src/HOL/Probability/Caratheodory.thy
 author wenzelm Mon Nov 09 19:42:33 2009 +0100 (2009-11-09) changeset 33536 fd28b7399f2b parent 33271 7be66dee1a5a child 35582 b16d99a72dc9 permissions -rw-r--r--
eliminated hard tabulators;
```     1 header {*Caratheodory Extension Theorem*}
```
```     2
```
```     3 theory Caratheodory
```
```     4   imports Sigma_Algebra SupInf SeriesPlus
```
```     5 begin
```
```     6
```
```     7 text{*From the Hurd/Coble measure theory development, translated by Lawrence Paulson.*}
```
```     8
```
```     9 subsection {* Measure Spaces *}
```
```    10
```
```    11 text {*A measure assigns a nonnegative real to every measurable set.
```
```    12        It is countably additive for disjoint sets.*}
```
```    13
```
```    14 record 'a measure_space = "'a algebra" +
```
```    15   measure:: "'a set \<Rightarrow> real"
```
```    16
```
```    17 definition
```
```    18   disjoint_family  where
```
```    19   "disjoint_family A \<longleftrightarrow> (\<forall>m n. m \<noteq> n \<longrightarrow> A m \<inter> A n = {})"
```
```    20
```
```    21 definition
```
```    22   positive  where
```
```    23   "positive M f \<longleftrightarrow> f {} = (0::real) & (\<forall>x \<in> sets M. 0 \<le> f x)"
```
```    24
```
```    25 definition
```
```    26   additive  where
```
```    27   "additive M f \<longleftrightarrow>
```
```    28     (\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<inter> y = {}
```
```    29     \<longrightarrow> f (x \<union> y) = f x + f y)"
```
```    30
```
```    31 definition
```
```    32   countably_additive  where
```
```    33   "countably_additive M f \<longleftrightarrow>
```
```    34     (\<forall>A. range A \<subseteq> sets M \<longrightarrow>
```
```    35          disjoint_family A \<longrightarrow>
```
```    36          (\<Union>i. A i) \<in> sets M \<longrightarrow>
```
```    37          (\<lambda>n. f (A n))  sums  f (\<Union>i. A i))"
```
```    38
```
```    39 definition
```
```    40   increasing  where
```
```    41   "increasing M f \<longleftrightarrow> (\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<subseteq> y \<longrightarrow> f x \<le> f y)"
```
```    42
```
```    43 definition
```
```    44   subadditive  where
```
```    45   "subadditive M f \<longleftrightarrow>
```
```    46     (\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<inter> y = {}
```
```    47     \<longrightarrow> f (x \<union> y) \<le> f x + f y)"
```
```    48
```
```    49 definition
```
```    50   countably_subadditive  where
```
```    51   "countably_subadditive M f \<longleftrightarrow>
```
```    52     (\<forall>A. range A \<subseteq> sets M \<longrightarrow>
```
```    53          disjoint_family A \<longrightarrow>
```
```    54          (\<Union>i. A i) \<in> sets M \<longrightarrow>
```
```    55          summable (f o A) \<longrightarrow>
```
```    56          f (\<Union>i. A i) \<le> suminf (\<lambda>n. f (A n)))"
```
```    57
```
```    58 definition
```
```    59   lambda_system where
```
```    60   "lambda_system M f =
```
```    61     {l. l \<in> sets M & (\<forall>x \<in> sets M. f (l \<inter> x) + f ((space M - l) \<inter> x) = f x)}"
```
```    62
```
```    63 definition
```
```    64   outer_measure_space where
```
```    65   "outer_measure_space M f  \<longleftrightarrow>
```
```    66      positive M f & increasing M f & countably_subadditive M f"
```
```    67
```
```    68 definition
```
```    69   measure_set where
```
```    70   "measure_set M f X =
```
```    71      {r . \<exists>A. range A \<subseteq> sets M & disjoint_family A & X \<subseteq> (\<Union>i. A i) & (f \<circ> A) sums r}"
```
```    72
```
```    73
```
```    74 locale measure_space = sigma_algebra +
```
```    75   assumes positive: "!!a. a \<in> sets M \<Longrightarrow> 0 \<le> measure M a"
```
```    76       and empty_measure [simp]: "measure M {} = (0::real)"
```
```    77       and ca: "countably_additive M (measure M)"
```
```    78
```
```    79 subsection {* Basic Lemmas *}
```
```    80
```
```    81 lemma positive_imp_0: "positive M f \<Longrightarrow> f {} = 0"
```
```    82   by (simp add: positive_def)
```
```    83
```
```    84 lemma positive_imp_pos: "positive M f \<Longrightarrow> x \<in> sets M \<Longrightarrow> 0 \<le> f x"
```
```    85   by (simp add: positive_def)
```
```    86
```
```    87 lemma increasingD:
```
```    88      "increasing M f \<Longrightarrow> x \<subseteq> y \<Longrightarrow> x\<in>sets M \<Longrightarrow> y\<in>sets M \<Longrightarrow> f x \<le> f y"
```
```    89   by (auto simp add: increasing_def)
```
```    90
```
```    91 lemma subadditiveD:
```
```    92      "subadditive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x\<in>sets M \<Longrightarrow> y\<in>sets M
```
```    93       \<Longrightarrow> f (x \<union> y) \<le> f x + f y"
```
```    94   by (auto simp add: subadditive_def)
```
```    95
```
```    96 lemma additiveD:
```
```    97      "additive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x\<in>sets M \<Longrightarrow> y\<in>sets M
```
```    98       \<Longrightarrow> f (x \<union> y) = f x + f y"
```
```    99   by (auto simp add: additive_def)
```
```   100
```
```   101 lemma countably_additiveD:
```
```   102   "countably_additive M f \<Longrightarrow> range A \<subseteq> sets M \<Longrightarrow> disjoint_family A
```
```   103    \<Longrightarrow> (\<Union>i. A i) \<in> sets M \<Longrightarrow> (\<lambda>n. f (A n))  sums  f (\<Union>i. A i)"
```
```   104   by (simp add: countably_additive_def)
```
```   105
```
```   106 lemma Int_Diff_disjoint: "A \<inter> B \<inter> (A - B) = {}"
```
```   107   by blast
```
```   108
```
```   109 lemma Int_Diff_Un: "A \<inter> B \<union> (A - B) = A"
```
```   110   by blast
```
```   111
```
```   112 lemma disjoint_family_subset:
```
```   113      "disjoint_family A \<Longrightarrow> (!!x. B x \<subseteq> A x) \<Longrightarrow> disjoint_family B"
```
```   114   by (force simp add: disjoint_family_def)
```
```   115
```
```   116 subsection {* A Two-Element Series *}
```
```   117
```
```   118 definition binaryset :: "'a set \<Rightarrow> 'a set \<Rightarrow> nat \<Rightarrow> 'a set "
```
```   119   where "binaryset A B = (\<lambda>\<^isup>x. {})(0 := A, Suc 0 := B)"
```
```   120
```
```   121 lemma range_binaryset_eq: "range(binaryset A B) = {A,B,{}}"
```
```   122   apply (simp add: binaryset_def)
```
```   123   apply (rule set_ext)
```
```   124   apply (auto simp add: image_iff)
```
```   125   done
```
```   126
```
```   127 lemma UN_binaryset_eq: "(\<Union>i. binaryset A B i) = A \<union> B"
```
```   128   by (simp add: UNION_eq_Union_image range_binaryset_eq)
```
```   129
```
```   130 lemma LIMSEQ_binaryset:
```
```   131   assumes f: "f {} = 0"
```
```   132   shows  "(\<lambda>n. \<Sum>i = 0..<n. f (binaryset A B i)) ----> f A + f B"
```
```   133 proof -
```
```   134   have "(\<lambda>n. \<Sum>i = 0..< Suc (Suc n). f (binaryset A B i)) = (\<lambda>n. f A + f B)"
```
```   135     proof
```
```   136       fix n
```
```   137       show "(\<Sum>i\<Colon>nat = 0\<Colon>nat..<Suc (Suc n). f (binaryset A B i)) = f A + f B"
```
```   138         by (induct n)  (auto simp add: binaryset_def f)
```
```   139     qed
```
```   140   moreover
```
```   141   have "... ----> f A + f B" by (rule LIMSEQ_const)
```
```   142   ultimately
```
```   143   have "(\<lambda>n. \<Sum>i = 0..< Suc (Suc n). f (binaryset A B i)) ----> f A + f B"
```
