src/HOL/Analysis/Caratheodory.thy
 author haftmann Sun Nov 18 18:07:51 2018 +0000 (8 months ago) changeset 69313 b021008c5397 parent 69260 0a9688695a1b child 69325 4b6ddc5989fc permissions -rw-r--r--
removed legacy input syntax
```     1 (*  Title:      HOL/Analysis/Caratheodory.thy
```
```     2     Author:     Lawrence C Paulson
```
```     3     Author:     Johannes Hölzl, TU München
```
```     4 *)
```
```     5
```
```     6 section%important \<open>Caratheodory Extension Theorem\<close>
```
```     7
```
```     8 theory Caratheodory
```
```     9   imports Measure_Space
```
```    10 begin
```
```    11
```
```    12 text \<open>
```
```    13   Originally from the Hurd/Coble measure theory development, translated by Lawrence Paulson.
```
```    14 \<close>
```
```    15
```
```    16 lemma%unimportant suminf_ennreal_2dimen:
```
```    17   fixes f:: "nat \<times> nat \<Rightarrow> ennreal"
```
```    18   assumes "\<And>m. g m = (\<Sum>n. f (m,n))"
```
```    19   shows "(\<Sum>i. f (prod_decode i)) = suminf g"
```
```    20 proof -
```
```    21   have g_def: "g = (\<lambda>m. (\<Sum>n. f (m,n)))"
```
```    22     using assms by (simp add: fun_eq_iff)
```
```    23   have reindex: "\<And>B. (\<Sum>x\<in>B. f (prod_decode x)) = sum f (prod_decode ` B)"
```
```    24     by (simp add: sum.reindex[OF inj_prod_decode] comp_def)
```
```    25   have "(SUP n. \<Sum>i<n. f (prod_decode i)) = (SUP p \<in> UNIV \<times> UNIV. \<Sum>i<fst p. \<Sum>n<snd p. f (i, n))"
```
```    26   proof (intro SUP_eq; clarsimp simp: sum.cartesian_product reindex)
```
```    27     fix n
```
```    28     let ?M = "\<lambda>f. Suc (Max (f ` prod_decode ` {..<n}))"
```
```    29     { fix a b x assume "x < n" and [symmetric]: "(a, b) = prod_decode x"
```
```    30       then have "a < ?M fst" "b < ?M snd"
```
```    31         by (auto intro!: Max_ge le_imp_less_Suc image_eqI) }
```
```    32     then have "sum f (prod_decode ` {..<n}) \<le> sum f ({..<?M fst} \<times> {..<?M snd})"
```
```    33       by (auto intro!: sum_mono2)
```
```    34     then show "\<exists>a b. sum f (prod_decode ` {..<n}) \<le> sum f ({..<a} \<times> {..<b})" by auto
```
```    35   next
```
```    36     fix a b
```
```    37     let ?M = "prod_decode ` {..<Suc (Max (prod_encode ` ({..<a} \<times> {..<b})))}"
```
```    38     { fix a' b' assume "a' < a" "b' < b" then have "(a', b') \<in> ?M"
```
```    39         by (auto intro!: Max_ge le_imp_less_Suc image_eqI[where x="prod_encode (a', b')"]) }
```
```    40     then have "sum f ({..<a} \<times> {..<b}) \<le> sum f ?M"
```
```    41       by (auto intro!: sum_mono2)
```
```    42     then show "\<exists>n. sum f ({..<a} \<times> {..<b}) \<le> sum f (prod_decode ` {..<n})"
```
```    43       by auto
```
```    44   qed
```
```    45   also have "\<dots> = (SUP p. \<Sum>i<p. \<Sum>n. f (i, n))"
```
```    46     unfolding suminf_sum[OF summableI, symmetric]
```
```    47     by (simp add: suminf_eq_SUP SUP_pair sum.swap[of _ "{..< fst _}"])
```
```    48   finally show ?thesis unfolding g_def
```
```    49     by (simp add: suminf_eq_SUP)
```
```    50 qed
```
```    51
```
```    52 subsection%important \<open>Characterizations of Measures\<close>
```
```    53
```
```    54 definition%important outer_measure_space where
```
```    55   "outer_measure_space M f \<longleftrightarrow> positive M f \<and> increasing M f \<and> countably_subadditive M f"
```
```    56
```
```    57 subsubsection%important \<open>Lambda Systems\<close>
```
```    58
```
```    59 definition%important lambda_system :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ennreal) \<Rightarrow> 'a set set"
```
```    60 where
```
```    61   "lambda_system \<Omega> M f = {l \<in> M. \<forall>x \<in> M. f (l \<inter> x) + f ((\<Omega> - l) \<inter> x) = f x}"
```
```    62
```
```    63 lemma%unimportant (in algebra) lambda_system_eq:
```
```    64   "lambda_system \<Omega> M f = {l \<in> M. \<forall>x \<in> M. f (x \<inter> l) + f (x - l) = f x}"
```
```    65 proof -
```
```    66   have [simp]: "\<And>l x. l \<in> M \<Longrightarrow> x \<in> M \<Longrightarrow> (\<Omega> - l) \<inter> x = x - l"
```
```    67     by (metis Int_Diff Int_absorb1 Int_commute sets_into_space)
```
```    68   show ?thesis
```
```    69     by (auto simp add: lambda_system_def) (metis Int_commute)+
```
```    70 qed
```
```    71
```
```    72 lemma%unimportant (in algebra) lambda_system_empty: "positive M f \<Longrightarrow> {} \<in> lambda_system \<Omega> M f"
```
```    73   by (auto simp add: positive_def lambda_system_eq)
```
```    74
```
```    75 lemma%unimportant lambda_system_sets: "x \<in> lambda_system \<Omega> M f \<Longrightarrow> x \<in> M"
```
```    76   by (simp add: lambda_system_def)
```
```    77
```
```    78 lemma%unimportant (in algebra) lambda_system_Compl:
```
```    79   fixes f:: "'a set \<Rightarrow> ennreal"
```
```    80   assumes x: "x \<in> lambda_system \<Omega> M f"
```
```    81   shows "\<Omega> - x \<in> lambda_system \<Omega> M f"
```
```    82 proof -
```
```    83   have "x \<subseteq> \<Omega>"
```
```    84     by (metis sets_into_space lambda_system_sets x)
```
```    85   hence "\<Omega> - (\<Omega> - x) = x"
```
```    86     by (metis double_diff equalityE)
```
```    87   with x show ?thesis
```
```    88     by (force simp add: lambda_system_def ac_simps)
```
```    89 qed
```
```    90
```
```    91 lemma%unimportant (in algebra) lambda_system_Int:
```
```    92   fixes f:: "'a set \<Rightarrow> ennreal"
```
```    93   assumes xl: "x \<in> lambda_system \<Omega> M f" and yl: "y \<in> lambda_system \<Omega> M f"
```
```    94   shows "x \<inter> y \<in> lambda_system \<Omega> M f"
```
```    95 proof -
```
```    96   from xl yl show ?thesis
```
```    97   proof (auto simp add: positive_def lambda_system_eq Int)
```
```    98     fix u
```
```    99     assume x: "x \<in> M" and y: "y \<in> M" and u: "u \<in> M"
```
```   100        and fx: "\<forall>z\<in>M. f (z \<inter> x) + f (z - x) = f z"
```
```   101        and fy: "\<forall>z\<in>M. f (z \<inter> y) + f (z - y) = f z"
```
```   102     have "u - x \<inter> y \<in> M"
```
```   103       by (metis Diff Diff_Int Un u x y)
```
```   104     moreover
```
```   105     have "(u - (x \<inter> y)) \<inter> y = u \<inter> y - x" by blast
```
```   106     moreover
```
```   107     have "u - x \<inter> y - y = u - y" by blast
```
```   108     ultimately
```
```   109     have ey: "f (u - x \<inter> y) = f (u \<inter> y - x) + f (u - y)" using fy
```
```   110       by force
```
```   111     have "f (u \<inter> (x \<inter> y)) + f (u - x \<inter> y)
```
```   112           = (f (u \<inter> (x \<inter> y)) + f (u \<inter> y - x)) + f (u - y)"
```
```   113       by (simp add: ey ac_simps)
```
```   114     also have "... =  (f ((u \<inter> y) \<inter> x) + f (u \<inter> y - x)) + f (u - y)"
```
```   115       by (simp add: Int_ac)
```
```   116     also have "... = f (u \<inter> y) + f (u - y)"
```
```   117       using fx [THEN bspec, of "u \<inter> y"] Int y u
```
```   118       by force
```
```   119     also have "... = f u"
```
```   120       by (metis fy u)
```
```   121     finally show "f (u \<inter> (x \<inter> y)) + f (u - x \<inter> y) = f u" .
