src/HOL/Probability/Caratheodory.thy
 author haftmann Wed Feb 17 21:51:56 2016 +0100 (2016-02-17) changeset 62343 24106dc44def parent 61969 e01015e49041 child 62390 842917225d56 permissions -rw-r--r--
prefer abbreviations for compound operators INFIMUM and SUPREMUM
```     1 (*  Title:      HOL/Probability/Caratheodory.thy
```
```     2     Author:     Lawrence C Paulson
```
```     3     Author:     Johannes Hölzl, TU München
```
```     4 *)
```
```     5
```
```     6 section \<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 suminf_ereal_2dimen:
```
```    17   fixes f:: "nat \<times> nat \<Rightarrow> ereal"
```
```    18   assumes pos: "\<And>p. 0 \<le> f p"
```
```    19   assumes "\<And>m. g m = (\<Sum>n. f (m,n))"
```
```    20   shows "(\<Sum>i. f (prod_decode i)) = suminf g"
```
```    21 proof -
```
```    22   have g_def: "g = (\<lambda>m. (\<Sum>n. f (m,n)))"
```
```    23     using assms by (simp add: fun_eq_iff)
```
```    24   have reindex: "\<And>B. (\<Sum>x\<in>B. f (prod_decode x)) = setsum f (prod_decode ` B)"
```
```    25     by (simp add: setsum.reindex[OF inj_prod_decode] comp_def)
```
```    26   { fix n
```
```    27     let ?M = "\<lambda>f. Suc (Max (f ` prod_decode ` {..<n}))"
```
```    28     { fix a b x assume "x < n" and [symmetric]: "(a, b) = prod_decode x"
```
```    29       then have "a < ?M fst" "b < ?M snd"
```
```    30         by (auto intro!: Max_ge le_imp_less_Suc image_eqI) }
```
```    31     then have "setsum f (prod_decode ` {..<n}) \<le> setsum f ({..<?M fst} \<times> {..<?M snd})"
```
```    32       by (auto intro!: setsum_mono3 simp: pos)
```
```    33     then have "\<exists>a b. setsum f (prod_decode ` {..<n}) \<le> setsum f ({..<a} \<times> {..<b})" by auto }
```
```    34   moreover
```
```    35   { fix a b
```
```    36     let ?M = "prod_decode ` {..<Suc (Max (prod_encode ` ({..<a} \<times> {..<b})))}"
```
```    37     { fix a' b' assume "a' < a" "b' < b" then have "(a', b') \<in> ?M"
```
```    38         by (auto intro!: Max_ge le_imp_less_Suc image_eqI[where x="prod_encode (a', b')"]) }
```
```    39     then have "setsum f ({..<a} \<times> {..<b}) \<le> setsum f ?M"
```
```    40       by (auto intro!: setsum_mono3 simp: pos) }
```
```    41   ultimately
```
```    42   show ?thesis unfolding g_def using pos
```
```    43     by (auto intro!: SUP_eq  simp: setsum.cartesian_product reindex SUP_upper2
```
```    44                      suminf_ereal_eq_SUP SUP_pair
```
```    45                      SUP_ereal_setsum[symmetric] incseq_setsumI setsum_nonneg)
```
```    46 qed
```
```    47
```
```    48 subsection \<open>Characterizations of Measures\<close>
```
```    49
```
```    50 definition subadditive where
```
```    51   "subadditive M f \<longleftrightarrow> (\<forall>x\<in>M. \<forall>y\<in>M. x \<inter> y = {} \<longrightarrow> f (x \<union> y) \<le> f x + f y)"
```
```    52
```
```    53 definition countably_subadditive where
```
```    54   "countably_subadditive M f \<longleftrightarrow>
```
```    55     (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow> (f (\<Union>i. A i) \<le> (\<Sum>i. f (A i))))"
```
```    56
```
```    57 definition outer_measure_space where
```
```    58   "outer_measure_space M f \<longleftrightarrow> positive M f \<and> increasing M f \<and> countably_subadditive M f"
```
```    59
```
```    60 lemma subadditiveD: "subadditive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x \<in> M \<Longrightarrow> y \<in> M \<Longrightarrow> f (x \<union> y) \<le> f x + f y"
```
```    61   by (auto simp add: subadditive_def)
```
```    62
```
```    63 subsubsection \<open>Lambda Systems\<close>
```
```    64
```
```    65 definition lambda_system where
```
```    66   "lambda_system \<Omega> M f = {l \<in> M. \<forall>x \<in> M. f (l \<inter> x) + f ((\<Omega> - l) \<inter> x) = f x}"
```
```    67
```
```    68 lemma (in algebra) lambda_system_eq:
```
```    69   "lambda_system \<Omega> M f = {l \<in> M. \<forall>x \<in> M. f (x \<inter> l) + f (x - l) = f x}"
```
```    70 proof -
```
```    71   have [simp]: "\<And>l x. l \<in> M \<Longrightarrow> x \<in> M \<Longrightarrow> (\<Omega> - l) \<inter> x = x - l"
```
```    72     by (metis Int_Diff Int_absorb1 Int_commute sets_into_space)
```
```    73   show ?thesis
```
```    74     by (auto simp add: lambda_system_def) (metis Int_commute)+
```
```    75 qed
```
```    76
```
```    77 lemma (in algebra) lambda_system_empty: "positive M f \<Longrightarrow> {} \<in> lambda_system \<Omega> M f"
```
```    78   by (auto simp add: positive_def lambda_system_eq)
```
```    79
```
```    80 lemma lambda_system_sets: "x \<in> lambda_system \<Omega> M f \<Longrightarrow> x \<in> M"
```
```    81   by (simp add: lambda_system_def)
```
```    82
```
```    83 lemma (in algebra) lambda_system_Compl:
```
```    84   fixes f:: "'a set \<Rightarrow> ereal"
```
```    85   assumes x: "x \<in> lambda_system \<Omega> M f"
```
```    86   shows "\<Omega> - x \<in> lambda_system \<Omega> M f"
```
```    87 proof -
```
```    88   have "x \<subseteq> \<Omega>"
```
```    89     by (metis sets_into_space lambda_system_sets x)
```
```    90   hence "\<Omega> - (\<Omega> - x) = x"
```
```    91     by (metis double_diff equalityE)
```
```    92   with x show ?thesis
```
```    93     by (force simp add: lambda_system_def ac_simps)
```
```    94 qed
```
```    95
```
```    96 lemma (in algebra) lambda_system_Int:
```
```    97   fixes f:: "'a set \<Rightarrow> ereal"
```
```    98   assumes xl: "x \<in> lambda_system \<Omega> M f" and yl: "y \<in> lambda_system \<Omega> M f"
```
```    99   shows "x \<inter> y \<in> lambda_system \<Omega> M f"
```
```   100 proof -
```
```   101   from xl yl show ?thesis
```
```   102   proof (auto simp add: positive_def lambda_system_eq Int)
```
```   103     fix u
```
```   104     assume x: "x \<in> M" and y: "y \<in> M" and u: "u \<in> M"
```
```   105        and fx: "\<forall>z\<in>M. f (z \<inter> x) + f (z - x) = f z"
```
```   106        and fy: "\<forall>z\<in>M. f (z \<inter> y) + f (z - y) = f z"
```
```   107     have "u - x \<inter> y \<in> M"
```
```   108       by (metis Diff Diff_Int Un u x y)
```
```   109     moreover
```
```   110     have "(u - (x \<inter> y)) \<inter> y = u \<inter> y - x" by blast
```
```   111     moreover
```
```   112     have "u - x \<inter> y - y = u - y" by blast
```
```   113     ultimately
```
```   114     have ey: "f (u - x \<inter> y) = f (u \<inter> y - x) + f (u - y)" using fy
```
```   115       by force
```
```   116     have "f (u \<inter> (x \<inter> y)) + f (u - x \<inter> y)
```
```   117           = (f (u \<inter> (x \<inter> y)) + f (u \<inter> y - x)) + f (u - y)"
```
```   118       by (simp add: ey ac_simps)
```
```   119     also have "... =  (f ((u \<inter> y) \<inter> x) + f (u \<inter> y - x)) + f (u - y)"
```
```   120       by (simp add: Int_ac)
```
```   121     also have "... = f (u \<inter> y) + f (u - y)"
```
```   122       using fx [THEN bspec, of "u \<inter> y"] Int y u
```
```   123       by force
```
```   124     also have "... = f u"
```
```   125       by (metis fy u)
```
```   126     finally show "f (u \<inter> (x \<inter> y)) + f (u - x \<inter> y) = f u" .