```   144     by metis
```
```   145   hence "(\<lambda>n. \<Sum>i = 0..< n+2. f (binaryset A B i)) ----> f A + f B"
```
```   146     by simp
```
```   147   thus ?thesis by (rule LIMSEQ_offset [where k=2])
```
```   148 qed
```
```   149
```
```   150 lemma binaryset_sums:
```
```   151   assumes f: "f {} = 0"
```
```   152   shows  "(\<lambda>n. f (binaryset A B n)) sums (f A + f B)"
```
```   153     by (simp add: sums_def LIMSEQ_binaryset [where f=f, OF f])
```
```   154
```
```   155 lemma suminf_binaryset_eq:
```
```   156      "f {} = 0 \<Longrightarrow> suminf (\<lambda>n. f (binaryset A B n)) = f A + f B"
```
```   157   by (metis binaryset_sums sums_unique)
```
```   158
```
```   159
```
```   160 subsection {* Lambda Systems *}
```
```   161
```
```   162 lemma (in algebra) lambda_system_eq:
```
```   163     "lambda_system M f =
```
```   164         {l. l \<in> sets M & (\<forall>x \<in> sets M. f (x \<inter> l) + f (x - l) = f x)}"
```
```   165 proof -
```
```   166   have [simp]: "!!l x. l \<in> sets M \<Longrightarrow> x \<in> sets M \<Longrightarrow> (space M - l) \<inter> x = x - l"
```
```   167     by (metis Diff_eq Int_Diff Int_absorb1 Int_commute sets_into_space)
```
```   168   show ?thesis
```
```   169     by (auto simp add: lambda_system_def) (metis Diff_Compl Int_commute)+
```
```   170 qed
```
```   171
```
```   172 lemma (in algebra) lambda_system_empty:
```
```   173     "positive M f \<Longrightarrow> {} \<in> lambda_system M f"
```
```   174   by (auto simp add: positive_def lambda_system_eq)
```
```   175
```
```   176 lemma lambda_system_sets:
```
```   177     "x \<in> lambda_system M f \<Longrightarrow> x \<in> sets M"
```
```   178   by (simp add:  lambda_system_def)
```
```   179
```
```   180 lemma (in algebra) lambda_system_Compl:
```
```   181   fixes f:: "'a set \<Rightarrow> real"
```
```   182   assumes x: "x \<in> lambda_system M f"
```
```   183   shows "space M - x \<in> lambda_system M f"
```
```   184   proof -
```
```   185     have "x \<subseteq> space M"
```
```   186       by (metis sets_into_space lambda_system_sets x)
```
```   187     hence "space M - (space M - x) = x"
```
```   188       by (metis double_diff equalityE)
```
```   189     with x show ?thesis
```
```   190       by (force simp add: lambda_system_def)
```
```   191   qed
```
```   192
```
```   193 lemma (in algebra) lambda_system_Int:
```
```   194   fixes f:: "'a set \<Rightarrow> real"
```
```   195   assumes xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f"
```
```   196   shows "x \<inter> y \<in> lambda_system M f"
```
```   197   proof -
```
```   198     from xl yl show ?thesis
```
```   199       proof (auto simp add: positive_def lambda_system_eq Int)
```
```   200         fix u
```
```   201         assume x: "x \<in> sets M" and y: "y \<in> sets M" and u: "u \<in> sets M"
```
```   202            and fx: "\<forall>z\<in>sets M. f (z \<inter> x) + f (z - x) = f z"
```
```   203            and fy: "\<forall>z\<in>sets M. f (z \<inter> y) + f (z - y) = f z"
```
```   204         have "u - x \<inter> y \<in> sets M"
```
```   205           by (metis Diff Diff_Int Un u x y)
```
```   206         moreover
```
```   207         have "(u - (x \<inter> y)) \<inter> y = u \<inter> y - x" by blast
```
```   208         moreover
```
```   209         have "u - x \<inter> y - y = u - y" by blast
```
```   210         ultimately
```
```   211         have ey: "f (u - x \<inter> y) = f (u \<inter> y - x) + f (u - y)" using fy
```
```   212           by force
```
```   213         have "f (u \<inter> (x \<inter> y)) + f (u - x \<inter> y)
```
```   214               = (f (u \<inter> (x \<inter> y)) + f (u \<inter> y - x)) + f (u - y)"
```
```   215           by (simp add: ey)
```
```   216         also have "... =  (f ((u \<inter> y) \<inter> x) + f (u \<inter> y - x)) + f (u - y)"
```
```   217           by (simp add: Int_ac)
```
```   218         also have "... = f (u \<inter> y) + f (u - y)"
```
```   219           using fx [THEN bspec, of "u \<inter> y"] Int y u
```
```   220           by force
```
```   221         also have "... = f u"
```
```   222           by (metis fy u)
```
```   223         finally show "f (u \<inter> (x \<inter> y)) + f (u - x \<inter> y) = f u" .
```
```   224       qed
```
```   225   qed
```
```   226
```
```   227
```
```   228 lemma (in algebra) lambda_system_Un:
```
```   229   fixes f:: "'a set \<Rightarrow> real"
```
```   230   assumes xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f"
```
```   231   shows "x \<union> y \<in> lambda_system M f"
```
```   232 proof -
```
```   233   have "(space M - x) \<inter> (space M - y) \<in> sets M"
```
```   234     by (metis Diff_Un Un compl_sets lambda_system_sets xl yl)
```
```   235   moreover
```
```   236   have "x \<union> y = space M - ((space M - x) \<inter> (space M - y))"
```
```   237     by auto  (metis subsetD lambda_system_sets sets_into_space xl yl)+
```
```   238   ultimately show ?thesis
```
```   239     by (metis lambda_system_Compl lambda_system_Int xl yl)
```
```   240 qed
```
```   241
```
```   242 lemma (in algebra) lambda_system_algebra:
```
```   243     "positive M f \<Longrightarrow> algebra (M (|sets := lambda_system M f|))"
```
```   244   apply (auto simp add: algebra_def)
```
```   245   apply (metis lambda_system_sets set_mp sets_into_space)
```
```   246   apply (metis lambda_system_empty)
```
```   247   apply (metis lambda_system_Compl)
```
```   248   apply (metis lambda_system_Un)
```
```   249   done
```
```   250
```
```   251 lemma (in algebra) lambda_system_strong_additive:
```
```   252   assumes z: "z \<in> sets M" and disj: "x \<inter> y = {}"
```
```   253       and xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f"
```
```   254   shows "f (z \<inter> (x \<union> y)) = f (z \<inter> x) + f (z \<inter> y)"
```
```   255   proof -
```
```   256     have "z \<inter> x = (z \<inter> (x \<union> y)) \<inter> x" using disj by blast
```
```   257     moreover
```
```   258     have "z \<inter> y = (z \<inter> (x \<union> y)) - x" using disj by blast
```
```   259     moreover
```
```   260     have "(z \<inter> (x \<union> y)) \<in> sets M"
```
```   261       by (metis Int Un lambda_system_sets xl yl z)
```
```   262     ultimately show ?thesis using xl yl
```
```   263       by (simp add: lambda_system_eq)
```
```   264   qed
```
```   265
```
```   266 lemma (in algebra) Int_space_eq1 [simp]: "x \<in> sets M \<Longrightarrow> space M \<inter> x = x"
```
```   267   by (metis Int_absorb1 sets_into_space)
```
```   268
```
```   269 lemma (in algebra) Int_space_eq2 [simp]: "x \<in> sets M \<Longrightarrow> x \<inter> space M = x"
```
```   270   by (metis Int_absorb2 sets_into_space)
```
```   271
```
```   272 lemma (in algebra) lambda_system_additive:
```
```   273      "additive (M (|sets := lambda_system M f|)) f"
```
```   274   proof (auto simp add: additive_def)
```
```   275     fix x and y
```
```   276     assume disj: "x \<inter> y = {}"
```
```   277        and xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f"
```
```   278     hence  "x \<in> sets M" "y \<in> sets M" by (blast intro: lambda_system_sets)+
```
```   279     thus "f (x \<union> y) = f x + f y"
```
```   280       using lambda_system_strong_additive [OF top disj xl yl]
```
```   281       by (simp add: Un)
```
```   282   qed
```
```   283
```
```   284
```
```   285 lemma (in algebra) countably_subadditive_subadditive:
