src/HOL/Analysis/Lebesgue_Measure.thy
 author immler Fri Mar 10 23:16:40 2017 +0100 (2017-03-10) changeset 65204 d23eded35a33 parent 64272 f76b6dda2e56 child 65585 a043de9ad41e permissions -rw-r--r--
modernized construction of type bcontfun; base explicit theorems on Uniform_Limit.thy; added some lemmas
```     1 (*  Title:      HOL/Analysis/Lebesgue_Measure.thy
```
```     2     Author:     Johannes Hölzl, TU München
```
```     3     Author:     Robert Himmelmann, TU München
```
```     4     Author:     Jeremy Avigad
```
```     5     Author:     Luke Serafin
```
```     6 *)
```
```     7
```
```     8 section \<open>Lebesgue measure\<close>
```
```     9
```
```    10 theory Lebesgue_Measure
```
```    11   imports Finite_Product_Measure Bochner_Integration Caratheodory Complete_Measure Summation_Tests Regularity
```
```    12 begin
```
```    13
```
```    14 lemma measure_eqI_lessThan:
```
```    15   fixes M N :: "real measure"
```
```    16   assumes sets: "sets M = sets borel" "sets N = sets borel"
```
```    17   assumes fin: "\<And>x. emeasure M {x <..} < \<infinity>"
```
```    18   assumes "\<And>x. emeasure M {x <..} = emeasure N {x <..}"
```
```    19   shows "M = N"
```
```    20 proof (rule measure_eqI_generator_eq_countable)
```
```    21   let ?LT = "\<lambda>a::real. {a <..}" let ?E = "range ?LT"
```
```    22   show "Int_stable ?E"
```
```    23     by (auto simp: Int_stable_def lessThan_Int_lessThan)
```
```    24
```
```    25   show "?E \<subseteq> Pow UNIV" "sets M = sigma_sets UNIV ?E" "sets N = sigma_sets UNIV ?E"
```
```    26     unfolding sets borel_Ioi by auto
```
```    27
```
```    28   show "?LT`Rats \<subseteq> ?E" "(\<Union>i\<in>Rats. ?LT i) = UNIV" "\<And>a. a \<in> ?LT`Rats \<Longrightarrow> emeasure M a \<noteq> \<infinity>"
```
```    29     using fin by (auto intro: Rats_no_bot_less simp: less_top)
```
```    30 qed (auto intro: assms countable_rat)
```
```    31
```
```    32 subsection \<open>Every right continuous and nondecreasing function gives rise to a measure\<close>
```
```    33
```
```    34 definition interval_measure :: "(real \<Rightarrow> real) \<Rightarrow> real measure" where
```
```    35   "interval_measure F = extend_measure UNIV {(a, b). a \<le> b} (\<lambda>(a, b). {a <.. b}) (\<lambda>(a, b). ennreal (F b - F a))"
```
```    36
```
```    37 lemma emeasure_interval_measure_Ioc:
```
```    38   assumes "a \<le> b"
```
```    39   assumes mono_F: "\<And>x y. x \<le> y \<Longrightarrow> F x \<le> F y"
```
```    40   assumes right_cont_F : "\<And>a. continuous (at_right a) F"
```
```    41   shows "emeasure (interval_measure F) {a <.. b} = F b - F a"
```
```    42 proof (rule extend_measure_caratheodory_pair[OF interval_measure_def \<open>a \<le> b\<close>])
```
```    43   show "semiring_of_sets UNIV {{a<..b} |a b :: real. a \<le> b}"
```
```    44   proof (unfold_locales, safe)
```
```    45     fix a b c d :: real assume *: "a \<le> b" "c \<le> d"
```
```    46     then show "\<exists>C\<subseteq>{{a<..b} |a b. a \<le> b}. finite C \<and> disjoint C \<and> {a<..b} - {c<..d} = \<Union>C"
```
```    47     proof cases
```
```    48       let ?C = "{{a<..b}}"
```
```    49       assume "b < c \<or> d \<le> a \<or> d \<le> c"
```
```    50       with * have "?C \<subseteq> {{a<..b} |a b. a \<le> b} \<and> finite ?C \<and> disjoint ?C \<and> {a<..b} - {c<..d} = \<Union>?C"
```
```    51         by (auto simp add: disjoint_def)
```
```    52       thus ?thesis ..
```
```    53     next
```
```    54       let ?C = "{{a<..c}, {d<..b}}"
```
```    55       assume "\<not> (b < c \<or> d \<le> a \<or> d \<le> c)"
```
```    56       with * have "?C \<subseteq> {{a<..b} |a b. a \<le> b} \<and> finite ?C \<and> disjoint ?C \<and> {a<..b} - {c<..d} = \<Union>?C"
```
```    57         by (auto simp add: disjoint_def Ioc_inj) (metis linear)+
```
```    58       thus ?thesis ..
```
```    59     qed
```
```    60   qed (auto simp: Ioc_inj, metis linear)
```
```    61 next
```
```    62   fix l r :: "nat \<Rightarrow> real" and a b :: real
```
```    63   assume l_r[simp]: "\<And>n. l n \<le> r n" and "a \<le> b" and disj: "disjoint_family (\<lambda>n. {l n<..r n})"
```
```    64   assume lr_eq_ab: "(\<Union>i. {l i<..r i}) = {a<..b}"
```
```    65
```
```    66   have [intro, simp]: "\<And>a b. a \<le> b \<Longrightarrow> F a \<le> F b"
```
```    67     by (auto intro!: l_r mono_F)
```
```    68
```
```    69   { fix S :: "nat set" assume "finite S"
```
```    70     moreover note \<open>a \<le> b\<close>
```
```    71     moreover have "\<And>i. i \<in> S \<Longrightarrow> {l i <.. r i} \<subseteq> {a <.. b}"
```
```    72       unfolding lr_eq_ab[symmetric] by auto
```
```    73     ultimately have "(\<Sum>i\<in>S. F (r i) - F (l i)) \<le> F b - F a"
```
```    74     proof (induction S arbitrary: a rule: finite_psubset_induct)
```
```    75       case (psubset S)
```
```    76       show ?case
```
```    77       proof cases
```
```    78         assume "\<exists>i\<in>S. l i < r i"
```
```    79         with \<open>finite S\<close> have "Min (l ` {i\<in>S. l i < r i}) \<in> l ` {i\<in>S. l i < r i}"
```
```    80           by (intro Min_in) auto
```
```    81         then obtain m where m: "m \<in> S" "l m < r m" "l m = Min (l ` {i\<in>S. l i < r i})"
```
```    82           by fastforce
```
```    83
```
```    84         have "(\<Sum>i\<in>S. F (r i) - F (l i)) = (F (r m) - F (l m)) + (\<Sum>i\<in>S - {m}. F (r i) - F (l i))"
```
```    85           using m psubset by (intro sum.remove) auto
```
```    86         also have "(\<Sum>i\<in>S - {m}. F (r i) - F (l i)) \<le> F b - F (r m)"
```
```    87         proof (intro psubset.IH)
```
```    88           show "S - {m} \<subset> S"
```
```    89             using \<open>m\<in>S\<close> by auto
```
```    90           show "r m \<le> b"
```
```    91             using psubset.prems(2)[OF \<open>m\<in>S\<close>] \<open>l m < r m\<close> by auto
```
```    92         next
```
```    93           fix i assume "i \<in> S - {m}"
```
```    94           then have i: "i \<in> S" "i \<noteq> m" by auto
```
```    95           { assume i': "l i < r i" "l i < r m"
```
```    96             with \<open>finite S\<close> i m have "l m \<le> l i"
```
```    97               by auto
```
```    98             with i' have "{l i <.. r i} \<inter> {l m <.. r m} \<noteq> {}"
```
```    99               by auto
```
```   100             then have False
```
```   101               using disjoint_family_onD[OF disj, of i m] i by auto }
```
```   102           then have "l i \<noteq> r i \<Longrightarrow> r m \<le> l i"
```
```   103             unfolding not_less[symmetric] using l_r[of i] by auto
```
```   104           then show "{l i <.. r i} \<subseteq> {r m <.. b}"
```
```   105             using psubset.prems(2)[OF \<open>i\<in>S\<close>] by auto
```
```   106         qed
```
```   107         also have "F (r m) - F (l m) \<le> F (r m) - F a"
```
```   108           using psubset.prems(2)[OF \<open>m \<in> S\<close>] \<open>l m < r m\<close>
```
```   109           by (auto simp add: Ioc_subset_iff intro!: mono_F)
```
```   110         finally show ?case
```
```   111           by (auto intro: add_mono)
```
```   112       qed (auto simp add: \<open>a \<le> b\<close> less_le)
```
```   113     qed }
```
```   114   note claim1 = this
```
```   115
```
```   116   (* second key induction: a lower bound on the measures of any finite collection of Ai's
```
```   117      that cover an interval {u..v} *)
```
```   118
```
```   119   { fix S u v and l r :: "nat \<Rightarrow> real"
```
```   120     assume "finite S" "\<And>i. i\<in>S \<Longrightarrow> l i < r i" "{u..v} \<subseteq> (\<Union>i\<in>S. {l i<..< r i})"
```
```   121     then have "F v - F u \<le> (\<Sum>i\<in>S. F (r i) - F (l i))"
```
```   122     proof (induction arbitrary: v u rule: finite_psubset_induct)
```
```   123       case (psubset S)
```
```   124       show ?case
```
```   125       proof cases
```
```   126         assume "S = {}" then show ?case
```
```   127           using psubset by (simp add: mono_F)
```
```   128       next
```
```   129         assume "S \<noteq> {}"
```
```   130         then obtain j where "j \<in> S"
```
```   131           by auto
```
```   132
```
```   133         let ?R = "r j < u \<or> l j > v \<or> (\<exists>i\<in>S-{j}. l i \<le> l j \<and> r j \<le> r i)"
```
```   134         show ?case
```
```   135         proof cases
```
```   136           assume "?R"
```
```   137           with \<open>j \<in> S\<close> psubset.prems have "{u..v} \<subseteq> (\<Union>i\<in>S-{j}. {l i<..< r i})"
```
```   138             apply (auto simp: subset_eq Ball_def)
```
```   139             apply (metis Diff_iff less_le_trans leD linear singletonD)
```
```   140             apply (metis Diff_iff less_le_trans leD linear singletonD)
```
```   141             apply (metis order_trans less_le_not_le linear)
```
```   142             done
```
```   143           with \<open>j \<in> S\<close> have "F v - F u \<le> (\<Sum>i\<in>S - {j}. F (r i) - F (l i))"
```
```   144             by (intro psubset) auto
```
```   145           also have "\<dots> \<le> (\<Sum>i\<in>S. F (r i) - F (l i))"
```
```   146             using psubset.prems
```
```   147             by (intro sum_mono2 psubset) (auto intro: less_imp_le)
```
```   148           finally show ?thesis .
