src/HOL/Probability/Caratheodory.thy
 author hoelzl Thu Sep 02 17:28:00 2010 +0200 (2010-09-02) changeset 39096 111756225292 parent 38656 d5d342611edb child 40859 de0b30e6c2d2 permissions -rw-r--r--
merged
```     1 header {*Caratheodory Extension Theorem*}
```
```     2
```
```     3 theory Caratheodory
```
```     4   imports Sigma_Algebra Positive_Infinite_Real
```
```     5 begin
```
```     6
```
```     7 text{*From the Hurd/Coble measure theory development, translated by Lawrence Paulson.*}
```
```     8
```
```     9 subsection {* Measure Spaces *}
```
```    10
```
```    11 definition "positive f \<longleftrightarrow> f {} = (0::pinfreal)" -- "Positive is enforced by the type"
```
```    12
```
```    13 definition
```
```    14   additive  where
```
```    15   "additive M f \<longleftrightarrow>
```
```    16     (\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<inter> y = {}
```
```    17     \<longrightarrow> f (x \<union> y) = f x + f y)"
```
```    18
```
```    19 definition
```
```    20   countably_additive  where
```
```    21   "countably_additive M f \<longleftrightarrow>
```
```    22     (\<forall>A. range A \<subseteq> sets M \<longrightarrow>
```
```    23          disjoint_family A \<longrightarrow>
```
```    24          (\<Union>i. A i) \<in> sets M \<longrightarrow>
```
```    25          (\<Sum>\<^isub>\<infinity> n. f (A n)) = f (\<Union>i. A i))"
```
```    26
```
```    27 definition
```
```    28   increasing  where
```
```    29   "increasing M f \<longleftrightarrow> (\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<subseteq> y \<longrightarrow> f x \<le> f y)"
```
```    30
```
```    31 definition
```
```    32   subadditive  where
```
```    33   "subadditive M f \<longleftrightarrow>
```
```    34     (\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<inter> y = {}
```
```    35     \<longrightarrow> f (x \<union> y) \<le> f x + f y)"
```
```    36
```
```    37 definition
```
```    38   countably_subadditive  where
```
```    39   "countably_subadditive M f \<longleftrightarrow>
```
```    40     (\<forall>A. range A \<subseteq> sets M \<longrightarrow>
```
```    41          disjoint_family A \<longrightarrow>
```
```    42          (\<Union>i. A i) \<in> sets M \<longrightarrow>
```
```    43          f (\<Union>i. A i) \<le> psuminf (\<lambda>n. f (A n)))"
```
```    44
```
```    45 definition
```
```    46   lambda_system where
```
```    47   "lambda_system M f =
```
```    48     {l. l \<in> sets M & (\<forall>x \<in> sets M. f (l \<inter> x) + f ((space M - l) \<inter> x) = f x)}"
```
```    49
```
```    50 definition
```
```    51   outer_measure_space where
```
```    52   "outer_measure_space M f  \<longleftrightarrow>
```
```    53      positive f \<and> increasing M f \<and> countably_subadditive M f"
```
```    54
```
```    55 definition
```
```    56   measure_set where
```
```    57   "measure_set M f X =
```
```    58      {r . \<exists>A. range A \<subseteq> sets M \<and> disjoint_family A \<and> X \<subseteq> (\<Union>i. A i) \<and> (\<Sum>\<^isub>\<infinity> i. f (A i)) = r}"
```
```    59
```
```    60 locale measure_space = sigma_algebra +
```
```    61   fixes \<mu> :: "'a set \<Rightarrow> pinfreal"
```
```    62   assumes empty_measure [simp]: "\<mu> {} = 0"
```
```    63       and ca: "countably_additive M \<mu>"
```
```    64
```
```    65 lemma increasingD:
```
```    66      "increasing M f \<Longrightarrow> x \<subseteq> y \<Longrightarrow> x\<in>sets M \<Longrightarrow> y\<in>sets M \<Longrightarrow> f x \<le> f y"
```
```    67   by (auto simp add: increasing_def)
```
```    68
```
```    69 lemma subadditiveD:
```
```    70      "subadditive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x\<in>sets M \<Longrightarrow> y\<in>sets M
```
```    71       \<Longrightarrow> f (x \<union> y) \<le> f x + f y"
```
```    72   by (auto simp add: subadditive_def)
```
```    73
```
```    74 lemma additiveD:
```
```    75      "additive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x\<in>sets M \<Longrightarrow> y\<in>sets M
```
```    76       \<Longrightarrow> f (x \<union> y) = f x + f y"
```
```    77   by (auto simp add: additive_def)
```
```    78
```
```    79 lemma countably_additiveD:
```
```    80   "countably_additive M f \<Longrightarrow> range A \<subseteq> sets M \<Longrightarrow> disjoint_family A
```
```    81    \<Longrightarrow> (\<Union>i. A i) \<in> sets M \<Longrightarrow> (\<Sum>\<^isub>\<infinity> n. f (A n)) = f (\<Union>i. A i)"
```
```    82   by (simp add: countably_additive_def)
```
```    83
```
```    84 section "Extend binary sets"
```
```    85
```
```    86 lemma LIMSEQ_binaryset:
```
```    87   assumes f: "f {} = 0"
```
```    88   shows  "(\<lambda>n. \<Sum>i = 0..<n. f (binaryset A B i)) ----> f A + f B"
```
```    89 proof -
```
```    90   have "(\<lambda>n. \<Sum>i = 0..< Suc (Suc n). f (binaryset A B i)) = (\<lambda>n. f A + f B)"
```
```    91     proof
```
```    92       fix n
```
```    93       show "(\<Sum>i\<Colon>nat = 0\<Colon>nat..<Suc (Suc n). f (binaryset A B i)) = f A + f B"
```
```    94         by (induct n)  (auto simp add: binaryset_def f)
```
```    95     qed
```
```    96   moreover
```
```    97   have "... ----> f A + f B" by (rule LIMSEQ_const)
```
```    98   ultimately
```
```    99   have "(\<lambda>n. \<Sum>i = 0..< Suc (Suc n). f (binaryset A B i)) ----> f A + f B"
```
```   100     by metis
```
```   101   hence "(\<lambda>n. \<Sum>i = 0..< n+2. f (binaryset A B i)) ----> f A + f B"
```
```   102     by simp
```
```   103   thus ?thesis by (rule LIMSEQ_offset [where k=2])
```
```   104 qed
```
```   105
```
```   106 lemma binaryset_sums:
```
```   107   assumes f: "f {} = 0"
```
```   108   shows  "(\<lambda>n. f (binaryset A B n)) sums (f A + f B)"
```
```   109     by (simp add: sums_def LIMSEQ_binaryset [where f=f, OF f])
```
```   110
```
```   111 lemma suminf_binaryset_eq:
```
```   112      "f {} = 0 \<Longrightarrow> suminf (\<lambda>n. f (binaryset A B n)) = f A + f B"
```
```   113   by (metis binaryset_sums sums_unique)
```
```   114
```
```   115 lemma binaryset_psuminf:
```
```   116   assumes "f {} = 0"
```
```   117   shows "(\<Sum>\<^isub>\<infinity> n. f (binaryset A B n)) = f A + f B" (is "?suminf = ?sum")
```
```   118 proof -
```
```   119   have *: "{..<2} = {0, 1::nat}" by auto
```
```   120   have "\<forall>n\<ge>2. f (binaryset A B n) = 0"
```
```   121     unfolding binaryset_def
```
```   122     using assms by auto
```
```   123   hence "?suminf = (\<Sum>N<2. f (binaryset A B N))"
```
```   124     by (rule psuminf_finite)
```
```   125   also have "... = ?sum" unfolding * binaryset_def
```
```   126     by simp
```
```   127   finally show ?thesis .