```
```   286   assumes f: "positive M f" and cs: "countably_subadditive M f"
```
```   287   shows  "subadditive M f"
```
```   288 proof (auto simp add: subadditive_def)
```
```   289   fix x y
```
```   290   assume x: "x \<in> sets M" and y: "y \<in> sets M" and "x \<inter> y = {}"
```
```   291   hence "disjoint_family (binaryset x y)"
```
```   292     by (auto simp add: disjoint_family_def binaryset_def)
```
```   293   hence "range (binaryset x y) \<subseteq> sets M \<longrightarrow>
```
```   294          (\<Union>i. binaryset x y i) \<in> sets M \<longrightarrow>
```
```   295          summable (f o (binaryset x y)) \<longrightarrow>
```
```   296          f (\<Union>i. binaryset x y i) \<le> suminf (\<lambda>n. f (binaryset x y n))"
```
```   297     using cs by (simp add: countably_subadditive_def)
```
```   298   hence "{x,y,{}} \<subseteq> sets M \<longrightarrow> x \<union> y \<in> sets M \<longrightarrow>
```
```   299          summable (f o (binaryset x y)) \<longrightarrow>
```
```   300          f (x \<union> y) \<le> suminf (\<lambda>n. f (binaryset x y n))"
```
```   301     by (simp add: range_binaryset_eq UN_binaryset_eq)
```
```   302   thus "f (x \<union> y) \<le>  f x + f y" using f x y binaryset_sums
```
```   303     by (auto simp add: Un sums_iff positive_def o_def)
```
```   304 qed
```
```   305
```
```   306
```
```   307 definition disjointed :: "(nat \<Rightarrow> 'a set) \<Rightarrow> nat \<Rightarrow> 'a set "
```
```   308   where "disjointed A n = A n - (\<Union>i\<in>{0..<n}. A i)"
```
```   309
```
```   310 lemma finite_UN_disjointed_eq: "(\<Union>i\<in>{0..<n}. disjointed A i) = (\<Union>i\<in>{0..<n}. A i)"
```
```   311 proof (induct n)
```
```   312   case 0 show ?case by simp
```
```   313 next
```
```   314   case (Suc n)
```
```   315   thus ?case by (simp add: atLeastLessThanSuc disjointed_def)
```
```   316 qed
```
```   317
```
```   318 lemma UN_disjointed_eq: "(\<Union>i. disjointed A i) = (\<Union>i. A i)"
```
```   319   apply (rule UN_finite2_eq [where k=0])
```
```   320   apply (simp add: finite_UN_disjointed_eq)
```
```   321   done
```
```   322
```
```   323 lemma less_disjoint_disjointed: "m<n \<Longrightarrow> disjointed A m \<inter> disjointed A n = {}"
```
```   324   by (auto simp add: disjointed_def)
```
```   325
```
```   326 lemma disjoint_family_disjointed: "disjoint_family (disjointed A)"
```
```   327   by (simp add: disjoint_family_def)
```
```   328      (metis neq_iff Int_commute less_disjoint_disjointed)
```
```   329
```
```   330 lemma disjointed_subset: "disjointed A n \<subseteq> A n"
```
```   331   by (auto simp add: disjointed_def)
```
```   332
```
```   333
```
```   334 lemma (in algebra) UNION_in_sets:
```
```   335   fixes A:: "nat \<Rightarrow> 'a set"
```
```   336   assumes A: "range A \<subseteq> sets M "
```
```   337   shows  "(\<Union>i\<in>{0..<n}. A i) \<in> sets M"
```
```   338 proof (induct n)
```
```   339   case 0 show ?case by simp
```
```   340 next
```
```   341   case (Suc n)
```
```   342   thus ?case
```
```   343     by (simp add: atLeastLessThanSuc) (metis A Un UNIV_I image_subset_iff)
```
```   344 qed
```
```   345
```
```   346 lemma (in algebra) range_disjointed_sets:
```
```   347   assumes A: "range A \<subseteq> sets M "
```
```   348   shows  "range (disjointed A) \<subseteq> sets M"
```
```   349 proof (auto simp add: disjointed_def)
```
```   350   fix n
```
```   351   show "A n - (\<Union>i\<in>{0..<n}. A i) \<in> sets M" using UNION_in_sets
```
```   352     by (metis A Diff UNIV_I disjointed_def image_subset_iff)
```
```   353 qed
```
```   354
```
```   355 lemma sigma_algebra_disjoint_iff:
```
```   356      "sigma_algebra M \<longleftrightarrow>
```
```   357       algebra M &
```
```   358       (\<forall>A. range A \<subseteq> sets M \<longrightarrow> disjoint_family A \<longrightarrow>
```
```   359            (\<Union>i::nat. A i) \<in> sets M)"
```
```   360 proof (auto simp add: sigma_algebra_iff)
```
```   361   fix A :: "nat \<Rightarrow> 'a set"
```
```   362   assume M: "algebra M"
```
```   363      and A: "range A \<subseteq> sets M"
```
```   364      and UnA: "\<forall>A. range A \<subseteq> sets M \<longrightarrow>
```
```   365                disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> sets M"
```
```   366   hence "range (disjointed A) \<subseteq> sets M \<longrightarrow>
```
```   367          disjoint_family (disjointed A) \<longrightarrow>
```
```   368          (\<Union>i. disjointed A i) \<in> sets M" by blast
```
```   369   hence "(\<Union>i. disjointed A i) \<in> sets M"
```
```   370     by (simp add: algebra.range_disjointed_sets M A disjoint_family_disjointed)
```
```   371   thus "(\<Union>i::nat. A i) \<in> sets M" by (simp add: UN_disjointed_eq)
```
```   372 qed
```
```   373
```
```   374
```
```   375 lemma (in algebra) additive_sum:
```
```   376   fixes A:: "nat \<Rightarrow> 'a set"
```
```   377   assumes f: "positive M f" and ad: "additive M f"
```
```   378       and A: "range A \<subseteq> sets M"
```
```   379       and disj: "disjoint_family A"
```
```   380   shows  "setsum (f o A) {0..<n} = f (\<Union>i\<in>{0..<n}. A i)"
```
```   381 proof (induct n)
```
```   382   case 0 show ?case using f by (simp add: positive_def)
```
```   383 next
```
```   384   case (Suc n)
```
```   385   have "A n \<inter> (\<Union>i\<in>{0..<n}. A i) = {}" using disj
```
```   386     by (auto simp add: disjoint_family_def neq_iff) blast
```
```   387   moreover
```
```   388   have "A n \<in> sets M" using A by blast
```
```   389   moreover have "(\<Union>i\<in>{0..<n}. A i) \<in> sets M"
```
```   390     by (metis A UNION_in_sets atLeast0LessThan)
```
```   391   moreover
```
```   392   ultimately have "f (A n \<union> (\<Union>i\<in>{0..<n}. A i)) = f (A n) + f(\<Union>i\<in>{0..<n}. A i)"
```
```   393     using ad UNION_in_sets A by (auto simp add: additive_def)
```
```   394   with Suc.hyps show ?case using ad
```
```   395     by (auto simp add: atLeastLessThanSuc additive_def)
```
```   396 qed
```
```   397
```
```   398
```
```   399 lemma countably_subadditiveD:
```
```   400   "countably_subadditive M f \<Longrightarrow> range A \<subseteq> sets M \<Longrightarrow> disjoint_family A \<Longrightarrow>
```
```   401    (\<Union>i. A i) \<in> sets M \<Longrightarrow> summable (f o A) \<Longrightarrow> f (\<Union>i. A i) \<le> suminf (f o A)"
```
```   402   by (auto simp add: countably_subadditive_def o_def)
```
```   403
```
```   404 lemma (in algebra) increasing_additive_summable:
```
```   405   fixes A:: "nat \<Rightarrow> 'a set"
```
```   406   assumes f: "positive M f" and ad: "additive M f"
```
```   407       and inc: "increasing M f"
```
```   408       and A: "range A \<subseteq> sets M"
```
```   409       and disj: "disjoint_family A"
```
```   410   shows  "summable (f o A)"
```
```   411 proof (rule pos_summable)
```
```   412   fix n
```
```   413   show "0 \<le> (f \<circ> A) n" using f A
```
```   414     by (force simp add: positive_def)
```
```   415   next
```
```   416   fix n
```
```   417   have "setsum (f \<circ> A) {0..<n} = f (\<Union>i\<in>{0..<n}. A i)"
```
```   418     by (rule additive_sum [OF f ad A disj])
```
```   419   also have "... \<le> f (space M)" using space_closed A
```
```   420     by (blast intro: increasingD [OF inc] UNION_in_sets top)
```
```   421   finally show "setsum (f \<circ> A) {0..<n} \<le> f (space M)" .