```
```   149         next
```
```   150           assume "\<not> ?R"
```
```   151           then have j: "u \<le> r j" "l j \<le> v" "\<And>i. i \<in> S - {j} \<Longrightarrow> r i < r j \<or> l i > l j"
```
```   152             by (auto simp: not_less)
```
```   153           let ?S1 = "{i \<in> S. l i < l j}"
```
```   154           let ?S2 = "{i \<in> S. r i > r j}"
```
```   155
```
```   156           have "(\<Sum>i\<in>S. F (r i) - F (l i)) \<ge> (\<Sum>i\<in>?S1 \<union> ?S2 \<union> {j}. F (r i) - F (l i))"
```
```   157             using \<open>j \<in> S\<close> \<open>finite S\<close> psubset.prems j
```
```   158             by (intro sum_mono2) (auto intro: less_imp_le)
```
```   159           also have "(\<Sum>i\<in>?S1 \<union> ?S2 \<union> {j}. F (r i) - F (l i)) =
```
```   160             (\<Sum>i\<in>?S1. F (r i) - F (l i)) + (\<Sum>i\<in>?S2 . F (r i) - F (l i)) + (F (r j) - F (l j))"
```
```   161             using psubset(1) psubset.prems(1) j
```
```   162             apply (subst sum.union_disjoint)
```
```   163             apply simp_all
```
```   164             apply (subst sum.union_disjoint)
```
```   165             apply auto
```
```   166             apply (metis less_le_not_le)
```
```   167             done
```
```   168           also (xtrans) have "(\<Sum>i\<in>?S1. F (r i) - F (l i)) \<ge> F (l j) - F u"
```
```   169             using \<open>j \<in> S\<close> \<open>finite S\<close> psubset.prems j
```
```   170             apply (intro psubset.IH psubset)
```
```   171             apply (auto simp: subset_eq Ball_def)
```
```   172             apply (metis less_le_trans not_le)
```
```   173             done
```
```   174           also (xtrans) have "(\<Sum>i\<in>?S2. F (r i) - F (l i)) \<ge> F v - F (r j)"
```
```   175             using \<open>j \<in> S\<close> \<open>finite S\<close> psubset.prems j
```
```   176             apply (intro psubset.IH psubset)
```
```   177             apply (auto simp: subset_eq Ball_def)
```
```   178             apply (metis le_less_trans not_le)
```
```   179             done
```
```   180           finally (xtrans) show ?case
```
```   181             by (auto simp: add_mono)
```
```   182         qed
```
```   183       qed
```
```   184     qed }
```
```   185   note claim2 = this
```
```   186
```
```   187   (* now prove the inequality going the other way *)
```
```   188   have "ennreal (F b - F a) \<le> (\<Sum>i. ennreal (F (r i) - F (l i)))"
```
```   189   proof (rule ennreal_le_epsilon)
```
```   190     fix epsilon :: real assume egt0: "epsilon > 0"
```
```   191     have "\<forall>i. \<exists>d>0. F (r i + d) < F (r i) + epsilon / 2^(i+2)"
```
```   192     proof
```
```   193       fix i
```
```   194       note right_cont_F [of "r i"]
```
```   195       thus "\<exists>d>0. F (r i + d) < F (r i) + epsilon / 2^(i+2)"
```
```   196         apply -
```
```   197         apply (subst (asm) continuous_at_right_real_increasing)
```
```   198         apply (rule mono_F, assumption)
```
```   199         apply (drule_tac x = "epsilon / 2 ^ (i + 2)" in spec)
```
```   200         apply (erule impE)
```
```   201         using egt0 by (auto simp add: field_simps)
```
```   202     qed
```
```   203     then obtain delta where
```
```   204         deltai_gt0: "\<And>i. delta i > 0" and
```
```   205         deltai_prop: "\<And>i. F (r i + delta i) < F (r i) + epsilon / 2^(i+2)"
```
```   206       by metis
```
```   207     have "\<exists>a' > a. F a' - F a < epsilon / 2"
```
```   208       apply (insert right_cont_F [of a])
```
```   209       apply (subst (asm) continuous_at_right_real_increasing)
```
```   210       using mono_F apply force
```
```   211       apply (drule_tac x = "epsilon / 2" in spec)
```
```   212       using egt0 unfolding mult.commute [of 2] by force
```
```   213     then obtain a' where a'lea [arith]: "a' > a" and
```
```   214       a_prop: "F a' - F a < epsilon / 2"
```
```   215       by auto
```
```   216     define S' where "S' = {i. l i < r i}"
```
```   217     obtain S :: "nat set" where
```
```   218       "S \<subseteq> S'" and finS: "finite S" and
```
```   219       Sprop: "{a'..b} \<subseteq> (\<Union>i \<in> S. {l i<..<r i + delta i})"
```
```   220     proof (rule compactE_image)
```
```   221       show "compact {a'..b}"
```
```   222         by (rule compact_Icc)
```
```   223       show "\<forall>i \<in> S'. open ({l i<..<r i + delta i})" by auto
```
```   224       have "{a'..b} \<subseteq> {a <.. b}"
```
```   225         by auto
```
```   226       also have "{a <.. b} = (\<Union>i\<in>S'. {l i<..r i})"
```
```   227         unfolding lr_eq_ab[symmetric] by (fastforce simp add: S'_def intro: less_le_trans)
```
```   228       also have "\<dots> \<subseteq> (\<Union>i \<in> S'. {l i<..<r i + delta i})"
```
```   229         apply (intro UN_mono)
```
```   230         apply (auto simp: S'_def)
```
```   231         apply (cut_tac i=i in deltai_gt0)
```
```   232         apply simp
```
```   233         done
```
```   234       finally show "{a'..b} \<subseteq> (\<Union>i \<in> S'. {l i<..<r i + delta i})" .
```
```   235     qed
```
```   236     with S'_def have Sprop2: "\<And>i. i \<in> S \<Longrightarrow> l i < r i" by auto
```
```   237     from finS have "\<exists>n. \<forall>i \<in> S. i \<le> n"
```
```   238       by (subst finite_nat_set_iff_bounded_le [symmetric])
```
```   239     then obtain n where Sbound [rule_format]: "\<forall>i \<in> S. i \<le> n" ..
```
```   240     have "F b - F a' \<le> (\<Sum>i\<in>S. F (r i + delta i) - F (l i))"
```
```   241       apply (rule claim2 [rule_format])
```
```   242       using finS Sprop apply auto
```
```   243       apply (frule Sprop2)
```
```   244       apply (subgoal_tac "delta i > 0")
```
```   245       apply arith
```
```   246       by (rule deltai_gt0)
```
```   247     also have "... \<le> (\<Sum>i \<in> S. F(r i) - F(l i) + epsilon / 2^(i+2))"
```
```   248       apply (rule sum_mono)
```
```   249       apply simp
```
```   250       apply (rule order_trans)
```
```   251       apply (rule less_imp_le)
```
```   252       apply (rule deltai_prop)
```
```   253       by auto
```
```   254     also have "... = (\<Sum>i \<in> S. F(r i) - F(l i)) +
```
```   255         (epsilon / 4) * (\<Sum>i \<in> S. (1 / 2)^i)" (is "_ = ?t + _")
```
```   256       by (subst sum.distrib) (simp add: field_simps sum_distrib_left)
```
```   257     also have "... \<le> ?t + (epsilon / 4) * (\<Sum> i < Suc n. (1 / 2)^i)"
```
```   258       apply (rule add_left_mono)
```
```   259       apply (rule mult_left_mono)
```
```   260       apply (rule sum_mono2)
```
```   261       using egt0 apply auto
```
```   262       by (frule Sbound, auto)
```
```   263     also have "... \<le> ?t + (epsilon / 2)"
```
```   264       apply (rule add_left_mono)
```
```   265       apply (subst geometric_sum)
```
```   266       apply auto
```
```   267       apply (rule mult_left_mono)
```
```   268       using egt0 apply auto
```
```   269       done
```
```   270     finally have aux2: "F b - F a' \<le> (\<Sum>i\<in>S. F (r i) - F (l i)) + epsilon / 2"
```
```   271       by simp
```
```   272
```
```   273     have "F b - F a = (F b - F a') + (F a' - F a)"
```
```   274       by auto
```
```   275     also have "... \<le> (F b - F a') + epsilon / 2"
```
```   276       using a_prop by (intro add_left_mono) simp
```
```   277     also have "... \<le> (\<Sum>i\<in>S. F (r i) - F (l i)) + epsilon / 2 + epsilon / 2"
```
```   278       apply (intro add_right_mono)
```
```   279       apply (rule aux2)
```
```   280       done
```
```   281     also have "... = (\<Sum>i\<in>S. F (r i) - F (l i)) + epsilon"
```
```   282       by auto
```
```   283     also have "... \<le> (\<Sum>i\<le>n. F (r i) - F (l i)) + epsilon"
```
```   284       using finS Sbound Sprop by (auto intro!: add_right_mono sum_mono3)
```
```   285     finally have "ennreal (F b - F a) \<le> (\<Sum>i\<le>n. ennreal (F (r i) - F (l i))) + epsilon"
```
```   286       using egt0 by (simp add: ennreal_plus[symmetric] sum_nonneg del: ennreal_plus)
```
```   287     then show "ennreal (F b - F a) \<le> (\<Sum>i. ennreal (F (r i) - F (l i))) + (epsilon :: real)"
```
```   288       by (rule order_trans) (auto intro!: add_mono sum_le_suminf simp del: sum_ennreal)
```
```   289   qed
```
```   290   moreover have "(\<Sum>i. ennreal (F (r i) - F (l i))) \<le> ennreal (F b - F a)"
```
```   291     using \<open>a \<le> b\<close> by (auto intro!: suminf_le_const ennreal_le_iff[THEN iffD2] claim1)
```
```   292   ultimately show "(\<Sum>n. ennreal (F (r n) - F (l n))) = ennreal (F b - F a)"
```
```   293     by (rule antisym[rotated])
```
```   294 qed (auto simp: Ioc_inj mono_F)
```
```   295
```
```   296 lemma measure_interval_measure_Ioc:
```
```   297   assumes "a \<le> b"
```