```
```   128 qed
```
```   129
```
```   130 subsection {* Lambda Systems *}
```
```   131
```
```   132 lemma (in algebra) lambda_system_eq:
```
```   133     "lambda_system M f =
```
```   134         {l. l \<in> sets M & (\<forall>x \<in> sets M. f (x \<inter> l) + f (x - l) = f x)}"
```
```   135 proof -
```
```   136   have [simp]: "!!l x. l \<in> sets M \<Longrightarrow> x \<in> sets M \<Longrightarrow> (space M - l) \<inter> x = x - l"
```
```   137     by (metis Int_Diff Int_absorb1 Int_commute sets_into_space)
```
```   138   show ?thesis
```
```   139     by (auto simp add: lambda_system_def) (metis Int_commute)+
```
```   140 qed
```
```   141
```
```   142 lemma (in algebra) lambda_system_empty:
```
```   143   "positive f \<Longrightarrow> {} \<in> lambda_system M f"
```
```   144   by (auto simp add: positive_def lambda_system_eq)
```
```   145
```
```   146 lemma lambda_system_sets:
```
```   147     "x \<in> lambda_system M f \<Longrightarrow> x \<in> sets M"
```
```   148   by (simp add:  lambda_system_def)
```
```   149
```
```   150 lemma (in algebra) lambda_system_Compl:
```
```   151   fixes f:: "'a set \<Rightarrow> pinfreal"
```
```   152   assumes x: "x \<in> lambda_system M f"
```
```   153   shows "space M - x \<in> lambda_system M f"
```
```   154   proof -
```
```   155     have "x \<subseteq> space M"
```
```   156       by (metis sets_into_space lambda_system_sets x)
```
```   157     hence "space M - (space M - x) = x"
```
```   158       by (metis double_diff equalityE)
```
```   159     with x show ?thesis
```
```   160       by (force simp add: lambda_system_def ac_simps)
```
```   161   qed
```
```   162
```
```   163 lemma (in algebra) lambda_system_Int:
```
```   164   fixes f:: "'a set \<Rightarrow> pinfreal"
```
```   165   assumes xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f"
```
```   166   shows "x \<inter> y \<in> lambda_system M f"
```
```   167   proof -
```
```   168     from xl yl show ?thesis
```
```   169       proof (auto simp add: positive_def lambda_system_eq Int)
```
```   170         fix u
```
```   171         assume x: "x \<in> sets M" and y: "y \<in> sets M" and u: "u \<in> sets M"
```
```   172            and fx: "\<forall>z\<in>sets M. f (z \<inter> x) + f (z - x) = f z"
```
```   173            and fy: "\<forall>z\<in>sets M. f (z \<inter> y) + f (z - y) = f z"
```
```   174         have "u - x \<inter> y \<in> sets M"
```
```   175           by (metis Diff Diff_Int Un u x y)
```
```   176         moreover
```
```   177         have "(u - (x \<inter> y)) \<inter> y = u \<inter> y - x" by blast
```
```   178         moreover
```
```   179         have "u - x \<inter> y - y = u - y" by blast
```
```   180         ultimately
```
```   181         have ey: "f (u - x \<inter> y) = f (u \<inter> y - x) + f (u - y)" using fy
```
```   182           by force
```
```   183         have "f (u \<inter> (x \<inter> y)) + f (u - x \<inter> y)
```
```   184               = (f (u \<inter> (x \<inter> y)) + f (u \<inter> y - x)) + f (u - y)"
```
```   185           by (simp add: ey ac_simps)
```
```   186         also have "... =  (f ((u \<inter> y) \<inter> x) + f (u \<inter> y - x)) + f (u - y)"
```
```   187           by (simp add: Int_ac)
```
```   188         also have "... = f (u \<inter> y) + f (u - y)"
```
```   189           using fx [THEN bspec, of "u \<inter> y"] Int y u
```
```   190           by force
```
```   191         also have "... = f u"
```
```   192           by (metis fy u)
```
```   193         finally show "f (u \<inter> (x \<inter> y)) + f (u - x \<inter> y) = f u" .