```
```   122   qed
```
```   123 qed
```
```   124
```
```   125 lemma%unimportant (in algebra) lambda_system_Un:
```
```   126   fixes f:: "'a set \<Rightarrow> ennreal"
```
```   127   assumes xl: "x \<in> lambda_system \<Omega> M f" and yl: "y \<in> lambda_system \<Omega> M f"
```
```   128   shows "x \<union> y \<in> lambda_system \<Omega> M f"
```
```   129 proof -
```
```   130   have "(\<Omega> - x) \<inter> (\<Omega> - y) \<in> M"
```
```   131     by (metis Diff_Un Un compl_sets lambda_system_sets xl yl)
```
```   132   moreover
```
```   133   have "x \<union> y = \<Omega> - ((\<Omega> - x) \<inter> (\<Omega> - y))"
```
```   134     by auto (metis subsetD lambda_system_sets sets_into_space xl yl)+
```
```   135   ultimately show ?thesis
```
```   136     by (metis lambda_system_Compl lambda_system_Int xl yl)
```
```   137 qed
```
```   138
```
```   139 lemma%unimportant (in algebra) lambda_system_algebra:
```
```   140   "positive M f \<Longrightarrow> algebra \<Omega> (lambda_system \<Omega> M f)"
```
```   141   apply (auto simp add: algebra_iff_Un)
```
```   142   apply (metis lambda_system_sets set_mp sets_into_space)
```
```   143   apply (metis lambda_system_empty)
```
```   144   apply (metis lambda_system_Compl)
```
```   145   apply (metis lambda_system_Un)
```
```   146   done
```
```   147
```
```   148 lemma%unimportant (in algebra) lambda_system_strong_additive:
```
```   149   assumes z: "z \<in> M" and disj: "x \<inter> y = {}"
```
```   150       and xl: "x \<in> lambda_system \<Omega> M f" and yl: "y \<in> lambda_system \<Omega> M f"
```
```   151   shows "f (z \<inter> (x \<union> y)) = f (z \<inter> x) + f (z \<inter> y)"
```
```   152 proof -
```
```   153   have "z \<inter> x = (z \<inter> (x \<union> y)) \<inter> x" using disj by blast
```
```   154   moreover
```
```   155   have "z \<inter> y = (z \<inter> (x \<union> y)) - x" using disj by blast
```
```   156   moreover
```
```   157   have "(z \<inter> (x \<union> y)) \<in> M"
```
```   158     by (metis Int Un lambda_system_sets xl yl z)
```
```   159   ultimately show ?thesis using xl yl
```
```   160     by (simp add: lambda_system_eq)
```
```   161 qed
```
```   162
```
```   163 lemma%unimportant (in algebra) lambda_system_additive: "additive (lambda_system \<Omega> M f) f"
```
```   164 proof (auto simp add: additive_def)
```
```   165   fix x and y
```
```   166   assume disj: "x \<inter> y = {}"
```
```   167      and xl: "x \<in> lambda_system \<Omega> M f" and yl: "y \<in> lambda_system \<Omega> M f"
```
```   168   hence  "x \<in> M" "y \<in> M" by (blast intro: lambda_system_sets)+
```
```   169   thus "f (x \<union> y) = f x + f y"
```
```   170     using lambda_system_strong_additive [OF top disj xl yl]
```
```   171     by (simp add: Un)
```
```   172 qed
```
```   173
```
```   174 lemma%unimportant lambda_system_increasing: "increasing M f \<Longrightarrow> increasing (lambda_system \<Omega> M f) f"
```
```   175   by (simp add: increasing_def lambda_system_def)
```
```   176
```
```   177 lemma%unimportant lambda_system_positive: "positive M f \<Longrightarrow> positive (lambda_system \<Omega> M f) f"
```
```   178   by (simp add: positive_def lambda_system_def)
```
```   179
```
```   180 lemma%unimportant (in algebra) lambda_system_strong_sum:
```
```   181   fixes A:: "nat \<Rightarrow> 'a set" and f :: "'a set \<Rightarrow> ennreal"
```
```   182   assumes f: "positive M f" and a: "a \<in> M"
```
```   183       and A: "range A \<subseteq> lambda_system \<Omega> M f"
```
```   184       and disj: "disjoint_family A"
```
```   185   shows  "(\<Sum>i = 0..<n. f (a \<inter>A i)) = f (a \<inter> (\<Union>i\<in>{0..<n}. A i))"
```
```   186 proof (induct n)
```
```   187   case 0 show ?case using f by (simp add: positive_def)
```
```   188 next
```
```   189   case (Suc n)
```
```   190   have 2: "A n \<inter> UNION {0..<n} A = {}" using disj
```
```   191     by (force simp add: disjoint_family_on_def neq_iff)
```
```   192   have 3: "A n \<in> lambda_system \<Omega> M f" using A
```
```   193     by blast
```
```   194   interpret l: algebra \<Omega> "lambda_system \<Omega> M f"
```
```   195     using f by (rule lambda_system_algebra)
```
```   196   have 4: "UNION {0..<n} A \<in> lambda_system \<Omega> M f"
```
```   197     using A l.UNION_in_sets by simp
```
```   198   from Suc.hyps show ?case
```
```   199     by (simp add: atLeastLessThanSuc lambda_system_strong_additive [OF a 2 3 4])
```
```   200 qed
```
```   201
```
```   202 lemma%important (in sigma_algebra) lambda_system_caratheodory:
```
```   203   assumes oms: "outer_measure_space M f"
```
```   204       and A: "range A \<subseteq> lambda_system \<Omega> M f"
```
```   205       and disj: "disjoint_family A"
```
```   206   shows  "(\<Union>i. A i) \<in> lambda_system \<Omega> M f \<and> (\<Sum>i. f (A i)) = f (\<Union>i. A i)"
```
```   207 proof%unimportant -
```
```   208   have pos: "positive M f" and inc: "increasing M f"
```
```   209    and csa: "countably_subadditive M f"
```
```   210     by (metis oms outer_measure_space_def)+
```
```   211   have sa: "subadditive M f"
```
```   212     by (metis countably_subadditive_subadditive csa pos)
```
```   213   have A': "\<And>S. A`S \<subseteq> (lambda_system \<Omega> M f)" using A
```
```   214     by auto
```
```   215   interpret ls: algebra \<Omega> "lambda_system \<Omega> M f"
```
```   216     using pos by (rule lambda_system_algebra)
```
```   217   have A'': "range A \<subseteq> M"
```
```   218      by (metis A image_subset_iff lambda_system_sets)
```
```   219
```
```   220   have U_in: "(\<Union>i. A i) \<in> M"
```
```   221     by (metis A'' countable_UN)
```
```   222   have U_eq: "f (\<Union>i. A i) = (\<Sum>i. f (A i))"
```
```   223   proof (rule antisym)
```
```   224     show "f (\<Union>i. A i) \<le> (\<Sum>i. f (A i))"
```
```   225       using csa[unfolded countably_subadditive_def] A'' disj U_in by auto
```
```   226     have dis: "\<And>N. disjoint_family_on A {..<N}" by (intro disjoint_family_on_mono[OF _ disj]) auto
```
```   227     show "(\<Sum>i. f (A i)) \<le> f (\<Union>i. A i)"
```
```   228       using ls.additive_sum [OF lambda_system_positive[OF pos] lambda_system_additive _ A' dis] A''
```
```   229       by (intro suminf_le_const[OF summableI]) (auto intro!: increasingD[OF inc] countable_UN)
```
```   230   qed
```
```   231   have "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) = f a"
```
```   232     if a [iff]: "a \<in> M" for a
```
```   233   proof (rule antisym)
```
```   234     have "range (\<lambda>i. a \<inter> A i) \<subseteq> M" using A''
```
```   235       by blast
```
```   236     moreover
```
```   237     have "disjoint_family (\<lambda>i. a \<inter> A i)" using disj
```
```   238       by (auto simp add: disjoint_family_on_def)
```
```   239     moreover
```
```   240     have "a \<inter> (\<Union>i. A i) \<in> M"
```
```   241       by (metis Int U_in a)
```
```   242     ultimately
```
```   243     have "f (a \<inter> (\<Union>i. A i)) \<le> (\<Sum>i. f (a \<inter> A i))"
```
```   244       using csa[unfolded countably_subadditive_def, rule_format, of "(\<lambda>i. a \<inter> A i)"]
```
```   245       by (simp add: o_def)
```
```   246     hence "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) \<le> (\<Sum>i. f (a \<inter> A i)) + f (a - (\<Union>i. A i))"
```
```   247       by (rule add_right_mono)
```
```   248     also have "\<dots> \<le> f a"
```
```   249     proof (intro ennreal_suminf_bound_add)
```
```   250       fix n
```
```   251       have UNION_in: "(\<Union>i\<in>{0..<n}. A i) \<in> M"
```
```   252         by (metis A'' UNION_in_sets)
```
```   253       have le_fa: "f (UNION {0..<n} A \<inter> a) \<le> f a" using A''
```
```   254         by (blast intro: increasingD [OF inc] A'' UNION_in_sets)
```
```   255       have ls: "(\<Union>i\<in>{0..<n}. A i) \<in> lambda_system \<Omega> M f"
```
```   256         using ls.UNION_in_sets by (simp add: A)
```
```   257       hence eq_fa: "f a = f (a \<inter> (\<Union>i\<in>{0..<n}. A i)) + f (a - (\<Union>i\<in>{0..<n}. A i))"
```
```   258         by (simp add: lambda_system_eq UNION_in)
```
```   259       have "f (a - (\<Union>i. A i)) \<le> f (a - (\<Union>i\<in>{0..<n}. A i))"
```
```   260         by (blast intro: increasingD [OF inc] UNION_in U_in)
```
```   261       thus "(\<Sum>i<n. f (a \<inter> A i)) + f (a - (\<Union>i. A i)) \<le> f a"
```
```   262         by (simp add: lambda_system_strong_sum pos A disj eq_fa add_left_mono atLeast0LessThan[symmetric])
```
```   263     qed
```
```   264     finally show "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) \<le> f a"
```
```   265       by simp
```
```   266   next
```
```   267     have "f a \<le> f (a \<inter> (\<Union>i. A i) \<union> (a - (\<Union>i. A i)))"
```
```   268       by (blast intro:  increasingD [OF inc] U_in)
```
```   269     also have "... \<le>  f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i))"
```
```   270       by (blast intro: subadditiveD [OF sa] U_in)
```
```   271     finally show "f a \<le> f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i))" .
