incseq_SucI real_arch_lt
```
```   809              simp: product_algebra_def)
```
```   810   then show ?thesis
```
```   811     unfolding lborel_def borel_eq_lessThan lebesgue_def sigma_def by simp
```
```   812 qed
```
```   813
```
```   814 lemma measurable_e2p:
```
```   815   "e2p \<in> measurable (borel::'a::ordered_euclidean_space algebra)
```
```   816                     (\<Pi>\<^isub>M i\<in>{..<DIM('a)}. (lborel :: real measure_space))"
```
```   817     (is "_ \<in> measurable ?E ?P")
```
```   818 proof -
```
```   819   let ?B = "\<lparr> space = UNIV::real set, sets = range lessThan, measure = lebesgue.\<mu> \<rparr>"
```
```   820   let ?G = "product_algebra_generator {..<DIM('a)} (\<lambda>_. ?B)"
```
```   821   have "e2p \<in> measurable ?E (sigma ?G)"
```
```   822   proof (rule borel.measurable_sigma)
```
```   823     show "e2p \<in> space ?E \<rightarrow> space ?G" by (auto simp: e2p_def)
```
```   824     fix A assume "A \<in> sets ?G"
```
```   825     then obtain E where A: "A = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. E i)"
```
```   826       and "E \<in> {..<DIM('a)} \<rightarrow> (range lessThan)"
```
```   827       by (auto elim!: product_algebraE simp: )
```
```   828     then have "\<forall>i\<in>{..<DIM('a)}. \<exists>xs. E i = {..< xs}" by auto
```
```   829     from this[THEN bchoice] guess xs ..
```
```   830     then have A_eq: "A = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. {..< xs i})"
```
```   831       using A by auto
```
```   832     have "e2p -` A = {..< (\<chi>\<chi> i. xs i) :: 'a}"
```
```   833       using DIM_positive by (auto simp add: Pi_iff set_eq_iff e2p_def A_eq
```
```   834         euclidean_eq[where 'a='a] eucl_less[where 'a='a])
```
```   835     then show "e2p -` A \<inter> space ?E \<in> sets ?E" by simp
```
```   836   qed (auto simp: product_algebra_generator_def)
```
```   837   with sets_product_borel[of "{..<DIM('a)}"] show ?thesis
```
```   838     unfolding measurable_def product_algebra_def by simp
```
```   839 qed
```
```   840
```
```   841 lemma measurable_p2e:
```
```   842   "p2e \<in> measurable (\<Pi>\<^isub>M i\<in>{..<DIM('a)}. (lborel :: real measure_space))
```
```   843     (borel :: 'a::ordered_euclidean_space algebra)"
```
```   844   (is "p2e \<in> measurable ?P _")
```
```   845   unfolding borel_eq_lessThan
```
```   846 proof (intro lborel_space.measurable_sigma)
```
```   847   let ?E = "\<lparr> space = UNIV :: 'a set, sets = range lessThan \<rparr>"
```
```   848   show "p2e \<in> space ?P \<rightarrow> space ?E" by simp
```
```   849   fix A assume "A \<in> sets ?E"
```
```   850   then obtain x where "A = {..<x}" by auto
```
```   851   then have "p2e -` A \<inter> space ?P = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. {..< x \$\$ i})"
```
```   852     using DIM_positive
```
```   853     by (auto simp: Pi_iff set_eq_iff p2e_def
```
```   854                    euclidean_eq[where 'a='a] eucl_less[where 'a='a])
```
```   855   then show "p2e -` A \<inter> space ?P \<in> sets ?P" by auto
```
```   856 qed simp
```
```   857
```
```   858 lemma Int_stable_cuboids:
```
```   859   fixes x::"'a::ordered_euclidean_space"
```
```   860   shows "Int_stable \<lparr>space = UNIV, sets = range (\<lambda>(a, b::'a). {a..b})\<rparr>"
```
```   861   by (auto simp: inter_interval Int_stable_def)
```
```   862
```
```   863 lemma lborel_eq_lborel_space:
```
```   864   fixes A :: "('a::ordered_euclidean_space) set"
```
```   865   assumes "A \<in> sets borel"
```
```   866   shows "lborel.\<mu> A = lborel_space.\<mu> DIM('a) (p2e -` A \<inter> (space (lborel_space.P DIM('a))))"
```
```   867     (is "_ = measure ?P (?T A)")
```
```   868 proof (rule measure_unique_Int_stable_vimage)
```
```   869   show "measure_space ?P" by default
```
```   870   show "measure_space lborel" by default
```
```   871
```
```   872   let ?E = "\<lparr> space = UNIV :: 'a set, sets = range (\<lambda>(a,b). {a..b}) \<rparr>"
```
```   873   show "Int_stable ?E" using Int_stable_cuboids .