```
```   194       qed
```
```   195   qed
```
```   196
```
```   197
```
```   198 lemma (in algebra) lambda_system_Un:
```
```   199   fixes f:: "'a set \<Rightarrow> pinfreal"
```
```   200   assumes xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f"
```
```   201   shows "x \<union> y \<in> lambda_system M f"
```
```   202 proof -
```
```   203   have "(space M - x) \<inter> (space M - y) \<in> sets M"
```
```   204     by (metis Diff_Un Un compl_sets lambda_system_sets xl yl)
```
```   205   moreover
```
```   206   have "x \<union> y = space M - ((space M - x) \<inter> (space M - y))"
```
```   207     by auto  (metis subsetD lambda_system_sets sets_into_space xl yl)+
```
```   208   ultimately show ?thesis
```
```   209     by (metis lambda_system_Compl lambda_system_Int xl yl)
```
```   210 qed
```
```   211
```
```   212 lemma (in algebra) lambda_system_algebra:
```
```   213   "positive f \<Longrightarrow> algebra (M (|sets := lambda_system M f|))"
```
```   214   apply (auto simp add: algebra_def)
```
```   215   apply (metis lambda_system_sets set_mp sets_into_space)
```
```   216   apply (metis lambda_system_empty)
```
```   217   apply (metis lambda_system_Compl)
```
```   218   apply (metis lambda_system_Un)
```
```   219   done
```
```   220
```
```   221 lemma (in algebra) lambda_system_strong_additive:
```
```   222   assumes z: "z \<in> sets M" and disj: "x \<inter> y = {}"
```
```   223       and xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f"
```
```   224   shows "f (z \<inter> (x \<union> y)) = f (z \<inter> x) + f (z \<inter> y)"
```
```   225   proof -
```
```   226     have "z \<inter> x = (z \<inter> (x \<union> y)) \<inter> x" using disj by blast
```
```   227     moreover
```
```   228     have "z \<inter> y = (z \<inter> (x \<union> y)) - x" using disj by blast
```
```   229     moreover
```
```   230     have "(z \<inter> (x \<union> y)) \<in> sets M"
```
```   231       by (metis Int Un lambda_system_sets xl yl z)
```
```   232     ultimately show ?thesis using xl yl
```
```   233       by (simp add: lambda_system_eq)
```
```   234   qed
```
```   235
```
```   236 lemma (in algebra) lambda_system_additive:
```
```   237      "additive (M (|sets := lambda_system M f|)) f"
```
```   238   proof (auto simp add: additive_def)
```
```   239     fix x and y
```
```   240     assume disj: "x \<inter> y = {}"
```
```   241        and xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f"
```
```   242     hence  "x \<in> sets M" "y \<in> sets M" by (blast intro: lambda_system_sets)+
```
```   243     thus "f (x \<union> y) = f x + f y"
```
```   244       using lambda_system_strong_additive [OF top disj xl yl]
```
```   245       by (simp add: Un)
```
```   246   qed
```
```   247
```
```   248
```
```   249 lemma (in algebra) countably_subadditive_subadditive:
```
```   250   assumes f: "positive f" and cs: "countably_subadditive M f"
```
```   251   shows  "subadditive M f"
```
```   252 proof (auto simp add: subadditive_def)
```
```   253   fix x y
```
```   254   assume x: "x \<in> sets M" and y: "y \<in> sets M" and "x \<inter> y = {}"
```
```   255   hence "disjoint_family (binaryset x y)"
```
```   256     by (auto simp add: disjoint_family_on_def binaryset_def)
```
```   257   hence "range (binaryset x y) \<subseteq> sets M \<longrightarrow>
```
```   258          (\<Union>i. binaryset x y i) \<in> sets M \<longrightarrow>
```
```   259          f (\<Union>i. binaryset x y i) \<le> (\<Sum>\<^isub>\<infinity> n. f (binaryset x y n))"
```
```   260     using cs by (simp add: countably_subadditive_def)
```
```   261   hence "{x,y,{}} \<subseteq> sets M \<longrightarrow> x \<union> y \<in> sets M \<longrightarrow>
```
```   262          f (x \<union> y) \<le> (\<Sum>\<^isub>\<infinity> n. f (binaryset x y n))"
```
```   263     by (simp add: range_binaryset_eq UN_binaryset_eq)
```
```   264   thus "f (x \<union> y) \<le>  f x + f y" using f x y
```
```   265     by (auto simp add: Un o_def binaryset_psuminf positive_def)
```
```   266 qed
```
```   267
```
```   268 lemma (in algebra) additive_sum:
```
```   269   fixes A:: "nat \<Rightarrow> 'a set"
```
```   270   assumes f: "positive f" and ad: "additive M f"
```
```   271       and A: "range A \<subseteq> sets M"
```
```   272       and disj: "disjoint_family A"
```
```   273   shows  "setsum (f \<circ> A) {0..<n} = f (\<Union>i\<in>{0..<n}. A i)"
```
```   274 proof (induct n)
```
```   275   case 0 show ?case using f by (simp add: positive_def)
```
```   276 next
```
```   277   case (Suc n)
```
```   278   have "A n \<inter> (\<Union>i\<in>{0..<n}. A i) = {}" using disj
```
```   279     by (auto simp add: disjoint_family_on_def neq_iff) blast
```
```   280   moreover
```
```   281   have "A n \<in> sets M" using A by blast
```
```   282   moreover have "(\<Union>i\<in>{0..<n}. A i) \<in> sets M"
```
```   283     by (metis A UNION_in_sets atLeast0LessThan)
```
```   284   moreover
```
```   285   ultimately have "f (A n \<union> (\<Union>i\<in>{0..<n}. A i)) = f (A n) + f(\<Union>i\<in>{0..<n}. A i)"
```
```   286     using ad UNION_in_sets A by (auto simp add: additive_def)
```
```   287   with Suc.hyps show ?case using ad
```
```   288     by (auto simp add: atLeastLessThanSuc additive_def)
```
```   289 qed
```
```   290
```
```   291
```
```   292 lemma countably_subadditiveD:
```
```   293   "countably_subadditive M f \<Longrightarrow> range A \<subseteq> sets M \<Longrightarrow> disjoint_family A \<Longrightarrow>
```
```   294    (\<Union>i. A i) \<in> sets M \<Longrightarrow> f (\<Union>i. A i) \<le> psuminf (f o A)"
```
```   295   by (auto simp add: countably_subadditive_def o_def)
```
```   296
```
```   297 lemma (in algebra) increasing_additive_bound:
```
```   298   fixes A:: "nat \<Rightarrow> 'a set" and  f :: "'a set \<Rightarrow> pinfreal"
```
```   299   assumes f: "positive f" and ad: "additive M f"
```
```   300       and inc: "increasing M f"
```
```   301       and A: "range A \<subseteq> sets M"
```
```   302       and disj: "disjoint_family A"
```
```   303   shows  "psuminf (f \<circ> A) \<le> f (space M)"
```
```   304 proof (safe intro!: psuminf_bound)
```
```   305   fix N
```
```   306   have "setsum (f \<circ> A) {0..<N} = f (\<Union>i\<in>{0..<N}. A i)"
```
```   307     by (rule additive_sum [OF f ad A disj])
```
```   308   also have "... \<le> f (space M)" using space_closed A
```
```   309     by (blast intro: increasingD [OF inc] UNION_in_sets top)
```
```   310   finally show "setsum (f \<circ> A) {..<N} \<le> f (space M)" by (simp add: atLeast0LessThan)
```
```   311 qed
```
```   312
```
```   313 lemma lambda_system_increasing:
```
```   314    "increasing M f \<Longrightarrow> increasing (M (|sets := lambda_system M f|)) f"
```
```   315   by (simp add: increasing_def lambda_system_def)
```
```   316
```
```   317 lemma (in algebra) lambda_system_strong_sum:
```
```   318   fixes A:: "nat \<Rightarrow> 'a set" and f :: "'a set \<Rightarrow> pinfreal"
```
```   319   assumes f: "positive f" and a: "a \<in> sets M"
```
```   320       and A: "range A \<subseteq> lambda_system M f"
```
```   321       and disj: "disjoint_family A"
```
```   322   shows  "(\<Sum>i = 0..<n. f (a \<inter>A i)) = f (a \<inter> (\<Union>i\<in>{0..<n}. A i))"
```
```   323 proof (induct n)
```
```   324   case 0 show ?case using f by (simp add: positive_def)
```
```   325 next
```
```   326   case (Suc n)
```
```   327   have 2: "A n \<inter> UNION {0..<n} A = {}" using disj
```
```   328     by (force simp add: disjoint_family_on_def neq_iff)
```
```   329   have 3: "A n \<in> lambda_system M f" using A
```
```   330     by blast
```
```   331   have 4: "UNION {0..<n} A \<in> lambda_system M f"
```
```   332     using A algebra.UNION_in_sets [OF local.lambda_system_algebra, of f, OF f]
```
```   333     by simp
```
```   334   from Suc.hyps show ?case
```
```   335     by (simp add: atLeastLessThanSuc lambda_system_strong_additive [OF a 2 3 4])
```
```   336 qed
```
```   337
```
```   338
```
```   339 lemma (in sigma_algebra) lambda_system_caratheodory:
```
```   340   assumes oms: "outer_measure_space M f"
```
```   341       and A: "range A \<subseteq> lambda_system M f"
```
```   342       and disj: "disjoint_family A"
```
```   343   shows  "(\<Union>i. A i) \<in> lambda_system M f \<and> psuminf (f \<circ> A) = f (\<Union>i. A i)"
```
```   344 proof -
```
```   345   have pos: "positive f" and inc: "increasing M f"
```
```   346    and csa: "countably_subadditive M f"
```
```   347     by (metis oms outer_measure_space_def)+
```
```   348   have sa: "subadditive M f"
```
```   349     by (metis countably_subadditive_subadditive csa pos)
```
```   350   have A': "range A \<subseteq> sets (M(|sets := lambda_system M f|))" using A
```
```   351     by simp
```
```   352   have alg_ls: "algebra (M(|sets := lambda_system M f|))"
```
```   353     by (rule lambda_system_algebra) (rule pos)
```
```   354   have A'': "range A \<subseteq> sets M"
```
```   355      by (metis A image_subset_iff lambda_system_sets)
```
```   356
```
```   357   have U_in: "(\<Union>i. A i) \<in> sets M"
```
```   358     by (metis A'' countable_UN)
```
```   359   have U_eq: "f (\<Union>i. A i) = psuminf (f o A)"
```
```   360     proof (rule antisym)
```
```   361       show "f (\<Union>i. A i) \<le> psuminf (f \<circ> A)"
```
```   362         by (rule countably_subadditiveD [OF csa A'' disj U_in])
```
```   363       show "psuminf (f \<circ> A) \<le> f (\<Union>i. A i)"
```
```   364         by (rule psuminf_bound, unfold atLeast0LessThan[symmetric])
```
```   365            (metis algebra.additive_sum [OF alg_ls] pos disj UN_Un Un_UNIV_right
```
```   366                   lambda_system_additive subset_Un_eq increasingD [OF inc]
```
```   367                   A' A'' UNION_in_sets U_in)
```
```   368     qed
```
```   369   {
```
```   370     fix a
```
```   371     assume a [iff]: "a \<in> sets M"
```
```   372     have "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) = f a"
```
```   373     proof -
```
```   374       show ?thesis
```
```   375       proof (rule antisym)
```
```   376         have "range (\<lambda>i. a \<inter> A i) \<subseteq> sets M" using A''
```
```   377           by blast
```
```   378         moreover
```
```   379         have "disjoint_family (\<lambda>i. a \<inter> A i)" using disj
```
```   380           by (auto simp add: disjoint_family_on_def)
```
```   381         moreover
```
```   382         have "a \<inter> (\<Union>i. A i) \<in> sets M"
```
```   383           by (metis Int U_in a)
```
```   384         ultimately
```
```   385         have "f (a \<inter> (\<Union>i. A i)) \<le> psuminf (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A)"
```
```   386           using countably_subadditiveD [OF csa, of "(\<lambda>i. a \<inter> A i)"]
```
```   387           by (simp add: o_def)
```
```   388         hence "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) \<le>
```
```   389             psuminf (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A) + f (a - (\<Union>i. A i))"
```
```   390           by (rule add_right_mono)
```
```   391         moreover
```
```   392         have "psuminf (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A) + f (a - (\<Union>i. A i)) \<le> f a"
```
```   393           proof (safe intro!: psuminf_bound_add)
```
```   394             fix n
```
```   395             have UNION_in: "(\<Union>i\<in>{0..<n}. A i) \<in> sets M"
```
```   396               by (metis A'' UNION_in_sets)
```
```   397             have le_fa: "f (UNION {0..<n} A \<inter> a) \<le> f a" using A''
```
```   398               by (blast intro: increasingD [OF inc] A'' UNION_in_sets)
```
```   399             have ls: "(\<Union>i\<in>{0..<n}. A i) \<in> lambda_system M f"
```
```   400               using algebra.UNION_in_sets [OF lambda_system_algebra [of f, OF pos]]
```
```   401               by (simp add: A)
```
```   402             hence eq_fa: "f a = f (a \<inter> (\<Union>i\<in>{0..<n}. A i)) + f (a - (\<Union>i\<in>{0..<n}. A i))"
```
```   403               by (simp add: lambda_system_eq UNION_in)
```
```   404             have "f (a - (\<Union>i. A i)) \<le> f (a - (\<Union>i\<in>{0..<n}. A i))"
```
```   405               by (blast intro: increasingD [OF inc] UNION_eq_Union_image
```
```   406                                UNION_in U_in)
```
```   407             thus "setsum (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A) {..<n} + f (a - (\<Union>i. A i)) \<le> f a"
```
```   408               by (simp add: lambda_system_strong_sum pos A disj eq_fa add_left_mono atLeast0LessThan[symmetric])
```
```   409           qed
```
```   410         ultimately show "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) \<le> f a"
```
```   411           by (rule order_trans)
```
```   412       next
```
```   413         have "f a \<le> f (a \<inter> (\<Union>i. A i) \<union> (a - (\<Union>i. A i)))"
```
```   414           by (blast intro:  increasingD [OF inc] U_in)
```
```   415         also have "... \<le>  f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i))"
```
```   416           by (blast intro: subadditiveD [OF sa] U_in)
```
```   417         finally show "f a \<le> f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i))" .