```
```   127   qed
```
```   128 qed
```
```   129
```
```   130 lemma (in algebra) lambda_system_Un:
```
```   131   fixes f:: "'a set \<Rightarrow> ereal"
```
```   132   assumes xl: "x \<in> lambda_system \<Omega> M f" and yl: "y \<in> lambda_system \<Omega> M f"
```
```   133   shows "x \<union> y \<in> lambda_system \<Omega> M f"
```
```   134 proof -
```
```   135   have "(\<Omega> - x) \<inter> (\<Omega> - y) \<in> M"
```
```   136     by (metis Diff_Un Un compl_sets lambda_system_sets xl yl)
```
```   137   moreover
```
```   138   have "x \<union> y = \<Omega> - ((\<Omega> - x) \<inter> (\<Omega> - y))"
```
```   139     by auto (metis subsetD lambda_system_sets sets_into_space xl yl)+
```
```   140   ultimately show ?thesis
```
```   141     by (metis lambda_system_Compl lambda_system_Int xl yl)
```
```   142 qed
```
```   143
```
```   144 lemma (in algebra) lambda_system_algebra:
```
```   145   "positive M f \<Longrightarrow> algebra \<Omega> (lambda_system \<Omega> M f)"
```
```   146   apply (auto simp add: algebra_iff_Un)
```
```   147   apply (metis lambda_system_sets set_mp sets_into_space)
```
```   148   apply (metis lambda_system_empty)
```
```   149   apply (metis lambda_system_Compl)
```
```   150   apply (metis lambda_system_Un)
```
```   151   done
```
```   152
```
```   153 lemma (in algebra) lambda_system_strong_additive:
```
```   154   assumes z: "z \<in> M" and disj: "x \<inter> y = {}"
```
```   155       and xl: "x \<in> lambda_system \<Omega> M f" and yl: "y \<in> lambda_system \<Omega> M f"
```
```   156   shows "f (z \<inter> (x \<union> y)) = f (z \<inter> x) + f (z \<inter> y)"
```
```   157 proof -
```
```   158   have "z \<inter> x = (z \<inter> (x \<union> y)) \<inter> x" using disj by blast
```
```   159   moreover
```
```   160   have "z \<inter> y = (z \<inter> (x \<union> y)) - x" using disj by blast
```
```   161   moreover
```
```   162   have "(z \<inter> (x \<union> y)) \<in> M"
```
```   163     by (metis Int Un lambda_system_sets xl yl z)
```
```   164   ultimately show ?thesis using xl yl
```
```   165     by (simp add: lambda_system_eq)
```
```   166 qed
```
```   167
```
```   168 lemma (in algebra) lambda_system_additive: "additive (lambda_system \<Omega> M f) f"
```
```   169 proof (auto simp add: additive_def)
```
```   170   fix x and y
```
```   171   assume disj: "x \<inter> y = {}"
```
```   172      and xl: "x \<in> lambda_system \<Omega> M f" and yl: "y \<in> lambda_system \<Omega> M f"
```
```   173   hence  "x \<in> M" "y \<in> M" by (blast intro: lambda_system_sets)+
```
```   174   thus "f (x \<union> y) = f x + f y"
```
```   175     using lambda_system_strong_additive [OF top disj xl yl]
```
```   176     by (simp add: Un)
```
```   177 qed
```
```   178
```
```   179 lemma (in ring_of_sets) countably_subadditive_subadditive:
```
```   180   assumes f: "positive M f" and cs: "countably_subadditive M f"
```
```   181   shows  "subadditive M f"
```
```   182 proof (auto simp add: subadditive_def)
```
```   183   fix x y
```
```   184   assume x: "x \<in> M" and y: "y \<in> M" and "x \<inter> y = {}"
```
```   185   hence "disjoint_family (binaryset x y)"
```
```   186     by (auto simp add: disjoint_family_on_def binaryset_def)
```
```   187   hence "range (binaryset x y) \<subseteq> M \<longrightarrow>
```
```   188          (\<Union>i. binaryset x y i) \<in> M \<longrightarrow>
```
```   189          f (\<Union>i. binaryset x y i) \<le> (\<Sum> n. f (binaryset x y n))"
```
```   190     using cs by (auto simp add: countably_subadditive_def)
```
```   191   hence "{x,y,{}} \<subseteq> M \<longrightarrow> x \<union> y \<in> M \<longrightarrow>
```
```   192          f (x \<union> y) \<le> (\<Sum> n. f (binaryset x y n))"
```
```   193     by (simp add: range_binaryset_eq UN_binaryset_eq)
```
```   194   thus "f (x \<union> y) \<le>  f x + f y" using f x y
```
```   195     by (auto simp add: Un o_def suminf_binaryset_eq positive_def)
```
```   196 qed
```
```   197
```
```   198 lemma lambda_system_increasing: "increasing M f \<Longrightarrow> increasing (lambda_system \<Omega> M f) f"
```
```   199   by (simp add: increasing_def lambda_system_def)
```
```   200
```
```   201 lemma lambda_system_positive: "positive M f \<Longrightarrow> positive (lambda_system \<Omega> M f) f"
```
```   202   by (simp add: positive_def lambda_system_def)
```
```   203
```
```   204 lemma (in algebra) lambda_system_strong_sum:
```
```   205   fixes A:: "nat \<Rightarrow> 'a set" and f :: "'a set \<Rightarrow> ereal"
```
```   206   assumes f: "positive M f" and a: "a \<in> M"
```
```   207       and A: "range A \<subseteq> lambda_system \<Omega> M f"
```
```   208       and disj: "disjoint_family A"
```
```   209   shows  "(\<Sum>i = 0..<n. f (a \<inter>A i)) = f (a \<inter> (\<Union>i\<in>{0..<n}. A i))"
```
```   210 proof (induct n)
```
```   211   case 0 show ?case using f by (simp add: positive_def)
```
```   212 next
```
```   213   case (Suc n)
```
```   214   have 2: "A n \<inter> UNION {0..<n} A = {}" using disj
```
```   215     by (force simp add: disjoint_family_on_def neq_iff)
```
```   216   have 3: "A n \<in> lambda_system \<Omega> M f" using A
```
```   217     by blast
```
```   218   interpret l: algebra \<Omega> "lambda_system \<Omega> M f"
```
```   219     using f by (rule lambda_system_algebra)
```
```   220   have 4: "UNION {0..<n} A \<in> lambda_system \<Omega> M f"
```
```   221     using A l.UNION_in_sets by simp
```
```   222   from Suc.hyps show ?case
```
```   223     by (simp add: atLeastLessThanSuc lambda_system_strong_additive [OF a 2 3 4])
```
```   224 qed
```
```   225
```
```   226 lemma (in sigma_algebra) lambda_system_caratheodory:
```
```   227   assumes oms: "outer_measure_space M f"
```
```   228       and A: "range A \<subseteq> lambda_system \<Omega> M f"
```
```   229       and disj: "disjoint_family A"
```
```   230   shows  "(\<Union>i. A i) \<in> lambda_system \<Omega> M f \<and> (\<Sum>i. f (A i)) = f (\<Union>i. A i)"
```
```   231 proof -
```
```   232   have pos: "positive M f" and inc: "increasing M f"
```
```   233    and csa: "countably_subadditive M f"
```
```   234     by (metis oms outer_measure_space_def)+
```
```   235   have sa: "subadditive M f"
```
```   236     by (metis countably_subadditive_subadditive csa pos)
```
```   237   have A': "\<And>S. A`S \<subseteq> (lambda_system \<Omega> M f)" using A
```
```   238     by auto
```
```   239   interpret ls: algebra \<Omega> "lambda_system \<Omega> M f"
```
```   240     using pos by (rule lambda_system_algebra)
```
```   241   have A'': "range A \<subseteq> M"
```
```   242      by (metis A image_subset_iff lambda_system_sets)
```
```   243
```
