src/HOL/Analysis/Measure_Space.thy
 author hoelzl Fri Sep 23 10:26:04 2016 +0200 (2016-09-23) changeset 63940 0d82c4c94014 parent 63658 7faa9bf9860b child 63958 02de4a58e210 permissions -rw-r--r--
prove HK-integrable implies Lebesgue measurable; prove HK-integral equals Lebesgue integral for nonneg functions
     1 (*  Title:      HOL/Analysis/Measure_Space.thy

     2     Author:     Lawrence C Paulson

     3     Author:     Johannes Hölzl, TU München

     4     Author:     Armin Heller, TU München

     5 *)

     6

     7 section \<open>Measure spaces and their properties\<close>

     8

     9 theory Measure_Space

    10 imports

    11   Measurable "~~/src/HOL/Library/Extended_Nonnegative_Real"

    12 begin

    13

    14 subsection "Relate extended reals and the indicator function"

    15

    16 lemma suminf_cmult_indicator:

    17   fixes f :: "nat \<Rightarrow> ennreal"

    18   assumes "disjoint_family A" "x \<in> A i"

    19   shows "(\<Sum>n. f n * indicator (A n) x) = f i"

    20 proof -

    21   have **: "\<And>n. f n * indicator (A n) x = (if n = i then f n else 0 :: ennreal)"

    22     using \<open>x \<in> A i\<close> assms unfolding disjoint_family_on_def indicator_def by auto

    23   then have "\<And>n. (\<Sum>j<n. f j * indicator (A j) x) = (if i < n then f i else 0 :: ennreal)"

    24     by (auto simp: setsum.If_cases)

    25   moreover have "(SUP n. if i < n then f i else 0) = (f i :: ennreal)"

    26   proof (rule SUP_eqI)

    27     fix y :: ennreal assume "\<And>n. n \<in> UNIV \<Longrightarrow> (if i < n then f i else 0) \<le> y"

    28     from this[of "Suc i"] show "f i \<le> y" by auto

    29   qed (insert assms, simp)

    30   ultimately show ?thesis using assms

    31     by (subst suminf_eq_SUP) (auto simp: indicator_def)

    32 qed

    33

    34 lemma suminf_indicator:

    35   assumes "disjoint_family A"

    36   shows "(\<Sum>n. indicator (A n) x :: ennreal) = indicator (\<Union>i. A i) x"

    37 proof cases

    38   assume *: "x \<in> (\<Union>i. A i)"

    39   then obtain i where "x \<in> A i" by auto

    40   from suminf_cmult_indicator[OF assms(1), OF \<open>x \<in> A i\<close>, of "\<lambda>k. 1"]

    41   show ?thesis using * by simp

    42 qed simp

    43

    44 lemma setsum_indicator_disjoint_family:

    45   fixes f :: "'d \<Rightarrow> 'e::semiring_1"

    46   assumes d: "disjoint_family_on A P" and "x \<in> A j" and "finite P" and "j \<in> P"

    47   shows "(\<Sum>i\<in>P. f i * indicator (A i) x) = f j"

    48 proof -

    49   have "P \<inter> {i. x \<in> A i} = {j}"

    50     using d \<open>x \<in> A j\<close> \<open>j \<in> P\<close> unfolding disjoint_family_on_def

    51     by auto

    52   thus ?thesis

    53     unfolding indicator_def

    54     by (simp add: if_distrib setsum.If_cases[OF \<open>finite P\<close>])

    55 qed

    56

    57 text \<open>

    58   The type for emeasure spaces is already defined in @{theory Sigma_Algebra}, as it is also used to

    59   represent sigma algebras (with an arbitrary emeasure).

    60 \<close>

    61

    62 subsection "Extend binary sets"

    63

    64 lemma LIMSEQ_binaryset:

    65   assumes f: "f {} = 0"

    66   shows  "(\<lambda>n. \<Sum>i<n. f (binaryset A B i)) \<longlonglongrightarrow> f A + f B"

    67 proof -

    68   have "(\<lambda>n. \<Sum>i < Suc (Suc n). f (binaryset A B i)) = (\<lambda>n. f A + f B)"

    69     proof

    70       fix n

    71       show "(\<Sum>i < Suc (Suc n). f (binaryset A B i)) = f A + f B"

    72         by (induct n)  (auto simp add: binaryset_def f)