```
```   874   show "range cube \<subseteq> sets ?E" unfolding cube_def_raw by auto
```
```   875   show "incseq cube" using cube_subset_Suc by (auto intro!: incseq_SucI)
```
```   876   { fix x have "\<exists>n. x \<in> cube n" using mem_big_cube[of x] by fastsimp }
```
```   877   then show "(\<Union>i. cube i) = space ?E" by auto
```
```   878   { fix i show "lborel.\<mu> (cube i) \<noteq> \<infinity>" unfolding cube_def by auto }
```
```   879   show "A \<in> sets (sigma ?E)" "sets (sigma ?E) = sets lborel" "space ?E = space lborel"
```
```   880     using assms by (simp_all add: borel_eq_atLeastAtMost)
```
```   881
```
```   882   show "p2e \<in> measurable ?P (lborel :: 'a measure_space)"
```
```   883     using measurable_p2e unfolding measurable_def by simp
```
```   884   { fix X assume "X \<in> sets ?E"
```
```   885     then obtain a b where X[simp]: "X = {a .. b}" by auto
```
```   886     have *: "?T X = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. {a \$\$ i .. b \$\$ i})"
```
```   887       by (auto simp: Pi_iff eucl_le[where 'a='a] p2e_def)
```
```   888     show "lborel.\<mu> X = measure ?P (?T X)"
```
```   889     proof cases
```
```   890       assume "X \<noteq> {}"
```
```   891       then have "a \<le> b"
```
```   892         by (simp add: interval_ne_empty eucl_le[where 'a='a])
```
```   893       then have "lborel.\<mu> X = (\<Prod>x<DIM('a). lborel.\<mu> {a \$\$ x .. b \$\$ x})"
```
```   894         by (auto simp: content_closed_interval eucl_le[where 'a='a]
```
```   895                  intro!: setprod_ereal[symmetric])
```
```   896       also have "\<dots> = measure ?P (?T X)"
```
```   897         unfolding * by (subst lborel_space.measure_times) auto
```
```   898       finally show ?thesis .
```
```   899     qed simp }
```
```   900 qed
```
```   901
```
```   902 lemma measure_preserving_p2e:
```
```   903   "p2e \<in> measure_preserving (\<Pi>\<^isub>M i\<in>{..<DIM('a)}. (lborel :: real measure_space))
```
```   904     (lborel::'a::ordered_euclidean_space measure_space)" (is "_ \<in> measure_preserving ?P ?E")
```
```   905 proof
```
```   906   show "p2e \<in> measurable ?P ?E"
```
```   907     using measurable_p2e by (simp add: measurable_def)
```
```   908   fix A :: "'a set" assume "A \<in> sets lborel"
```
```   909   then show "lborel_space.\<mu> DIM('a) (p2e -` A \<inter> (space (lborel_space.P DIM('a)))) = lborel.\<mu> A"
```
```   910     by (intro lborel_eq_lborel_space[symmetric]) simp
```
```   911 qed
```
```   912
```
```   913 lemma lebesgue_eq_lborel_space_in_borel:
```
```   914   fixes A :: "('a::ordered_euclidean_space) set"
```
```   915   assumes A: "A \<in> sets borel"
```
```   916   shows "lebesgue.\<mu> A = lborel_space.\<mu> DIM('a) (p2e -` A \<inter> (space (lborel_space.P DIM('a))))"
```
```   917   using lborel_eq_lborel_space[OF A] by simp
```
```   918
```
```   919 lemma borel_fubini_positiv_integral:
```
```   920   fixes f :: "'a::ordered_euclidean_space \<Rightarrow> ereal"
```
```   921   assumes f: "f \<in> borel_measurable borel"
```
```   922   shows "integral\<^isup>P lborel f = \<integral>\<^isup>+x. f (p2e x) \<partial>(lborel_space.P DIM('a))"
```
```   923 proof (rule lborel_space.positive_integral_vimage[OF _ measure_preserving_p2e _])
```
```   924   show "f \<in> borel_measurable lborel"
```
```   925     using f by (simp_all add: measurable_def)
```
```   926 qed default
```
```   927
```
```   928 lemma borel_fubini_integrable:
```
```   929   fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real"
```
```   930   shows "integrable lborel f \<longleftrightarrow>
```
```   931     integrable (lborel_space.P DIM('a)) (\<lambda>x. f (p2e x))"
```
```   932     (is "_ \<longleftrightarrow> integrable ?B ?f")