```   244   have U_in: "(\<Union>i. A i) \<in> M"
```
```   245     by (metis A'' countable_UN)
```
```   246   have U_eq: "f (\<Union>i. A i) = (\<Sum>i. f (A i))"
```
```   247   proof (rule antisym)
```
```   248     show "f (\<Union>i. A i) \<le> (\<Sum>i. f (A i))"
```
```   249       using csa[unfolded countably_subadditive_def] A'' disj U_in by auto
```
```   250     have *: "\<And>i. 0 \<le> f (A i)" using pos A'' unfolding positive_def by auto
```
```   251     have dis: "\<And>N. disjoint_family_on A {..<N}" by (intro disjoint_family_on_mono[OF _ disj]) auto
```
```   252     show "(\<Sum>i. f (A i)) \<le> f (\<Union>i. A i)"
```
```   253       using ls.additive_sum [OF lambda_system_positive[OF pos] lambda_system_additive _ A' dis] A''
```
```   254       by (intro suminf_bound[OF _ *]) (auto intro!: increasingD[OF inc] countable_UN)
```
```   255   qed
```
```   256   have "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) = f a"
```
```   257     if a [iff]: "a \<in> M" for a
```
```   258   proof (rule antisym)
```
```   259     have "range (\<lambda>i. a \<inter> A i) \<subseteq> M" using A''
```
```   260       by blast
```
```   261     moreover
```
```   262     have "disjoint_family (\<lambda>i. a \<inter> A i)" using disj
```
```   263       by (auto simp add: disjoint_family_on_def)
```
```   264     moreover
```
```   265     have "a \<inter> (\<Union>i. A i) \<in> M"
```
```   266       by (metis Int U_in a)
```
```   267     ultimately
```
```   268     have "f (a \<inter> (\<Union>i. A i)) \<le> (\<Sum>i. f (a \<inter> A i))"
```
```   269       using csa[unfolded countably_subadditive_def, rule_format, of "(\<lambda>i. a \<inter> A i)"]
```
```   270       by (simp add: o_def)
```
```   271     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))"
```
```   272       by (rule add_right_mono)
```
```   273     also have "\<dots> \<le> f a"
```
```   274     proof (intro suminf_bound_add allI)
```
```   275       fix n
```
```   276       have UNION_in: "(\<Union>i\<in>{0..<n}. A i) \<in> M"
```
```   277         by (metis A'' UNION_in_sets)
```
```   278       have le_fa: "f (UNION {0..<n} A \<inter> a) \<le> f a" using A''
```
```   279         by (blast intro: increasingD [OF inc] A'' UNION_in_sets)
```
```   280       have ls: "(\<Union>i\<in>{0..<n}. A i) \<in> lambda_system \<Omega> M f"
```
```   281         using ls.UNION_in_sets by (simp add: A)
```
```   282       hence eq_fa: "f a = f (a \<inter> (\<Union>i\<in>{0..<n}. A i)) + f (a - (\<Union>i\<in>{0..<n}. A i))"
```
```   283         by (simp add: lambda_system_eq UNION_in)
```
```   284       have "f (a - (\<Union>i. A i)) \<le> f (a - (\<Union>i\<in>{0..<n}. A i))"
```
```   285         by (blast intro: increasingD [OF inc] UNION_in U_in)
```
```   286       thus "(\<Sum>i<n. f (a \<inter> A i)) + f (a - (\<Union>i. A i)) \<le> f a"
```
```   287         by (simp add: lambda_system_strong_sum pos A disj eq_fa add_left_mono atLeast0LessThan[symmetric])
```
```   288     next
```
```   289       have "\<And>i. a \<inter> A i \<in> M" using A'' by auto
```
```   290       then show "\<And>i. 0 \<le> f (a \<inter> A i)" using pos[unfolded positive_def] by auto
```
```   291       have "\<And>i. a - (\<Union>i. A i) \<in> M" using A'' by auto
```
```   292       then have "\<And>i. 0 \<le> f (a - (\<Union>i. A i))" using pos[unfolded positive_def] by auto
```
```   293       then show "f (a - (\<Union>i. A i)) \<noteq> -\<infinity>" by auto
```
```   294     qed
```
```   295     finally show "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) \<le> f a" .
```
```   296   next
```
```   297     have "f a \<le> f (a \<inter> (\<Union>i. A i) \<union> (a - (\<Union>i. A i)))"
```
```   298       by (blast intro:  increasingD [OF inc] U_in)
```
```   299     also have "... \<le>  f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i))"
```
```   300       by (blast intro: subadditiveD [OF sa] U_in)
```
```   301     finally show "f a \<le> f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i))" .
```
```   302   qed
```
```   303   thus  ?thesis
```
```   304     by (simp add: lambda_system_eq sums_iff U_eq U_in)
```
```   305 qed
```
```   306
```
```   307 lemma (in sigma_algebra) caratheodory_lemma:
```
```   308   assumes oms: "outer_measure_space M f"
```
```   309   defines "L \<equiv> lambda_system \<Omega> M f"
```
```   310   shows "measure_space \<Omega> L f"
```
```   311 proof -
```
```   312   have pos: "positive M f"
```
```   313     by (metis oms outer_measure_space_def)
```
```   314   have alg: "algebra \<Omega> L"
```
```   315     using lambda_system_algebra [of f, OF pos]
```
```   316     by (simp add: algebra_iff_Un L_def)
```
```   317   then
```
```   318   have "sigma_algebra \<Omega> L"
```
```   319     using lambda_system_caratheodory [OF oms]
```
```   320     by (simp add: sigma_algebra_disjoint_iff L_def)
```
```   321   moreover
```
```   322   have "countably_additive L f" "positive L f"
```
```   323     using pos lambda_system_caratheodory [OF oms]
```
```   324     by (auto simp add: lambda_system_sets L_def countably_additive_def positive_def)
```
```   325   ultimately
```
```   326   show ?thesis
```
```   327     using pos by (simp add: measure_space_def)
```
```   328 qed
```
```   329
```
```   330 definition outer_measure :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> 'a set \<Rightarrow> ereal" where
```
```   331    "outer_measure M f X =
```
```   332      (INF A:{A. range A \<subseteq> M \<and> disjoint_family A \<and> X \<subseteq> (\<Union>i. A i)}. \<Sum>i. f (A i))"
```
```   333
```
```   334 lemma (in ring_of_sets) outer_measure_agrees:
```
```   335   assumes posf: "positive M f" and ca: "countably_additive M f" and s: "s \<in> M"
```
```   336   shows "outer_measure M f s = f s"
```
```   337   unfolding outer_measure_def
```
```   338 proof (safe intro!: antisym INF_greatest)
```
```   339   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)"
```
```   340   have inc: "increasing M f"
```
```   341     by (metis additive_increasing ca countably_additive_additive posf)
```
```   342   have "f s = f (\<Union>i. A i \<inter> s)"
```
```   343     using sA by (auto simp: Int_absorb1)
```
```   344   also have "\<dots> = (\<Sum>i. f (A i \<inter> s))"
```
```   345     using sA dA A s
```
```   346     by (intro ca[unfolded countably_additive_def, rule_format, symmetric])
```
```   347        (auto simp: Int_absorb1 disjoint_family_on_def)
```
```   348   also have "... \<le> (\<Sum>i. f (A i))"
```
```   349     using A s by (intro suminf_le_pos increasingD[OF inc] positiveD2[OF posf]) auto
```
```   350   finally show "f s \<le> (\<Sum>i. f (A i))" .