```   298   assumes mono_F: "\<And>x y. x \<le> y \<Longrightarrow> F x \<le> F y"
```
```   299   assumes right_cont_F : "\<And>a. continuous (at_right a) F"
```
```   300   shows "measure (interval_measure F) {a <.. b} = F b - F a"
```
```   301   unfolding measure_def
```
```   302   apply (subst emeasure_interval_measure_Ioc)
```
```   303   apply fact+
```
```   304   apply (simp add: assms)
```
```   305   done
```
```   306
```
```   307 lemma emeasure_interval_measure_Ioc_eq:
```
```   308   "(\<And>x y. x \<le> y \<Longrightarrow> F x \<le> F y) \<Longrightarrow> (\<And>a. continuous (at_right a) F) \<Longrightarrow>
```
```   309     emeasure (interval_measure F) {a <.. b} = (if a \<le> b then F b - F a else 0)"
```
```   310   using emeasure_interval_measure_Ioc[of a b F] by auto
```
```   311
```
```   312 lemma sets_interval_measure [simp, measurable_cong]: "sets (interval_measure F) = sets borel"
```
```   313   apply (simp add: sets_extend_measure interval_measure_def borel_sigma_sets_Ioc)
```
```   314   apply (rule sigma_sets_eqI)
```
```   315   apply auto
```
```   316   apply (case_tac "a \<le> ba")
```
```   317   apply (auto intro: sigma_sets.Empty)
```
```   318   done
```
```   319
```
```   320 lemma space_interval_measure [simp]: "space (interval_measure F) = UNIV"
```
```   321   by (simp add: interval_measure_def space_extend_measure)
```
```   322
```
```   323 lemma emeasure_interval_measure_Icc:
```
```   324   assumes "a \<le> b"
```
```   325   assumes mono_F: "\<And>x y. x \<le> y \<Longrightarrow> F x \<le> F y"
```
```   326   assumes cont_F : "continuous_on UNIV F"
```
```   327   shows "emeasure (interval_measure F) {a .. b} = F b - F a"
```
```   328 proof (rule tendsto_unique)
```
```   329   { fix a b :: real assume "a \<le> b" then have "emeasure (interval_measure F) {a <.. b} = F b - F a"
```
```   330       using cont_F
```
```   331       by (subst emeasure_interval_measure_Ioc)
```
```   332          (auto intro: mono_F continuous_within_subset simp: continuous_on_eq_continuous_within) }
```
```   333   note * = this
```
```   334
```
```   335   let ?F = "interval_measure F"
```
```   336   show "((\<lambda>a. F b - F a) \<longlongrightarrow> emeasure ?F {a..b}) (at_left a)"
```
```   337   proof (rule tendsto_at_left_sequentially)
```
```   338     show "a - 1 < a" by simp
```
```   339     fix X assume "\<And>n. X n < a" "incseq X" "X \<longlonglongrightarrow> a"
```
```   340     with \<open>a \<le> b\<close> have "(\<lambda>n. emeasure ?F {X n<..b}) \<longlonglongrightarrow> emeasure ?F (\<Inter>n. {X n <..b})"
```
```   341       apply (intro Lim_emeasure_decseq)
```
```   342       apply (auto simp: decseq_def incseq_def emeasure_interval_measure_Ioc *)
```
```   343       apply force
```
```   344       apply (subst (asm ) *)
```
```   345       apply (auto intro: less_le_trans less_imp_le)
```
```   346       done
```
```   347     also have "(\<Inter>n. {X n <..b}) = {a..b}"
```
```   348       using \<open>\<And>n. X n < a\<close>
```
```   349       apply auto
```
```   350       apply (rule LIMSEQ_le_const2[OF \<open>X \<longlonglongrightarrow> a\<close>])
```
```   351       apply (auto intro: less_imp_le)
```
```   352       apply (auto intro: less_le_trans)
```
```   353       done
```
```   354     also have "(\<lambda>n. emeasure ?F {X n<..b}) = (\<lambda>n. F b - F (X n))"
```
```   355       using \<open>\<And>n. X n < a\<close> \<open>a \<le> b\<close> by (subst *) (auto intro: less_imp_le less_le_trans)
```
```   356     finally show "(\<lambda>n. F b - F (X n)) \<longlonglongrightarrow> emeasure ?F {a..b}" .
```
```   357   qed
```
```   358   show "((\<lambda>a. ennreal (F b - F a)) \<longlongrightarrow> F b - F a) (at_left a)"
```
```   359     by (rule continuous_on_tendsto_compose[where g="\<lambda>x. x" and s=UNIV])
```
```   360        (auto simp: continuous_on_ennreal continuous_on_diff cont_F continuous_on_const)
```
```   361 qed (rule trivial_limit_at_left_real)
```
```   362
```
```   363 lemma sigma_finite_interval_measure:
```
```   364   assumes mono_F: "\<And>x y. x \<le> y \<Longrightarrow> F x \<le> F y"
```
```   365   assumes right_cont_F : "\<And>a. continuous (at_right a) F"
```
```   366   shows "sigma_finite_measure (interval_measure F)"
```
```   367   apply unfold_locales
```
```   368   apply (intro exI[of _ "(\<lambda>(a, b). {a <.. b}) ` (\<rat> \<times> \<rat>)"])
```
```   369   apply (auto intro!: Rats_no_top_le Rats_no_bot_less countable_rat simp: emeasure_interval_measure_Ioc_eq[OF assms])
```
```   370   done
```
```   371
```
```   372 subsection \<open>Lebesgue-Borel measure\<close>
```
```   373
```
```   374 definition lborel :: "('a :: euclidean_space) measure" where
```
```   375   "lborel = distr (\<Pi>\<^sub>M b\<in>Basis. interval_measure (\<lambda>x. x)) borel (\<lambda>f. \<Sum>b\<in>Basis. f b *\<^sub>R b)"
```
```   376
```
```   377 abbreviation lebesgue :: "'a::euclidean_space measure"
```
```   378   where "lebesgue \<equiv> completion lborel"
```
```   379
```
```   380 abbreviation lebesgue_on :: "'a set \<Rightarrow> 'a::euclidean_space measure"
```
```   381   where "lebesgue_on \<Omega> \<equiv> restrict_space (completion lborel) \<Omega>"
```
```   382
```
```   383 lemma
```
```   384   shows sets_lborel[simp, measurable_cong]: "sets lborel = sets borel"
```
```   385     and space_lborel[simp]: "space lborel = space borel"
```
```   386     and measurable_lborel1[simp]: "measurable M lborel = measurable M borel"
```
```   387     and measurable_lborel2[simp]: "measurable lborel M = measurable borel M"
```
```   388   by (simp_all add: lborel_def)
```
```   389
```
```   390 context
```
```   391 begin
```
```   392
```
```   393 interpretation sigma_finite_measure "interval_measure (\<lambda>x. x)"
```
```   394   by (rule sigma_finite_interval_measure) auto
```
```   395 interpretation finite_product_sigma_finite "\<lambda>_. interval_measure (\<lambda>x. x)" Basis
```
```   396   proof qed simp
```
```   397
```
```   398 lemma lborel_eq_real: "lborel = interval_measure (\<lambda>x. x)"
```
```   399   unfolding lborel_def Basis_real_def
```
```   400   using distr_id[of "interval_measure (\<lambda>x. x)"]
```
```   401   by (subst distr_component[symmetric])
```
```   402      (simp_all add: distr_distr comp_def del: distr_id cong: distr_cong)
```
```   403
```
```   404 lemma lborel_eq: "lborel = distr (\<Pi>\<^sub>M b\<in>Basis. lborel) borel (\<lambda>f. \<Sum>b\<in>Basis. f b *\<^sub>R b)"
```
```   405   by (subst lborel_def) (simp add: lborel_eq_real)
```
```   406
```
```   407 lemma nn_integral_lborel_prod:
```
```   408   assumes [measurable]: "\<And>b. b \<in> Basis \<Longrightarrow> f b \<in> borel_measurable borel"
```
```   409   assumes nn[simp]: "\<And>b x. b \<in> Basis \<Longrightarrow> 0 \<le> f b x"
```
```   410   shows "(\<integral>\<^sup>+x. (\<Prod>b\<in>Basis. f b (x \<bullet> b)) \<partial>lborel) = (\<Prod>b\<in>Basis. (\<integral>\<^sup>+x. f b x \<partial>lborel))"
```
```   411   by (simp add: lborel_def nn_integral_distr product_nn_integral_prod
```
```   412                 product_nn_integral_singleton)
```
```   413
```
```   414 lemma emeasure_lborel_Icc[simp]:
```
```   415   fixes l u :: real
```
```   416   assumes [simp]: "l \<le> u"
```
```   417   shows "emeasure lborel {l .. u} = u - l"
```
```   418 proof -
```
```   419   have "((\<lambda>f. f 1) -` {l..u} \<inter> space (Pi\<^sub>M {1} (\<lambda>b. interval_measure (\<lambda>x. x)))) = {1::real} \<rightarrow>\<^sub>E {l..u}"
```
```   420     by (auto simp: space_PiM)
```
```   421   then show ?thesis
```
```   422     by (simp add: lborel_def emeasure_distr emeasure_PiM emeasure_interval_measure_Icc continuous_on_id)
```
```   423 qed
```
```   424
```
```   425 lemma emeasure_lborel_Icc_eq: "emeasure lborel {l .. u} = ennreal (if l \<le> u then u - l else 0)"
```
```   426   by simp
```
```   427
```
```   428 lemma emeasure_lborel_cbox[simp]:
```
```   429   assumes [simp]: "\<And>b. b \<in> Basis \<Longrightarrow> l \<bullet> b \<le> u \<bullet> b"
```
```   430   shows "emeasure lborel (cbox l u) = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
```
```   431 proof -
```
```   432   have "(\<lambda>x. \<Prod>b\<in>Basis. indicator {l\<bullet>b .. u\<bullet>b} (x \<bullet> b) :: ennreal) = indicator (cbox l u)"
```
```   433     by (auto simp: fun_eq_iff cbox_def split: split_indicator)
```
```   434   then have "emeasure lborel (cbox l u) = (\<integral>\<^sup>+x. (\<Prod>b\<in>Basis. indicator {l\<bullet>b .. u\<bullet>b} (x \<bullet> b)) \<partial>lborel)"
```
```   435     by simp
```
```   436   also have "\<dots> = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
```
```   437     by (subst nn_integral_lborel_prod) (simp_all add: prod_ennreal inner_diff_left)
```
```   438   finally show ?thesis .