    73     qed

    74   moreover

    75   have "... \<longlonglongrightarrow> f A + f B" by (rule tendsto_const)

    76   ultimately

    77   have "(\<lambda>n. \<Sum>i< Suc (Suc n). f (binaryset A B i)) \<longlonglongrightarrow> f A + f B"

    78     by metis

    79   hence "(\<lambda>n. \<Sum>i< n+2. f (binaryset A B i)) \<longlonglongrightarrow> f A + f B"

    80     by simp

    81   thus ?thesis by (rule LIMSEQ_offset [where k=2])

    82 qed

    83

    84 lemma binaryset_sums:

    85   assumes f: "f {} = 0"

    86   shows  "(\<lambda>n. f (binaryset A B n)) sums (f A + f B)"

    87     by (simp add: sums_def LIMSEQ_binaryset [where f=f, OF f] atLeast0LessThan)

    88

    89 lemma suminf_binaryset_eq:

    90   fixes f :: "'a set \<Rightarrow> 'b::{comm_monoid_add, t2_space}"

    91   shows "f {} = 0 \<Longrightarrow> (\<Sum>n. f (binaryset A B n)) = f A + f B"

    92   by (metis binaryset_sums sums_unique)

    93

    94 subsection \<open>Properties of a premeasure @{term \<mu>}\<close>

    95

    96 text \<open>

    97   The definitions for @{const positive} and @{const countably_additive} should be here, by they are

    98   necessary to define @{typ "'a measure"} in @{theory Sigma_Algebra}.

    99 \<close>

   100

   101 definition subadditive where

   102   "subadditive M f \<longleftrightarrow> (\<forall>x\<in>M. \<forall>y\<in>M. x \<inter> y = {} \<longrightarrow> f (x \<union> y) \<le> f x + f y)"

   103

   104 lemma subadditiveD: "subadditive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x \<in> M \<Longrightarrow> y \<in> M \<Longrightarrow> f (x \<union> y) \<le> f x + f y"

   105   by (auto simp add: subadditive_def)

   106

   107 definition countably_subadditive where

   108   "countably_subadditive M f \<longleftrightarrow>

   109     (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow> (f (\<Union>i. A i) \<le> (\<Sum>i. f (A i))))"

   110

   111 lemma (in ring_of_sets) countably_subadditive_subadditive:

   112   fixes f :: "'a set \<Rightarrow> ennreal"

   113   assumes f: "positive M f" and cs: "countably_subadditive M f"

   114   shows  "subadditive M f"

   115 proof (auto simp add: subadditive_def)

   116   fix x y

   117   assume x: "x \<in> M" and y: "y \<in> M" and "x \<inter> y = {}"

   118   hence "disjoint_family (binaryset x y)"

   119     by (auto simp add: disjoint_family_on_def binaryset_def)

   120   hence "range (binaryset x y) \<subseteq> M \<longrightarrow>

   121          (\<Union>i. binaryset x y i) \<in> M \<longrightarrow>

   122          f (\<Union>i. binaryset x y i) \<le> (\<Sum> n. f (binaryset x y n))"

   123     using cs by (auto simp add: countably_subadditive_def)

   124   hence "{x,y,{}} \<subseteq> M \<longrightarrow> x \<union> y \<in> M \<longrightarrow>

   125          f (x \<union> y) \<le> (\<Sum> n. f (binaryset x y n))"

   126     by (simp add: range_binaryset_eq UN_binaryset_eq)

   127   thus "f (x \<union> y) \<le>  f x + f y" using f x y

   128     by (auto simp add: Un o_def suminf_binaryset_eq positive_def)

   129 qed

   130

   131 definition additive where

   132   "additive M \<mu> \<longleftrightarrow> (\<forall>x\<in>M. \<forall>y\<in>M. x \<inter> y = {} \<longrightarrow> \<mu> (x \<union> y) = \<mu> x + \<mu> y)"

   133

   134 definition increasing where

   135   "increasing M \<mu> \<longleftrightarrow> (\<forall>x\<in>M. \<forall>y\<in>M. x \<subseteq> y \<longrightarrow> \<mu> x \<le> \<mu> y)"

   136

   137 lemma positiveD1: "positive M f \<Longrightarrow> f {} = 0" by (auto simp: positive_def)

   138

   139 lemma positiveD_empty:

   140   "positive M f \<Longrightarrow> f {} = 0"

   141   by (auto simp add: positive_def)

   142

   143 lemma additiveD:

   144   "additive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x \<in> M \<Longrightarrow> y \<in> M \<Longrightarrow> f (x \<union> y) = f x + f y"

   145   by (auto simp add: additive_def)

   146

   147 lemma increasingD:

   148   "increasing M f \<Longrightarrow> x \<subseteq> y \<Longrightarrow> x\<in>M \<Longrightarrow> y\<in>M \<Longrightarrow> f x \<le> f y"

   149   by (auto simp add: increasing_def)

   150

   151 lemma countably_additiveI[case_names countably]:

   152   "(\<And>A. range A \<subseteq> M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i. A i) \<in> M \<Longrightarrow> (\<Sum>i. f (A i)) = f (\<Union>i. A i))

   153   \<Longrightarrow> countably_additive M f"

   154   by (simp add: countably_additive_def)

   155

   156 lemma (in ring_of_sets) disjointed_additive:

   157   assumes f: "positive M f" "additive M f" and A: "range A \<subseteq> M" "incseq A"

   158   shows "(\<Sum>i\<le>n. f (disjointed A i)) = f (A n)"

   159 proof (induct n)

   160   case (Suc n)

   161   then have "(\<Sum>i\<le>Suc n. f (disjointed A i)) = f (A n) + f (disjointed A (Suc n))"

   162     by simp

   163   also have "\<dots> = f (A n \<union> disjointed A (Suc n))"

   164     using A by (subst f(2)[THEN additiveD]) (auto simp: disjointed_mono)

   165   also have "A n \<union> disjointed A (Suc n) = A (Suc n)"

   166     using \<open>incseq A\<close> by (auto dest: incseq_SucD simp: disjointed_mono)

   167   finally show ?case .

   168 qed simp

   169

   170 lemma (in ring_of_sets) additive_sum:

   171   fixes A:: "'i \<Rightarrow> 'a set"

   172   assumes f: "positive M f" and ad: "additive M f" and "finite S"

   173       and A: "AS \<subseteq> M"

   174       and disj: "disjoint_family_on A S"

   175   shows  "(\<Sum>i\<in>S. f (A i)) = f (\<Union>i\<in>S. A i)"

   176   using \<open>finite S\<close> disj A

   177 proof induct

   178   case empty show ?case using f by (simp add: positive_def)

   179 next

   180   case (insert s S)

   181   then have "A s \<inter> (\<Union>i\<in>S. A i) = {}"

   182     by (auto simp add: disjoint_family_on_def neq_iff)

   183   moreover

   184   have "A s \<in> M" using insert by blast

   185   moreover have "(\<Union>i\<in>S. A i) \<in> M"

   186     using insert \<open>finite S\<close> by auto

   187   ultimately have "f (A s \<union> (\<Union>i\<in>S. A i)) = f (A s) + f(\<Union>i\<in>S. A i)"

   188     using ad UNION_in_sets A by (auto simp add: additive_def)

   189   with insert show ?case using ad disjoint_family_on_mono[of S "insert s S" A]

   190     by (auto simp add: additive_def subset_insertI)

   191 qed

   192

   193 lemma (in ring_of_sets) additive_increasing:

   194   fixes f :: "'a set \<Rightarrow> ennreal"

   195   assumes posf: "positive M f" and addf: "additive M f"

   196   shows "increasing M f"

   197 proof (auto simp add: increasing_def)

   198   fix x y

   199   assume xy: "x \<in> M" "y \<in> M" "x \<subseteq> y"

   200   then have "y - x \<in> M" by auto

   201   then have "f x + 0 \<le> f x + f (y-x)" by (intro add_left_mono zero_le)

   202   also have "... = f (x \<union> (y-x))" using addf

   203     by (auto simp add: additive_def) (metis Diff_disjoint Un_Diff_cancel Diff xy(1,2))

   204   also have "... = f y"

   205     by (metis Un_Diff_cancel Un_absorb1 xy(3))

   206   finally show "f x \<le> f y" by simp

   207 qed

   208

   209 lemma (in ring_of_sets) subadditive:

   210   fixes f :: "'a set \<Rightarrow> ennreal"

   211   assumes f: "positive M f" "additive M f" and A: "AS \<subseteq> M" and S: "finite S"