```
```   272   qed
```
```   273   thus  ?thesis
```
```   274     by (simp add: lambda_system_eq sums_iff U_eq U_in)
```
```   275 qed
```
```   276
```
```   277 lemma%important (in sigma_algebra) caratheodory_lemma:
```
```   278   assumes oms: "outer_measure_space M f"
```
```   279   defines "L \<equiv> lambda_system \<Omega> M f"
```
```   280   shows "measure_space \<Omega> L f"
```
```   281 proof%unimportant -
```
```   282   have pos: "positive M f"
```
```   283     by (metis oms outer_measure_space_def)
```
```   284   have alg: "algebra \<Omega> L"
```
```   285     using lambda_system_algebra [of f, OF pos]
```
```   286     by (simp add: algebra_iff_Un L_def)
```
```   287   then
```
```   288   have "sigma_algebra \<Omega> L"
```
```   289     using lambda_system_caratheodory [OF oms]
```
```   290     by (simp add: sigma_algebra_disjoint_iff L_def)
```
```   291   moreover
```
```   292   have "countably_additive L f" "positive L f"
```
```   293     using pos lambda_system_caratheodory [OF oms]
```
```   294     by (auto simp add: lambda_system_sets L_def countably_additive_def positive_def)
```
```   295   ultimately
```
```   296   show ?thesis
```
```   297     using pos by (simp add: measure_space_def)
```
```   298 qed
```
```   299
```
```   300 definition%important outer_measure :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ennreal) \<Rightarrow> 'a set \<Rightarrow> ennreal" where
```
```   301    "outer_measure M f X =
```
```   302      (INF A\<in>{A. range A \<subseteq> M \<and> disjoint_family A \<and> X \<subseteq> (\<Union>i. A i)}. \<Sum>i. f (A i))"
```
```   303
```
```   304 lemma%unimportant (in ring_of_sets) outer_measure_agrees:
```
```   305   assumes posf: "positive M f" and ca: "countably_additive M f" and s: "s \<in> M"
```
```   306   shows "outer_measure M f s = f s"
```
```   307   unfolding outer_measure_def
```
```   308 proof (safe intro!: antisym INF_greatest)
```
```   309   fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" and dA: "disjoint_family A" and sA: "s \<subseteq> (\<Union>x. A x)"
```
```   310   have inc: "increasing M f"
```
```   311     by (metis additive_increasing ca countably_additive_additive posf)
```
```   312   have "f s = f (\<Union>i. A i \<inter> s)"
```
```   313     using sA by (auto simp: Int_absorb1)
```
```   314   also have "\<dots> = (\<Sum>i. f (A i \<inter> s))"
```
```   315     using sA dA A s
```
```   316     by (intro ca[unfolded countably_additive_def, rule_format, symmetric])
```
```   317        (auto simp: Int_absorb1 disjoint_family_on_def)
```
```   318   also have "... \<le> (\<Sum>i. f (A i))"
```
```   319     using A s by (auto intro!: suminf_le increasingD[OF inc])
```
```   320   finally show "f s \<le> (\<Sum>i. f (A i))" .
```
```   321 next
```
```   322   have "(\<Sum>i. f (if i = 0 then s else {})) \<le> f s"
```
```   323     using positiveD1[OF posf] by (subst suminf_finite[of "{0}"]) auto
```
```   324   with s show "(INF A\<in>{A. range A \<subseteq> M \<and> disjoint_family A \<and> s \<subseteq> \<Union>(A ` UNIV)}. \<Sum>i. f (A i)) \<le> f s"
```
```   325     by (intro INF_lower2[of "\<lambda>i. if i = 0 then s else {}"])
```
```   326        (auto simp: disjoint_family_on_def)
```
```   327 qed
```
```   328
```
```   329 lemma%unimportant outer_measure_empty:
```
```   330   "positive M f \<Longrightarrow> {} \<in> M \<Longrightarrow> outer_measure M f {} = 0"
```
```   331   unfolding outer_measure_def
```
```   332   by (intro antisym INF_lower2[of  "\<lambda>_. {}"]) (auto simp: disjoint_family_on_def positive_def)
```
```   333
```
```   334 lemma%unimportant (in ring_of_sets) positive_outer_measure:
```
```   335   assumes "positive M f" shows "positive (Pow \<Omega>) (outer_measure M f)"
```
```   336   unfolding positive_def by (auto simp: assms outer_measure_empty)
```
```   337
```
```   338 lemma%unimportant (in ring_of_sets) increasing_outer_measure: "increasing (Pow \<Omega>) (outer_measure M f)"
```
```   339   by (force simp: increasing_def outer_measure_def intro!: INF_greatest intro: INF_lower)
```
```   340
```
```   341 lemma%unimportant (in ring_of_sets) outer_measure_le:
```
```   342   assumes pos: "positive M f" and inc: "increasing M f" and A: "range A \<subseteq> M" and X: "X \<subseteq> (\<Union>i. A i)"
```
```   343   shows "outer_measure M f X \<le> (\<Sum>i. f (A i))"
```
```   344   unfolding outer_measure_def
```
```   345 proof (safe intro!: INF_lower2[of "disjointed A"] del: subsetI)
```
```   346   show dA: "range (disjointed A) \<subseteq> M"
```
```   347     by (auto intro!: A range_disjointed_sets)
```
```   348   have "\<forall>n. f (disjointed A n) \<le> f (A n)"
```
```   349     by (metis increasingD [OF inc] UNIV_I dA image_subset_iff disjointed_subset A)
```
```   350   then show "(\<Sum>i. f (disjointed A i)) \<le> (\<Sum>i. f (A i))"
```
```   351     by (blast intro!: suminf_le)
```
```   352 qed (auto simp: X UN_disjointed_eq disjoint_family_disjointed)
```
```   353
```
```   354 lemma%unimportant (in ring_of_sets) outer_measure_close:
```
```   355   "outer_measure M f X < e \<Longrightarrow> \<exists>A. range A \<subseteq> M \<and> disjoint_family A \<and> X \<subseteq> (\<Union>i. A i) \<and> (\<Sum>i. f (A i)) < e"
```
```   356   unfolding outer_measure_def INF_less_iff by auto
```
```   357
```
```   358 lemma%unimportant (in ring_of_sets) countably_subadditive_outer_measure:
```
```   359   assumes posf: "positive M f" and inc: "increasing M f"
```
```   360   shows "countably_subadditive (Pow \<Omega>) (outer_measure M f)"
```
```   361 proof (simp add: countably_subadditive_def, safe)
```
```   362   fix A :: "nat \<Rightarrow> _" assume A: "range A \<subseteq> Pow (\<Omega>)" and sb: "(\<Union>i. A i) \<subseteq> \<Omega>"
```
```   363   let ?O = "outer_measure M f"
```
```   364   show "?O (\<Union>i. A i) \<le> (\<Sum>n. ?O (A n))"
```
```   365   proof (rule ennreal_le_epsilon)
```
```   366     fix b and e :: real assume "0 < e" "(\<Sum>n. outer_measure M f (A n)) < top"