```
```   439 qed
```
```   440
```
```   441 lemma AE_lborel_singleton: "AE x in lborel::'a::euclidean_space measure. x \<noteq> c"
```
```   442   using SOME_Basis AE_discrete_difference [of "{c}" lborel] emeasure_lborel_cbox [of c c]
```
```   443   by (auto simp add: cbox_sing prod_constant power_0_left)
```
```   444
```
```   445 lemma emeasure_lborel_Ioo[simp]:
```
```   446   assumes [simp]: "l \<le> u"
```
```   447   shows "emeasure lborel {l <..< u} = ennreal (u - l)"
```
```   448 proof -
```
```   449   have "emeasure lborel {l <..< u} = emeasure lborel {l .. u}"
```
```   450     using AE_lborel_singleton[of u] AE_lborel_singleton[of l] by (intro emeasure_eq_AE) auto
```
```   451   then show ?thesis
```
```   452     by simp
```
```   453 qed
```
```   454
```
```   455 lemma emeasure_lborel_Ioc[simp]:
```
```   456   assumes [simp]: "l \<le> u"
```
```   457   shows "emeasure lborel {l <.. u} = ennreal (u - l)"
```
```   458 proof -
```
```   459   have "emeasure lborel {l <.. u} = emeasure lborel {l .. u}"
```
```   460     using AE_lborel_singleton[of u] AE_lborel_singleton[of l] by (intro emeasure_eq_AE) auto
```
```   461   then show ?thesis
```
```   462     by simp
```
```   463 qed
```
```   464
```
```   465 lemma emeasure_lborel_Ico[simp]:
```
```   466   assumes [simp]: "l \<le> u"
```
```   467   shows "emeasure lborel {l ..< u} = ennreal (u - l)"
```
```   468 proof -
```
```   469   have "emeasure lborel {l ..< u} = emeasure lborel {l .. u}"
```
```   470     using AE_lborel_singleton[of u] AE_lborel_singleton[of l] by (intro emeasure_eq_AE) auto
```
```   471   then show ?thesis
```
```   472     by simp
```
```   473 qed
```
```   474
```
```   475 lemma emeasure_lborel_box[simp]:
```
```   476   assumes [simp]: "\<And>b. b \<in> Basis \<Longrightarrow> l \<bullet> b \<le> u \<bullet> b"
```
```   477   shows "emeasure lborel (box l u) = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
```
```   478 proof -
```
```   479   have "(\<lambda>x. \<Prod>b\<in>Basis. indicator {l\<bullet>b <..< u\<bullet>b} (x \<bullet> b) :: ennreal) = indicator (box l u)"
```
```   480     by (auto simp: fun_eq_iff box_def split: split_indicator)
```
```   481   then have "emeasure lborel (box l u) = (\<integral>\<^sup>+x. (\<Prod>b\<in>Basis. indicator {l\<bullet>b <..< u\<bullet>b} (x \<bullet> b)) \<partial>lborel)"
```
```   482     by simp
```
```   483   also have "\<dots> = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
```
```   484     by (subst nn_integral_lborel_prod) (simp_all add: prod_ennreal inner_diff_left)
```
```   485   finally show ?thesis .
```
```   486 qed
```
```   487
```
```   488 lemma emeasure_lborel_cbox_eq:
```
```   489   "emeasure lborel (cbox l u) = (if \<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b then \<Prod>b\<in>Basis. (u - l) \<bullet> b else 0)"
```
```   490   using box_eq_empty(2)[THEN iffD2, of u l] by (auto simp: not_le)
```
```   491
```
```   492 lemma emeasure_lborel_box_eq:
```
```   493   "emeasure lborel (box l u) = (if \<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b then \<Prod>b\<in>Basis. (u - l) \<bullet> b else 0)"
```
```   494   using box_eq_empty(1)[THEN iffD2, of u l] by (auto simp: not_le dest!: less_imp_le) force
```
```   495
```
```   496 lemma emeasure_lborel_singleton[simp]: "emeasure lborel {x} = 0"
```
```   497   using emeasure_lborel_cbox[of x x] nonempty_Basis
```
```   498   by (auto simp del: emeasure_lborel_cbox nonempty_Basis simp add: cbox_sing prod_constant)
```
```   499
```
```   500 lemma
```
```   501   fixes l u :: real
```
```   502   assumes [simp]: "l \<le> u"
```
```   503   shows measure_lborel_Icc[simp]: "measure lborel {l .. u} = u - l"
```
```   504     and measure_lborel_Ico[simp]: "measure lborel {l ..< u} = u - l"
```
```   505     and measure_lborel_Ioc[simp]: "measure lborel {l <.. u} = u - l"
```
```   506     and measure_lborel_Ioo[simp]: "measure lborel {l <..< u} = u - l"
```
```   507   by (simp_all add: measure_def)
```
```   508
```
```   509 lemma
```
```   510   assumes [simp]: "\<And>b. b \<in> Basis \<Longrightarrow> l \<bullet> b \<le> u \<bullet> b"
```
```   511   shows measure_lborel_box[simp]: "measure lborel (box l u) = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
```
```   512     and measure_lborel_cbox[simp]: "measure lborel (cbox l u) = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
```
```   513   by (simp_all add: measure_def inner_diff_left prod_nonneg)
```
```   514
```
```   515 lemma measure_lborel_cbox_eq:
```
```   516   "measure lborel (cbox l u) = (if \<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b then \<Prod>b\<in>Basis. (u - l) \<bullet> b else 0)"
```
```   517   using box_eq_empty(2)[THEN iffD2, of u l] by (auto simp: not_le)
```
```   518
```
```   519 lemma measure_lborel_box_eq:
```
```   520   "measure lborel (box l u) = (if \<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b then \<Prod>b\<in>Basis. (u - l) \<bullet> b else 0)"
```
```   521   using box_eq_empty(1)[THEN iffD2, of u l] by (auto simp: not_le dest!: less_imp_le) force
```
```   522
```
```   523 lemma measure_lborel_singleton[simp]: "measure lborel {x} = 0"
```
```   524   by (simp add: measure_def)
```
```   525
```
```   526 lemma sigma_finite_lborel: "sigma_finite_measure lborel"
```
```   527 proof
```
```   528   show "\<exists>A::'a set set. countable A \<and> A \<subseteq> sets lborel \<and> \<Union>A = space lborel \<and> (\<forall>a\<in>A. emeasure lborel a \<noteq> \<infinity>)"
```
```   529     by (intro exI[of _ "range (\<lambda>n::nat. box (- real n *\<^sub>R One) (real n *\<^sub>R One))"])
```
```   530        (auto simp: emeasure_lborel_cbox_eq UN_box_eq_UNIV)
```
```   531 qed
```
```   532
```
```   533 end
```
```   534
```
```   535 lemma emeasure_lborel_UNIV: "emeasure lborel (UNIV::'a::euclidean_space set) = \<infinity>"
```
```   536 proof -
```
```   537   { fix n::nat
```
```   538     let ?Ba = "Basis :: 'a set"
```
```   539     have "real n \<le> (2::real) ^ card ?Ba * real n"
```
```   540       by (simp add: mult_le_cancel_right1)
```
```   541     also
```
```   542     have "... \<le> (2::real) ^ card ?Ba * real (Suc n) ^ card ?Ba"
```
```   543       apply (rule mult_left_mono)
```
```   544       apply (metis DIM_positive One_nat_def less_eq_Suc_le less_imp_le of_nat_le_iff of_nat_power self_le_power zero_less_Suc)
```
```   545       apply (simp add: DIM_positive)
```
```   546       done
```
```   547     finally have "real n \<le> (2::real) ^ card ?Ba * real (Suc n) ^ card ?Ba" .