```
```   933 proof
```
```   934   assume "integrable lborel f"
```
```   935   moreover then have f: "f \<in> borel_measurable borel"
```
```   936     by auto
```
```   937   moreover with measurable_p2e
```
```   938   have "f \<circ> p2e \<in> borel_measurable ?B"
```
```   939     by (rule measurable_comp)
```
```   940   ultimately show "integrable ?B ?f"
```
```   941     by (simp add: comp_def borel_fubini_positiv_integral integrable_def)
```
```   942 next
```
```   943   assume "integrable ?B ?f"
```
```   944   moreover then
```
```   945   have "?f \<circ> e2p \<in> borel_measurable (borel::'a algebra)"
```
```   946     by (auto intro!: measurable_e2p measurable_comp)
```
```   947   then have "f \<in> borel_measurable borel"
```
```   948     by (simp cong: measurable_cong)
```
```   949   ultimately show "integrable lborel f"
```
```   950     by (simp add: borel_fubini_positiv_integral integrable_def)
```
```   951 qed
```
```   952
```
```   953 lemma borel_fubini:
```
```   954   fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real"
```
```   955   assumes f: "f \<in> borel_measurable borel"
```
```   956   shows "integral\<^isup>L lborel f = \<integral>x. f (p2e x) \<partial>(lborel_space.P DIM('a))"
```
```   957   using f by (simp add: borel_fubini_positiv_integral lebesgue_integral_def)
```
```   958
```
```   959
```
```   960 lemma Int_stable_atLeastAtMost:
```
```   961   "Int_stable \<lparr>space = UNIV, sets = range (\<lambda>(a,b). {a::'a::ordered_euclidean_space .. b}) \<rparr>"
```
```   962 proof (simp add: Int_stable_def image_iff, intro allI)
```
```   963   fix a1 b1 a2 b2 :: 'a
```
```   964   have "\<forall>i<DIM('a). \<exists>a b. {a1\$\$i..b1\$\$i} \<inter> {a2\$\$i..b2\$\$i} = {a..b}" by auto
```
```   965   then have "\<exists>a b. \<forall>i<DIM('a). {a1\$\$i..b1\$\$i} \<inter> {a2\$\$i..b2\$\$i} = {a i..b i}"
```
```   966     unfolding choice_iff' .
```
```   967   then guess a b by safe
```
```   968   then have "{a1..b1} \<inter> {a2..b2} = {(\<chi>\<chi> i. a i) .. (\<chi>\<chi> i. b i)}"
```
```   969     by (simp add: set_eq_iff eucl_le[where 'a='a] all_conj_distrib[symmetric]) blast
```
```   970   then show "\<exists>a' b'. {a1..b1} \<inter> {a2..b2} = {a'..b'}" by blast
```
```   971 qed
```
```   972
```
```   973 lemma (in sigma_algebra) borel_measurable_sets:
```
```   974   assumes "f \<in> measurable borel M" "A \<in> sets M"
```
```   975   shows "f -` A \<in> sets borel"
```
```   976   using measurable_sets[OF assms] by simp
```
```   977
```
```   978 lemma (in sigma_algebra) measurable_identity[intro,simp]:
```
```   979   "(\<lambda>x. x) \<in> measurable M M"
```
```   980   unfolding measurable_def by auto
```
```   981
```
```   982 lemma lebesgue_real_affine:
```
```   983   fixes X :: "real set"
```
```   984   assumes "X \<in> sets borel" and "c \<noteq> 0"
```
```   985   shows "measure lebesgue X = ereal \<bar>c\<bar> * measure lebesgue ((\<lambda>x. t + c * x) -` X)"
```
```   986     (is "_ = ?\<nu> X")
```
```   987 proof -
```
```   988   let ?E = "\<lparr>space = UNIV, sets = range (\<lambda>(a,b). {a::real .. b})\<rparr> :: real algebra"
```
```   989   let "?M \<nu>" = "\<lparr>space = space ?E, sets = sets (sigma ?E), measure = \<nu>\<rparr> :: real measure_space"
```
```   990   have *: "?M (measure lebesgue) = lborel"
```
```   991     unfolding borel_eq_atLeastAtMost[symmetric]
```
```   992     by (simp add: lborel_def lebesgue_def)
```
```   993   have **: "?M ?\<nu> = lborel \<lparr> measure := ?\<nu> \<rparr>"
```
```   994     unfolding borel_eq_atLeastAtMost[symmetric]
```
```   995     by (simp add: lborel_def lebesgue_def)
```
```   996   show ?thesis
```