```
```   422 qed
```
```   423
```
```   424 lemma lambda_system_positive:
```
```   425      "positive M f \<Longrightarrow> positive (M (|sets := lambda_system M f|)) f"
```
```   426   by (simp add: positive_def lambda_system_def)
```
```   427
```
```   428 lemma lambda_system_increasing:
```
```   429    "increasing M f \<Longrightarrow> increasing (M (|sets := lambda_system M f|)) f"
```
```   430   by (simp add: increasing_def lambda_system_def)
```
```   431
```
```   432 lemma (in algebra) lambda_system_strong_sum:
```
```   433   fixes A:: "nat \<Rightarrow> 'a set"
```
```   434   assumes f: "positive M f" and a: "a \<in> sets M"
```
```   435       and A: "range A \<subseteq> lambda_system M f"
```
```   436       and disj: "disjoint_family A"
```
```   437   shows  "(\<Sum>i = 0..<n. f (a \<inter>A i)) = f (a \<inter> (\<Union>i\<in>{0..<n}. A i))"
```
```   438 proof (induct n)
```
```   439   case 0 show ?case using f by (simp add: positive_def)
```
```   440 next
```
```   441   case (Suc n)
```
```   442   have 2: "A n \<inter> UNION {0..<n} A = {}" using disj
```
```   443     by (force simp add: disjoint_family_def neq_iff)
```
```   444   have 3: "A n \<in> lambda_system M f" using A
```
```   445     by blast
```
```   446   have 4: "UNION {0..<n} A \<in> lambda_system M f"
```
```   447     using A algebra.UNION_in_sets [OF local.lambda_system_algebra [OF f]]
```
```   448     by simp
```
```   449   from Suc.hyps show ?case
```
```   450     by (simp add: atLeastLessThanSuc lambda_system_strong_additive [OF a 2 3 4])
```
```   451 qed
```
```   452
```
```   453
```
```   454 lemma (in sigma_algebra) lambda_system_caratheodory:
```
```   455   assumes oms: "outer_measure_space M f"
```
```   456       and A: "range A \<subseteq> lambda_system M f"
```
```   457       and disj: "disjoint_family A"
```
```   458   shows  "(\<Union>i. A i) \<in> lambda_system M f & (f \<circ> A)  sums  f (\<Union>i. A i)"
```
```   459 proof -
```
```   460   have pos: "positive M f" and inc: "increasing M f"
```
```   461    and csa: "countably_subadditive M f"
```
```   462     by (metis oms outer_measure_space_def)+
```
```   463   have sa: "subadditive M f"
```
```   464     by (metis countably_subadditive_subadditive csa pos)
```
```   465   have A': "range A \<subseteq> sets (M(|sets := lambda_system M f|))" using A
```
```   466     by simp
```
```   467   have alg_ls: "algebra (M(|sets := lambda_system M f|))"
```
```   468     by (rule lambda_system_algebra [OF pos])
```
```   469   have A'': "range A \<subseteq> sets M"
```
```   470      by (metis A image_subset_iff lambda_system_sets)
```
```   471   have sumfA: "summable (f \<circ> A)"
```
```   472     by (metis algebra.increasing_additive_summable [OF alg_ls]
```
```   473           lambda_system_positive lambda_system_additive lambda_system_increasing
```
```   474           A' oms outer_measure_space_def disj)
```
```   475   have U_in: "(\<Union>i. A i) \<in> sets M"
```
```   476     by (metis A countable_UN image_subset_iff lambda_system_sets)
```
```   477   have U_eq: "f (\<Union>i. A i) = suminf (f o A)"
```
```   478     proof (rule antisym)
```
```   479       show "f (\<Union>i. A i) \<le> suminf (f \<circ> A)"
```
```   480         by (rule countably_subadditiveD [OF csa A'' disj U_in sumfA])
```
```   481       show "suminf (f \<circ> A) \<le> f (\<Union>i. A i)"
```
```   482         by (rule suminf_le [OF sumfA])
```
```   483            (metis algebra.additive_sum [OF alg_ls] pos disj UN_Un Un_UNIV_right
```
```   484                   lambda_system_positive lambda_system_additive
```
```   485                   subset_Un_eq increasingD [OF inc] A' A'' UNION_in_sets U_in)
```
```   486     qed
```
```   487   {
```
```   488     fix a
```
```   489     assume a [iff]: "a \<in> sets M"
```
```   490     have "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) = f a"
```
```   491     proof -
```
```   492       have summ: "summable (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A)" using pos A''
```
```   493         apply -
```
```   494         apply (rule summable_comparison_test [OF _ sumfA])
```
```   495         apply (rule_tac x="0" in exI)
```
```   496         apply (simp add: positive_def)
```
```   497         apply (auto simp add: )
```
```   498         apply (subst abs_of_nonneg)
```
```   499         apply (metis A'' Int UNIV_I a image_subset_iff)
```
```   500         apply (blast intro:  increasingD [OF inc] a)
```
```   501         done
```
```   502       show ?thesis
```
```   503       proof (rule antisym)
```
```   504         have "range (\<lambda>i. a \<inter> A i) \<subseteq> sets M" using A''
```
```   505           by blast
```
```   506         moreover
```
```   507         have "disjoint_family (\<lambda>i. a \<inter> A i)" using disj
```
```   508           by (auto simp add: disjoint_family_def)
```
```   509         moreover
```
```   510         have "a \<inter> (\<Union>i. A i) \<in> sets M"
```
```   511           by (metis Int U_in a)
```
```   512         ultimately
```
```   513         have "f (a \<inter> (\<Union>i. A i)) \<le> suminf (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A)"
```
```   514           using countably_subadditiveD [OF csa, of "(\<lambda>i. a \<inter> A i)"] summ
```
```   515           by (simp add: o_def)
```
```   516         moreover
```
```   517         have "suminf (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A)  \<le> f a - f (a - (\<Union>i. A i))"
```
```   518           proof (rule suminf_le [OF summ])
```
```   519             fix n
```
```   520             have UNION_in: "(\<Union>i\<in>{0..<n}. A i) \<in> sets M"
```
```   521               by (metis A'' UNION_in_sets)
```
```   522             have le_fa: "f (UNION {0..<n} A \<inter> a) \<le> f a" using A''
```
```   523               by (blast intro: increasingD [OF inc] A'' Int UNION_in_sets a)
```
```   524             have ls: "(\<Union>i\<in>{0..<n}. A i) \<in> lambda_system M f"
```
```   525               using algebra.UNION_in_sets [OF lambda_system_algebra [OF pos]]
```
```   526               by (simp add: A)
```
```   527             hence eq_fa: "f (a \<inter> (\<Union>i\<in>{0..<n}. A i)) + f (a - (\<Union>i\<in>{0..<n}. A i)) = f a"
```
```   528               by (simp add: lambda_system_eq UNION_in Diff_Compl a)
```
```   529             have "f (a - (\<Union>i. A i)) \<le> f (a - (\<Union>i\<in>{0..<n}. A i))"
```
```   530               by (blast intro: increasingD [OF inc] Diff UNION_eq_Union_image
```
```   531                                UNION_in U_in a)
```
```   532             thus "setsum (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A) {0..<n} \<le> f a - f (a - (\<Union>i. A i))"
```
```   533               using eq_fa
```
```   534               by (simp add: suminf_le [OF summ] lambda_system_strong_sum pos
```
```   535                             a A disj)
```
```   536           qed
```
```   537         ultimately show "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) \<le> f a"
```
```   538           by arith
```
```   539       next
```
```   540         have "f a \<le> f (a \<inter> (\<Union>i. A i) \<union> (a - (\<Union>i. A i)))"
```
```   541           by (blast intro:  increasingD [OF inc] a U_in)
```
```   542         also have "... \<le>  f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i))"
```
```   543           by (blast intro: subadditiveD [OF sa] Int Diff U_in)
```
```   544         finally show "f a \<le> f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i))" .