   212   shows "f (\<Union>i\<in>S. A i) \<le> (\<Sum>i\<in>S. f (A i))"

   213 using S A

   214 proof (induct S)

   215   case empty thus ?case using f by (auto simp: positive_def)

   216 next

   217   case (insert x F)

   218   hence in_M: "A x \<in> M" "(\<Union>i\<in>F. A i) \<in> M" "(\<Union>i\<in>F. A i) - A x \<in> M" using A by force+

   219   have subs: "(\<Union>i\<in>F. A i) - A x \<subseteq> (\<Union>i\<in>F. A i)" by auto

   220   have "(\<Union>i\<in>(insert x F). A i) = A x \<union> ((\<Union>i\<in>F. A i) - A x)" by auto

   221   hence "f (\<Union>i\<in>(insert x F). A i) = f (A x \<union> ((\<Union>i\<in>F. A i) - A x))"

   222     by simp

   223   also have "\<dots> = f (A x) + f ((\<Union>i\<in>F. A i) - A x)"

   224     using f(2) by (rule additiveD) (insert in_M, auto)

   225   also have "\<dots> \<le> f (A x) + f (\<Union>i\<in>F. A i)"

   226     using additive_increasing[OF f] in_M subs by (auto simp: increasing_def intro: add_left_mono)

   227   also have "\<dots> \<le> f (A x) + (\<Sum>i\<in>F. f (A i))" using insert by (auto intro: add_left_mono)

   228   finally show "f (\<Union>i\<in>(insert x F). A i) \<le> (\<Sum>i\<in>(insert x F). f (A i))" using insert by simp

   229 qed

   230

   231 lemma (in ring_of_sets) countably_additive_additive:

   232   fixes f :: "'a set \<Rightarrow> ennreal"

   233   assumes posf: "positive M f" and ca: "countably_additive M f"

   234   shows "additive M f"

   235 proof (auto simp add: additive_def)

   236   fix x y

   237   assume x: "x \<in> M" and y: "y \<in> M" and "x \<inter> y = {}"

   238   hence "disjoint_family (binaryset x y)"

   239     by (auto simp add: disjoint_family_on_def binaryset_def)

   240   hence "range (binaryset x y) \<subseteq> M \<longrightarrow>

   241          (\<Union>i. binaryset x y i) \<in> M \<longrightarrow>

   242          f (\<Union>i. binaryset x y i) = (\<Sum> n. f (binaryset x y n))"

   243     using ca

   244     by (simp add: countably_additive_def)

   245   hence "{x,y,{}} \<subseteq> M \<longrightarrow> x \<union> y \<in> M \<longrightarrow>

   246          f (x \<union> y) = (\<Sum>n. f (binaryset x y n))"

   247     by (simp add: range_binaryset_eq UN_binaryset_eq)

   248   thus "f (x \<union> y) = f x + f y" using posf x y

   249     by (auto simp add: Un suminf_binaryset_eq positive_def)

   250 qed

   251

   252 lemma (in algebra) increasing_additive_bound:

   253   fixes A:: "nat \<Rightarrow> 'a set" and  f :: "'a set \<Rightarrow> ennreal"

   254   assumes f: "positive M f" and ad: "additive M f"

   255       and inc: "increasing M f"

   256       and A: "range A \<subseteq> M"

   257       and disj: "disjoint_family A"

   258   shows  "(\<Sum>i. f (A i)) \<le> f \<Omega>"

   259 proof (safe intro!: suminf_le_const)

   260   fix N

   261   note disj_N = disjoint_family_on_mono[OF _ disj, of "{..<N}"]

   262   have "(\<Sum>i<N. f (A i)) = f (\<Union>i\<in>{..<N}. A i)"

   263     using A by (intro additive_sum [OF f ad _ _]) (auto simp: disj_N)

   264   also have "... \<le> f \<Omega>" using space_closed A

   265     by (intro increasingD[OF inc] finite_UN) auto

   266   finally show "(\<Sum>i<N. f (A i)) \<le> f \<Omega>" by simp

   267 qed (insert f A, auto simp: positive_def)

   268

   269 lemma (in ring_of_sets) countably_additiveI_finite:

   270   fixes \<mu> :: "'a set \<Rightarrow> ennreal"