```
```   367     then have *: "\<And>n. outer_measure M f (A n) < outer_measure M f (A n) + e * (1/2)^Suc n"
```
```   368       by (auto simp add: less_top dest!: ennreal_suminf_lessD)
```
```   369     obtain B
```
```   370       where B: "\<And>n. range (B n) \<subseteq> M"
```
```   371       and sbB: "\<And>n. A n \<subseteq> (\<Union>i. B n i)"
```
```   372       and Ble: "\<And>n. (\<Sum>i. f (B n i)) \<le> ?O (A n) + e * (1/2)^(Suc n)"
```
```   373       by (metis less_imp_le outer_measure_close[OF *])
```
```   374
```
```   375     define C where "C = case_prod B o prod_decode"
```
```   376     from B have B_in_M: "\<And>i j. B i j \<in> M"
```
```   377       by (rule range_subsetD)
```
```   378     then have C: "range C \<subseteq> M"
```
```   379       by (auto simp add: C_def split_def)
```
```   380     have A_C: "(\<Union>i. A i) \<subseteq> (\<Union>i. C i)"
```
```   381       using sbB by (auto simp add: C_def subset_eq) (metis prod.case prod_encode_inverse)
```
```   382
```
```   383     have "?O (\<Union>i. A i) \<le> ?O (\<Union>i. C i)"
```
```   384       using A_C A C by (intro increasing_outer_measure[THEN increasingD]) (auto dest!: sets_into_space)
```
```   385     also have "\<dots> \<le> (\<Sum>i. f (C i))"
```
```   386       using C by (intro outer_measure_le[OF posf inc]) auto
```
```   387     also have "\<dots> = (\<Sum>n. \<Sum>i. f (B n i))"
```
```   388       using B_in_M unfolding C_def comp_def by (intro suminf_ennreal_2dimen) auto
```
```   389     also have "\<dots> \<le> (\<Sum>n. ?O (A n) + e * (1/2) ^ Suc n)"
```
```   390       using B_in_M by (intro suminf_le suminf_nonneg allI Ble) auto
```
```   391     also have "... = (\<Sum>n. ?O (A n)) + (\<Sum>n. ennreal e * ennreal ((1/2) ^ Suc n))"
```
```   392       using \<open>0 < e\<close> by (subst suminf_add[symmetric])
```
```   393                        (auto simp del: ennreal_suminf_cmult simp add: ennreal_mult[symmetric])
```
```   394     also have "\<dots> = (\<Sum>n. ?O (A n)) + e"
```
```   395       unfolding ennreal_suminf_cmult
```
```   396       by (subst suminf_ennreal_eq[OF zero_le_power power_half_series]) auto
```
```   397     finally show "?O (\<Union>i. A i) \<le> (\<Sum>n. ?O (A n)) + e" .
```
```   398   qed
```
```   399 qed
```
```   400
```
```   401 lemma%unimportant (in ring_of_sets) outer_measure_space_outer_measure:
```
```   402   "positive M f \<Longrightarrow> increasing M f \<Longrightarrow> outer_measure_space (Pow \<Omega>) (outer_measure M f)"
```
```   403   by (simp add: outer_measure_space_def
```
```   404     positive_outer_measure increasing_outer_measure countably_subadditive_outer_measure)
```
```   405
```
```   406 lemma%unimportant (in ring_of_sets) algebra_subset_lambda_system:
```
```   407   assumes posf: "positive M f" and inc: "increasing M f"
```
```   408       and add: "additive M f"
```
```   409   shows "M \<subseteq> lambda_system \<Omega> (Pow \<Omega>) (outer_measure M f)"
```
```   410 proof (auto dest: sets_into_space
```
```   411             simp add: algebra.lambda_system_eq [OF algebra_Pow])
```
```   412   fix x s assume x: "x \<in> M" and s: "s \<subseteq> \<Omega>"
```
```   413   have [simp]: "\<And>x. x \<in> M \<Longrightarrow> s \<inter> (\<Omega> - x) = s - x" using s
```
```   414     by blast
```
```   415   have "outer_measure M f (s \<inter> x) + outer_measure M f (s - x) \<le> outer_measure M f s"
```
```   416     unfolding outer_measure_def[of M f s]
```
```   417   proof (safe intro!: INF_greatest)
```
```   418     fix A :: "nat \<Rightarrow> 'a set" assume A: "disjoint_family A" "range A \<subseteq> M" "s \<subseteq> (\<Union>i. A i)"
```
```   419     have "outer_measure M f (s \<inter> x) \<le> (\<Sum>i. f (A i \<inter> x))"
```
```   420       unfolding outer_measure_def
```
```   421     proof (safe intro!: INF_lower2[of "\<lambda>i. A i \<inter> x"])
```
```   422       from A(1) show "disjoint_family (\<lambda>i. A i \<inter> x)"
```
```   423         by (rule disjoint_family_on_bisimulation) auto
```
```   424     qed (insert x A, auto)
```
```   425     moreover
```
```   426     have "outer_measure M f (s - x) \<le> (\<Sum>i. f (A i - x))"
```
```   427       unfolding outer_measure_def
```
```   428     proof (safe intro!: INF_lower2[of "\<lambda>i. A i - x"])
```
```   429       from A(1) show "disjoint_family (\<lambda>i. A i - x)"
```
```   430         by (rule disjoint_family_on_bisimulation) auto
```
```   431     qed (insert x A, auto)
```
```   432     ultimately have "outer_measure M f (s \<inter> x) + outer_measure M f (s - x) \<le>
```
```   433         (\<Sum>i. f (A i \<inter> x)) + (\<Sum>i. f (A i - x))" by (rule add_mono)
```
```   434     also have "\<dots> = (\<Sum>i. f (A i \<inter> x) + f (A i - x))"
```
```   435       using A(2) x posf by (subst suminf_add) (auto simp: positive_def)
```
```   436     also have "\<dots> = (\<Sum>i. f (A i))"
```
```   437       using A x
```
```   438       by (subst add[THEN additiveD, symmetric])
```
```   439          (auto intro!: arg_cong[where f=suminf] arg_cong[where f=f])
```
```   440     finally show "outer_measure M f (s \<inter> x) + outer_measure M f (s - x) \<le> (\<Sum>i. f (A i))" .
```
```   441   qed
```
```   442   moreover
```
```   443   have "outer_measure M f s \<le> outer_measure M f (s \<inter> x) + outer_measure M f (s - x)"
```
```   444   proof -
```
```   445     have "outer_measure M f s = outer_measure M f ((s \<inter> x) \<union> (s - x))"
```
```   446       by (metis Un_Diff_Int Un_commute)
```
```   447     also have "... \<le> outer_measure M f (s \<inter> x) + outer_measure M f (s - x)"
```
```   448       apply (rule subadditiveD)
```
```   449       apply (rule ring_of_sets.countably_subadditive_subadditive [OF ring_of_sets_Pow])
```
```   450       apply (simp add: positive_def outer_measure_empty[OF posf])
```
```   451       apply (rule countably_subadditive_outer_measure)
```
```   452       using s by (auto intro!: posf inc)
```
```   453     finally show ?thesis .