```
```   351 next
```
```   352   have "(\<Sum>i. f (if i = 0 then s else {})) \<le> f s"
```
```   353     using positiveD1[OF posf] by (subst suminf_finite[of "{0}"]) auto
```
```   354   with s show "(INF A:{A. range A \<subseteq> M \<and> disjoint_family A \<and> s \<subseteq> UNION UNIV A}. \<Sum>i. f (A i)) \<le> f s"
```
```   355     by (intro INF_lower2[of "\<lambda>i. if i = 0 then s else {}"])
```
```   356        (auto simp: disjoint_family_on_def)
```
```   357 qed
```
```   358
```
```   359 lemma outer_measure_nonneg: "positive M f \<Longrightarrow> 0 \<le> outer_measure M f X"
```
```   360   by (auto intro!: INF_greatest suminf_0_le intro: positiveD2 simp: outer_measure_def)
```
```   361
```
```   362 lemma outer_measure_empty:
```
```   363   assumes posf: "positive M f" and "{} \<in> M"
```
```   364   shows "outer_measure M f {} = 0"
```
```   365 proof (rule antisym)
```
```   366   show "outer_measure M f {} \<le> 0"
```
```   367     using assms by (auto intro!: INF_lower2[of "\<lambda>_. {}"] simp: outer_measure_def disjoint_family_on_def positive_def)
```
```   368 qed (intro outer_measure_nonneg posf)
```
```   369
```
```   370 lemma (in ring_of_sets) positive_outer_measure:
```
```   371   assumes "positive M f" shows "positive (Pow \<Omega>) (outer_measure M f)"
```
```   372   unfolding positive_def by (auto simp: assms outer_measure_empty outer_measure_nonneg)
```
```   373
```
```   374 lemma (in ring_of_sets) increasing_outer_measure: "increasing (Pow \<Omega>) (outer_measure M f)"
```
```   375   by (force simp: increasing_def outer_measure_def intro!: INF_greatest intro: INF_lower)
```
```   376
```
```   377 lemma (in ring_of_sets) outer_measure_le:
```
```   378   assumes pos: "positive M f" and inc: "increasing M f" and A: "range A \<subseteq> M" and X: "X \<subseteq> (\<Union>i. A i)"
```
```   379   shows "outer_measure M f X \<le> (\<Sum>i. f (A i))"
```
```   380   unfolding outer_measure_def
```
```   381 proof (safe intro!: INF_lower2[of "disjointed A"] del: subsetI)
```
```   382   show dA: "range (disjointed A) \<subseteq> M"
```
```   383     by (auto intro!: A range_disjointed_sets)
```
```   384   have "\<forall>n. f (disjointed A n) \<le> f (A n)"
```
```   385     by (metis increasingD [OF inc] UNIV_I dA image_subset_iff disjointed_subset A)
```
```   386   moreover have "\<forall>i. 0 \<le> f (disjointed A i)"
```
```   387     using pos dA unfolding positive_def by auto
```
```   388   ultimately show "(\<Sum>i. f (disjointed A i)) \<le> (\<Sum>i. f (A i))"
```
```   389     by (blast intro!: suminf_le_pos)
```
```   390 qed (auto simp: X UN_disjointed_eq disjoint_family_disjointed)
```
```   391
```
```   392 lemma (in ring_of_sets) outer_measure_close:
```
```   393   assumes posf: "positive M f" and "0 < e" and "outer_measure M f X \<noteq> \<infinity>"
```
```   394   shows "\<exists>A. range A \<subseteq> M \<and> disjoint_family A \<and> X \<subseteq> (\<Union>i. A i) \<and> (\<Sum>i. f (A i)) \<le> outer_measure M f X + e"
```
```   395 proof -
```
```   396   from \<open>outer_measure M f X \<noteq> \<infinity>\<close> have fin: "\<bar>outer_measure M f X\<bar> \<noteq> \<infinity>"
```
```   397     using outer_measure_nonneg[OF posf, of X] by auto
```
```   398   show ?thesis
```
```   399     using Inf_ereal_close [OF fin [unfolded outer_measure_def], OF \<open>0 < e\<close>]
```
```   400     by (auto intro: less_imp_le simp add: outer_measure_def)
```
```   401 qed
```
```   402
```
```   403 lemma (in ring_of_sets) countably_subadditive_outer_measure:
```
```   404   assumes posf: "positive M f" and inc: "increasing M f"
```
```   405   shows "countably_subadditive (Pow \<Omega>) (outer_measure M f)"
```
```   406 proof (simp add: countably_subadditive_def, safe)
```
```   407   fix A :: "nat \<Rightarrow> _" assume A: "range A \<subseteq> Pow (\<Omega>)" and sb: "(\<Union>i. A i) \<subseteq> \<Omega>"
```
```   408   let ?O = "outer_measure M f"
```
```   409
```
```   410   { fix e :: ereal assume e: "0 < e" and "\<forall>i. ?O (A i) \<noteq> \<infinity>"
```
```   411     hence "\<exists>B. \<forall>n. range (B n) \<subseteq> M \<and> disjoint_family (B n) \<and> A n \<subseteq> (\<Union>i. B n i) \<and>
```
```   412         (\<Sum>i. f (B n i)) \<le> ?O (A n) + e * (1/2)^(Suc n)"
```
```   413       using e sb by (auto intro!: choice outer_measure_close [of f, OF posf] simp: ereal_zero_less_0_iff one_ereal_def)
```
```   414     then obtain B
```
```   415       where B: "\<And>n. range (B n) \<subseteq> M"
```
```   416       and sbB: "\<And>n. A n \<subseteq> (\<Union>i. B n i)"
```
```   417       and Ble: "\<And>n. (\<Sum>i. f (B n i)) \<le> ?O (A n) + e * (1/2)^(Suc n)"
```
```   418       by auto blast
```
```   419
```
```   420     def C \<equiv> "case_prod B o prod_decode"
```
```   421     from B have B_in_M: "\<And>i j. B i j \<in> M"
```
```   422       by (rule range_subsetD)
```
```   423     then have C: "range C \<subseteq> M"
```
```   424       by (auto simp add: C_def split_def)
```
```   425     have A_C: "(\<Union>i. A i) \<subseteq> (\<Union>i. C i)"
```
```   426       using sbB by (auto simp add: C_def subset_eq) (metis prod.case prod_encode_inverse)
```
```   427
```
```   428     have "?O (\<Union>i. A i) \<le> ?O (\<Union>i. C i)"
```
```   429       using A_C A C by (intro increasing_outer_measure[THEN increasingD]) (auto dest!: sets_into_space)
```
```   430     also have "\<dots> \<le> (\<Sum>i. f (C i))"
```
```   431       using C by (intro outer_measure_le[OF posf inc]) auto
```
```   432     also have "\<dots> = (\<Sum>n. \<Sum>i. f (B n i))"
```
```   433       using B_in_M unfolding C_def comp_def by (intro suminf_ereal_2dimen positiveD2[OF posf]) auto
```
```   434     also have "\<dots> \<le> (\<Sum>n. ?O (A n) + e*(1/2) ^ Suc n)"
```
```   435       using B_in_M by (intro suminf_le_pos[OF Ble] suminf_0_le posf[THEN positiveD2]) auto
```
```   436     also have "... = (\<Sum>n. ?O (A n)) + (\<Sum>n. e*(1/2) ^ Suc n)"
```
```   437       using e by (subst suminf_add_ereal) (auto simp add: ereal_zero_le_0_iff outer_measure_nonneg posf)
```
```   438     also have "(\<Sum>n. e*(1/2) ^ Suc n) = e"
```
```   439       using suminf_half_series_ereal e by (simp add: ereal_zero_le_0_iff suminf_cmult_ereal)
```
```   440     finally have "?O (\<Union>i. A i) \<le> (\<Sum>n. ?O (A n)) + e" . }
```
```   441   note * = this
```
```   442
```
```   443   show "?O (\<Union>i. A i) \<le> (\<Sum>n. ?O (A n))"
```
```   444   proof cases
```
```   445     assume "\<forall>i. ?O (A i) \<noteq> \<infinity>" with * show ?thesis
```
```   446       by (intro ereal_le_epsilon) auto
```
```   447   qed (metis suminf_PInfty[OF outer_measure_nonneg, OF posf] ereal_less_eq(1))
```
```   448 qed
```
```   449
```
```   450 lemma (in ring_of_sets) outer_measure_space_outer_measure:
```
```   451   "positive M f \<Longrightarrow> increasing M f \<Longrightarrow> outer_measure_space (Pow \<Omega>) (outer_measure M f)"
```
```   452   by (simp add: outer_measure_space_def
```
```   453     positive_outer_measure increasing_outer_measure countably_subadditive_outer_measure)
```
```   454
```
```   455 lemma (in ring_of_sets) algebra_subset_lambda_system:
```
```   456   assumes posf: "positive M f" and inc: "increasing M f"
```
```   457       and add: "additive M f"
```
```   458   shows "M \<subseteq> lambda_system \<Omega> (Pow \<Omega>) (outer_measure M f)"
```
```   459 proof (auto dest: sets_into_space
```
```   460             simp add: algebra.lambda_system_eq [OF algebra_Pow])
```
```   461   fix x s assume x: "x \<in> M" and s: "s \<subseteq> \<Omega>"
```
```   462   have [simp]: "\<And>x. x \<in> M \<Longrightarrow> s \<inter> (\<Omega> - x) = s - x" using s
```
```   463     by blast
```
```   464   have "outer_measure M f (s \<inter> x) + outer_measure M f (s - x) \<le> outer_measure M f s"
```
```   465     unfolding outer_measure_def[of M f s]
```
```   466   proof (safe intro!: INF_greatest)
```
```   467     fix A :: "nat \<Rightarrow> 'a set" assume A: "disjoint_family A" "range A \<subseteq> M" "s \<subseteq> (\<Union>i. A i)"
```
```   468     have "outer_measure M f (s \<inter> x) \<le> (\<Sum>i. f (A i \<inter> x))"
```
```   469       unfolding outer_measure_def
```
```   470     proof (safe intro!: INF_lower2[of "\<lambda>i. A i \<inter> x"])
```
```   471       from A(1) show "disjoint_family (\<lambda>i. A i \<inter> x)"
```
```   472         by (rule disjoint_family_on_bisimulation) auto
```
```   473     qed (insert x A, auto)
```
```   474     moreover
```
```   475     have "outer_measure M f (s - x) \<le> (\<Sum>i. f (A i - x))"
```
```   476       unfolding outer_measure_def
```
```   477     proof (safe intro!: INF_lower2[of "\<lambda>i. A i - x"])
```
```   478       from A(1) show "disjoint_family (\<lambda>i. A i - x)"
```
```   479         by (rule disjoint_family_on_bisimulation) auto
```
```   480     qed (insert x A, auto)
```
```   481     ultimately have "outer_measure M f (s \<inter> x) + outer_measure M f (s - x) \<le>
```
```   482         (\<Sum>i. f (A i \<inter> x)) + (\<Sum>i. f (A i - x))" by (rule add_mono)
```
```   483     also have "\<dots> = (\<Sum>i. f (A i \<inter> x) + f (A i - x))"
```
```   484       using A(2) x posf by (subst suminf_add_ereal) (auto simp: positive_def)
```
```   485     also have "\<dots> = (\<Sum>i. f (A i))"
```
```   486       using A x
```
```   487       by (subst add[THEN additiveD, symmetric])
```
```   488          (auto intro!: arg_cong[where f=suminf] arg_cong[where f=f])
```
```   489     finally show "outer_measure M f (s \<inter> x) + outer_measure M f (s - x) \<le> (\<Sum>i. f (A i))" .