   271   assumes "finite \<Omega>" "positive M \<mu>" "additive M \<mu>"

   272   shows "countably_additive M \<mu>"

   273 proof (rule countably_additiveI)

   274   fix F :: "nat \<Rightarrow> 'a set" assume F: "range F \<subseteq> M" "(\<Union>i. F i) \<in> M" and disj: "disjoint_family F"

   275

   276   have "\<forall>i\<in>{i. F i \<noteq> {}}. \<exists>x. x \<in> F i" by auto

   277   from bchoice[OF this] obtain f where f: "\<And>i. F i \<noteq> {} \<Longrightarrow> f i \<in> F i" by auto

   278

   279   have inj_f: "inj_on f {i. F i \<noteq> {}}"

   280   proof (rule inj_onI, simp)

   281     fix i j a b assume *: "f i = f j" "F i \<noteq> {}" "F j \<noteq> {}"

   282     then have "f i \<in> F i" "f j \<in> F j" using f by force+

   283     with disj * show "i = j" by (auto simp: disjoint_family_on_def)

   284   qed

   285   have "finite (\<Union>i. F i)"

   286     by (metis F(2) assms(1) infinite_super sets_into_space)

   287

   288   have F_subset: "{i. \<mu> (F i) \<noteq> 0} \<subseteq> {i. F i \<noteq> {}}"

   289     by (auto simp: positiveD_empty[OF \<open>positive M \<mu>\<close>])

   290   moreover have fin_not_empty: "finite {i. F i \<noteq> {}}"

   291   proof (rule finite_imageD)

   292     from f have "f{i. F i \<noteq> {}} \<subseteq> (\<Union>i. F i)" by auto

   293     then show "finite (f{i. F i \<noteq> {}})"

   294       by (rule finite_subset) fact

   295   qed fact

   296   ultimately have fin_not_0: "finite {i. \<mu> (F i) \<noteq> 0}"

   297     by (rule finite_subset)

   298

   299   have disj_not_empty: "disjoint_family_on F {i. F i \<noteq> {}}"

   300     using disj by (auto simp: disjoint_family_on_def)

   301

   302   from fin_not_0 have "(\<Sum>i. \<mu> (F i)) = (\<Sum>i | \<mu> (F i) \<noteq> 0. \<mu> (F i))"

   303     by (rule suminf_finite) auto

   304   also have "\<dots> = (\<Sum>i | F i \<noteq> {}. \<mu> (F i))"

   305     using fin_not_empty F_subset by (rule setsum.mono_neutral_left) auto

   306   also have "\<dots> = \<mu> (\<Union>i\<in>{i. F i \<noteq> {}}. F i)"

   307     using \<open>positive M \<mu>\<close> \<open>additive M \<mu>\<close> fin_not_empty disj_not_empty F by (intro additive_sum) auto

   308   also have "\<dots> = \<mu> (\<Union>i. F i)"

   309     by (rule arg_cong[where f=\<mu>]) auto

   310   finally show "(\<Sum>i. \<mu> (F i)) = \<mu> (\<Union>i. F i)" .

   311 qed

   312

   313 lemma (in ring_of_sets) countably_additive_iff_continuous_from_below:

   314   fixes f :: "'a set \<Rightarrow> ennreal"

   315   assumes f: "positive M f" "additive M f"

   316   shows "countably_additive M f \<longleftrightarrow>

   317     (\<forall>A. range A \<subseteq> M \<longrightarrow> incseq A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> f (\<Union>i. A i))"

   318   unfolding countably_additive_def

   319 proof safe

   320   assume count_sum: "\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> UNION UNIV A \<in> M \<longrightarrow> (\<Sum>i. f (A i)) = f (UNION UNIV A)"

   321   fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "incseq A" "(\<Union>i. A i) \<in> M"

   322   then have dA: "range (disjointed A) \<subseteq> M" by (auto simp: range_disjointed_sets)

   323   with count_sum[THEN spec, of "disjointed A"] A(3)

   324   have f_UN: "(\<Sum>i. f (disjointed A i)) = f (\<Union>i. A i)"

   325     by (auto simp: UN_disjointed_eq disjoint_family_disjointed)

   326   moreover have "(\<lambda>n. (\<Sum>i<n. f (disjointed A i))) \<longlonglongrightarrow> (\<Sum>i. f (disjointed A i))"

   327     using f(1)[unfolded positive_def] dA

   328     by (auto intro!: summable_LIMSEQ)

   329   from LIMSEQ_Suc[OF this]

   330   have "(\<lambda>n. (\<Sum>i\<le>n. f (disjointed A i))) \<longlonglongrightarrow> (\<Sum>i. f (disjointed A i))"

   331     unfolding lessThan_Suc_atMost .