```
```   454   qed
```
```   455   ultimately
```
```   456   show "outer_measure M f (s \<inter> x) + outer_measure M f (s - x) = outer_measure M f s"
```
```   457     by (rule order_antisym)
```
```   458 qed
```
```   459
```
```   460 lemma%unimportant measure_down: "measure_space \<Omega> N \<mu> \<Longrightarrow> sigma_algebra \<Omega> M \<Longrightarrow> M \<subseteq> N \<Longrightarrow> measure_space \<Omega> M \<mu>"
```
```   461   by (auto simp add: measure_space_def positive_def countably_additive_def subset_eq)
```
```   462
```
```   463 subsection%important \<open>Caratheodory's theorem\<close>
```
```   464
```
```   465 theorem%important (in ring_of_sets) caratheodory':
```
```   466   assumes posf: "positive M f" and ca: "countably_additive M f"
```
```   467   shows "\<exists>\<mu> :: 'a set \<Rightarrow> ennreal. (\<forall>s \<in> M. \<mu> s = f s) \<and> measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"
```
```   468 proof%unimportant -
```
```   469   have inc: "increasing M f"
```
```   470     by (metis additive_increasing ca countably_additive_additive posf)
```
```   471   let ?O = "outer_measure M f"
```
```   472   define ls where "ls = lambda_system \<Omega> (Pow \<Omega>) ?O"
```
```   473   have mls: "measure_space \<Omega> ls ?O"
```
```   474     using sigma_algebra.caratheodory_lemma
```
```   475             [OF sigma_algebra_Pow outer_measure_space_outer_measure [OF posf inc]]
```
```   476     by (simp add: ls_def)
```
```   477   hence sls: "sigma_algebra \<Omega> ls"
```
```   478     by (simp add: measure_space_def)
```
```   479   have "M \<subseteq> ls"
```
```   480     by (simp add: ls_def)
```
```   481        (metis ca posf inc countably_additive_additive algebra_subset_lambda_system)
```
```   482   hence sgs_sb: "sigma_sets (\<Omega>) (M) \<subseteq> ls"
```
```   483     using sigma_algebra.sigma_sets_subset [OF sls, of "M"]
```
```   484     by simp
```
```   485   have "measure_space \<Omega> (sigma_sets \<Omega> M) ?O"
```
```   486     by (rule measure_down [OF mls], rule sigma_algebra_sigma_sets)
```
```   487        (simp_all add: sgs_sb space_closed)
```
```   488   thus ?thesis using outer_measure_agrees [OF posf ca]
```
```   489     by (intro exI[of _ ?O]) auto
```
```   490 qed
```
```   491
```
```   492 lemma%important (in ring_of_sets) caratheodory_empty_continuous:
```
```   493   assumes f: "positive M f" "additive M f" and fin: "\<And>A. A \<in> M \<Longrightarrow> f A \<noteq> \<infinity>"
```
```   494   assumes cont: "\<And>A. range A \<subseteq> M \<Longrightarrow> decseq A \<Longrightarrow> (\<Inter>i. A i) = {} \<Longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> 0"
```
```   495   shows "\<exists>\<mu> :: 'a set \<Rightarrow> ennreal. (\<forall>s \<in> M. \<mu> s = f s) \<and> measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"
```
```   496 proof%unimportant (intro caratheodory' empty_continuous_imp_countably_additive f)
```
```   497   show "\<forall>A\<in>M. f A \<noteq> \<infinity>" using fin by auto
```
```   498 qed (rule cont)
```
```   499
```
```   500 subsection%important \<open>Volumes\<close>
```
```   501
```
```   502 definition%important volume :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ennreal) \<Rightarrow> bool" where
```
```   503   "volume M f \<longleftrightarrow>
```
```   504   (f {} = 0) \<and> (\<forall>a\<in>M. 0 \<le> f a) \<and>
```
```   505   (\<forall>C\<subseteq>M. disjoint C \<longrightarrow> finite C \<longrightarrow> \<Union>C \<in> M \<longrightarrow> f (\<Union>C) = (\<Sum>c\<in>C. f c))"
```
```   506
```
```   507 lemma%unimportant volumeI:
```
```   508   assumes "f {} = 0"
```
```   509   assumes "\<And>a. a \<in> M \<Longrightarrow> 0 \<le> f a"
```
```   510   assumes "\<And>C. C \<subseteq> M \<Longrightarrow> disjoint C \<Longrightarrow> finite C \<Longrightarrow> \<Union>C \<in> M \<Longrightarrow> f (\<Union>C) = (\<Sum>c\<in>C. f c)"
```
```   511   shows "volume M f"
```
```   512   using assms by (auto simp: volume_def)
```
```   513
```
```   514 lemma%unimportant volume_positive:
```
```   515   "volume M f \<Longrightarrow> a \<in> M \<Longrightarrow> 0 \<le> f a"
```
```   516   by (auto simp: volume_def)
```
```   517
```
```   518 lemma%unimportant volume_empty:
```
```   519   "volume M f \<Longrightarrow> f {} = 0"
```
```   520   by (auto simp: volume_def)
```
```   521
```
```   522 lemma%unimportant volume_finite_additive:
```
```   523   assumes "volume M f"
```
```   524   assumes A: "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M" "disjoint_family_on A I" "finite I" "\<Union>(A ` I) \<in> M"
```
```   525   shows "f (\<Union>(A ` I)) = (\<Sum>i\<in>I. f (A i))"
```
```   526 proof -
```
```   527   have "A`I \<subseteq> M" "disjoint (A`I)" "finite (A`I)" "\<Union>(A`I) \<in> M"
```
```   528     using A by (auto simp: disjoint_family_on_disjoint_image)
```
```   529   with \<open>volume M f\<close> have "f (\<Union>(A`I)) = (\<Sum>a\<in>A`I. f a)"
```
```   530     unfolding volume_def by blast
```
```   531   also have "\<dots> = (\<Sum>i\<in>I. f (A i))"
```
```   532   proof (subst sum.reindex_nontrivial)
```
```   533     fix i j assume "i \<in> I" "j \<in> I" "i \<noteq> j" "A i = A j"
```
```   534     with \<open>disjoint_family_on A I\<close> have "A i = {}"
```
```   535       by (auto simp: disjoint_family_on_def)
```
```   536     then show "f (A i) = 0"
```
```   537       using volume_empty[OF \<open>volume M f\<close>] by simp
```
```   538   qed (auto intro: \<open>finite I\<close>)
```
```   539   finally show "f (\<Union>(A ` I)) = (\<Sum>i\<in>I. f (A i))"
```
```   540     by simp
```
```   541 qed
```
```   542
```
```   543 lemma%unimportant (in ring_of_sets) volume_additiveI:
```
```   544   assumes pos: "\<And>a. a \<in> M \<Longrightarrow> 0 \<le> \<mu> a"
```
```   545   assumes [simp]: "\<mu> {} = 0"
```
```   546   assumes add: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<inter> b = {} \<Longrightarrow> \<mu> (a \<union> b) = \<mu> a + \<mu> b"
```
```   547   shows "volume M \<mu>"
```
```   548 proof (unfold volume_def, safe)
```
```   549   fix C assume "finite C" "C \<subseteq> M" "disjoint C"
```
```   550   then show "\<mu> (\<Union>C) = sum \<mu> C"
```
```   551   proof (induct C)
```
```   552     case (insert c C)
```
```   553     from insert(1,2,4,5) have "\<mu> (\<Union>insert c C) = \<mu> c + \<mu> (\<Union>C)"
```
```   554       by (auto intro!: add simp: disjoint_def)
```
```   555     with insert show ?case
```
```   556       by (simp add: disjoint_def)
```
```   557   qed simp
```
```   558 qed fact+
```
```   559
```
```   560 lemma%important (in semiring_of_sets) extend_volume:
```
```   561   assumes "volume M \<mu>"
```
```   562   shows "\<exists>\<mu>'. volume generated_ring \<mu>' \<and> (\<forall>a\<in>M. \<mu>' a = \<mu> a)"
```
```   563 proof%unimportant -
```
```   564   let ?R = generated_ring
```
```   565   have "\<forall>a\<in>?R. \<exists>m. \<exists>C\<subseteq>M. a = \<Union>C \<and> finite C \<and> disjoint C \<and> m = (\<Sum>c\<in>C. \<mu> c)"
```
```   566     by (auto simp: generated_ring_def)
```
```   567   from bchoice[OF this] guess \<mu>' .. note \<mu>'_spec = this
```
```   568
```
```   569   { fix C assume C: "C \<subseteq> M" "finite C" "disjoint C"
```