```
```   418         qed
```
```   419      qed
```
```   420   }
```
```   421   thus  ?thesis
```
```   422     by (simp add: lambda_system_eq sums_iff U_eq U_in)
```
```   423 qed
```
```   424
```
```   425 lemma (in sigma_algebra) caratheodory_lemma:
```
```   426   assumes oms: "outer_measure_space M f"
```
```   427   shows "measure_space (|space = space M, sets = lambda_system M f|) f"
```
```   428 proof -
```
```   429   have pos: "positive f"
```
```   430     by (metis oms outer_measure_space_def)
```
```   431   have alg: "algebra (|space = space M, sets = lambda_system M f|)"
```
```   432     using lambda_system_algebra [of f, OF pos]
```
```   433     by (simp add: algebra_def)
```
```   434   then moreover
```
```   435   have "sigma_algebra (|space = space M, sets = lambda_system M f|)"
```
```   436     using lambda_system_caratheodory [OF oms]
```
```   437     by (simp add: sigma_algebra_disjoint_iff)
```
```   438   moreover
```
```   439   have "measure_space_axioms (|space = space M, sets = lambda_system M f|) f"
```
```   440     using pos lambda_system_caratheodory [OF oms]
```
```   441     by (simp add: measure_space_axioms_def positive_def lambda_system_sets
```
```   442                   countably_additive_def o_def)
```
```   443   ultimately
```
```   444   show ?thesis
```
```   445     by intro_locales (auto simp add: sigma_algebra_def)
```
```   446 qed
```
```   447
```
```   448 lemma (in algebra) additive_increasing:
```
```   449   assumes posf: "positive f" and addf: "additive M f"
```
```   450   shows "increasing M f"
```
```   451 proof (auto simp add: increasing_def)
```
```   452   fix x y
```
```   453   assume xy: "x \<in> sets M" "y \<in> sets M" "x \<subseteq> y"
```
```   454   have "f x \<le> f x + f (y-x)" ..
```
```   455   also have "... = f (x \<union> (y-x))" using addf
```
```   456     by (auto simp add: additive_def) (metis Diff_disjoint Un_Diff_cancel Diff xy(1,2))
```
```   457   also have "... = f y"
```
```   458     by (metis Un_Diff_cancel Un_absorb1 xy(3))
```
```   459   finally show "f x \<le> f y" .
```
```   460 qed
```
```   461
```
```   462 lemma (in algebra) countably_additive_additive:
```
```   463   assumes posf: "positive f" and ca: "countably_additive M f"
```
```   464   shows "additive M f"
```
```   465 proof (auto simp add: additive_def)
```
```   466   fix x y
```
```   467   assume x: "x \<in> sets M" and y: "y \<in> sets M" and "x \<inter> y = {}"
```
```   468   hence "disjoint_family (binaryset x y)"
```
```   469     by (auto simp add: disjoint_family_on_def binaryset_def)
```
```   470   hence "range (binaryset x y) \<subseteq> sets M \<longrightarrow>
```
```   471          (\<Union>i. binaryset x y i) \<in> sets M \<longrightarrow>
```
```   472          f (\<Union>i. binaryset x y i) = (\<Sum>\<^isub>\<infinity> n. f (binaryset x y n))"
```
```   473     using ca
```
```   474     by (simp add: countably_additive_def)
```
```   475   hence "{x,y,{}} \<subseteq> sets M \<longrightarrow> x \<union> y \<in> sets M \<longrightarrow>
```
```   476          f (x \<union> y) = (\<Sum>\<^isub>\<infinity> n. f (binaryset x y n))"
```
```   477     by (simp add: range_binaryset_eq UN_binaryset_eq)
```
```   478   thus "f (x \<union> y) = f x + f y" using posf x y
```
```   479     by (auto simp add: Un binaryset_psuminf positive_def)
```
```   480 qed
```
```   481
```
```   482 lemma inf_measure_nonempty:
```
```   483   assumes f: "positive f" and b: "b \<in> sets M" and a: "a \<subseteq> b" "{} \<in> sets M"
```
```   484   shows "f b \<in> measure_set M f a"
```
```   485 proof -
```
```   486   have "psuminf (f \<circ> (\<lambda>i. {})(0 := b)) = setsum (f \<circ> (\<lambda>i. {})(0 := b)) {..<1::nat}"
```
```   487     by (rule psuminf_finite) (simp add: f[unfolded positive_def])
```
```   488   also have "... = f b"
```
```   489     by simp
```
```   490   finally have "psuminf (f \<circ> (\<lambda>i. {})(0 := b)) = f b" .