```
```   545         qed
```
```   546      qed
```
```   547   }
```
```   548   thus  ?thesis
```
```   549     by (simp add: lambda_system_eq sums_iff U_eq U_in sumfA)
```
```   550 qed
```
```   551
```
```   552 lemma (in sigma_algebra) caratheodory_lemma:
```
```   553   assumes oms: "outer_measure_space M f"
```
```   554   shows "measure_space (|space = space M, sets = lambda_system M f, measure = f|)"
```
```   555 proof -
```
```   556   have pos: "positive M f"
```
```   557     by (metis oms outer_measure_space_def)
```
```   558   have alg: "algebra (|space = space M, sets = lambda_system M f, measure = f|)"
```
```   559     using lambda_system_algebra [OF pos]
```
```   560     by (simp add: algebra_def)
```
```   561   then moreover
```
```   562   have "sigma_algebra (|space = space M, sets = lambda_system M f, measure = f|)"
```
```   563     using lambda_system_caratheodory [OF oms]
```
```   564     by (simp add: sigma_algebra_disjoint_iff)
```
```   565   moreover
```
```   566   have "measure_space_axioms (|space = space M, sets = lambda_system M f, measure = f|)"
```
```   567     using pos lambda_system_caratheodory [OF oms]
```
```   568     by (simp add: measure_space_axioms_def positive_def lambda_system_sets
```
```   569                   countably_additive_def o_def)
```
```   570   ultimately
```
```   571   show ?thesis
```
```   572     by intro_locales (auto simp add: sigma_algebra_def)
```
```   573 qed
```
```   574
```
```   575
```
```   576 lemma (in algebra) inf_measure_nonempty:
```
```   577   assumes f: "positive M f" and b: "b \<in> sets M" and a: "a \<subseteq> b"
```
```   578   shows "f b \<in> measure_set M f a"
```
```   579 proof -
```
```   580   have "(f \<circ> (\<lambda>i. {})(0 := b)) sums setsum (f \<circ> (\<lambda>i. {})(0 := b)) {0..<1::nat}"
```
```   581     by (rule series_zero)  (simp add: positive_imp_0 [OF f])
```
```   582   also have "... = f b"
```
```   583     by simp
```
```   584   finally have "(f \<circ> (\<lambda>i. {})(0 := b)) sums f b" .
```
```   585   thus ?thesis using a
```
```   586     by (auto intro!: exI [of _ "(\<lambda>i. {})(0 := b)"]
```
```   587              simp add: measure_set_def disjoint_family_def b split_if_mem2)
```
```   588 qed
```
```   589
```
```   590 lemma (in algebra) inf_measure_pos0:
```
```   591      "positive M f \<Longrightarrow> x \<in> measure_set M f a \<Longrightarrow> 0 \<le> x"
```
```   592 apply (auto simp add: positive_def measure_set_def sums_iff intro!: suminf_ge_zero)
```
```   593 apply blast
```
```   594 done
```
```   595
```
```   596 lemma (in algebra) inf_measure_pos:
```
```   597   shows "positive M f \<Longrightarrow> x \<subseteq> space M \<Longrightarrow> 0 \<le> Inf (measure_set M f x)"
```
```   598 apply (rule Inf_greatest)
```
```   599 apply (metis emptyE inf_measure_nonempty top)
```
```   600 apply (metis inf_measure_pos0)
```
```   601 done
```
```   602
```
```   603 lemma (in algebra) additive_increasing:
```
```   604   assumes posf: "positive M f" and addf: "additive M f"
```
```   605   shows "increasing M f"
```
```   606 proof (auto simp add: increasing_def)
```
```   607   fix x y
```
```   608   assume xy: "x \<in> sets M" "y \<in> sets M" "x \<subseteq> y"
```
```   609   have "f x \<le> f x + f (y-x)" using posf
```
```   610     by (simp add: positive_def) (metis Diff xy)
```
```   611   also have "... = f (x \<union> (y-x))" using addf
```
```   612     by (auto simp add: additive_def) (metis Diff_disjoint Un_Diff_cancel Diff xy)
```
```   613   also have "... = f y"
```
```   614     by (metis Un_Diff_cancel Un_absorb1 xy)
```
```   615   finally show "f x \<le> f y" .