```   570     fix D assume D: "D \<subseteq> M" "finite D" "disjoint D"
```
```   571     assume "\<Union>C = \<Union>D"
```
```   572     have "(\<Sum>d\<in>D. \<mu> d) = (\<Sum>d\<in>D. \<Sum>c\<in>C. \<mu> (c \<inter> d))"
```
```   573     proof (intro sum.cong refl)
```
```   574       fix d assume "d \<in> D"
```
```   575       have Un_eq_d: "(\<Union>c\<in>C. c \<inter> d) = d"
```
```   576         using \<open>d \<in> D\<close> \<open>\<Union>C = \<Union>D\<close> by auto
```
```   577       moreover have "\<mu> (\<Union>c\<in>C. c \<inter> d) = (\<Sum>c\<in>C. \<mu> (c \<inter> d))"
```
```   578       proof (rule volume_finite_additive)
```
```   579         { fix c assume "c \<in> C" then show "c \<inter> d \<in> M"
```
```   580             using C D \<open>d \<in> D\<close> by auto }
```
```   581         show "(\<Union>a\<in>C. a \<inter> d) \<in> M"
```
```   582           unfolding Un_eq_d using \<open>d \<in> D\<close> D by auto
```
```   583         show "disjoint_family_on (\<lambda>a. a \<inter> d) C"
```
```   584           using \<open>disjoint C\<close> by (auto simp: disjoint_family_on_def disjoint_def)
```
```   585       qed fact+
```
```   586       ultimately show "\<mu> d = (\<Sum>c\<in>C. \<mu> (c \<inter> d))" by simp
```
```   587     qed }
```
```   588   note split_sum = this
```
```   589
```
```   590   { fix C assume C: "C \<subseteq> M" "finite C" "disjoint C"
```
```   591     fix D assume D: "D \<subseteq> M" "finite D" "disjoint D"
```
```   592     assume "\<Union>C = \<Union>D"
```
```   593     with split_sum[OF C D] split_sum[OF D C]
```
```   594     have "(\<Sum>d\<in>D. \<mu> d) = (\<Sum>c\<in>C. \<mu> c)"
```
```   595       by (simp, subst sum.swap, simp add: ac_simps) }
```
```   596   note sum_eq = this
```
```   597
```
```   598   { fix C assume C: "C \<subseteq> M" "finite C" "disjoint C"
```
```   599     then have "\<Union>C \<in> ?R" by (auto simp: generated_ring_def)
```
```   600     with \<mu>'_spec[THEN bspec, of "\<Union>C"]
```
```   601     obtain D where
```
```   602       D: "D \<subseteq> M" "finite D" "disjoint D" "\<Union>C = \<Union>D" and "\<mu>' (\<Union>C) = (\<Sum>d\<in>D. \<mu> d)"
```
```   603       by auto
```
```   604     with sum_eq[OF C D] have "\<mu>' (\<Union>C) = (\<Sum>c\<in>C. \<mu> c)" by simp }
```
```   605   note \<mu>' = this
```
```   606
```
```   607   show ?thesis
```
```   608   proof (intro exI conjI ring_of_sets.volume_additiveI[OF generating_ring] ballI)
```
```   609     fix a assume "a \<in> M" with \<mu>'[of "{a}"] show "\<mu>' a = \<mu> a"
```
```   610       by (simp add: disjoint_def)
```
```   611   next
```
```   612     fix a assume "a \<in> ?R" then guess Ca .. note Ca = this
```
```   613     with \<mu>'[of Ca] \<open>volume M \<mu>\<close>[THEN volume_positive]
```
```   614     show "0 \<le> \<mu>' a"
```
```   615       by (auto intro!: sum_nonneg)
```
```   616   next
```
```   617     show "\<mu>' {} = 0" using \<mu>'[of "{}"] by auto
```
```   618   next
```
```   619     fix a assume "a \<in> ?R" then guess Ca .. note Ca = this
```
```   620     fix b assume "b \<in> ?R" then guess Cb .. note Cb = this
```
```   621     assume "a \<inter> b = {}"
```
```   622     with Ca Cb have "Ca \<inter> Cb \<subseteq> {{}}" by auto
```
```   623     then have C_Int_cases: "Ca \<inter> Cb = {{}} \<or> Ca \<inter> Cb = {}" by auto
```
```   624
```
```   625     from \<open>a \<inter> b = {}\<close> have "\<mu>' (\<Union>(Ca \<union> Cb)) = (\<Sum>c\<in>Ca \<union> Cb. \<mu> c)"
```
```   626       using Ca Cb by (intro \<mu>') (auto intro!: disjoint_union)
```
```   627     also have "\<dots> = (\<Sum>c\<in>Ca \<union> Cb. \<mu> c) + (\<Sum>c\<in>Ca \<inter> Cb. \<mu> c)"
```
```   628       using C_Int_cases volume_empty[OF \<open>volume M \<mu>\<close>] by (elim disjE) simp_all
```
```   629     also have "\<dots> = (\<Sum>c\<in>Ca. \<mu> c) + (\<Sum>c\<in>Cb. \<mu> c)"
```
```   630       using Ca Cb by (simp add: sum.union_inter)
```
```   631     also have "\<dots> = \<mu>' a + \<mu>' b"
```
```   632       using Ca Cb by (simp add: \<mu>')
```
```   633     finally show "\<mu>' (a \<union> b) = \<mu>' a + \<mu>' b"
```
```   634       using Ca Cb by simp
```
```   635   qed
```
```   636 qed
```
```   637
```
```   638 subsubsection%important \<open>Caratheodory on semirings\<close>
```
```   639
```
```   640 theorem%important (in semiring_of_sets) caratheodory:
```
```   641   assumes pos: "positive M \<mu>" and ca: "countably_additive M \<mu>"
```
```   642   shows "\<exists>\<mu>' :: 'a set \<Rightarrow> ennreal. (\<forall>s \<in> M. \<mu>' s = \<mu> s) \<and> measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>'"
```
```   643 proof%unimportant -
```
```   644   have "volume M \<mu>"
```
```   645   proof (rule volumeI)
```
```   646     { fix a assume "a \<in> M" then show "0 \<le> \<mu> a"
```
```   647         using pos unfolding positive_def by auto }
```
```   648     note p = this
```
```   649
```
```   650     fix C assume sets_C: "C \<subseteq> M" "\<Union>C \<in> M" and "disjoint C" "finite C"
```
```   651     have "\<exists>F'. bij_betw F' {..<card C} C"
```
```   652       by (rule finite_same_card_bij[OF _ \<open>finite C\<close>]) auto
```
```   653     then guess F' .. note F' = this
```
```   654     then have F': "C = F' ` {..< card C}" "inj_on F' {..< card C}"
```
```   655       by (auto simp: bij_betw_def)
```
```   656     { fix i j assume *: "i < card C" "j < card C" "i \<noteq> j"
```
```   657       with F' have "F' i \<in> C" "F' j \<in> C" "F' i \<noteq> F' j"
```
```   658         unfolding inj_on_def by auto
```
```   659       with \<open>disjoint C\<close>[THEN disjointD]
```
```   660       have "F' i \<inter> F' j = {}"
```
```   661         by auto }
```
```   662     note F'_disj = this
```
```   663     define F where "F i = (if i < card C then F' i else {})" for i
```
```   664     then have "disjoint_family F"
```
```   665       using F'_disj by (auto simp: disjoint_family_on_def)
```
```   666     moreover from F' have "(\<Union>i. F i) = \<Union>C"
```
```   667       by (auto simp add: F_def split: if_split_asm) blast
```
```   668     moreover have sets_F: "\<And>i. F i \<in> M"
```
```   669       using F' sets_C by (auto simp: F_def)
```
```   670     moreover note sets_C
```
```   671     ultimately have "\<mu> (\<Union>C) = (\<Sum>i. \<mu> (F i))"
```
```   672       using ca[unfolded countably_additive_def, THEN spec, of F] by auto
```
```   673     also have "\<dots> = (\<Sum>i<card C. \<mu> (F' i))"
```
```   674     proof -
```
```   675       have "(\<lambda>i. if i \<in> {..< card C} then \<mu> (F' i) else 0) sums (\<Sum>i<card C. \<mu> (F' i))"
```
```   676         by (rule sums_If_finite_set) auto
```
```   677       also have "(\<lambda>i. if i \<in> {..< card C} then \<mu> (F' i) else 0) = (\<lambda>i. \<mu> (F i))"
```
```   678         using pos by (auto simp: positive_def F_def)
```
```   679       finally show "(\<Sum>i. \<mu> (F i)) = (\<Sum>i<card C. \<mu> (F' i))"
```
```   680         by (simp add: sums_iff)
```
```   681     qed
```
```   682     also have "\<dots> = (\<Sum>c\<in>C. \<mu> c)"
```
```   683       using F'(2) by (subst (2) F') (simp add: sum.reindex)
```
```   684     finally show "\<mu> (\<Union>C) = (\<Sum>c\<in>C. \<mu> c)" .