```
```   491   thus ?thesis using assms
```
```   492     by (auto intro!: exI [of _ "(\<lambda>i. {})(0 := b)"]
```
```   493              simp: measure_set_def disjoint_family_on_def split_if_mem2 comp_def)
```
```   494 qed
```
```   495
```
```   496 lemma (in algebra) inf_measure_agrees:
```
```   497   assumes posf: "positive f" and ca: "countably_additive M f"
```
```   498       and s: "s \<in> sets M"
```
```   499   shows "Inf (measure_set M f s) = f s"
```
```   500   unfolding Inf_pinfreal_def
```
```   501 proof (safe intro!: Greatest_equality)
```
```   502   fix z
```
```   503   assume z: "z \<in> measure_set M f s"
```
```   504   from this obtain A where
```
```   505     A: "range A \<subseteq> sets M" and disj: "disjoint_family A"
```
```   506     and "s \<subseteq> (\<Union>x. A x)" and si: "psuminf (f \<circ> A) = z"
```
```   507     by (auto simp add: measure_set_def comp_def)
```
```   508   hence seq: "s = (\<Union>i. A i \<inter> s)" by blast
```
```   509   have inc: "increasing M f"
```
```   510     by (metis additive_increasing ca countably_additive_additive posf)
```
```   511   have sums: "psuminf (\<lambda>i. f (A i \<inter> s)) = f (\<Union>i. A i \<inter> s)"
```
```   512     proof (rule countably_additiveD [OF ca])
```
```   513       show "range (\<lambda>n. A n \<inter> s) \<subseteq> sets M" using A s
```
```   514         by blast
```
```   515       show "disjoint_family (\<lambda>n. A n \<inter> s)" using disj
```
```   516         by (auto simp add: disjoint_family_on_def)
```
```   517       show "(\<Union>i. A i \<inter> s) \<in> sets M" using A s
```
```   518         by (metis UN_extend_simps(4) s seq)
```
```   519     qed
```
```   520   hence "f s = psuminf (\<lambda>i. f (A i \<inter> s))"
```
```   521     using seq [symmetric] by (simp add: sums_iff)
```
```   522   also have "... \<le> psuminf (f \<circ> A)"
```
```   523     proof (rule psuminf_le)
```
```   524       fix n show "f (A n \<inter> s) \<le> (f \<circ> A) n" using A s
```
```   525         by (force intro: increasingD [OF inc])
```
```   526     qed
```
```   527   also have "... = z" by (rule si)
```
```   528   finally show "f s \<le> z" .
```
```   529 next
```
```   530   fix y
```
```   531   assume y: "\<forall>u \<in> measure_set M f s. y \<le> u"
```
```   532   thus "y \<le> f s"
```
```   533     by (blast intro: inf_measure_nonempty [of f, OF posf s subset_refl])
```
```   534 qed
```
```   535
```
```   536 lemma (in algebra) inf_measure_empty:
```
```   537   assumes posf: "positive f"  "{} \<in> sets M"
```
```   538   shows "Inf (measure_set M f {}) = 0"
```
```   539 proof (rule antisym)
```
```   540   show "Inf (measure_set M f {}) \<le> 0"
```
```   541     by (metis complete_lattice_class.Inf_lower `{} \<in> sets M` inf_measure_nonempty[OF posf] subset_refl posf[unfolded positive_def])
```
```   542 qed simp
```
```   543
```
```   544 lemma (in algebra) inf_measure_positive:
```
```   545   "positive f \<Longrightarrow>
```
```   546    positive (\<lambda>x. Inf (measure_set M f x))"
```
```   547   by (simp add: positive_def inf_measure_empty)
```
```   548
```
```   549 lemma (in algebra) inf_measure_increasing:
```
```   550   assumes posf: "positive f"
```
```   551   shows "increasing (| space = space M, sets = Pow (space M) |)
```
```   552                     (\<lambda>x. Inf (measure_set M f x))"
```
```   553 apply (auto simp add: increasing_def)
```
```   554 apply (rule complete_lattice_class.Inf_greatest)
```
```   555 apply (rule complete_lattice_class.Inf_lower)
```
```   556 apply (clarsimp simp add: measure_set_def, rule_tac x=A in exI, blast)
```
```   557 done
```
```   558
```
```   559
```
```   560 lemma (in algebra) inf_measure_le:
```
```   561   assumes posf: "positive f" and inc: "increasing M f"
```
```   562       and x: "x \<in> {r . \<exists>A. range A \<subseteq> sets M \<and> s \<subseteq> (\<Union>i. A i) \<and> psuminf (f \<circ> A) = r}"
```
```   563   shows "Inf (measure_set M f s) \<le> x"
```
```   564 proof -
```
```   565   from x
```
```   566   obtain A where A: "range A \<subseteq> sets M" and ss: "s \<subseteq> (\<Union>i. A i)"
```
```   567              and xeq: "psuminf (f \<circ> A) = x"
```
```   568     by auto
```
```   569   have dA: "range (disjointed A) \<subseteq> sets M"
```
```   570     by (metis A range_disjointed_sets)
```
```   571   have "\<forall>n.(f o disjointed A) n \<le> (f \<circ> A) n" unfolding comp_def
```
```   572     by (metis increasingD [OF inc] UNIV_I dA image_subset_iff disjointed_subset A comp_def)
```
```   573   hence sda: "psuminf (f o disjointed A) \<le> psuminf (f \<circ> A)"
```
```   574     by (blast intro: psuminf_le)
```
```   575   hence ley: "psuminf (f o disjointed A) \<le> x"
```
```   576     by (metis xeq)
```
```   577   hence y: "psuminf (f o disjointed A) \<in> measure_set M f s"
```
```   578     apply (auto simp add: measure_set_def)
```
```   579     apply (rule_tac x="disjointed A" in exI)
```
```   580     apply (simp add: disjoint_family_disjointed UN_disjointed_eq ss dA comp_def)
```
```   581     done
```
```   582   show ?thesis