```
```   490   qed
```
```   491   moreover
```
```   492   have "outer_measure M f s \<le> outer_measure M f (s \<inter> x) + outer_measure M f (s - x)"
```
```   493   proof -
```
```   494     have "outer_measure M f s = outer_measure M f ((s \<inter> x) \<union> (s - x))"
```
```   495       by (metis Un_Diff_Int Un_commute)
```
```   496     also have "... \<le> outer_measure M f (s \<inter> x) + outer_measure M f (s - x)"
```
```   497       apply (rule subadditiveD)
```
```   498       apply (rule ring_of_sets.countably_subadditive_subadditive [OF ring_of_sets_Pow])
```
```   499       apply (simp add: positive_def outer_measure_empty[OF posf] outer_measure_nonneg[OF posf])
```
```   500       apply (rule countably_subadditive_outer_measure)
```
```   501       using s by (auto intro!: posf inc)
```
```   502     finally show ?thesis .
```
```   503   qed
```
```   504   ultimately
```
```   505   show "outer_measure M f (s \<inter> x) + outer_measure M f (s - x) = outer_measure M f s"
```
```   506     by (rule order_antisym)
```
```   507 qed
```
```   508
```
```   509 lemma measure_down: "measure_space \<Omega> N \<mu> \<Longrightarrow> sigma_algebra \<Omega> M \<Longrightarrow> M \<subseteq> N \<Longrightarrow> measure_space \<Omega> M \<mu>"
```
```   510   by (auto simp add: measure_space_def positive_def countably_additive_def subset_eq)
```
```   511
```
```   512 subsection \<open>Caratheodory's theorem\<close>
```
```   513
```
```   514 theorem (in ring_of_sets) caratheodory':
```
```   515   assumes posf: "positive M f" and ca: "countably_additive M f"
```
```   516   shows "\<exists>\<mu> :: 'a set \<Rightarrow> ereal. (\<forall>s \<in> M. \<mu> s = f s) \<and> measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"
```
```   517 proof -
```
```   518   have inc: "increasing M f"
```
```   519     by (metis additive_increasing ca countably_additive_additive posf)
```
```   520   let ?O = "outer_measure M f"
```
```   521   def ls \<equiv> "lambda_system \<Omega> (Pow \<Omega>) ?O"
```
```   522   have mls: "measure_space \<Omega> ls ?O"
```
```   523     using sigma_algebra.caratheodory_lemma
```
```   524             [OF sigma_algebra_Pow outer_measure_space_outer_measure [OF posf inc]]
```
```   525     by (simp add: ls_def)
```
```   526   hence sls: "sigma_algebra \<Omega> ls"
```
```   527     by (simp add: measure_space_def)
```
```   528   have "M \<subseteq> ls"
```
```   529     by (simp add: ls_def)
```
```   530        (metis ca posf inc countably_additive_additive algebra_subset_lambda_system)
```
```   531   hence sgs_sb: "sigma_sets (\<Omega>) (M) \<subseteq> ls"
```
```   532     using sigma_algebra.sigma_sets_subset [OF sls, of "M"]
```
```   533     by simp
```
```   534   have "measure_space \<Omega> (sigma_sets \<Omega> M) ?O"
```
```   535     by (rule measure_down [OF mls], rule sigma_algebra_sigma_sets)
```
```   536        (simp_all add: sgs_sb space_closed)
```
```   537   thus ?thesis using outer_measure_agrees [OF posf ca]
```
```   538     by (intro exI[of _ ?O]) auto
```
```   539 qed
```
```   540
```
```   541 lemma (in ring_of_sets) caratheodory_empty_continuous:
```
```   542   assumes f: "positive M f" "additive M f" and fin: "\<And>A. A \<in> M \<Longrightarrow> f A \<noteq> \<infinity>"
```
```   543   assumes cont: "\<And>A. range A \<subseteq> M \<Longrightarrow> decseq A \<Longrightarrow> (\<Inter>i. A i) = {} \<Longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> 0"
```
```   544   shows "\<exists>\<mu> :: 'a set \<Rightarrow> ereal. (\<forall>s \<in> M. \<mu> s = f s) \<and> measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"
```
```   545 proof (intro caratheodory' empty_continuous_imp_countably_additive f)
```
```   546   show "\<forall>A\<in>M. f A \<noteq> \<infinity>" using fin by auto
```
```   547 qed (rule cont)
```
```   548
```
```   549 subsection \<open>Volumes\<close>
```
```   550
```
```   551 definition volume :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
```
```   552   "volume M f \<longleftrightarrow>
```
```   553   (f {} = 0) \<and> (\<forall>a\<in>M. 0 \<le> f a) \<and>
```
```   554   (\<forall>C\<subseteq>M. disjoint C \<longrightarrow> finite C \<longrightarrow> \<Union>C \<in> M \<longrightarrow> f (\<Union>C) = (\<Sum>c\<in>C. f c))"
```
```   555
```
```   556 lemma volumeI:
```
```   557   assumes "f {} = 0"
```
```   558   assumes "\<And>a. a \<in> M \<Longrightarrow> 0 \<le> f a"
```
```   559   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)"
```
```   560   shows "volume M f"
```
```   561   using assms by (auto simp: volume_def)
```
```   562
```
```   563 lemma volume_positive:
```
```   564   "volume M f \<Longrightarrow> a \<in> M \<Longrightarrow> 0 \<le> f a"
```
```   565   by (auto simp: volume_def)
```
```   566
```
```   567 lemma volume_empty:
```
```   568   "volume M f \<Longrightarrow> f {} = 0"
```
```   569   by (auto simp: volume_def)
```
```   570
```
```   571 lemma volume_finite_additive:
```
```   572   assumes "volume M f"
```
```   573   assumes A: "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M" "disjoint_family_on A I" "finite I" "UNION I A \<in> M"
```
```   574   shows "f (UNION I A) = (\<Sum>i\<in>I. f (A i))"
```
```   575 proof -
```
```   576   have "A`I \<subseteq> M" "disjoint (A`I)" "finite (A`I)" "\<Union>(A`I) \<in> M"
```
```   577     using A by (auto simp: disjoint_family_on_disjoint_image)
```
```   578   with \<open>volume M f\<close> have "f (\<Union>(A`I)) = (\<Sum>a\<in>A`I. f a)"
```
```   579     unfolding volume_def by blast
```
```   580   also have "\<dots> = (\<Sum>i\<in>I. f (A i))"
```
```   581   proof (subst setsum.reindex_nontrivial)
```
```   582     fix i j assume "i \<in> I" "j \<in> I" "i \<noteq> j" "A i = A j"
```
```   583     with \<open>disjoint_family_on A I\<close> have "A i = {}"
```
```   584       by (auto simp: disjoint_family_on_def)
```
```   585     then show "f (A i) = 0"
```
```   586       using volume_empty[OF \<open>volume M f\<close>] by simp
```
```   587   qed (auto intro: \<open>finite I\<close>)
```
```   588   finally show "f (UNION I A) = (\<Sum>i\<in>I. f (A i))"
```
```   589     by simp
```
```   590 qed
```
```   591
```
```   592 lemma (in ring_of_sets) volume_additiveI:
```
```   593   assumes pos: "\<And>a. a \<in> M \<Longrightarrow> 0 \<le> \<mu> a"
```
```   594   assumes [simp]: "\<mu> {} = 0"
```
```   595   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"
```
```   596   shows "volume M \<mu>"
```
```   597 proof (unfold volume_def, safe)
```
```   598   fix C assume "finite C" "C \<subseteq> M" "disjoint C"
```
```   599   then show "\<mu> (\<Union>C) = setsum \<mu> C"
```
```   600   proof (induct C)
```
```   601     case (insert c C)
```
```   602     from insert(1,2,4,5) have "\<mu> (\<Union>insert c C) = \<mu> c + \<mu> (\<Union>C)"
```
```   603       by (auto intro!: add simp: disjoint_def)
```
```   604     with insert show ?case
```
```   605       by (simp add: disjoint_def)
```
```   606   qed simp
```
```   607 qed fact+
```
```   608
```
```   609 lemma (in semiring_of_sets) extend_volume:
```
```   610   assumes "volume M \<mu>"
```
```   611   shows "\<exists>\<mu>'. volume generated_ring \<mu>' \<and> (\<forall>a\<in>M. \<mu>' a = \<mu> a)"
```
```   612 proof -
```
```   613   let ?R = generated_ring
```
```   614   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)"
```
```   615     by (auto simp: generated_ring_def)
```