```
```   685   next
```
```   686     show "\<mu> {} = 0"
```
```   687       using \<open>positive M \<mu>\<close> by (rule positiveD1)
```
```   688   qed
```
```   689   from extend_volume[OF this] obtain \<mu>_r where
```
```   690     V: "volume generated_ring \<mu>_r" "\<And>a. a \<in> M \<Longrightarrow> \<mu> a = \<mu>_r a"
```
```   691     by auto
```
```   692
```
```   693   interpret G: ring_of_sets \<Omega> generated_ring
```
```   694     by (rule generating_ring)
```
```   695
```
```   696   have pos: "positive generated_ring \<mu>_r"
```
```   697     using V unfolding positive_def by (auto simp: positive_def intro!: volume_positive volume_empty)
```
```   698
```
```   699   have "countably_additive generated_ring \<mu>_r"
```
```   700   proof (rule countably_additiveI)
```
```   701     fix A' :: "nat \<Rightarrow> 'a set" assume A': "range A' \<subseteq> generated_ring" "disjoint_family A'"
```
```   702       and Un_A: "(\<Union>i. A' i) \<in> generated_ring"
```
```   703
```
```   704     from generated_ringE[OF Un_A] guess C' . note C' = this
```
```   705
```
```   706     { fix c assume "c \<in> C'"
```
```   707       moreover define A where [abs_def]: "A i = A' i \<inter> c" for i
```
```   708       ultimately have A: "range A \<subseteq> generated_ring" "disjoint_family A"
```
```   709         and Un_A: "(\<Union>i. A i) \<in> generated_ring"
```
```   710         using A' C'
```
```   711         by (auto intro!: G.Int G.finite_Union intro: generated_ringI_Basic simp: disjoint_family_on_def)
```
```   712       from A C' \<open>c \<in> C'\<close> have UN_eq: "(\<Union>i. A i) = c"
```
```   713         by (auto simp: A_def)
```
```   714
```
```   715       have "\<forall>i::nat. \<exists>f::nat \<Rightarrow> 'a set. \<mu>_r (A i) = (\<Sum>j. \<mu>_r (f j)) \<and> disjoint_family f \<and> \<Union>range f = A i \<and> (\<forall>j. f j \<in> M)"
```
```   716         (is "\<forall>i. ?P i")
```
```   717       proof
```
```   718         fix i
```
```   719         from A have Ai: "A i \<in> generated_ring" by auto
```
```   720         from generated_ringE[OF this] guess C . note C = this
```
```   721
```
```   722         have "\<exists>F'. bij_betw F' {..<card C} C"
```
```   723           by (rule finite_same_card_bij[OF _ \<open>finite C\<close>]) auto
```
```   724         then guess F .. note F = this
```
```   725         define f where [abs_def]: "f i = (if i < card C then F i else {})" for i
```
```   726         then have f: "bij_betw f {..< card C} C"
```
```   727           by (intro bij_betw_cong[THEN iffD1, OF _ F]) auto
```
```   728         with C have "\<forall>j. f j \<in> M"
```
```   729           by (auto simp: Pi_iff f_def dest!: bij_betw_imp_funcset)
```
```   730         moreover
```
```   731         from f C have d_f: "disjoint_family_on f {..<card C}"
```
```   732           by (intro disjoint_image_disjoint_family_on) (auto simp: bij_betw_def)
```
```   733         then have "disjoint_family f"
```
```   734           by (auto simp: disjoint_family_on_def f_def)
```
```   735         moreover
```
```   736         have Ai_eq: "A i = (\<Union>x<card C. f x)"
```
```   737           using f C Ai unfolding bij_betw_def by auto
```
```   738         then have "\<Union>range f = A i"
```
```   739           using f C Ai unfolding bij_betw_def
```
```   740             by (auto simp add: f_def cong del: SUP_cong_strong)
```
```   741         moreover
```
```   742         { have "(\<Sum>j. \<mu>_r (f j)) = (\<Sum>j. if j \<in> {..< card C} then \<mu>_r (f j) else 0)"
```
```   743             using volume_empty[OF V(1)] by (auto intro!: arg_cong[where f=suminf] simp: f_def)
```
```   744           also have "\<dots> = (\<Sum>j<card C. \<mu>_r (f j))"
```
```   745             by (rule sums_If_finite_set[THEN sums_unique, symmetric]) simp
```
```   746           also have "\<dots> = \<mu>_r (A i)"
```
```   747             using C f[THEN bij_betw_imp_funcset] unfolding Ai_eq
```
```   748             by (intro volume_finite_additive[OF V(1) _ d_f, symmetric])
```
```   749                (auto simp: Pi_iff Ai_eq intro: generated_ringI_Basic)
```
```   750           finally have "\<mu>_r (A i) = (\<Sum>j. \<mu>_r (f j))" .. }
```
```   751         ultimately show "?P i"
```
```   752           by blast
```
```   753       qed
```
```   754       from choice[OF this] guess f .. note f = this
```
```   755       then have UN_f_eq: "(\<Union>i. case_prod f (prod_decode i)) = (\<Union>i. A i)"
```
```   756         unfolding UN_extend_simps surj_prod_decode by (auto simp: set_eq_iff)
```
```   757
```
```   758       have d: "disjoint_family (\<lambda>i. case_prod f (prod_decode i))"
```
```   759         unfolding disjoint_family_on_def
```
```   760       proof (intro ballI impI)
```
```   761         fix m n :: nat assume "m \<noteq> n"
```
```   762         then have neq: "prod_decode m \<noteq> prod_decode n"
```
```   763           using inj_prod_decode[of UNIV] by (auto simp: inj_on_def)
```
```   764         show "case_prod f (prod_decode m) \<inter> case_prod f (prod_decode n) = {}"
```
```   765         proof cases
```
```   766           assume "fst (prod_decode m) = fst (prod_decode n)"
```
```   767           then show ?thesis
```
```   768             using neq f by (fastforce simp: disjoint_family_on_def)
```
```   769         next
```
```   770           assume neq: "fst (prod_decode m) \<noteq> fst (prod_decode n)"
```
```   771           have "case_prod f (prod_decode m) \<subseteq> A (fst (prod_decode m))"
```
```   772             "case_prod f (prod_decode n) \<subseteq> A (fst (prod_decode n))"
```
```   773             using f[THEN spec, of "fst (prod_decode m)"]
```
```   774             using f[THEN spec, of "fst (prod_decode n)"]
```
```   775             by (auto simp: set_eq_iff)
```
```   776           with f A neq show ?thesis
```
```   777             by (fastforce simp: disjoint_family_on_def subset_eq set_eq_iff)
```
```   778         qed
```
```   779       qed
```
```   780       from f have "(\<Sum>n. \<mu>_r (A n)) = (\<Sum>n. \<mu>_r (case_prod f (prod_decode n)))"
```
```   781         by (intro suminf_ennreal_2dimen[symmetric] generated_ringI_Basic)
```
```   782          (auto split: prod.split)
```
```   783       also have "\<dots> = (\<Sum>n. \<mu> (case_prod f (prod_decode n)))"
```
```   784         using f V(2) by (auto intro!: arg_cong[where f=suminf] split: prod.split)
```
```   785       also have "\<dots> = \<mu> (\<Union>i. case_prod f (prod_decode i))"
```
```   786         using f \<open>c \<in> C'\<close> C'
```
```   787         by (intro ca[unfolded countably_additive_def, rule_format])
```
```   788            (auto split: prod.split simp: UN_f_eq d UN_eq)
```
```   789       finally have "(\<Sum>n. \<mu>_r (A' n \<inter> c)) = \<mu> c"
```
```   790         using UN_f_eq UN_eq by (simp add: A_def) }
```
```   791     note eq = this
```
```   792
```
```   793     have "(\<Sum>n. \<mu>_r (A' n)) = (\<Sum>n. \<Sum>c\<in>C'. \<mu>_r (A' n \<inter> c))"
```
```   794       using C' A'
```
```   795       by (subst volume_finite_additive[symmetric, OF V(1)])
```
```   796          (auto simp: disjoint_def disjoint_family_on_def
```
```   797                intro!: G.Int G.finite_Union arg_cong[where f="\<lambda>X. suminf (\<lambda>i. \<mu>_r (X i))"] ext
```
```   798                intro: generated_ringI_Basic)
```
```   799     also have "\<dots> = (\<Sum>c\<in>C'. \<Sum>n. \<mu>_r (A' n \<inter> c))"
```
```   800       using C' A'
```
```   801       by (intro suminf_sum G.Int G.finite_Union) (auto intro: generated_ringI_Basic)
```
```   802     also have "\<dots> = (\<Sum>c\<in>C'. \<mu>_r c)"
```
```   803       using eq V C' by (auto intro!: sum.cong)
```
```   804     also have "\<dots> = \<mu>_r (\<Union>C')"
```
```   805       using C' Un_A
```
```   806       by (subst volume_finite_additive[symmetric, OF V(1)])
```
```   807          (auto simp: disjoint_family_on_def disjoint_def
```
```   808                intro: generated_ringI_Basic)
```
```   809     finally show "(\<Sum>n. \<mu>_r (A' n)) = \<mu>_r (\<Union>i. A' i)"
```
```   810       using C' by simp
```
```   811   qed
```
```   812   from G.caratheodory'[OF \<open>positive generated_ring \<mu>_r\<close> \<open>countably_additive generated_ring \<mu>_r\<close>]
```
```   813   guess \<mu>' ..