```
```   548   } note [intro!] = this
```
```   549   show ?thesis
```
```   550     unfolding UN_box_eq_UNIV[symmetric]
```
```   551     apply (subst SUP_emeasure_incseq[symmetric])
```
```   552     apply (auto simp: incseq_def subset_box inner_add_left prod_constant
```
```   553       simp del: Sup_eq_top_iff SUP_eq_top_iff
```
```   554       intro!: ennreal_SUP_eq_top)
```
```   555     done
```
```   556 qed
```
```   557
```
```   558 lemma emeasure_lborel_countable:
```
```   559   fixes A :: "'a::euclidean_space set"
```
```   560   assumes "countable A"
```
```   561   shows "emeasure lborel A = 0"
```
```   562 proof -
```
```   563   have "A \<subseteq> (\<Union>i. {from_nat_into A i})" using from_nat_into_surj assms by force
```
```   564   then have "emeasure lborel A \<le> emeasure lborel (\<Union>i. {from_nat_into A i})"
```
```   565     by (intro emeasure_mono) auto
```
```   566   also have "emeasure lborel (\<Union>i. {from_nat_into A i}) = 0"
```
```   567     by (rule emeasure_UN_eq_0) auto
```
```   568   finally show ?thesis
```
```   569     by (auto simp add: )
```
```   570 qed
```
```   571
```
```   572 lemma countable_imp_null_set_lborel: "countable A \<Longrightarrow> A \<in> null_sets lborel"
```
```   573   by (simp add: null_sets_def emeasure_lborel_countable sets.countable)
```
```   574
```
```   575 lemma finite_imp_null_set_lborel: "finite A \<Longrightarrow> A \<in> null_sets lborel"
```
```   576   by (intro countable_imp_null_set_lborel countable_finite)
```
```   577
```
```   578 lemma lborel_neq_count_space[simp]: "lborel \<noteq> count_space (A::('a::ordered_euclidean_space) set)"
```
```   579 proof
```
```   580   assume asm: "lborel = count_space A"
```
```   581   have "space lborel = UNIV" by simp
```
```   582   hence [simp]: "A = UNIV" by (subst (asm) asm) (simp only: space_count_space)
```
```   583   have "emeasure lborel {undefined::'a} = 1"
```
```   584       by (subst asm, subst emeasure_count_space_finite) auto
```
```   585   moreover have "emeasure lborel {undefined} \<noteq> 1" by simp
```
```   586   ultimately show False by contradiction
```
```   587 qed
```
```   588
```
```   589 lemma mem_closed_if_AE_lebesgue_open:
```
```   590   assumes "open S" "closed C"
```
```   591   assumes "AE x \<in> S in lebesgue. x \<in> C"
```
```   592   assumes "x \<in> S"
```
```   593   shows "x \<in> C"
```
```   594 proof (rule ccontr)
```
```   595   assume xC: "x \<notin> C"
```
```   596   with openE[of "S - C"] assms
```
```   597   obtain e where e: "0 < e" "ball x e \<subseteq> S - C"
```
```   598     by blast
```
```   599   then obtain a b where box: "x \<in> box a b" "box a b \<subseteq> S - C"
```
```   600     by (metis rational_boxes order_trans)
```
```   601   then have "0 < emeasure lebesgue (box a b)"
```
```   602     by (auto simp: emeasure_lborel_box_eq mem_box algebra_simps intro!: prod_pos)
```
```   603   also have "\<dots> \<le> emeasure lebesgue (S - C)"
```
```   604     using assms box
```
```   605     by (auto intro!: emeasure_mono)
```
```   606   also have "\<dots> = 0"
```
```   607     using assms
```
```   608     by (auto simp: eventually_ae_filter completion.complete2 set_diff_eq null_setsD1)
```
```   609   finally show False by simp
```
```   610 qed
```
```   611
```
```   612 lemma mem_closed_if_AE_lebesgue: "closed C \<Longrightarrow> (AE x in lebesgue. x \<in> C) \<Longrightarrow> x \<in> C"
```
```   613   using mem_closed_if_AE_lebesgue_open[OF open_UNIV] by simp
```
```   614
```
```   615
```
```   616 subsection \<open>Affine transformation on the Lebesgue-Borel\<close>
```
```   617
```
```   618 lemma lborel_eqI:
```
```   619   fixes M :: "'a::euclidean_space measure"
```
```   620   assumes emeasure_eq: "\<And>l u. (\<And>b. b \<in> Basis \<Longrightarrow> l \<bullet> b \<le> u \<bullet> b) \<Longrightarrow> emeasure M (box l u) = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
```
```   621   assumes sets_eq: "sets M = sets borel"
```
```   622   shows "lborel = M"
```
```   623 proof (rule measure_eqI_generator_eq)
```
```   624   let ?E = "range (\<lambda>(a, b). box a b::'a set)"
```
```   625   show "Int_stable ?E"
```
```   626     by (auto simp: Int_stable_def box_Int_box)
```
```   627
```
```   628   show "?E \<subseteq> Pow UNIV" "sets lborel = sigma_sets UNIV ?E" "sets M = sigma_sets UNIV ?E"
```
```   629     by (simp_all add: borel_eq_box sets_eq)
```
```   630
```
```   631   let ?A = "\<lambda>n::nat. box (- (real n *\<^sub>R One)) (real n *\<^sub>R One) :: 'a set"
```
```   632   show "range ?A \<subseteq> ?E" "(\<Union>i. ?A i) = UNIV"
```
```   633     unfolding UN_box_eq_UNIV by auto
```
```   634
```
```   635   { fix i show "emeasure lborel (?A i) \<noteq> \<infinity>" by auto }
```
```   636   { fix X assume "X \<in> ?E" then show "emeasure lborel X = emeasure M X"
```
```   637       apply (auto simp: emeasure_eq emeasure_lborel_box_eq)
```
```   638       apply (subst box_eq_empty(1)[THEN iffD2])
```
```   639       apply (auto intro: less_imp_le simp: not_le)
```
```   640       done }
```
```   641 qed
```
```   642
```
```   643 lemma lborel_affine_euclidean:
```
```   644   fixes c :: "'a::euclidean_space \<Rightarrow> real" and t
```
```   645   defines "T x \<equiv> t + (\<Sum>j\<in>Basis. (c j * (x \<bullet> j)) *\<^sub>R j)"
```
```   646   assumes c: "\<And>j. j \<in> Basis \<Longrightarrow> c j \<noteq> 0"
```
```   647   shows "lborel = density (distr lborel borel T) (\<lambda>_. (\<Prod>j\<in>Basis. \<bar>c j\<bar>))" (is "_ = ?D")
```
```   648 proof (rule lborel_eqI)
```
```   649   let ?B = "Basis :: 'a set"
```
```   650   fix l u assume le: "\<And>b. b \<in> ?B \<Longrightarrow> l \<bullet> b \<le> u \<bullet> b"
```
```   651   have [measurable]: "T \<in> borel \<rightarrow>\<^sub>M borel"
```
```   652     by (simp add: T_def[abs_def])
```
```   653   have eq: "T -` box l u = box
```
```   654     (\<Sum>j\<in>Basis. (((if 0 < c j then l - t else u - t) \<bullet> j) / c j) *\<^sub>R j)
```
```   655     (\<Sum>j\<in>Basis. (((if 0 < c j then u - t else l - t) \<bullet> j) / c j) *\<^sub>R j)"
```
```   656     using c by (auto simp: box_def T_def field_simps inner_simps divide_less_eq)
```
```   657   with le c show "emeasure ?D (box l u) = (\<Prod>b\<in>?B. (u - l) \<bullet> b)"
```
```   658     by (auto simp: emeasure_density emeasure_distr nn_integral_multc emeasure_lborel_box_eq inner_simps
```
```   659                    field_simps divide_simps ennreal_mult'[symmetric] prod_nonneg prod.distrib[symmetric]
```
```   660              intro!: prod.cong)
```
```   661 qed simp
```
```   662
```
```   663 lemma lborel_affine:
```
```   664   fixes t :: "'a::euclidean_space"
```
```   665   shows "c \<noteq> 0 \<Longrightarrow> lborel = density (distr lborel borel (\<lambda>x. t + c *\<^sub>R x)) (\<lambda>_. \<bar>c\<bar>^DIM('a))"
```
```   666   using lborel_affine_euclidean[where c="\<lambda>_::'a. c" and t=t]
```
```   667   unfolding scaleR_scaleR[symmetric] scaleR_sum_right[symmetric] euclidean_representation prod_constant by simp
```
```   668
```
```   669 lemma lborel_real_affine:
```
```   670   "c \<noteq> 0 \<Longrightarrow> lborel = density (distr lborel borel (\<lambda>x. t + c * x)) (\<lambda>_. ennreal (abs c))"
```
```   671   using lborel_affine[of c t] by simp
```
```   672
```
```   673 lemma AE_borel_affine:
```
```   674   fixes P :: "real \<Rightarrow> bool"
```
```   675   shows "c \<noteq> 0 \<Longrightarrow> Measurable.pred borel P \<Longrightarrow> AE x in lborel. P x \<Longrightarrow> AE x in lborel. P (t + c * x)"
```
```   676   by (subst lborel_real_affine[where t="- t / c" and c="1 / c"])
```
```   677      (simp_all add: AE_density AE_distr_iff field_simps)
```
```   678
```
```   679 lemma nn_integral_real_affine:
```
```   680   fixes c :: real assumes [measurable]: "f \<in> borel_measurable borel" and c: "c \<noteq> 0"
```
```   681   shows "(\<integral>\<^sup>+x. f x \<partial>lborel) = \<bar>c\<bar> * (\<integral>\<^sup>+x. f (t + c * x) \<partial>lborel)"
```
```   682   by (subst lborel_real_affine[OF c, of t])
```
```   683      (simp add: nn_integral_density nn_integral_distr nn_integral_cmult)
```
```   684
```
```   685 lemma lborel_integrable_real_affine:
```
```   686   fixes f :: "real \<Rightarrow> 'a :: {banach, second_countable_topology}"
```
```   687   assumes f: "integrable lborel f"
```
```   688   shows "c \<noteq> 0 \<Longrightarrow> integrable lborel (\<lambda>x. f (t + c * x))"
```
```   689   using f f[THEN borel_measurable_integrable] unfolding integrable_iff_bounded
```
```   690   by (subst (asm) nn_integral_real_affine[where c=c and t=t]) (auto simp: ennreal_mult_less_top)
```
```   691
```
```   692 lemma lborel_integrable_real_affine_iff:
```
```   693   fixes f :: "real \<Rightarrow> 'a :: {banach, second_countable_topology}"
```
```   694   shows "c \<noteq> 0 \<Longrightarrow> integrable lborel (\<lambda>x. f (t + c * x)) \<longleftrightarrow> integrable lborel f"
```
```   695   using
```
```   696     lborel_integrable_real_affine[of f c t]
```
```   697     lborel_integrable_real_affine[of "\<lambda>x. f (t + c * x)" "1/c" "-t/c"]
```
```   698   by (auto simp add: field_simps)
```
```   699
```
```   700 lemma lborel_integral_real_affine:
```
```   701   fixes f :: "real \<Rightarrow> 'a :: {banach, second_countable_topology}" and c :: real
```
```   702   assumes c: "c \<noteq> 0" shows "(\<integral>x. f x \<partial> lborel) = \<bar>c\<bar> *\<^sub>R (\<integral>x. f (t + c * x) \<partial>lborel)"
```
```   703 proof cases
```
```   704   assume f[measurable]: "integrable lborel f" then show ?thesis
```
```   705     using c f f[THEN borel_measurable_integrable] f[THEN lborel_integrable_real_affine, of c t]
```
```   706     by (subst lborel_real_affine[OF c, of t])
```
```   707        (simp add: integral_density integral_distr)
```
```   708 next
```
```   709   assume "\<not> integrable lborel f" with c show ?thesis
```
```   710     by (simp add: lborel_integrable_real_affine_iff not_integrable_integral_eq)
```
```   711 qed
```
```   712
```
```   713 lemma
```
```   714   fixes c :: "'a::euclidean_space \<Rightarrow> real" and t
```
```   715   assumes c: "\<And>j. j \<in> Basis \<Longrightarrow> c j \<noteq> 0"
```
```   716   defines "T == (\<lambda>x. t + (\<Sum>j\<in>Basis. (c j * (x \<bullet> j)) *\<^sub>R j))"
```
```   717   shows lebesgue_affine_euclidean: "lebesgue = density (distr lebesgue lebesgue T) (\<lambda>_. (\<Prod>j\<in>Basis. \<bar>c j\<bar>))" (is "_ = ?D")
```
```   718     and lebesgue_affine_measurable: "T \<in> lebesgue \<rightarrow>\<^sub>M lebesgue"
```
```   719 proof -
```
```   720   have T_borel[measurable]: "T \<in> borel \<rightarrow>\<^sub>M borel"
```
```   721     by (auto simp: T_def[abs_def])
```
```   722   { fix A :: "'a set" assume A: "A \<in> sets borel"
```
```   723     then have "emeasure lborel A = 0 \<longleftrightarrow> emeasure (density (distr lborel borel T) (\<lambda>_. (\<Prod>j\<in>Basis. \<bar>c j\<bar>))) A = 0"
```
```   724       unfolding T_def using c by (subst lborel_affine_euclidean[symmetric]) auto
```
```   725     also have "\<dots> \<longleftrightarrow> emeasure (distr lebesgue lborel T) A = 0"
```
```   726       using A c by (simp add: distr_completion emeasure_density nn_integral_cmult prod_nonneg cong: distr_cong)
```
```   727     finally have "emeasure lborel A = 0 \<longleftrightarrow> emeasure (distr lebesgue lborel T) A = 0" . }
```
```   728   then have eq: "null_sets lborel = null_sets (distr lebesgue lborel T)"
```
```   729     by (auto simp: null_sets_def)
```
```   730
```
```   731   show "T \<in> lebesgue \<rightarrow>\<^sub>M lebesgue"
```
```   732     by (rule completion.measurable_completion2) (auto simp: eq measurable_completion)
```
```   733
```
```   734   have "lebesgue = completion (density (distr lborel borel T) (\<lambda>_. (\<Prod>j\<in>Basis. \<bar>c j\<bar>)))"
```
```   735     using c by (subst lborel_affine_euclidean[of c t]) (simp_all add: T_def[abs_def])
```
```   736   also have "\<dots> = density (completion (distr lebesgue lborel T)) (\<lambda>_. (\<Prod>j\<in>Basis. \<bar>c j\<bar>))"
```
```   737     using c by (auto intro!: always_eventually prod_pos completion_density_eq simp: distr_completion cong: distr_cong)
```
```   738   also have "\<dots> = density (distr lebesgue lebesgue T) (\<lambda>_. (\<Prod>j\<in>Basis. \<bar>c j\<bar>))"
```
```   739     by (subst completion.completion_distr_eq) (auto simp: eq measurable_completion)
```
```   740   finally show "lebesgue = density (distr lebesgue lebesgue T) (\<lambda>_. (\<Prod>j\<in>Basis. \<bar>c j\<bar>))" .