```   616   from bchoice[OF this] guess \<mu>' .. note \<mu>'_spec = this
```
```   617
```
```   618   { fix C assume C: "C \<subseteq> M" "finite C" "disjoint C"
```
```   619     fix D assume D: "D \<subseteq> M" "finite D" "disjoint D"
```
```   620     assume "\<Union>C = \<Union>D"
```
```   621     have "(\<Sum>d\<in>D. \<mu> d) = (\<Sum>d\<in>D. \<Sum>c\<in>C. \<mu> (c \<inter> d))"
```
```   622     proof (intro setsum.cong refl)
```
```   623       fix d assume "d \<in> D"
```
```   624       have Un_eq_d: "(\<Union>c\<in>C. c \<inter> d) = d"
```
```   625         using \<open>d \<in> D\<close> \<open>\<Union>C = \<Union>D\<close> by auto
```
```   626       moreover have "\<mu> (\<Union>c\<in>C. c \<inter> d) = (\<Sum>c\<in>C. \<mu> (c \<inter> d))"
```
```   627       proof (rule volume_finite_additive)
```
```   628         { fix c assume "c \<in> C" then show "c \<inter> d \<in> M"
```
```   629             using C D \<open>d \<in> D\<close> by auto }
```
```   630         show "(\<Union>a\<in>C. a \<inter> d) \<in> M"
```
```   631           unfolding Un_eq_d using \<open>d \<in> D\<close> D by auto
```
```   632         show "disjoint_family_on (\<lambda>a. a \<inter> d) C"
```
```   633           using \<open>disjoint C\<close> by (auto simp: disjoint_family_on_def disjoint_def)
```
```   634       qed fact+
```
```   635       ultimately show "\<mu> d = (\<Sum>c\<in>C. \<mu> (c \<inter> d))" by simp
```
```   636     qed }
```
```   637   note split_sum = this
```
```   638
```
```   639   { fix C assume C: "C \<subseteq> M" "finite C" "disjoint C"
```
```   640     fix D assume D: "D \<subseteq> M" "finite D" "disjoint D"
```
```   641     assume "\<Union>C = \<Union>D"
```
```   642     with split_sum[OF C D] split_sum[OF D C]
```
```   643     have "(\<Sum>d\<in>D. \<mu> d) = (\<Sum>c\<in>C. \<mu> c)"
```
```   644       by (simp, subst setsum.commute, simp add: ac_simps) }
```
```   645   note sum_eq = this
```
```   646
```
```   647   { fix C assume C: "C \<subseteq> M" "finite C" "disjoint C"
```
```   648     then have "\<Union>C \<in> ?R" by (auto simp: generated_ring_def)
```
```   649     with \<mu>'_spec[THEN bspec, of "\<Union>C"]
```
```   650     obtain D where
```
```   651       D: "D \<subseteq> M" "finite D" "disjoint D" "\<Union>C = \<Union>D" and "\<mu>' (\<Union>C) = (\<Sum>d\<in>D. \<mu> d)"
```
```   652       by auto
```
```   653     with sum_eq[OF C D] have "\<mu>' (\<Union>C) = (\<Sum>c\<in>C. \<mu> c)" by simp }
```
```   654   note \<mu>' = this
```
```   655
```
```   656   show ?thesis
```
```   657   proof (intro exI conjI ring_of_sets.volume_additiveI[OF generating_ring] ballI)
```
```   658     fix a assume "a \<in> M" with \<mu>'[of "{a}"] show "\<mu>' a = \<mu> a"
```
```   659       by (simp add: disjoint_def)
```
```   660   next
```
```   661     fix a assume "a \<in> ?R" then guess Ca .. note Ca = this
```
```   662     with \<mu>'[of Ca] \<open>volume M \<mu>\<close>[THEN volume_positive]
```
```   663     show "0 \<le> \<mu>' a"
```
```   664       by (auto intro!: setsum_nonneg)
```
```   665   next
```
```   666     show "\<mu>' {} = 0" using \<mu>'[of "{}"] by auto
```
```   667   next
```
```   668     fix a assume "a \<in> ?R" then guess Ca .. note Ca = this
```
```   669     fix b assume "b \<in> ?R" then guess Cb .. note Cb = this
```
```   670     assume "a \<inter> b = {}"
```
```   671     with Ca Cb have "Ca \<inter> Cb \<subseteq> {{}}" by auto
```
```   672     then have C_Int_cases: "Ca \<inter> Cb = {{}} \<or> Ca \<inter> Cb = {}" by auto
```
```   673
```
```   674     from \<open>a \<inter> b = {}\<close> have "\<mu>' (\<Union>(Ca \<union> Cb)) = (\<Sum>c\<in>Ca \<union> Cb. \<mu> c)"
```
```   675       using Ca Cb by (intro \<mu>') (auto intro!: disjoint_union)
```
```   676     also have "\<dots> = (\<Sum>c\<in>Ca \<union> Cb. \<mu> c) + (\<Sum>c\<in>Ca \<inter> Cb. \<mu> c)"
```
```   677       using C_Int_cases volume_empty[OF \<open>volume M \<mu>\<close>] by (elim disjE) simp_all
```
```   678     also have "\<dots> = (\<Sum>c\<in>Ca. \<mu> c) + (\<Sum>c\<in>Cb. \<mu> c)"
```
```   679       using Ca Cb by (simp add: setsum.union_inter)
```
```   680     also have "\<dots> = \<mu>' a + \<mu>' b"
```
```   681       using Ca Cb by (simp add: \<mu>')
```
```   682     finally show "\<mu>' (a \<union> b) = \<mu>' a + \<mu>' b"
```
```   683       using Ca Cb by simp
```
```   684   qed
```
```   685 qed
```
```   686
```
```   687 subsubsection \<open>Caratheodory on semirings\<close>
```
```   688
```
```   689 theorem (in semiring_of_sets) caratheodory:
```
```   690   assumes pos: "positive M \<mu>" and ca: "countably_additive M \<mu>"
```
```   691   shows "\<exists>\<mu>' :: 'a set \<Rightarrow> ereal. (\<forall>s \<in> M. \<mu>' s = \<mu> s) \<and> measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>'"
```
```   692 proof -
```
```   693   have "volume M \<mu>"
```
```   694   proof (rule volumeI)
```
```   695     { fix a assume "a \<in> M" then show "0 \<le> \<mu> a"
```
```   696         using pos unfolding positive_def by auto }
```
```   697     note p = this
```
```   698
```
```   699     fix C assume sets_C: "C \<subseteq> M" "\<Union>C \<in> M" and "disjoint C" "finite C"
```
```   700     have "\<exists>F'. bij_betw F' {..<card C} C"
```
```   701       by (rule finite_same_card_bij[OF _ \<open>finite C\<close>]) auto
```
```   702     then guess F' .. note F' = this
```
```   703     then have F': "C = F' ` {..< card C}" "inj_on F' {..< card C}"
```
```   704       by (auto simp: bij_betw_def)
```
```   705     { fix i j assume *: "i < card C" "j < card C" "i \<noteq> j"
```
```   706       with F' have "F' i \<in> C" "F' j \<in> C" "F' i \<noteq> F' j"
```
```   707         unfolding inj_on_def by auto
```
```   708       with \<open>disjoint C\<close>[THEN disjointD]
```
```   709       have "F' i \<inter> F' j = {}"
```
```   710         by auto }
```
```   711     note F'_disj = this
```
```   712     def F \<equiv> "\<lambda>i. if i < card C then F' i else {}"
```
```   713     then have "disjoint_family F"
```
```   714       using F'_disj by (auto simp: disjoint_family_on_def)
```
```   715     moreover from F' have "(\<Union>i. F i) = \<Union>C"
```
```   716       by (auto simp add: F_def split: split_if_asm) blast
```
```   717     moreover have sets_F: "\<And>i. F i \<in> M"
```
```   718       using F' sets_C by (auto simp: F_def)
```
```   719     moreover note sets_C
```
```   720     ultimately have "\<mu> (\<Union>C) = (\<Sum>i. \<mu> (F i))"
```
```   721       using ca[unfolded countably_additive_def, THEN spec, of F] by auto
```
```   722     also have "\<dots> = (\<Sum>i<card C. \<mu> (F' i))"
```
```   723     proof -
```
```   724       have "(\<lambda>i. if i \<in> {..< card C} then \<mu> (F' i) else 0) sums (\<Sum>i<card C. \<mu> (F' i))"
```
```   725         by (rule sums_If_finite_set) auto
```
```   726       also have "(\<lambda>i. if i \<in> {..< card C} then \<mu> (F' i) else 0) = (\<lambda>i. \<mu> (F i))"
```
```   727         using pos by (auto simp: positive_def F_def)
```
```   728       finally show "(\<Sum>i. \<mu> (F i)) = (\<Sum>i<card C. \<mu> (F' i))"
```
```   729         by (simp add: sums_iff)
```
```   730     qed
```
```   731     also have "\<dots> = (\<Sum>c\<in>C. \<mu> c)"
```
```   732       using F'(2) by (subst (2) F') (simp add: setsum.reindex)
```
```   733     finally show "\<mu> (\<Union>C) = (\<Sum>c\<in>C. \<mu> c)" .