```   997   proof (rule measure_unique_Int_stable[where X=X, OF Int_stable_atLeastAtMost], unfold * **)
```
```   998     show "X \<in> sets (sigma ?E)"
```
```   999       unfolding borel_eq_atLeastAtMost[symmetric] by fact
```
```  1000     have "\<And>x. \<exists>xa. x \<in> cube xa" apply(rule_tac x=x in mem_big_cube) by fastsimp
```
```  1001     then show "(\<Union>i. cube i) = space ?E" by auto
```
```  1002     show "incseq cube" by (intro incseq_SucI cube_subset_Suc)
```
```  1003     show "range cube \<subseteq> sets ?E"
```
```  1004       unfolding cube_def_raw by auto
```
```  1005     show "\<And>i. measure lebesgue (cube i) \<noteq> \<infinity>"
```
```  1006       by (simp add: cube_def)
```
```  1007     show "measure_space lborel" by default
```
```  1008     then interpret sigma_algebra "lborel\<lparr>measure := ?\<nu>\<rparr>"
```
```  1009       by (auto simp add: measure_space_def)
```
```  1010     show "measure_space (lborel\<lparr>measure := ?\<nu>\<rparr>)"
```
```  1011     proof
```
```  1012       show "positive (lborel\<lparr>measure := ?\<nu>\<rparr>) (measure (lborel\<lparr>measure := ?\<nu>\<rparr>))"
```
```  1013         by (auto simp: positive_def intro!: ereal_0_le_mult borel.borel_measurable_sets)
```
```  1014       show "countably_additive (lborel\<lparr>measure := ?\<nu>\<rparr>) (measure (lborel\<lparr>measure := ?\<nu>\<rparr>))"
```
```  1015       proof (simp add: countably_additive_def, safe)
```
```  1016         fix A :: "nat \<Rightarrow> real set" assume A: "range A \<subseteq> sets borel" "disjoint_family A"
```
```  1017         then have Ai: "\<And>i. A i \<in> sets borel" by auto
```
```  1018         have "(\<Sum>n. measure lebesgue ((\<lambda>x. t + c * x) -` A n)) = measure lebesgue (\<Union>n. (\<lambda>x. t + c * x) -` A n)"
```
```  1019         proof (intro lborel.measure_countably_additive)
```
```  1020           { fix n have "(\<lambda>x. t + c * x) -` A n \<inter> space borel \<in> sets borel"
```
```  1021               using A borel.measurable_ident unfolding id_def
```
```  1022               by (intro measurable_sets[where A=borel] borel.borel_measurable_add[OF _ borel.borel_measurable_times]) auto }
```
```  1023           then show "range (\<lambda>i. (\<lambda>x. t + c * x) -` A i) \<subseteq> sets borel" by auto
```
```  1024           from `disjoint_family A`
```
```  1025           show "disjoint_family (\<lambda>i. (\<lambda>x. t + c * x) -` A i)"
```
```  1026             by (rule disjoint_family_on_bisimulation) auto
```
```  1027         qed
```
```  1028         with Ai show "(\<Sum>n. ?\<nu> (A n)) = ?\<nu> (UNION UNIV A)"
```
```  1029           by (subst suminf_cmult_ereal)
```
```  1030              (auto simp: vimage_UN borel.borel_measurable_sets)
```
```  1031       qed
```
```  1032     qed
```
```  1033     fix X assume "X \<in> sets ?E"
```
```  1034     then obtain a b where [simp]: "X = {a .. b}" by auto
```
```  1035     show "measure lebesgue X = ?\<nu> X"
```
```  1036     proof cases
```
```  1037       assume "0 < c"
```
```  1038       then have "(\<lambda>x. t + c * x) -` {a..b} = {(a - t) / c .. (b - t) / c}"
```
```  1039         by (auto simp: field_simps)
```
```  1040       with `0 < c` show ?thesis
```
```  1041         by (cases "a \<le> b") (auto simp: field_simps)
```
```  1042     next
```
```  1043       assume "\<not> 0 < c" with `c \<noteq> 0` have "c < 0" by auto
```
```  1044       then have *: "(\<lambda>x. t + c * x) -` {a..b} = {(b - t) / c .. (a - t) / c}"
```
```  1045         by (auto simp: field_simps)
```
```  1046       with `c < 0` show ?thesis
```
```  1047         by (cases "a \<le> b") (auto simp: field_simps)
```
```  1048     qed
```
```  1049   qed
```
```  1050 qed
```
```  1051
```
```  1052 end
```