```
```   616 qed
```
```   617
```
```   618 lemma (in algebra) countably_additive_additive:
```
```   619   assumes posf: "positive M f" and ca: "countably_additive M f"
```
```   620   shows "additive M f"
```
```   621 proof (auto simp add: additive_def)
```
```   622   fix x y
```
```   623   assume x: "x \<in> sets M" and y: "y \<in> sets M" and "x \<inter> y = {}"
```
```   624   hence "disjoint_family (binaryset x y)"
```
```   625     by (auto simp add: disjoint_family_def binaryset_def)
```
```   626   hence "range (binaryset x y) \<subseteq> sets M \<longrightarrow>
```
```   627          (\<Union>i. binaryset x y i) \<in> sets M \<longrightarrow>
```
```   628          f (\<Union>i. binaryset x y i) = suminf (\<lambda>n. f (binaryset x y n))"
```
```   629     using ca
```
```   630     by (simp add: countably_additive_def) (metis UN_binaryset_eq sums_unique)
```
```   631   hence "{x,y,{}} \<subseteq> sets M \<longrightarrow> x \<union> y \<in> sets M \<longrightarrow>
```
```   632          f (x \<union> y) = suminf (\<lambda>n. f (binaryset x y n))"
```
```   633     by (simp add: range_binaryset_eq UN_binaryset_eq)
```
```   634   thus "f (x \<union> y) = f x + f y" using posf x y
```
```   635     by (simp add: Un suminf_binaryset_eq positive_def)
```
```   636 qed
```
```   637
```
```   638 lemma (in algebra) inf_measure_agrees:
```
```   639   assumes posf: "positive M f" and ca: "countably_additive M f"
```
```   640       and s: "s \<in> sets M"
```
```   641   shows "Inf (measure_set M f s) = f s"
```
```   642 proof (rule Inf_eq)
```
```   643   fix z
```
```   644   assume z: "z \<in> measure_set M f s"
```
```   645   from this obtain A where
```
```   646     A: "range A \<subseteq> sets M" and disj: "disjoint_family A"
```
```   647     and "s \<subseteq> (\<Union>x. A x)" and sm: "summable (f \<circ> A)"
```
```   648     and si: "suminf (f \<circ> A) = z"
```
```   649     by (auto simp add: measure_set_def sums_iff)
```
```   650   hence seq: "s = (\<Union>i. A i \<inter> s)" by blast
```
```   651   have inc: "increasing M f"
```
```   652     by (metis additive_increasing ca countably_additive_additive posf)
```
```   653   have sums: "(\<lambda>i. f (A i \<inter> s)) sums f (\<Union>i. A i \<inter> s)"
```
```   654     proof (rule countably_additiveD [OF ca])
```
```   655       show "range (\<lambda>n. A n \<inter> s) \<subseteq> sets M" using A s
```
```   656         by blast
```
```   657       show "disjoint_family (\<lambda>n. A n \<inter> s)" using disj
```
```   658         by (auto simp add: disjoint_family_def)
```
```   659       show "(\<Union>i. A i \<inter> s) \<in> sets M" using A s
```
```   660         by (metis UN_extend_simps(4) s seq)
```
```   661     qed
```
```   662   hence "f s = suminf (\<lambda>i. f (A i \<inter> s))"
```
```   663     by (metis Int_commute UN_simps(4) seq sums_iff)
```
```   664   also have "... \<le> suminf (f \<circ> A)"
```
```   665     proof (rule summable_le [OF _ _ sm])
```
```   666       show "\<forall>n. f (A n \<inter> s) \<le> (f \<circ> A) n" using A s
```
```   667         by (force intro: increasingD [OF inc])
```
```   668       show "summable (\<lambda>i. f (A i \<inter> s))" using sums
```
```   669         by (simp add: sums_iff)
```
```   670     qed
```
```   671   also have "... = z" by (rule si)
```
```   672   finally show "f s \<le> z" .
```
```   673 next
```
```   674   fix y
```
```   675   assume y: "!!u. u \<in> measure_set M f s \<Longrightarrow> y \<le> u"
```
```   676   thus "y \<le> f s"
```
```   677     by (blast intro: inf_measure_nonempty [OF posf s subset_refl])
```
```   678 qed
```
```   679
```
```   680 lemma (in algebra) inf_measure_empty:
```
```   681   assumes posf: "positive M f"
```
```   682   shows "Inf (measure_set M f {}) = 0"
```
```   683 proof (rule antisym)
```
```   684   show "0 \<le> Inf (measure_set M f {})"
```
```   685     by (metis empty_subsetI inf_measure_pos posf)
```
```   686   show "Inf (measure_set M f {}) \<le> 0"
```
```   687     by (metis Inf_lower empty_sets inf_measure_pos0 inf_measure_nonempty posf
```
```   688               positive_imp_0 subset_refl)
```
```   689 qed
```
```   690
```
```   691 lemma (in algebra) inf_measure_positive:
```
```   692   "positive M f \<Longrightarrow>
```
```   693    positive (| space = space M, sets = Pow (space M) |)
```
```   694                   (\<lambda>x. Inf (measure_set M f x))"
```
```   695   by (simp add: positive_def inf_measure_empty inf_measure_pos)
```
```   696
```
```   697 lemma (in algebra) inf_measure_increasing:
```
```   698   assumes posf: "positive M f"
```
```   699   shows "increasing (| space = space M, sets = Pow (space M) |)
```
```   700                     (\<lambda>x. Inf (measure_set M f x))"
```
```   701 apply (auto simp add: increasing_def)
```
```   702 apply (rule Inf_greatest, metis emptyE inf_measure_nonempty top posf)
```
```   703 apply (rule Inf_lower)
```
```   704 apply (clarsimp simp add: measure_set_def, blast)
```
```   705 apply (blast intro: inf_measure_pos0 posf)
```
```   706 done
```
```   707
```
```   708
```
```   709 lemma (in algebra) inf_measure_le:
```
```   710   assumes posf: "positive M f" and inc: "increasing M f"
```
```   711       and x: "x \<in> {r . \<exists>A. range A \<subseteq> sets M & s \<subseteq> (\<Union>i. A i) & (f \<circ> A) sums r}"
```
```   712   shows "Inf (measure_set M f s) \<le> x"
```
```   713 proof -
```
```   714   from x
```
```   715   obtain A where A: "range A \<subseteq> sets M" and ss: "s \<subseteq> (\<Union>i. A i)"
```
```   716              and sm: "summable (f \<circ> A)" and xeq: "suminf (f \<circ> A) = x"
```
```   717     by (auto simp add: sums_iff)
```
```   718   have dA: "range (disjointed A) \<subseteq> sets M"
```
```   719     by (metis A range_disjointed_sets)
```
```   720   have "\<forall>n. \<bar>(f o disjointed A) n\<bar> \<le> (f \<circ> A) n"
```
```   721     proof (auto)
```
```   722       fix n
```
```   723       have "\<bar>f (disjointed A n)\<bar> = f (disjointed A n)" using posf dA
```
```   724         by (auto simp add: positive_def image_subset_iff)
```
```   725       also have "... \<le> f (A n)"
```
```   726         by (metis increasingD [OF inc] UNIV_I dA image_subset_iff disjointed_subset A)
```
```   727       finally show "\<bar>f (disjointed A n)\<bar> \<le> f (A n)" .
```
```   728     qed
```
```   729   from Series.summable_le2 [OF this sm]
```
```   730   have sda:  "summable (f o disjointed A)"
```
```   731              "suminf (f o disjointed A) \<le> suminf (f \<circ> A)"
```
```   732     by blast+
```
```   733   hence ley: "suminf (f o disjointed A) \<le> x"
```
```   734     by (metis xeq)
```
```   735   from sda have "(f \<circ> disjointed A) sums suminf (f \<circ> disjointed A)"
```
```   736     by (simp add: sums_iff)
```
```   737   hence y: "suminf (f o disjointed A) \<in> measure_set M f s"
```
```   738     apply (auto simp add: measure_set_def)
```
```   739     apply (rule_tac x="disjointed A" in exI)
```
```   740     apply (simp add: disjoint_family_disjointed UN_disjointed_eq ss dA)