   332   moreover have "\<And>n. (\<Sum>i\<le>n. f (disjointed A i)) = f (A n)"

   333     using disjointed_additive[OF f A(1,2)] .

   334   ultimately show "(\<lambda>i. f (A i)) \<longlonglongrightarrow> f (\<Union>i. A i)" by simp

   335 next

   336   assume cont: "\<forall>A. range A \<subseteq> M \<longrightarrow> incseq A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> f (\<Union>i. A i)"

   337   fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "disjoint_family A" "(\<Union>i. A i) \<in> M"

   338   have *: "(\<Union>n. (\<Union>i<n. A i)) = (\<Union>i. A i)" by auto

   339   have "(\<lambda>n. f (\<Union>i<n. A i)) \<longlonglongrightarrow> f (\<Union>i. A i)"

   340   proof (unfold *[symmetric], intro cont[rule_format])

   341     show "range (\<lambda>i. \<Union>i<i. A i) \<subseteq> M" "(\<Union>i. \<Union>i<i. A i) \<in> M"

   342       using A * by auto

   343   qed (force intro!: incseq_SucI)

   344   moreover have "\<And>n. f (\<Union>i<n. A i) = (\<Sum>i<n. f (A i))"

   345     using A

   346     by (intro additive_sum[OF f, of _ A, symmetric])

   347        (auto intro: disjoint_family_on_mono[where B=UNIV])

   348   ultimately

   349   have "(\<lambda>i. f (A i)) sums f (\<Union>i. A i)"

   350     unfolding sums_def by simp

   351   from sums_unique[OF this]

   352   show "(\<Sum>i. f (A i)) = f (\<Union>i. A i)" by simp

   353 qed

   354

   355 lemma (in ring_of_sets) continuous_from_above_iff_empty_continuous:

   356   fixes f :: "'a set \<Rightarrow> ennreal"

   357   assumes f: "positive M f" "additive M f"

   358   shows "(\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) \<in> M \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> f (\<Inter>i. A i))

   359      \<longleftrightarrow> (\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> 0)"

   360 proof safe

   361   assume cont: "(\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) \<in> M \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> f (\<Inter>i. A i))"

   362   fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "decseq A" "(\<Inter>i. A i) = {}" "\<forall>i. f (A i) \<noteq> \<infinity>"

   363   with cont[THEN spec, of A] show "(\<lambda>i. f (A i)) \<longlonglongrightarrow> 0"

   364     using \<open>positive M f\<close>[unfolded positive_def] by auto

   365 next

   366   assume cont: "\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> 0"

   367   fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "decseq A" "(\<Inter>i. A i) \<in> M" "\<forall>i. f (A i) \<noteq> \<infinity>"

   368

   369   have f_mono: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<subseteq> b \<Longrightarrow> f a \<le> f b"

   370     using additive_increasing[OF f] unfolding increasing_def by simp

   371

   372   have decseq_fA: "decseq (\<lambda>i. f (A i))"

   373     using A by (auto simp: decseq_def intro!: f_mono)

   374   have decseq: "decseq (\<lambda>i. A i - (\<Inter>i. A i))"

   375     using A by (auto simp: decseq_def)

   376   then have decseq_f: "decseq (\<lambda>i. f (A i - (\<Inter>i. A i)))"

   377     using A unfolding decseq_def by (auto intro!: f_mono Diff)

   378   have "f (\<Inter>x. A x) \<le> f (A 0)"

   379     using A by (auto intro!: f_mono)

   380   then have f_Int_fin: "f (\<Inter>x. A x) \<noteq> \<infinity>"

   381     using A by (auto simp: top_unique)

   382   { fix i

   383     have "f (A i - (\<Inter>i. A i)) \<le> f (A i)" using A by (auto intro!: f_mono)

   384     then have "f (A i - (\<Inter>i. A i)) \<noteq> \<infinity>"

   385       using A by (auto simp: top_unique) }

   386   note f_fin = this

   387   have "(\<lambda>i. f (A i - (\<Inter>i. A i))) \<longlonglongrightarrow> 0"

   388   proof (intro cont[rule_format, OF _ decseq _ f_fin])

   389     show "range (\<lambda>i. A i - (\<Inter>i. A i)) \<subseteq> M" "(\<Inter>i. A i - (\<Inter>i. A i)) = {}"

   390       using A by auto

   391   qed

   392   from INF_Lim_ereal[OF decseq_f this]

   393   have "(INF n. f (A n - (\<Inter>i. A i))) = 0" .