```
```   741 qed
```
```   742
```
```   743 lemma lebesgue_measurable_scaling[measurable]: "op *\<^sub>R x \<in> lebesgue \<rightarrow>\<^sub>M lebesgue"
```
```   744 proof cases
```
```   745   assume "x = 0"
```
```   746   then have "op *\<^sub>R x = (\<lambda>x. 0::'a)"
```
```   747     by (auto simp: fun_eq_iff)
```
```   748   then show ?thesis by auto
```
```   749 next
```
```   750   assume "x \<noteq> 0" then show ?thesis
```
```   751     using lebesgue_affine_measurable[of "\<lambda>_. x" 0]
```
```   752     unfolding scaleR_scaleR[symmetric] scaleR_sum_right[symmetric] euclidean_representation
```
```   753     by (auto simp add: ac_simps)
```
```   754 qed
```
```   755
```
```   756 lemma
```
```   757   fixes m :: real and \<delta> :: "'a::euclidean_space"
```
```   758   defines "T r d x \<equiv> r *\<^sub>R x + d"
```
```   759   shows emeasure_lebesgue_affine: "emeasure lebesgue (T m \<delta> ` S) = \<bar>m\<bar> ^ DIM('a) * emeasure lebesgue S" (is ?e)
```
```   760     and measure_lebesgue_affine: "measure lebesgue (T m \<delta> ` S) = \<bar>m\<bar> ^ DIM('a) * measure lebesgue S" (is ?m)
```
```   761 proof -
```
```   762   show ?e
```
```   763   proof cases
```
```   764     assume "m = 0" then show ?thesis
```
```   765       by (simp add: image_constant_conv T_def[abs_def])
```
```   766   next
```
```   767     let ?T = "T m \<delta>" and ?T' = "T (1 / m) (- ((1/m) *\<^sub>R \<delta>))"
```
```   768     assume "m \<noteq> 0"
```
```   769     then have s_comp_s: "?T' \<circ> ?T = id" "?T \<circ> ?T' = id"
```
```   770       by (auto simp: T_def[abs_def] fun_eq_iff scaleR_add_right scaleR_diff_right)
```
```   771     then have "inv ?T' = ?T" "bij ?T'"
```
```   772       by (auto intro: inv_unique_comp o_bij)
```
```   773     then have eq: "T m \<delta> ` S = T (1 / m) ((-1/m) *\<^sub>R \<delta>) -` S \<inter> space lebesgue"
```
```   774       using bij_vimage_eq_inv_image[OF \<open>bij ?T'\<close>, of S] by auto
```
```   775
```
```   776     have trans_eq_T: "(\<lambda>x. \<delta> + (\<Sum>j\<in>Basis. (m * (x \<bullet> j)) *\<^sub>R j)) = T m \<delta>" for m \<delta>
```
```   777       unfolding T_def[abs_def] scaleR_scaleR[symmetric] scaleR_sum_right[symmetric]
```
```   778       by (auto simp add: euclidean_representation ac_simps)
```
```   779
```
```   780     have T[measurable]: "T r d \<in> lebesgue \<rightarrow>\<^sub>M lebesgue" for r d
```
```   781       using lebesgue_affine_measurable[of "\<lambda>_. r" d]
```
```   782       by (cases "r = 0") (auto simp: trans_eq_T T_def[abs_def])
```
```   783
```
```   784     show ?thesis
```
```   785     proof cases
```
```   786       assume "S \<in> sets lebesgue" with \<open>m \<noteq> 0\<close> show ?thesis
```
```   787         unfolding eq
```
```   788         apply (subst lebesgue_affine_euclidean[of "\<lambda>_. m" \<delta>])
```
```   789         apply (simp_all add: emeasure_density trans_eq_T nn_integral_cmult emeasure_distr
```
```   790                         del: space_completion emeasure_completion)
```
```   791         apply (simp add: vimage_comp s_comp_s prod_constant)
```
```   792         done
```
```   793     next
```
```   794       assume "S \<notin> sets lebesgue"
```
```   795       moreover have "?T ` S \<notin> sets lebesgue"
```
```   796       proof
```
```   797         assume "?T ` S \<in> sets lebesgue"
```
```   798         then have "?T -` (?T ` S) \<inter> space lebesgue \<in> sets lebesgue"
```
```   799           by (rule measurable_sets[OF T])
```
```   800         also have "?T -` (?T ` S) \<inter> space lebesgue = S"
```
```   801           by (simp add: vimage_comp s_comp_s eq)
```
```   802         finally show False using \<open>S \<notin> sets lebesgue\<close> by auto
```
```   803       qed
```
```   804       ultimately show ?thesis
```
```   805         by (simp add: emeasure_notin_sets)
```
```   806     qed
```
```   807   qed
```
```   808   show ?m
```
```   809     unfolding measure_def \<open>?e\<close> by (simp add: enn2real_mult prod_nonneg)
```
```   810 qed
```
```   811
```
```   812 lemma divideR_right:
```
```   813   fixes x y :: "'a::real_normed_vector"
```
```   814   shows "r \<noteq> 0 \<Longrightarrow> y = x /\<^sub>R r \<longleftrightarrow> r *\<^sub>R y = x"
```
```   815   using scaleR_cancel_left[of r y "x /\<^sub>R r"] by simp
```
```   816
```
```   817 lemma lborel_has_bochner_integral_real_affine_iff:
```
```   818   fixes x :: "'a :: {banach, second_countable_topology}"
```
```   819   shows "c \<noteq> 0 \<Longrightarrow>
```
```   820     has_bochner_integral lborel f x \<longleftrightarrow>
```
```   821     has_bochner_integral lborel (\<lambda>x. f (t + c * x)) (x /\<^sub>R \<bar>c\<bar>)"
```
```   822   unfolding has_bochner_integral_iff lborel_integrable_real_affine_iff
```
```   823   by (simp_all add: lborel_integral_real_affine[symmetric] divideR_right cong: conj_cong)
```
```   824
```
```   825 lemma lborel_distr_uminus: "distr lborel borel uminus = (lborel :: real measure)"
```
```   826   by (subst lborel_real_affine[of "-1" 0])
```
```   827      (auto simp: density_1 one_ennreal_def[symmetric])
```
```   828
```
```   829 lemma lborel_distr_mult:
```
```   830   assumes "(c::real) \<noteq> 0"
```
```   831   shows "distr lborel borel (op * c) = density lborel (\<lambda>_. inverse \<bar>c\<bar>)"
```
```   832 proof-
```
```   833   have "distr lborel borel (op * c) = distr lborel lborel (op * c)" by (simp cong: distr_cong)
```
```   834   also from assms have "... = density lborel (\<lambda>_. inverse \<bar>c\<bar>)"
```
```   835     by (subst lborel_real_affine[of "inverse c" 0]) (auto simp: o_def distr_density_distr)
```
```   836   finally show ?thesis .
```
```   837 qed
```
```   838
```
```   839 lemma lborel_distr_mult':
```
```   840   assumes "(c::real) \<noteq> 0"
```
```   841   shows "lborel = density (distr lborel borel (op * c)) (\<lambda>_. \<bar>c\<bar>)"
```
```   842 proof-
```
```   843   have "lborel = density lborel (\<lambda>_. 1)" by (rule density_1[symmetric])
```
```   844   also from assms have "(\<lambda>_. 1 :: ennreal) = (\<lambda>_. inverse \<bar>c\<bar> * \<bar>c\<bar>)" by (intro ext) simp
```
```   845   also have "density lborel ... = density (density lborel (\<lambda>_. inverse \<bar>c\<bar>)) (\<lambda>_. \<bar>c\<bar>)"
```
```   846     by (subst density_density_eq) (auto simp: ennreal_mult)
```
```   847   also from assms have "density lborel (\<lambda>_. inverse \<bar>c\<bar>) = distr lborel borel (op * c)"
```
```   848     by (rule lborel_distr_mult[symmetric])
```
```   849   finally show ?thesis .