```
```   583     by (blast intro: y order_trans [OF _ ley] posf complete_lattice_class.Inf_lower)
```
```   584 qed
```
```   585
```
```   586 lemma (in algebra) inf_measure_close:
```
```   587   assumes posf: "positive f" and e: "0 < e" and ss: "s \<subseteq> (space M)"
```
```   588   shows "\<exists>A. range A \<subseteq> sets M \<and> disjoint_family A \<and> s \<subseteq> (\<Union>i. A i) \<and>
```
```   589                psuminf (f \<circ> A) \<le> Inf (measure_set M f s) + e"
```
```   590 proof (cases "Inf (measure_set M f s) = \<omega>")
```
```   591   case False
```
```   592   obtain l where "l \<in> measure_set M f s" "l \<le> Inf (measure_set M f s) + e"
```
```   593     using Inf_close[OF False e] by auto
```
```   594   thus ?thesis
```
```   595     by (auto intro!: exI[of _ l] simp: measure_set_def comp_def)
```
```   596 next
```
```   597   case True
```
```   598   have "measure_set M f s \<noteq> {}"
```
```   599     by (metis emptyE ss inf_measure_nonempty [of f, OF posf top _ empty_sets])
```
```   600   then obtain l where "l \<in> measure_set M f s" by auto
```
```   601   moreover from True have "l \<le> Inf (measure_set M f s) + e" by simp
```
```   602   ultimately show ?thesis
```
```   603     by (auto intro!: exI[of _ l] simp: measure_set_def comp_def)
```
```   604 qed
```
```   605
```
```   606 lemma (in algebra) inf_measure_countably_subadditive:
```
```   607   assumes posf: "positive f" and inc: "increasing M f"
```
```   608   shows "countably_subadditive (| space = space M, sets = Pow (space M) |)
```
```   609                   (\<lambda>x. Inf (measure_set M f x))"
```
```   610   unfolding countably_subadditive_def o_def
```
```   611 proof (safe, simp, rule pinfreal_le_epsilon)
```
```   612   fix A :: "nat \<Rightarrow> 'a set" and e :: pinfreal
```
```   613
```
```   614   let "?outer n" = "Inf (measure_set M f (A n))"
```
```   615   assume A: "range A \<subseteq> Pow (space M)"
```
```   616      and disj: "disjoint_family A"
```
```   617      and sb: "(\<Union>i. A i) \<subseteq> space M"
```
```   618      and e: "0 < e"
```
```   619   hence "\<exists>BB. \<forall>n. range (BB n) \<subseteq> sets M \<and> disjoint_family (BB n) \<and>
```
```   620                    A n \<subseteq> (\<Union>i. BB n i) \<and>
```
```   621                    psuminf (f o BB n) \<le> ?outer n + e * (1/2)^(Suc n)"
```
```   622     apply (safe intro!: choice inf_measure_close [of f, OF posf _])
```
```   623     using e sb by (cases e, auto simp add: not_le mult_pos_pos)
```
```   624   then obtain BB
```
```   625     where BB: "\<And>n. (range (BB n) \<subseteq> sets M)"
```
```   626       and disjBB: "\<And>n. disjoint_family (BB n)"
```
```   627       and sbBB: "\<And>n. A n \<subseteq> (\<Union>i. BB n i)"
```
```   628       and BBle: "\<And>n. psuminf (f o BB n) \<le> ?outer n + e * (1/2)^(Suc n)"
```
```   629     by auto blast
```
```   630   have sll: "(\<Sum>\<^isub>\<infinity> n. psuminf (f o BB n)) \<le> psuminf ?outer + e"
```
```   631     proof -
```
```   632       have "(\<Sum>\<^isub>\<infinity> n. psuminf (f o BB n)) \<le> (\<Sum>\<^isub>\<infinity> n. ?outer n + e*(1/2) ^ Suc n)"
```
```   633         by (rule psuminf_le[OF BBle])
```
```   634       also have "... = psuminf ?outer + e"
```
```   635         using psuminf_half_series by simp
```
```   636       finally show ?thesis .
```
```   637     qed
```
```   638   def C \<equiv> "(split BB) o prod_decode"
```
```   639   have C: "!!n. C n \<in> sets M"
```
```   640     apply (rule_tac p="prod_decode n" in PairE)
```
```   641     apply (simp add: C_def)
```
```   642     apply (metis BB subsetD rangeI)
```
```   643     done
```
```   644   have sbC: "(\<Union>i. A i) \<subseteq> (\<Union>i. C i)"
```
```   645     proof (auto simp add: C_def)
```
```   646       fix x i
```
```   647       assume x: "x \<in> A i"
```
```   648       with sbBB [of i] obtain j where "x \<in> BB i j"
```
```   649         by blast
```
```   650       thus "\<exists>i. x \<in> split BB (prod_decode i)"
```
```   651         by (metis prod_encode_inverse prod.cases)
```
```   652     qed
```
```   653   have "(f \<circ> C) = (f \<circ> (\<lambda>(x, y). BB x y)) \<circ> prod_decode"
```
```   654     by (rule ext)  (auto simp add: C_def)
```
```   655   moreover have "psuminf ... = (\<Sum>\<^isub>\<infinity> n. psuminf (f o BB n))" using BBle
```
```   656     by (force intro!: psuminf_2dimen simp: o_def)
```
```   657   ultimately have Csums: "psuminf (f \<circ> C) = (\<Sum>\<^isub>\<infinity> n. psuminf (f o BB n))" by simp
```
```   658   have "Inf (measure_set M f (\<Union>i. A i)) \<le> (\<Sum>\<^isub>\<infinity> n. psuminf (f o BB n))"
```
```   659     apply (rule inf_measure_le [OF posf(1) inc], auto)
```
```   660     apply (rule_tac x="C" in exI)
```
```   661     apply (auto simp add: C sbC Csums)
```
```   662     done
```
```   663   also have "... \<le> (\<Sum>\<^isub>\<infinity>n. Inf (measure_set M f (A n))) + e" using sll
```
```   664     by blast
```
```   665   finally show "Inf (measure_set M f (\<Union>i. A i)) \<le> psuminf ?outer + e" .