```
```   814   with V show ?thesis
```
```   815     unfolding sigma_sets_generated_ring_eq
```
```   816     by (intro exI[of _ \<mu>']) (auto intro: generated_ringI_Basic)
```
```   817 qed
```
```   818
```
```   819 lemma%important extend_measure_caratheodory:
```
```   820   fixes G :: "'i \<Rightarrow> 'a set"
```
```   821   assumes M: "M = extend_measure \<Omega> I G \<mu>"
```
```   822   assumes "i \<in> I"
```
```   823   assumes "semiring_of_sets \<Omega> (G ` I)"
```
```   824   assumes empty: "\<And>i. i \<in> I \<Longrightarrow> G i = {} \<Longrightarrow> \<mu> i = 0"
```
```   825   assumes inj: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> G i = G j \<Longrightarrow> \<mu> i = \<mu> j"
```
```   826   assumes nonneg: "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> \<mu> i"
```
```   827   assumes add: "\<And>A::nat \<Rightarrow> 'i. \<And>j. A \<in> UNIV \<rightarrow> I \<Longrightarrow> j \<in> I \<Longrightarrow> disjoint_family (G \<circ> A) \<Longrightarrow>
```
```   828     (\<Union>i. G (A i)) = G j \<Longrightarrow> (\<Sum>n. \<mu> (A n)) = \<mu> j"
```
```   829   shows "emeasure M (G i) = \<mu> i"
```
```   830
```
```   831 proof%unimportant -
```
```   832   interpret semiring_of_sets \<Omega> "G ` I"
```
```   833     by fact
```
```   834   have "\<forall>g\<in>G`I. \<exists>i\<in>I. g = G i"
```
```   835     by auto
```
```   836   then obtain sel where sel: "\<And>g. g \<in> G ` I \<Longrightarrow> sel g \<in> I" "\<And>g. g \<in> G ` I \<Longrightarrow> G (sel g) = g"
```
```   837     by metis
```
```   838
```
```   839   have "\<exists>\<mu>'. (\<forall>s\<in>G ` I. \<mu>' s = \<mu> (sel s)) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G ` I)) \<mu>'"
```
```   840   proof (rule caratheodory)
```
```   841     show "positive (G ` I) (\<lambda>s. \<mu> (sel s))"
```
```   842       by (auto simp: positive_def intro!: empty sel nonneg)
```
```   843     show "countably_additive (G ` I) (\<lambda>s. \<mu> (sel s))"
```
```   844     proof (rule countably_additiveI)
```
```   845       fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> G ` I" "disjoint_family A" "(\<Union>i. A i) \<in> G ` I"
```
```   846       then show "(\<Sum>i. \<mu> (sel (A i))) = \<mu> (sel (\<Union>i. A i))"
```
```   847         by (intro add) (auto simp: sel image_subset_iff_funcset comp_def Pi_iff intro!: sel)
```
```   848     qed
```
```   849   qed
```
```   850   then obtain \<mu>' where \<mu>': "\<forall>s\<in>G ` I. \<mu>' s = \<mu> (sel s)" "measure_space \<Omega> (sigma_sets \<Omega> (G ` I)) \<mu>'"
```
```   851     by metis
```
```   852
```
```   853   show ?thesis
```
```   854   proof (rule emeasure_extend_measure[OF M])
```
```   855     { fix i assume "i \<in> I" then show "\<mu>' (G i) = \<mu> i"
```
```   856       using \<mu>' by (auto intro!: inj sel) }
```
```   857     show "G ` I \<subseteq> Pow \<Omega>"
```
```   858       by (rule space_closed)
```
```   859     then show "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'"
```
```   860       using \<mu>' by (simp_all add: M sets_extend_measure measure_space_def)
```
```   861   qed fact
```
```   862 qed
```
```   863
```
```   864 lemma%important extend_measure_caratheodory_pair:
```
```   865   fixes G :: "'i \<Rightarrow> 'j \<Rightarrow> 'a set"
```
```   866   assumes M: "M = extend_measure \<Omega> {(a, b). P a b} (\<lambda>(a, b). G a b) (\<lambda>(a, b). \<mu> a b)"
```
```   867   assumes "P i j"
```
```   868   assumes semiring: "semiring_of_sets \<Omega> {G a b | a b. P a b}"
```
```   869   assumes empty: "\<And>i j. P i j \<Longrightarrow> G i j = {} \<Longrightarrow> \<mu> i j = 0"
```
```   870   assumes inj: "\<And>i j k l. P i j \<Longrightarrow> P k l \<Longrightarrow> G i j = G k l \<Longrightarrow> \<mu> i j = \<mu> k l"
```
```   871   assumes nonneg: "\<And>i j. P i j \<Longrightarrow> 0 \<le> \<mu> i j"
```
```   872   assumes add: "\<And>A::nat \<Rightarrow> 'i. \<And>B::nat \<Rightarrow> 'j. \<And>j k.
```
```   873     (\<And>n. P (A n) (B n)) \<Longrightarrow> P j k \<Longrightarrow> disjoint_family (\<lambda>n. G (A n) (B n)) \<Longrightarrow>
```
```   874     (\<Union>i. G (A i) (B i)) = G j k \<Longrightarrow> (\<Sum>n. \<mu> (A n) (B n)) = \<mu> j k"
```
```   875   shows "emeasure M (G i j) = \<mu> i j"
```
```   876 proof%unimportant -
```
```   877   have "emeasure M ((\<lambda>(a, b). G a b) (i, j)) = (\<lambda>(a, b). \<mu> a b) (i, j)"
```
```   878   proof (rule extend_measure_caratheodory[OF M])
```
```   879     show "semiring_of_sets \<Omega> ((\<lambda>(a, b). G a b) ` {(a, b). P a b})"
```
```   880       using semiring by (simp add: image_def conj_commute)
```
```   881   next
```
```   882     fix A :: "nat \<Rightarrow> ('i \<times> 'j)" and j assume "A \<in> UNIV \<rightarrow> {(a, b). P a b}" "j \<in> {(a, b). P a b}"
```
```   883       "disjoint_family ((\<lambda>(a, b). G a b) \<circ> A)"
```
```   884       "(\<Union>i. case A i of (a, b) \<Rightarrow> G a b) = (case j of (a, b) \<Rightarrow> G a b)"
```
```   885     then show "(\<Sum>n. case A n of (a, b) \<Rightarrow> \<mu> a b) = (case j of (a, b) \<Rightarrow> \<mu> a b)"
```
```   886       using add[of "\<lambda>i. fst (A i)" "\<lambda>i. snd (A i)" "fst j" "snd j"]
```
```   887       by (simp add: split_beta' comp_def Pi_iff)
```
```   888   qed (auto split: prod.splits intro: assms)
```
```   889   then show ?thesis by simp
```
```   890 qed
```
```   891
```
```   892 end
```