   394   moreover have "(INF n. f (\<Inter>i. A i)) = f (\<Inter>i. A i)"

   395     by auto

   396   ultimately have "(INF n. f (A n - (\<Inter>i. A i)) + f (\<Inter>i. A i)) = 0 + f (\<Inter>i. A i)"

   397     using A(4) f_fin f_Int_fin

   398     by (subst INF_ennreal_add_const) (auto simp: decseq_f)

   399   moreover {

   400     fix n

   401     have "f (A n - (\<Inter>i. A i)) + f (\<Inter>i. A i) = f ((A n - (\<Inter>i. A i)) \<union> (\<Inter>i. A i))"

   402       using A by (subst f(2)[THEN additiveD]) auto

   403     also have "(A n - (\<Inter>i. A i)) \<union> (\<Inter>i. A i) = A n"

   404       by auto

   405     finally have "f (A n - (\<Inter>i. A i)) + f (\<Inter>i. A i) = f (A n)" . }

   406   ultimately have "(INF n. f (A n)) = f (\<Inter>i. A i)"

   407     by simp

   408   with LIMSEQ_INF[OF decseq_fA]

   409   show "(\<lambda>i. f (A i)) \<longlonglongrightarrow> f (\<Inter>i. A i)" by simp

   410 qed

   411

   412 lemma (in ring_of_sets) empty_continuous_imp_continuous_from_below:

   413   fixes f :: "'a set \<Rightarrow> ennreal"

   414   assumes f: "positive M f" "additive M f" "\<forall>A\<in>M. f A \<noteq> \<infinity>"

   415   assumes cont: "\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> 0"

   416   assumes A: "range A \<subseteq> M" "incseq A" "(\<Union>i. A i) \<in> M"

   417   shows "(\<lambda>i. f (A i)) \<longlonglongrightarrow> f (\<Union>i. A i)"

   418 proof -

   419   from A have "(\<lambda>i. f ((\<Union>i. A i) - A i)) \<longlonglongrightarrow> 0"

   420     by (intro cont[rule_format]) (auto simp: decseq_def incseq_def)

   421   moreover

   422   { fix i

   423     have "f ((\<Union>i. A i) - A i \<union> A i) = f ((\<Union>i. A i) - A i) + f (A i)"

   424       using A by (intro f(2)[THEN additiveD]) auto

   425     also have "((\<Union>i. A i) - A i) \<union> A i = (\<Union>i. A i)"

   426       by auto

   427     finally have "f ((\<Union>i. A i) - A i) = f (\<Union>i. A i) - f (A i)"

   428       using f(3)[rule_format, of "A i"] A by (auto simp: ennreal_add_diff_cancel subset_eq) }

   429   moreover have "\<forall>\<^sub>F i in sequentially. f (A i) \<le> f (\<Union>i. A i)"

   430     using increasingD[OF additive_increasing[OF f(1, 2)], of "A _" "\<Union>i. A i"] A

   431     by (auto intro!: always_eventually simp: subset_eq)

   432   ultimately show "(\<lambda>i. f (A i)) \<longlonglongrightarrow> f (\<Union>i. A i)"

   433     by (auto intro: ennreal_tendsto_const_minus)

   434 qed

   435

   436 lemma (in ring_of_sets) empty_continuous_imp_countably_additive:

   437   fixes f :: "'a set \<Rightarrow> ennreal"

   438   assumes f: "positive M f" "additive M f" and fin: "\<forall>A\<in>M. f A \<noteq> \<infinity>"

   439   assumes cont: "\<And>A. range A \<subseteq> M \<Longrightarrow> decseq A \<Longrightarrow> (\<Inter>i. A i) = {} \<Longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> 0"

   440   shows "countably_additive M f"