```
```   734   next
```
```   735     show "\<mu> {} = 0"
```
```   736       using \<open>positive M \<mu>\<close> by (rule positiveD1)
```
```   737   qed
```
```   738   from extend_volume[OF this] obtain \<mu>_r where
```
```   739     V: "volume generated_ring \<mu>_r" "\<And>a. a \<in> M \<Longrightarrow> \<mu> a = \<mu>_r a"
```
```   740     by auto
```
```   741
```
```   742   interpret G: ring_of_sets \<Omega> generated_ring
```
```   743     by (rule generating_ring)
```
```   744
```
```   745   have pos: "positive generated_ring \<mu>_r"
```
```   746     using V unfolding positive_def by (auto simp: positive_def intro!: volume_positive volume_empty)
```
```   747
```
```   748   have "countably_additive generated_ring \<mu>_r"
```
```   749   proof (rule countably_additiveI)
```
```   750     fix A' :: "nat \<Rightarrow> 'a set" assume A': "range A' \<subseteq> generated_ring" "disjoint_family A'"
```
```   751       and Un_A: "(\<Union>i. A' i) \<in> generated_ring"
```
```   752
```
```   753     from generated_ringE[OF Un_A] guess C' . note C' = this
```
```   754
```
```   755     { fix c assume "c \<in> C'"
```
```   756       moreover def A \<equiv> "\<lambda>i. A' i \<inter> c"
```
```   757       ultimately have A: "range A \<subseteq> generated_ring" "disjoint_family A"
```
```   758         and Un_A: "(\<Union>i. A i) \<in> generated_ring"
```
```   759         using A' C'
```
```   760         by (auto intro!: G.Int G.finite_Union intro: generated_ringI_Basic simp: disjoint_family_on_def)
```
```   761       from A C' \<open>c \<in> C'\<close> have UN_eq: "(\<Union>i. A i) = c"
```
```   762         by (auto simp: A_def)
```
```   763
```
```   764       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)"
```
```   765         (is "\<forall>i. ?P i")
```
```   766       proof
```
```   767         fix i
```
```   768         from A have Ai: "A i \<in> generated_ring" by auto
```
```   769         from generated_ringE[OF this] guess C . note C = this
```
```   770
```
```   771         have "\<exists>F'. bij_betw F' {..<card C} C"
```
```   772           by (rule finite_same_card_bij[OF _ \<open>finite C\<close>]) auto
```
```   773         then guess F .. note F = this
```
```   774         def f \<equiv> "\<lambda>i. if i < card C then F i else {}"
```
```   775         then have f: "bij_betw f {..< card C} C"
```
```   776           by (intro bij_betw_cong[THEN iffD1, OF _ F]) auto
```
```   777         with C have "\<forall>j. f j \<in> M"
```
```   778           by (auto simp: Pi_iff f_def dest!: bij_betw_imp_funcset)
```
```   779         moreover
```
```   780         from f C have d_f: "disjoint_family_on f {..<card C}"
```
```   781           by (intro disjoint_image_disjoint_family_on) (auto simp: bij_betw_def)
```
```   782         then have "disjoint_family f"
```
```   783           by (auto simp: disjoint_family_on_def f_def)
```
```   784         moreover
```
```   785         have Ai_eq: "A i = (\<Union>x<card C. f x)"
```
```   786           using f C Ai unfolding bij_betw_def by auto
```
```   787         then have "\<Union>range f = A i"
```
```   788           using f C Ai unfolding bij_betw_def
```
```   789             by (auto simp add: f_def cong del: strong_SUP_cong)
```
```   790         moreover
```
```   791         { have "(\<Sum>j. \<mu>_r (f j)) = (\<Sum>j. if j \<in> {..< card C} then \<mu>_r (f j) else 0)"
```
```   792             using volume_empty[OF V(1)] by (auto intro!: arg_cong[where f=suminf] simp: f_def)
```
```   793           also have "\<dots> = (\<Sum>j<card C. \<mu>_r (f j))"
```
```   794             by (rule sums_If_finite_set[THEN sums_unique, symmetric]) simp
```
```   795           also have "\<dots> = \<mu>_r (A i)"
```
```   796             using C f[THEN bij_betw_imp_funcset] unfolding Ai_eq
```
```   797             by (intro volume_finite_additive[OF V(1) _ d_f, symmetric])
```
```   798                (auto simp: Pi_iff Ai_eq intro: generated_ringI_Basic)
```
```   799           finally have "\<mu>_r (A i) = (\<Sum>j. \<mu>_r (f j))" .. }
```
```   800         ultimately show "?P i"
```
```   801           by blast
```
```   802       qed
```
```   803       from choice[OF this] guess f .. note f = this
```
```   804       then have UN_f_eq: "(\<Union>i. case_prod f (prod_decode i)) = (\<Union>i. A i)"
```
```   805         unfolding UN_extend_simps surj_prod_decode by (auto simp: set_eq_iff)
```
```   806
```
```   807       have d: "disjoint_family (\<lambda>i. case_prod f (prod_decode i))"
```
```   808         unfolding disjoint_family_on_def
```
```   809       proof (intro ballI impI)
```
```   810         fix m n :: nat assume "m \<noteq> n"
```
```   811         then have neq: "prod_decode m \<noteq> prod_decode n"
```
```   812           using inj_prod_decode[of UNIV] by (auto simp: inj_on_def)
```
```   813         show "case_prod f (prod_decode m) \<inter> case_prod f (prod_decode n) = {}"
```
```   814         proof cases
```
```   815           assume "fst (prod_decode m) = fst (prod_decode n)"
```
```   816           then show ?thesis
```
```   817             using neq f by (fastforce simp: disjoint_family_on_def)
```
```   818         next
```
```   819           assume neq: "fst (prod_decode m) \<noteq> fst (prod_decode n)"
```
```   820           have "case_prod f (prod_decode m) \<subseteq> A (fst (prod_decode m))"
```
```   821             "case_prod f (prod_decode n) \<subseteq> A (fst (prod_decode n))"
```
```   822             using f[THEN spec, of "fst (prod_decode m)"]
```
```   823             using f[THEN spec, of "fst (prod_decode n)"]
```
```   824             by (auto simp: set_eq_iff)
```
```   825           with f A neq show ?thesis
```
```   826             by (fastforce simp: disjoint_family_on_def subset_eq set_eq_iff)
```
```   827         qed
```
```   828       qed
```
```   829       from f have "(\<Sum>n. \<mu>_r (A n)) = (\<Sum>n. \<mu>_r (case_prod f (prod_decode n)))"
```
```   830         by (intro suminf_ereal_2dimen[symmetric] positiveD2[OF pos] generated_ringI_Basic)
```
```   831          (auto split: prod.split)
```
```   832       also have "\<dots> = (\<Sum>n. \<mu> (case_prod f (prod_decode n)))"
```
```   833         using f V(2) by (auto intro!: arg_cong[where f=suminf] split: prod.split)
```
```   834       also have "\<dots> = \<mu> (\<Union>i. case_prod f (prod_decode i))"
```
```   835         using f \<open>c \<in> C'\<close> C'
```
```   836         by (intro ca[unfolded countably_additive_def, rule_format])
```
```   837            (auto split: prod.split simp: UN_f_eq d UN_eq)
```
```   838       finally have "(\<Sum>n. \<mu>_r (A' n \<inter> c)) = \<mu> c"