```
```   741     done
```
```   742   show ?thesis
```
```   743     by (blast intro: Inf_lower y order_trans [OF _ ley] inf_measure_pos0 posf)
```
```   744 qed
```
```   745
```
```   746 lemma (in algebra) inf_measure_close:
```
```   747   assumes posf: "positive M f" and e: "0 < e" and ss: "s \<subseteq> (space M)"
```
```   748   shows "\<exists>A l. range A \<subseteq> sets M & disjoint_family A & s \<subseteq> (\<Union>i. A i) &
```
```   749                (f \<circ> A) sums l & l \<le> Inf (measure_set M f s) + e"
```
```   750 proof -
```
```   751   have " measure_set M f s \<noteq> {}"
```
```   752     by (metis emptyE ss inf_measure_nonempty [OF posf top])
```
```   753   hence "\<exists>l \<in> measure_set M f s. l < Inf (measure_set M f s) + e"
```
```   754     by (rule Inf_close [OF _ e])
```
```   755   thus ?thesis
```
```   756     by (auto simp add: measure_set_def, rule_tac x=" A" in exI, auto)
```
```   757 qed
```
```   758
```
```   759 lemma (in algebra) inf_measure_countably_subadditive:
```
```   760   assumes posf: "positive M f" and inc: "increasing M f"
```
```   761   shows "countably_subadditive (| space = space M, sets = Pow (space M) |)
```
```   762                   (\<lambda>x. Inf (measure_set M f x))"
```
```   763 proof (auto simp add: countably_subadditive_def o_def, rule field_le_epsilon)
```
```   764   fix A :: "nat \<Rightarrow> 'a set" and e :: real
```
```   765     assume A: "range A \<subseteq> Pow (space M)"
```
```   766        and disj: "disjoint_family A"
```
```   767        and sb: "(\<Union>i. A i) \<subseteq> space M"
```
```   768        and sum1: "summable (\<lambda>n. Inf (measure_set M f (A n)))"
```
```   769        and e: "0 < e"
```
```   770     have "!!n. \<exists>B l. range B \<subseteq> sets M \<and> disjoint_family B \<and> A n \<subseteq> (\<Union>i. B i) \<and>
```
```   771                     (f o B) sums l \<and>
```
```   772                     l \<le> Inf (measure_set M f (A n)) + e * (1/2)^(Suc n)"
```
```   773       apply (rule inf_measure_close [OF posf])
```
```   774       apply (metis e half mult_pos_pos zero_less_power)
```
```   775       apply (metis UNIV_I UN_subset_iff sb)
```
```   776       done
```
```   777     hence "\<exists>BB ll. \<forall>n. range (BB n) \<subseteq> sets M \<and> disjoint_family (BB n) \<and>
```
```   778                        A n \<subseteq> (\<Union>i. BB n i) \<and> (f o BB n) sums ll n \<and>
```
```   779                        ll n \<le> Inf (measure_set M f (A n)) + e * (1/2)^(Suc n)"
```
```   780       by (rule choice2)
```
```   781     then obtain BB ll
```
```   782       where BB: "!!n. (range (BB n) \<subseteq> sets M)"
```
```   783         and disjBB: "!!n. disjoint_family (BB n)"
```
```   784         and sbBB: "!!n. A n \<subseteq> (\<Union>i. BB n i)"
```
```   785         and BBsums: "!!n. (f o BB n) sums ll n"
```
```   786         and ll: "!!n. ll n \<le> Inf (measure_set M f (A n)) + e * (1/2)^(Suc n)"
```
```   787       by auto blast
```
```   788     have llpos: "!!n. 0 \<le> ll n"
```
```   789         by (metis BBsums sums_iff o_apply posf positive_imp_pos suminf_ge_zero
```
```   790               range_subsetD BB)
```
```   791     have sll: "summable ll &
```
```   792                suminf ll \<le> suminf (\<lambda>n. Inf (measure_set M f (A n))) + e"
```
```   793       proof -
```
```   794         have "(\<lambda>n. e * (1/2)^(Suc n)) sums (e*1)"
```
```   795           by (rule sums_mult [OF power_half_series])
```
```   796         hence sum0: "summable (\<lambda>n. e * (1 / 2) ^ Suc n)"
```
```   797           and eqe:  "(\<Sum>n. e * (1 / 2) ^ n / 2) = e"
```
```   798           by (auto simp add: sums_iff)
```
```   799         have 0: "suminf (\<lambda>n. Inf (measure_set M f (A n))) +
```
```   800                  suminf (\<lambda>n. e * (1/2)^(Suc n)) =
```
```   801                  suminf (\<lambda>n. Inf (measure_set M f (A n)) + e * (1/2)^(Suc n))"
```
```   802           by (rule suminf_add [OF sum1 sum0])
```
```   803         have 1: "\<forall>n. \<bar>ll n\<bar> \<le> Inf (measure_set M f (A n)) + e * (1/2) ^ Suc n"
```
```   804           by (metis ll llpos abs_of_nonneg)
```
```   805         have 2: "summable (\<lambda>n. Inf (measure_set M f (A n)) + e*(1/2)^(Suc n))"
```
```   806           by (rule summable_add [OF sum1 sum0])
```
```   807         have "suminf ll \<le> (\<Sum>n. Inf (measure_set M f (A n)) + e*(1/2) ^ Suc n)"
```
```   808           using Series.summable_le2 [OF 1 2] by auto
```
```   809         also have "... = (\<Sum>n. Inf (measure_set M f (A n))) +
```
```   810                          (\<Sum>n. e * (1 / 2) ^ Suc n)"
```
```   811           by (metis 0)
```
```   812         also have "... = (\<Sum>n. Inf (measure_set M f (A n))) + e"
```
```   813           by (simp add: eqe)
```
```   814         finally show ?thesis using  Series.summable_le2 [OF 1 2] by auto
```
```   815       qed
```
```   816     def C \<equiv> "(split BB) o nat_to_nat2"
```
```   817     have C: "!!n. C n \<in> sets M"
```
```   818       apply (rule_tac p="nat_to_nat2 n" in PairE)
```
```   819       apply (simp add: C_def)
```
```   820       apply (metis BB subsetD rangeI)
```
```   821       done
```
```   822     have sbC: "(\<Union>i. A i) \<subseteq> (\<Union>i. C i)"
```
```   823       proof (auto simp add: C_def)
```
```   824         fix x i
```
```   825         assume x: "x \<in> A i"
```
```   826         with sbBB [of i] obtain j where "x \<in> BB i j"
```
```   827           by blast
```
```   828         thus "\<exists>i. x \<in> split BB (nat_to_nat2 i)"
```
```   829           by (metis nat_to_nat2_surj internal_split_def prod.cases
```
```   830                 prod_case_split surj_f_inv_f)
```
```   831       qed
```
```   832     have "(f \<circ> C) = (f \<circ> (\<lambda>(x, y). BB x y)) \<circ> nat_to_nat2"
```
```   833       by (rule ext)  (auto simp add: C_def)
```
```   834     also have "... sums suminf ll"
```
```   835       proof (rule suminf_2dimen)
```
```   836         show "\<And>m n. 0 \<le> (f \<circ> (\<lambda>(x, y). BB x y)) (m, n)" using posf BB
```
```   837           by (force simp add: positive_def)
```
```   838         show "\<And>m. (\<lambda>n. (f \<circ> (\<lambda>(x, y). BB x y)) (m, n)) sums ll m"using BBsums BB
```
```   839           by (force simp add: o_def)
```
```   840         show "summable ll" using sll
```
```   841           by auto
```
```   842       qed
```
```   843     finally have Csums: "(f \<circ> C) sums suminf ll" .
```
```   844     have "Inf (measure_set M f (\<Union>i. A i)) \<le> suminf ll"
```
```   845       apply (rule inf_measure_le [OF posf inc], auto)
```
```   846       apply (rule_tac x="C" in exI)
```
```   847       apply (auto simp add: C sbC Csums)
```
```   848       done
```
```   849     also have "... \<le> (\<Sum>n. Inf (measure_set M f (A n))) + e" using sll
```
```   850       by blast
```
```   851     finally show "Inf (measure_set M f (\<Union>i. A i)) \<le>
```
```   852           (\<Sum>n. Inf (measure_set M f (A n))) + e" .