```
```   850 qed
```
```   851
```
```   852 lemma lborel_distr_plus: "distr lborel borel (op + c) = (lborel :: real measure)"
```
```   853   by (subst lborel_real_affine[of 1 c]) (auto simp: density_1 one_ennreal_def[symmetric])
```
```   854
```
```   855 interpretation lborel: sigma_finite_measure lborel
```
```   856   by (rule sigma_finite_lborel)
```
```   857
```
```   858 interpretation lborel_pair: pair_sigma_finite lborel lborel ..
```
```   859
```
```   860 lemma lborel_prod:
```
```   861   "lborel \<Otimes>\<^sub>M lborel = (lborel :: ('a::euclidean_space \<times> 'b::euclidean_space) measure)"
```
```   862 proof (rule lborel_eqI[symmetric], clarify)
```
```   863   fix la ua :: 'a and lb ub :: 'b
```
```   864   assume lu: "\<And>a b. (a, b) \<in> Basis \<Longrightarrow> (la, lb) \<bullet> (a, b) \<le> (ua, ub) \<bullet> (a, b)"
```
```   865   have [simp]:
```
```   866     "\<And>b. b \<in> Basis \<Longrightarrow> la \<bullet> b \<le> ua \<bullet> b"
```
```   867     "\<And>b. b \<in> Basis \<Longrightarrow> lb \<bullet> b \<le> ub \<bullet> b"
```
```   868     "inj_on (\<lambda>u. (u, 0)) Basis" "inj_on (\<lambda>u. (0, u)) Basis"
```
```   869     "(\<lambda>u. (u, 0)) ` Basis \<inter> (\<lambda>u. (0, u)) ` Basis = {}"
```
```   870     "box (la, lb) (ua, ub) = box la ua \<times> box lb ub"
```
```   871     using lu[of _ 0] lu[of 0] by (auto intro!: inj_onI simp add: Basis_prod_def ball_Un box_def)
```
```   872   show "emeasure (lborel \<Otimes>\<^sub>M lborel) (box (la, lb) (ua, ub)) =
```
```   873       ennreal (prod (op \<bullet> ((ua, ub) - (la, lb))) Basis)"
```
```   874     by (simp add: lborel.emeasure_pair_measure_Times Basis_prod_def prod.union_disjoint
```
```   875                   prod.reindex ennreal_mult inner_diff_left prod_nonneg)
```
```   876 qed (simp add: borel_prod[symmetric])
```
```   877
```
```   878 (* FIXME: conversion in measurable prover *)
```
```   879 lemma lborelD_Collect[measurable (raw)]: "{x\<in>space borel. P x} \<in> sets borel \<Longrightarrow> {x\<in>space lborel. P x} \<in> sets lborel" by simp
```
```   880 lemma lborelD[measurable (raw)]: "A \<in> sets borel \<Longrightarrow> A \<in> sets lborel" by simp
```
```   881
```
```   882 lemma emeasure_bounded_finite:
```
```   883   assumes "bounded A" shows "emeasure lborel A < \<infinity>"
```
```   884 proof -
```
```   885   from bounded_subset_cbox[OF \<open>bounded A\<close>] obtain a b where "A \<subseteq> cbox a b"
```
```   886     by auto
```
```   887   then have "emeasure lborel A \<le> emeasure lborel (cbox a b)"
```
```   888     by (intro emeasure_mono) auto
```
```   889   then show ?thesis
```
```   890     by (auto simp: emeasure_lborel_cbox_eq prod_nonneg less_top[symmetric] top_unique split: if_split_asm)
```
```   891 qed
```
```   892
```
```   893 lemma emeasure_compact_finite: "compact A \<Longrightarrow> emeasure lborel A < \<infinity>"
```
```   894   using emeasure_bounded_finite[of A] by (auto intro: compact_imp_bounded)
```
```   895
```
```   896 lemma borel_integrable_compact:
```
```   897   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::{banach, second_countable_topology}"
```
```   898   assumes "compact S" "continuous_on S f"
```
```   899   shows "integrable lborel (\<lambda>x. indicator S x *\<^sub>R f x)"
```
```   900 proof cases
```
```   901   assume "S \<noteq> {}"
```
```   902   have "continuous_on S (\<lambda>x. norm (f x))"
```
```   903     using assms by (intro continuous_intros)
```
```   904   from continuous_attains_sup[OF \<open>compact S\<close> \<open>S \<noteq> {}\<close> this]
```
```   905   obtain M where M: "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> M"
```
```   906     by auto
```
```   907
```
```   908   show ?thesis
```
```   909   proof (rule integrable_bound)
```
```   910     show "integrable lborel (\<lambda>x. indicator S x * M)"
```
```   911       using assms by (auto intro!: emeasure_compact_finite borel_compact integrable_mult_left)
```
```   912     show "(\<lambda>x. indicator S x *\<^sub>R f x) \<in> borel_measurable lborel"
```
```   913       using assms by (auto intro!: borel_measurable_continuous_on_indicator borel_compact)
```
```   914     show "AE x in lborel. norm (indicator S x *\<^sub>R f x) \<le> norm (indicator S x * M)"
```
```   915       by (auto split: split_indicator simp: abs_real_def dest!: M)
```
```   916   qed
```
```   917 qed simp
```
```   918
```
```   919 lemma borel_integrable_atLeastAtMost:
```
```   920   fixes f :: "real \<Rightarrow> real"
```
```   921   assumes f: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> isCont f x"
```
```   922   shows "integrable lborel (\<lambda>x. f x * indicator {a .. b} x)" (is "integrable _ ?f")
```
```   923 proof -
```
```   924   have "integrable lborel (\<lambda>x. indicator {a .. b} x *\<^sub>R f x)"
```
```   925   proof (rule borel_integrable_compact)
```
```   926     from f show "continuous_on {a..b} f"
```
```   927       by (auto intro: continuous_at_imp_continuous_on)
```
```   928   qed simp
```
```   929   then show ?thesis
```
```   930     by (auto simp: mult.commute)
```
```   931 qed
```
```   932
```
```   933 abbreviation lmeasurable :: "'a::euclidean_space set set"
```
```   934 where
```
```   935   "lmeasurable \<equiv> fmeasurable lebesgue"
```
```   936
```
```   937 lemma lmeasurable_iff_integrable:
```
```   938   "S \<in> lmeasurable \<longleftrightarrow> integrable lebesgue (indicator S :: 'a::euclidean_space \<Rightarrow> real)"
```
```   939   by (auto simp: fmeasurable_def integrable_iff_bounded borel_measurable_indicator_iff ennreal_indicator)
```
```   940
```
```   941 lemma lmeasurable_cbox [iff]: "cbox a b \<in> lmeasurable"
```
```   942   and lmeasurable_box [iff]: "box a b \<in> lmeasurable"
```
```   943   by (auto simp: fmeasurable_def emeasure_lborel_box_eq emeasure_lborel_cbox_eq)
```
```   944
```
```   945 lemma lmeasurable_compact: "compact S \<Longrightarrow> S \<in> lmeasurable"
```
```   946   using emeasure_compact_finite[of S] by (intro fmeasurableI) (auto simp: borel_compact)
```
```   947
```
```   948 lemma lmeasurable_open: "bounded S \<Longrightarrow> open S \<Longrightarrow> S \<in> lmeasurable"
```
```   949   using emeasure_bounded_finite[of S] by (intro fmeasurableI) (auto simp: borel_open)
```
```   950
```
```   951 lemma lmeasurable_ball: "ball a r \<in> lmeasurable"
```
```   952   by (simp add: lmeasurable_open)
```
```   953
```
```   954 lemma lmeasurable_interior: "bounded S \<Longrightarrow> interior S \<in> lmeasurable"
```
```   955   by (simp add: bounded_interior lmeasurable_open)
```
```   956
```
```   957 lemma null_sets_cbox_Diff_box: "cbox a b - box a b \<in> null_sets lborel"
```
```   958 proof -
```
```   959   have "emeasure lborel (cbox a b - box a b) = 0"
```
```   960     by (subst emeasure_Diff) (auto simp: emeasure_lborel_cbox_eq emeasure_lborel_box_eq box_subset_cbox)
```
```   961   then have "cbox a b - box a b \<in> null_sets lborel"
```
```   962     by (auto simp: null_sets_def)
```
```   963   then show ?thesis
```
```   964     by (auto dest!: AE_not_in)
```
```   965 qed
```
```   966 subsection\<open> A nice lemma for negligibility proofs.\<close>
```
```   967
```
```   968 lemma summable_iff_suminf_neq_top: "(\<And>n. f n \<ge> 0) \<Longrightarrow> \<not> summable f \<Longrightarrow> (\<Sum>i. ennreal (f i)) = top"
```
```   969   by (metis summable_suminf_not_top)
```
```   970
```
```   971 proposition starlike_negligible_bounded_gmeasurable:
```
```   972   fixes S :: "'a :: euclidean_space set"
```
```   973   assumes S: "S \<in> sets lebesgue" and "bounded S"
```
```   974       and eq1: "\<And>c x. \<lbrakk>(c *\<^sub>R x) \<in> S; 0 \<le> c; x \<in> S\<rbrakk> \<Longrightarrow> c = 1"
```
```   975     shows "S \<in> null_sets lebesgue"
```
```   976 proof -
```
```   977   obtain M where "0 < M" "S \<subseteq> ball 0 M"
```
```   978     using \<open>bounded S\<close> by (auto dest: bounded_subset_ballD)
```
```   979
```
```   980   let ?f = "\<lambda>n. root DIM('a) (Suc n)"
```
```   981
```
```   982   have vimage_eq_image: "op *\<^sub>R (?f n) -` S = op *\<^sub>R (1 / ?f n) ` S" for n
```
```   983     apply safe
```
```   984     subgoal for x by (rule image_eqI[of _ _ "?f n *\<^sub>R x"]) auto
```
```   985     subgoal by auto
```
```   986     done
```
```   987
```
```   988   have eq: "(1 / ?f n) ^ DIM('a) = 1 / Suc n" for n
```
```   989     by (simp add: field_simps)