```
```   839         using UN_f_eq UN_eq by (simp add: A_def) }
```
```   840     note eq = this
```
```   841
```
```   842     have "(\<Sum>n. \<mu>_r (A' n)) = (\<Sum>n. \<Sum>c\<in>C'. \<mu>_r (A' n \<inter> c))"
```
```   843       using C' A'
```
```   844       by (subst volume_finite_additive[symmetric, OF V(1)])
```
```   845          (auto simp: disjoint_def disjoint_family_on_def
```
```   846                intro!: G.Int G.finite_Union arg_cong[where f="\<lambda>X. suminf (\<lambda>i. \<mu>_r (X i))"] ext
```
```   847                intro: generated_ringI_Basic)
```
```   848     also have "\<dots> = (\<Sum>c\<in>C'. \<Sum>n. \<mu>_r (A' n \<inter> c))"
```
```   849       using C' A'
```
```   850       by (intro suminf_setsum_ereal positiveD2[OF pos] G.Int G.finite_Union)
```
```   851          (auto intro: generated_ringI_Basic)
```
```   852     also have "\<dots> = (\<Sum>c\<in>C'. \<mu>_r c)"
```
```   853       using eq V C' by (auto intro!: setsum.cong)
```
```   854     also have "\<dots> = \<mu>_r (\<Union>C')"
```
```   855       using C' Un_A
```
```   856       by (subst volume_finite_additive[symmetric, OF V(1)])
```
```   857          (auto simp: disjoint_family_on_def disjoint_def
```
```   858                intro: generated_ringI_Basic)
```
```   859     finally show "(\<Sum>n. \<mu>_r (A' n)) = \<mu>_r (\<Union>i. A' i)"
```
```   860       using C' by simp
```
```   861   qed
```
```   862   from G.caratheodory'[OF \<open>positive generated_ring \<mu>_r\<close> \<open>countably_additive generated_ring \<mu>_r\<close>]
```
```   863   guess \<mu>' ..
```
```   864   with V show ?thesis
```
```   865     unfolding sigma_sets_generated_ring_eq
```
```   866     by (intro exI[of _ \<mu>']) (auto intro: generated_ringI_Basic)
```
```   867 qed
```
```   868
```
```   869 lemma extend_measure_caratheodory:
```
```   870   fixes G :: "'i \<Rightarrow> 'a set"
```
```   871   assumes M: "M = extend_measure \<Omega> I G \<mu>"
```
```   872   assumes "i \<in> I"
```
```   873   assumes "semiring_of_sets \<Omega> (G ` I)"
```
```   874   assumes empty: "\<And>i. i \<in> I \<Longrightarrow> G i = {} \<Longrightarrow> \<mu> i = 0"
```
```   875   assumes inj: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> G i = G j \<Longrightarrow> \<mu> i = \<mu> j"
```
```   876   assumes nonneg: "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> \<mu> i"
```
```   877   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>
```
```   878     (\<Union>i. G (A i)) = G j \<Longrightarrow> (\<Sum>n. \<mu> (A n)) = \<mu> j"
```
```   879   shows "emeasure M (G i) = \<mu> i"
```
```   880 proof -
```
```   881   interpret semiring_of_sets \<Omega> "G ` I"
```
```   882     by fact
```
```   883   have "\<forall>g\<in>G`I. \<exists>i\<in>I. g = G i"
```
```   884     by auto
```
```   885   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"
```
```   886     by metis
```
```   887
```
```   888   have "\<exists>\<mu>'. (\<forall>s\<in>G ` I. \<mu>' s = \<mu> (sel s)) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G ` I)) \<mu>'"
```
```   889   proof (rule caratheodory)
```
```   890     show "positive (G ` I) (\<lambda>s. \<mu> (sel s))"
```
```   891       by (auto simp: positive_def intro!: empty sel nonneg)
```
```   892     show "countably_additive (G ` I) (\<lambda>s. \<mu> (sel s))"
```
```   893     proof (rule countably_additiveI)
```
```   894       fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> G ` I" "disjoint_family A" "(\<Union>i. A i) \<in> G ` I"
```
```   895       then show "(\<Sum>i. \<mu> (sel (A i))) = \<mu> (sel (\<Union>i. A i))"
```
```   896         by (intro add) (auto simp: sel image_subset_iff_funcset comp_def Pi_iff intro!: sel)
```
```   897     qed
```
```   898   qed
```
```   899   then obtain \<mu>' where \<mu>': "\<forall>s\<in>G ` I. \<mu>' s = \<mu> (sel s)" "measure_space \<Omega> (sigma_sets \<Omega> (G ` I)) \<mu>'"
```
```   900     by metis
```
```   901
```
```   902   show ?thesis
```
```   903   proof (rule emeasure_extend_measure[OF M])
```
```   904     { fix i assume "i \<in> I" then show "\<mu>' (G i) = \<mu> i"
```
```   905       using \<mu>' by (auto intro!: inj sel) }
```
```   906     show "G ` I \<subseteq> Pow \<Omega>"
```
```   907       by fact
```
```   908     then show "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'"
```
```   909       using \<mu>' by (simp_all add: M sets_extend_measure measure_space_def)
```
```   910   qed fact
```
```   911 qed
```
```   912
```
```   913 lemma extend_measure_caratheodory_pair:
```
```   914   fixes G :: "'i \<Rightarrow> 'j \<Rightarrow> 'a set"
```
```   915   assumes M: "M = extend_measure \<Omega> {(a, b). P a b} (\<lambda>(a, b). G a b) (\<lambda>(a, b). \<mu> a b)"
```
```   916   assumes "P i j"
```
```   917   assumes semiring: "semiring_of_sets \<Omega> {G a b | a b. P a b}"
```
```   918   assumes empty: "\<And>i j. P i j \<Longrightarrow> G i j = {} \<Longrightarrow> \<mu> i j = 0"
```
```   919   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"
```
```   920   assumes nonneg: "\<And>i j. P i j \<Longrightarrow> 0 \<le> \<mu> i j"
```
```   921   assumes add: "\<And>A::nat \<Rightarrow> 'i. \<And>B::nat \<Rightarrow> 'j. \<And>j k.
```
```   922     (\<And>n. P (A n) (B n)) \<Longrightarrow> P j k \<Longrightarrow> disjoint_family (\<lambda>n. G (A n) (B n)) \<Longrightarrow>
```
```   923     (\<Union>i. G (A i) (B i)) = G j k \<Longrightarrow> (\<Sum>n. \<mu> (A n) (B n)) = \<mu> j k"
```
```   924   shows "emeasure M (G i j) = \<mu> i j"
```
```   925 proof -
```
```   926   have "emeasure M ((\<lambda>(a, b). G a b) (i, j)) = (\<lambda>(a, b). \<mu> a b) (i, j)"
```
```   927   proof (rule extend_measure_caratheodory[OF M])
```
```   928     show "semiring_of_sets \<Omega> ((\<lambda>(a, b). G a b) ` {(a, b). P a b})"
```
```   929       using semiring by (simp add: image_def conj_commute)
```
```   930   next
```
```   931     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}"
```
```   932       "disjoint_family ((\<lambda>(a, b). G a b) \<circ> A)"
```
```   933       "(\<Union>i. case A i of (a, b) \<Rightarrow> G a b) = (case j of (a, b) \<Rightarrow> G a b)"
```
```   934     then show "(\<Sum>n. case A n of (a, b) \<Rightarrow> \<mu> a b) = (case j of (a, b) \<Rightarrow> \<mu> a b)"
```
```   935       using add[of "\<lambda>i. fst (A i)" "\<lambda>i. snd (A i)" "fst j" "snd j"]
```
```   936       by (simp add: split_beta' comp_def Pi_iff)
```
```   937   qed (auto split: prod.splits intro: assms)
```
```   938   then show ?thesis by simp
```
```   939 qed
```
```   940
```
```   941 end
```