```
```   666 qed
```
```   667
```
```   668 lemma (in algebra) inf_measure_outer:
```
```   669   "\<lbrakk> positive f ; increasing M f \<rbrakk>
```
```   670    \<Longrightarrow> outer_measure_space (| space = space M, sets = Pow (space M) |)
```
```   671                           (\<lambda>x. Inf (measure_set M f x))"
```
```   672   by (simp add: outer_measure_space_def inf_measure_empty
```
```   673                 inf_measure_increasing inf_measure_countably_subadditive positive_def)
```
```   674
```
```   675 (*MOVE UP*)
```
```   676
```
```   677 lemma (in algebra) algebra_subset_lambda_system:
```
```   678   assumes posf: "positive f" and inc: "increasing M f"
```
```   679       and add: "additive M f"
```
```   680   shows "sets M \<subseteq> lambda_system (| space = space M, sets = Pow (space M) |)
```
```   681                                 (\<lambda>x. Inf (measure_set M f x))"
```
```   682 proof (auto dest: sets_into_space
```
```   683             simp add: algebra.lambda_system_eq [OF algebra_Pow])
```
```   684   fix x s
```
```   685   assume x: "x \<in> sets M"
```
```   686      and s: "s \<subseteq> space M"
```
```   687   have [simp]: "!!x. x \<in> sets M \<Longrightarrow> s \<inter> (space M - x) = s-x" using s
```
```   688     by blast
```
```   689   have "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))
```
```   690         \<le> Inf (measure_set M f s)"
```
```   691     proof (rule pinfreal_le_epsilon)
```
```   692       fix e :: pinfreal
```
```   693       assume e: "0 < e"
```
```   694       from inf_measure_close [of f, OF posf e s]
```
```   695       obtain A where A: "range A \<subseteq> sets M" and disj: "disjoint_family A"
```
```   696                  and sUN: "s \<subseteq> (\<Union>i. A i)"
```
```   697                  and l: "psuminf (f \<circ> A) \<le> Inf (measure_set M f s) + e"
```
```   698         by auto
```
```   699       have [simp]: "!!x. x \<in> sets M \<Longrightarrow>
```
```   700                       (f o (\<lambda>z. z \<inter> (space M - x)) o A) = (f o (\<lambda>z. z - x) o A)"
```
```   701         by (rule ext, simp, metis A Int_Diff Int_space_eq2 range_subsetD)
```
```   702       have  [simp]: "!!n. f (A n \<inter> x) + f (A n - x) = f (A n)"
```
```   703         by (subst additiveD [OF add, symmetric])
```
```   704            (auto simp add: x range_subsetD [OF A] Int_Diff_Un Int_Diff_disjoint)
```
```   705       { fix u
```
```   706         assume u: "u \<in> sets M"
```
```   707         have [simp]: "\<And>n. f (A n \<inter> u) \<le> f (A n)"
```
```   708           by (simp add: increasingD [OF inc] u Int range_subsetD [OF A])
```
```   709         have 2: "Inf (measure_set M f (s \<inter> u)) \<le> psuminf (f \<circ> (\<lambda>z. z \<inter> u) \<circ> A)"
```
```   710           proof (rule complete_lattice_class.Inf_lower)
```
```   711             show "psuminf (f \<circ> (\<lambda>z. z \<inter> u) \<circ> A) \<in> measure_set M f (s \<inter> u)"
```
```   712               apply (simp add: measure_set_def)
```
```   713               apply (rule_tac x="(\<lambda>z. z \<inter> u) o A" in exI)
```
```   714               apply (auto simp add: disjoint_family_subset [OF disj] o_def)
```
```   715               apply (blast intro: u range_subsetD [OF A])
```
```   716               apply (blast dest: subsetD [OF sUN])
```
```   717               done
```
```   718           qed
```
```   719       } note lesum = this
```
```   720       have inf1: "Inf (measure_set M f (s\<inter>x)) \<le> psuminf (f o (\<lambda>z. z\<inter>x) o A)"
```
```   721         and inf2: "Inf (measure_set M f (s \<inter> (space M - x)))
```
```   722                    \<le> psuminf (f o (\<lambda>z. z \<inter> (space M - x)) o A)"
```
```   723         by (metis Diff lesum top x)+
```
```   724       hence "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))
```
```   725            \<le> psuminf (f o (\<lambda>s. s\<inter>x) o A) + psuminf (f o (\<lambda>s. s-x) o A)"
```
```   726         by (simp add: x add_mono)
```
```   727       also have "... \<le> psuminf (f o A)"
```
```   728         by (simp add: x psuminf_add[symmetric] o_def)
```
```   729       also have "... \<le> Inf (measure_set M f s) + e"
```
```   730         by (rule l)
```
```   731       finally show "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))
```
```   732         \<le> Inf (measure_set M f s) + e" .
```
```   733     qed
```
```   734   moreover
```
```   735   have "Inf (measure_set M f s)
```
```   736        \<le> Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))"
```
```   737     proof -
```
```   738     have "Inf (measure_set M f s) = Inf (measure_set M f ((s\<inter>x) \<union> (s-x)))"
```
```   739       by (metis Un_Diff_Int Un_commute)
```
```   740     also have "... \<le> Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))"
```
```   741       apply (rule subadditiveD)
```
```   742       apply (iprover intro: algebra.countably_subadditive_subadditive algebra_Pow
```
```   743                inf_measure_positive inf_measure_countably_subadditive posf inc)
```
```   744       apply (auto simp add: subsetD [OF s])
```
```   745       done
```
```   746     finally show ?thesis .
```
```   747     qed
```
```   748   ultimately
```
```   749   show "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))
```
```   750         = Inf (measure_set M f s)"
```
```   751     by (rule order_antisym)
```
```   752 qed
```
```   753
```
```   754 lemma measure_down:
```
```   755      "measure_space N \<mu> \<Longrightarrow> sigma_algebra M \<Longrightarrow> sets M \<subseteq> sets N \<Longrightarrow>
```
```   756       (\<nu> = \<mu>) \<Longrightarrow> measure_space M \<nu>"
```
```   757   by (simp add: measure_space_def measure_space_axioms_def positive_def
```
```   758                 countably_additive_def)
```
```   759      blast
```
```   760
```
```   761 theorem (in algebra) caratheodory:
```
```   762   assumes posf: "positive f" and ca: "countably_additive M f"
```
```   763   shows "\<exists>\<mu> :: 'a set \<Rightarrow> pinfreal. (\<forall>s \<in> sets M. \<mu> s = f s) \<and> measure_space (sigma (space M) (sets M)) \<mu>"
```
```   764   proof -
```
```   765     have inc: "increasing M f"
```
```   766       by (metis additive_increasing ca countably_additive_additive posf)
```
```   767     let ?infm = "(\<lambda>x. Inf (measure_set M f x))"
```
```   768     def ls \<equiv> "lambda_system (|space = space M, sets = Pow (space M)|) ?infm"
```
```   769     have mls: "measure_space \<lparr>space = space M, sets = ls\<rparr> ?infm"
```
```   770       using sigma_algebra.caratheodory_lemma
```
```   771               [OF sigma_algebra_Pow  inf_measure_outer [OF posf inc]]
```
```   772       by (simp add: ls_def)
```
```   773     hence sls: "sigma_algebra (|space = space M, sets = ls|)"
```
```   774       by (simp add: measure_space_def)
```
```   775     have "sets M \<subseteq> ls"
```
```   776       by (simp add: ls_def)
```
```   777          (metis ca posf inc countably_additive_additive algebra_subset_lambda_system)
```
```   778     hence sgs_sb: "sigma_sets (space M) (sets M) \<subseteq> ls"
```
```   779       using sigma_algebra.sigma_sets_subset [OF sls, of "sets M"]
```
```   780       by simp
```
```   781     have "measure_space (sigma (space M) (sets M)) ?infm"
```
```   782       unfolding sigma_def
```
```   783       by (rule measure_down [OF mls], rule sigma_algebra_sigma_sets)
```
```   784          (simp_all add: sgs_sb space_closed)
```
```   785     thus ?thesis using inf_measure_agrees [OF posf ca] by (auto intro!: exI[of _ ?infm])
```
```   786   qed
```
```   787
```
```   788 end
```