```
```   853 qed
```
```   854
```
```   855 lemma (in algebra) inf_measure_outer:
```
```   856   "positive M f \<Longrightarrow> increasing M f
```
```   857    \<Longrightarrow> outer_measure_space (| space = space M, sets = Pow (space M) |)
```
```   858                           (\<lambda>x. Inf (measure_set M f x))"
```
```   859   by (simp add: outer_measure_space_def inf_measure_positive
```
```   860                 inf_measure_increasing inf_measure_countably_subadditive)
```
```   861
```
```   862 (*MOVE UP*)
```
```   863
```
```   864 lemma (in algebra) algebra_subset_lambda_system:
```
```   865   assumes posf: "positive M f" and inc: "increasing M f"
```
```   866       and add: "additive M f"
```
```   867   shows "sets M \<subseteq> lambda_system (| space = space M, sets = Pow (space M) |)
```
```   868                                 (\<lambda>x. Inf (measure_set M f x))"
```
```   869 proof (auto dest: sets_into_space
```
```   870             simp add: algebra.lambda_system_eq [OF algebra_Pow])
```
```   871   fix x s
```
```   872   assume x: "x \<in> sets M"
```
```   873      and s: "s \<subseteq> space M"
```
```   874   have [simp]: "!!x. x \<in> sets M \<Longrightarrow> s \<inter> (space M - x) = s-x" using s
```
```   875     by blast
```
```   876   have "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))
```
```   877         \<le> Inf (measure_set M f s)"
```
```   878     proof (rule field_le_epsilon)
```
```   879       fix e :: real
```
```   880       assume e: "0 < e"
```
```   881       from inf_measure_close [OF posf e s]
```
```   882       obtain A l where A: "range A \<subseteq> sets M" and disj: "disjoint_family A"
```
```   883                    and sUN: "s \<subseteq> (\<Union>i. A i)" and fsums: "(f \<circ> A) sums l"
```
```   884                    and l: "l \<le> Inf (measure_set M f s) + e"
```
```   885         by auto
```
```   886       have [simp]: "!!x. x \<in> sets M \<Longrightarrow>
```
```   887                       (f o (\<lambda>z. z \<inter> (space M - x)) o A) = (f o (\<lambda>z. z - x) o A)"
```
```   888         by (rule ext, simp, metis A Int_Diff Int_space_eq2 range_subsetD)
```
```   889       have  [simp]: "!!n. f (A n \<inter> x) + f (A n - x) = f (A n)"
```
```   890         by (subst additiveD [OF add, symmetric])
```
```   891            (auto simp add: x range_subsetD [OF A] Int_Diff_Un Int_Diff_disjoint)
```
```   892       have fsumb: "summable (f \<circ> A)"
```
```   893         by (metis fsums sums_iff)
```
```   894       { fix u
```
```   895         assume u: "u \<in> sets M"
```
```   896         have [simp]: "\<And>n. \<bar>f (A n \<inter> u)\<bar> \<le> f (A n)"
```
```   897           by (simp add: positive_imp_pos [OF posf]  increasingD [OF inc]
```
```   898                         u Int  range_subsetD [OF A])
```
```   899         have 1: "summable (f o (\<lambda>z. z\<inter>u) o A)"
```
```   900           by (rule summable_comparison_test [OF _ fsumb]) simp
```
```   901         have 2: "Inf (measure_set M f (s\<inter>u)) \<le> suminf (f o (\<lambda>z. z\<inter>u) o A)"
```
```   902           proof (rule Inf_lower)
```
```   903             show "suminf (f \<circ> (\<lambda>z. z \<inter> u) \<circ> A) \<in> measure_set M f (s \<inter> u)"
```
```   904               apply (simp add: measure_set_def)
```
```   905               apply (rule_tac x="(\<lambda>z. z \<inter> u) o A" in exI)
```
```   906               apply (auto simp add: disjoint_family_subset [OF disj])
```
```   907               apply (blast intro: u range_subsetD [OF A])
```
```   908               apply (blast dest: subsetD [OF sUN])
```
```   909               apply (metis 1 o_assoc sums_iff)
```
```   910               done
```
```   911           next
```
```   912             show "\<And>x. x \<in> measure_set M f (s \<inter> u) \<Longrightarrow> 0 \<le> x"
```
```   913               by (blast intro: inf_measure_pos0 [OF posf])
```
```   914             qed
```
```   915           note 1 2
```
```   916       } note lesum = this
```
```   917       have sum1: "summable (f o (\<lambda>z. z\<inter>x) o A)"
```
```   918         and inf1: "Inf (measure_set M f (s\<inter>x)) \<le> suminf (f o (\<lambda>z. z\<inter>x) o A)"
```
```   919         and sum2: "summable (f o (\<lambda>z. z \<inter> (space M - x)) o A)"
```
```   920         and inf2: "Inf (measure_set M f (s \<inter> (space M - x)))
```
```   921                    \<le> suminf (f o (\<lambda>z. z \<inter> (space M - x)) o A)"
```
```   922         by (metis Diff lesum top x)+
```
```   923       hence "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))
```
```   924            \<le> suminf (f o (\<lambda>s. s\<inter>x) o A) + suminf (f o (\<lambda>s. s-x) o A)"
```
```   925         by (simp add: x)
```
```   926       also have "... \<le> suminf (f o A)" using suminf_add [OF sum1 sum2]
```
```   927         by (simp add: x) (simp add: o_def)
```
```   928       also have "... \<le> Inf (measure_set M f s) + e"
```
```   929         by (metis fsums l sums_unique)
```
```   930       finally show "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))
```
```   931         \<le> Inf (measure_set M f s) + e" .
```
```   932     qed
```
```   933   moreover
```
```   934   have "Inf (measure_set M f s)
```
```   935        \<le> Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))"
```
```   936     proof -
```
```   937     have "Inf (measure_set M f s) = Inf (measure_set M f ((s\<inter>x) \<union> (s-x)))"
```
```   938       by (metis Un_Diff_Int Un_commute)
```
```   939     also have "... \<le> Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))"
```
```   940       apply (rule subadditiveD)
```
```   941       apply (iprover intro: algebra.countably_subadditive_subadditive algebra_Pow
```
```   942                inf_measure_positive inf_measure_countably_subadditive posf inc)
```
```   943       apply (auto simp add: subsetD [OF s])
```
```   944       done
```
```   945     finally show ?thesis .
```
```   946     qed
```
```   947   ultimately
```
```   948   show "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))
```
```   949         = Inf (measure_set M f s)"
```
```   950     by (rule order_antisym)
```
```   951 qed
```
```   952
```
```   953 lemma measure_down:
```
```   954      "measure_space N \<Longrightarrow> sigma_algebra M \<Longrightarrow> sets M \<subseteq> sets N \<Longrightarrow>
```
```   955       (measure M = measure N) \<Longrightarrow> measure_space M"
```
```   956   by (simp add: measure_space_def measure_space_axioms_def positive_def
```
```   957                 countably_additive_def)
```
```   958      blast
```
```   959
```
```   960 theorem (in algebra) caratheodory:
```
```   961   assumes posf: "positive M f" and ca: "countably_additive M f"
```
```   962   shows "\<exists>MS :: 'a measure_space.
```
```   963              (\<forall>s \<in> sets M. measure MS s = f s) \<and>
```
```   964              ((|space = space MS, sets = sets MS|) = sigma (space M) (sets M)) \<and>
```
```   965              measure_space MS"
```
```   966   proof -
```
```   967     have inc: "increasing M f"
```
```   968       by (metis additive_increasing ca countably_additive_additive posf)
```
```   969     let ?infm = "(\<lambda>x. Inf (measure_set M f x))"
```
```   970     def ls \<equiv> "lambda_system (|space = space M, sets = Pow (space M)|) ?infm"
```
```   971     have mls: "measure_space (|space = space M, sets = ls, measure = ?infm|)"
```
```   972       using sigma_algebra.caratheodory_lemma
```
```   973               [OF sigma_algebra_Pow  inf_measure_outer [OF posf inc]]
```
```   974       by (simp add: ls_def)
```
```   975     hence sls: "sigma_algebra (|space = space M, sets = ls, measure = ?infm|)"
```
```   976       by (simp add: measure_space_def)
```
```   977     have "sets M \<subseteq> ls"
```
```   978       by (simp add: ls_def)
```
```   979          (metis ca posf inc countably_additive_additive algebra_subset_lambda_system)
```
```   980     hence sgs_sb: "sigma_sets (space M) (sets M) \<subseteq> ls"
```
```   981       using sigma_algebra.sigma_sets_subset [OF sls, of "sets M"]
```
```   982       by simp
```
```   983     have "measure_space (|space = space M,
```
```   984                           sets = sigma_sets (space M) (sets M),
```
```   985                           measure = ?infm|)"
```
```   986       by (rule measure_down [OF mls], rule sigma_algebra_sigma_sets)
```
```   987          (simp_all add: sgs_sb space_closed)
```
```   988     thus ?thesis
```
```   989       by (force simp add: sigma_def inf_measure_agrees [OF posf ca])
```
```   990 qed
```
```   991
```
```   992 end
```