```
```   990
```
```   991   { fix n x assume x: "root DIM('a) (1 + real n) *\<^sub>R x \<in> S"
```
```   992     have "1 * norm x \<le> root DIM('a) (1 + real n) * norm x"
```
```   993       by (rule mult_mono) auto
```
```   994     also have "\<dots> < M"
```
```   995       using x \<open>S \<subseteq> ball 0 M\<close> by auto
```
```   996     finally have "norm x < M" by simp }
```
```   997   note less_M = this
```
```   998
```
```   999   have "(\<Sum>n. ennreal (1 / Suc n)) = top"
```
```  1000     using not_summable_harmonic[where 'a=real] summable_Suc_iff[where f="\<lambda>n. 1 / (real n)"]
```
```  1001     by (intro summable_iff_suminf_neq_top) (auto simp add: inverse_eq_divide)
```
```  1002   then have "top * emeasure lebesgue S = (\<Sum>n. (1 / ?f n)^DIM('a) * emeasure lebesgue S)"
```
```  1003     unfolding ennreal_suminf_multc eq by simp
```
```  1004   also have "\<dots> = (\<Sum>n. emeasure lebesgue (op *\<^sub>R (?f n) -` S))"
```
```  1005     unfolding vimage_eq_image using emeasure_lebesgue_affine[of "1 / ?f n" 0 S for n] by simp
```
```  1006   also have "\<dots> = emeasure lebesgue (\<Union>n. op *\<^sub>R (?f n) -` S)"
```
```  1007   proof (intro suminf_emeasure)
```
```  1008     show "disjoint_family (\<lambda>n. op *\<^sub>R (?f n) -` S)"
```
```  1009       unfolding disjoint_family_on_def
```
```  1010     proof safe
```
```  1011       fix m n :: nat and x assume "m \<noteq> n" "?f m *\<^sub>R x \<in> S" "?f n *\<^sub>R x \<in> S"
```
```  1012       with eq1[of "?f m / ?f n" "?f n *\<^sub>R x"] show "x \<in> {}"
```
```  1013         by auto
```
```  1014     qed
```
```  1015     have "op *\<^sub>R (?f i) -` S \<in> sets lebesgue" for i
```
```  1016       using measurable_sets[OF lebesgue_measurable_scaling[of "?f i"] S] by auto
```
```  1017     then show "range (\<lambda>i. op *\<^sub>R (?f i) -` S) \<subseteq> sets lebesgue"
```
```  1018       by auto
```
```  1019   qed
```
```  1020   also have "\<dots> \<le> emeasure lebesgue (ball 0 M :: 'a set)"
```
```  1021     using less_M by (intro emeasure_mono) auto
```
```  1022   also have "\<dots> < top"
```
```  1023     using lmeasurable_ball by (auto simp: fmeasurable_def)
```
```  1024   finally have "emeasure lebesgue S = 0"
```
```  1025     by (simp add: ennreal_top_mult split: if_split_asm)
```
```  1026   then show "S \<in> null_sets lebesgue"
```
```  1027     unfolding null_sets_def using \<open>S \<in> sets lebesgue\<close> by auto
```
```  1028 qed
```
```  1029
```
```  1030 corollary starlike_negligible_compact:
```
```  1031   "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"
```
```  1032   using starlike_negligible_bounded_gmeasurable[of S] by (auto simp: compact_eq_bounded_closed)
```
```  1033
```
```  1034 lemma outer_regular_lborel:
```
```  1035   assumes B: "B \<in> fmeasurable lborel" "0 < (e::real)"
```
```  1036   shows "\<exists>U. open U \<and> B \<subseteq> U \<and> emeasure lborel U \<le> emeasure lborel B + e"
```
```  1037 proof -
```
```  1038   let ?\<mu> = "emeasure lborel"
```
```  1039   let ?B = "\<lambda>n::nat. ball 0 n :: 'a set"
```
```  1040   have B[measurable]: "B \<in> sets borel"
```
```  1041     using B by auto
```
```  1042   let ?e = "\<lambda>n. e*((1/2)^Suc n)"
```
```  1043   have "\<forall>n. \<exists>U. open U \<and> ?B n \<inter> B \<subseteq> U \<and> ?\<mu> (U - B) < ?e n"
```
```  1044   proof
```
```  1045     fix n :: nat
```
```  1046     let ?A = "density lborel (indicator (?B n))"
```
```  1047     have emeasure_A: "X \<in> sets borel \<Longrightarrow> emeasure ?A X = ?\<mu> (?B n \<inter> X)" for X
```
```  1048       by (auto simp add: emeasure_density borel_measurable_indicator indicator_inter_arith[symmetric])
```
```  1049
```
```  1050     have finite_A: "emeasure ?A (space ?A) \<noteq> \<infinity>"
```
```  1051       using emeasure_bounded_finite[of "?B n"] by (auto simp add: emeasure_A)
```
```  1052     interpret A: finite_measure ?A
```
```  1053       by rule fact
```
```  1054     have "emeasure ?A B + ?e n > (INF U:{U. B \<subseteq> U \<and> open U}. emeasure ?A U)"
```
```  1055       using \<open>0<e\<close> by (auto simp: outer_regular[OF _ finite_A B, symmetric])
```
```  1056     then obtain U where U: "B \<subseteq> U" "open U" "?\<mu> (?B n \<inter> B) + ?e n > ?\<mu> (?B n \<inter> U)"
```
```  1057       unfolding INF_less_iff by (auto simp: emeasure_A)
```
```  1058     moreover
```
```  1059     { have "?\<mu> ((?B n \<inter> U) - B) = ?\<mu> ((?B n \<inter> U) - (?B n \<inter> B))"
```
```  1060         using U by (intro arg_cong[where f="?\<mu>"]) auto
```
```  1061       also have "\<dots> = ?\<mu> (?B n \<inter> U) - ?\<mu> (?B n \<inter> B)"
```
```  1062         using U A.emeasure_finite[of B]
```
```  1063         by (intro emeasure_Diff) (auto simp del: A.emeasure_finite simp: emeasure_A)
```
```  1064       also have "\<dots> < ?e n"
```
```  1065         using U(1,2,3) A.emeasure_finite[of B]
```
```  1066         by (subst minus_less_iff_ennreal)
```
```  1067           (auto simp del: A.emeasure_finite simp: emeasure_A less_top ac_simps intro!: emeasure_mono)
```
```  1068       finally have "?\<mu> ((?B n \<inter> U) - B) < ?e n" . }
```
```  1069     ultimately show "\<exists>U. open U \<and> ?B n \<inter> B \<subseteq> U \<and> ?\<mu> (U - B) < ?e n"
```
```  1070       by (intro exI[of _ "?B n \<inter> U"]) auto
```
```  1071   qed
```
```  1072   then obtain U
```
```  1073     where U: "\<And>n. open (U n)" "\<And>n. ?B n \<inter> B \<subseteq> U n" "\<And>n. ?\<mu> (U n - B) < ?e n"
```
```  1074     by metis
```
```  1075   then show ?thesis
```
```  1076   proof (intro exI conjI)
```
```  1077     { fix x assume "x \<in> B"
```
```  1078       moreover
```
```  1079       have "\<exists>n. norm x < real n"
```
```  1080         by (simp add: reals_Archimedean2)
```
```  1081       then guess n ..
```
```  1082       ultimately have "x \<in> (\<Union>n. U n)"
```
```  1083         using U(2)[of n] by auto }
```
```  1084     note * = this
```
```  1085     then show "open (\<Union>n. U n)" "B \<subseteq> (\<Union>n. U n)"
```
```  1086       using U(1,2) by auto
```
```  1087     have "?\<mu> (\<Union>n. U n) = ?\<mu> (B \<union> (\<Union>n. U n - B))"
```
```  1088       using * U(2) by (intro arg_cong[where ?f="?\<mu>"]) auto
```
```  1089     also have "\<dots> = ?\<mu> B + ?\<mu> (\<Union>n. U n - B)"
```
```  1090       using U(1) by (intro plus_emeasure[symmetric]) auto
```
```  1091     also have "\<dots> \<le> ?\<mu> B + (\<Sum>n. ?\<mu> (U n - B))"
```
```  1092       using U(1) by (intro add_mono emeasure_subadditive_countably) auto
```
```  1093     also have "\<dots> \<le> ?\<mu> B + (\<Sum>n. ennreal (?e n))"
```
```  1094       using U(3) by (intro add_mono suminf_le) (auto intro: less_imp_le)
```
```  1095     also have "(\<Sum>n. ennreal (?e n)) = ennreal (e * 1)"
```
```  1096       using \<open>0<e\<close> by (intro suminf_ennreal_eq sums_mult power_half_series) auto
```
```  1097     finally show "emeasure lborel (\<Union>n. U n) \<le> emeasure lborel B + ennreal e"
```
```  1098       by simp
```
```  1099   qed
```
```  1100 qed
```
```  1101
```
```  1102 lemma lmeasurable_outer_open:
```
```  1103   assumes S: "S \<in> lmeasurable" and "0 < e"
```
```  1104   obtains T where "open T" "S \<subseteq> T" "T \<in> lmeasurable" "measure lebesgue T \<le> measure lebesgue S + e"
```
```  1105 proof -
```
```  1106   obtain S' where S': "S \<subseteq> S'" "S' \<in> sets borel" "emeasure lborel S' = emeasure lebesgue S"
```
```  1107     using completion_upper[of S lborel] S by auto
```
```  1108   then have f_S': "S' \<in> fmeasurable lborel"
```
```  1109     using S by (auto simp: fmeasurable_def)
```
```  1110   from outer_regular_lborel[OF this \<open>0<e\<close>] guess U .. note U = this
```
```  1111   show thesis
```
```  1112   proof (rule that)
```
```  1113     show "open U" "S \<subseteq> U" "U \<in> lmeasurable"
```
```  1114       using f_S' U S' by (auto simp: fmeasurable_def less_top[symmetric] top_unique)
```
```  1115     then have "U \<in> fmeasurable lborel"
```
```  1116       by (auto simp: fmeasurable_def)
```
```  1117     with S U \<open>0<e\<close> show "measure lebesgue U \<le> measure lebesgue S + e"
```
```  1118       unfolding S'(3) by (simp add: emeasure_eq_measure2 ennreal_plus[symmetric] del: ennreal_plus)
```
```  1119   qed
```
```  1120 qed
```
```  1121
```
```  1122 end
```