   441   using countably_additive_iff_continuous_from_below[OF f]

   442   using empty_continuous_imp_continuous_from_below[OF f fin] cont

   443   by blast

   444

   445 subsection \<open>Properties of @{const emeasure}\<close>

   446

   447 lemma emeasure_positive: "positive (sets M) (emeasure M)"

   448   by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)

   449

   450 lemma emeasure_empty[simp, intro]: "emeasure M {} = 0"

   451   using emeasure_positive[of M] by (simp add: positive_def)

   452

   453 lemma emeasure_single_in_space: "emeasure M {x} \<noteq> 0 \<Longrightarrow> x \<in> space M"

   454   using emeasure_notin_sets[of "{x}" M] by (auto dest: sets.sets_into_space zero_less_iff_neq_zero[THEN iffD2])

   455

   456 lemma emeasure_countably_additive: "countably_additive (sets M) (emeasure M)"

   457   by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)

   458

   459 lemma suminf_emeasure:

   460   "range A \<subseteq> sets M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Sum>i. emeasure M (A i)) = emeasure M (\<Union>i. A i)"

   461   using sets.countable_UN[of A UNIV M] emeasure_countably_additive[of M]

   462   by (simp add: countably_additive_def)

   463

   464 lemma sums_emeasure:

   465   "disjoint_family F \<Longrightarrow> (\<And>i. F i \<in> sets M) \<Longrightarrow> (\<lambda>i. emeasure M (F i)) sums emeasure M (\<Union>i. F i)"

   466   unfolding sums_iff by (intro conjI suminf_emeasure) auto

   467

   468 lemma emeasure_additive: "additive (sets M) (emeasure M)"

   469   by (metis sets.countably_additive_additive emeasure_positive emeasure_countably_additive)

   470

   471 lemma plus_emeasure:

   472   "a \<in> sets M \<Longrightarrow> b \<in> sets M \<Longrightarrow> a \<inter> b = {} \<Longrightarrow> emeasure M a + emeasure M b = emeasure M (a \<union> b)"

   473   using additiveD[OF emeasure_additive] ..

   474

   475 lemma setsum_emeasure:

   476   "FI \<subseteq> sets M \<Longrightarrow> disjoint_family_on F I \<Longrightarrow> finite I \<Longrightarrow>

   477     (\<Sum>i\<in>I. emeasure M (F i)) = emeasure M (\<Union>i\<in>I. F i)"

   478   by (metis sets.additive_sum emeasure_positive emeasure_additive)

   479

   480 lemma emeasure_mono:

   481   "a \<subseteq> b \<Longrightarrow> b \<in> sets M \<Longrightarrow> emeasure M a \<le> emeasure M b"

   482   by (metis zero_le sets.additive_increasing emeasure_additive emeasure_notin_sets emeasure_positive increasingD)

   483

   484 lemma emeasure_space:

   485   "emeasure M A \<le> emeasure M (space M)"

   486   by (metis emeasure_mono emeasure_notin_sets sets.sets_into_space sets.top zero_le)

   487

   488 lemma emeasure_Diff:

   489   assumes finite: "emeasure M B \<noteq> \<infinity>"

   490   and [measurable]: "A \<in> sets M" "B \<in> sets M" and "B \<subseteq> A"

   491   shows "emeasure M (A - B) = emeasure M A - emeasure M B"

   492 proof -

   493   have "(A - B) \<union> B = A" using \<open>B \<subseteq> A\<close> by auto

   494   then have "emeasure M A = emeasure M ((A - B) \<union> B)" by simp

   495   also have "\<dots> = emeasure M (A - B) + emeasure M B"

   496     by (subst plus_emeasure[symmetric]) auto

   497   finally show "emeasure M (A - B) = emeasure M A - emeasure M B"

   498     using finite by simp

   499 qed

   500

   501 lemma emeasure_compl:

   502   "s \<in> sets M \<Longrightarrow> emeasure M s \<noteq> \<infinity> \<Longrightarrow> emeasure M (space M - s) = emeasure M (space M) - emeasure M s"

   503   by (rule emeasure_Diff) (auto dest: sets.sets_into_space)

   504

   505 lemma Lim_emeasure_incseq:

   506   "range A \<subseteq> sets M \<Longrightarrow> incseq A \<Longrightarrow> (\<lambda>i. (emeasure M (A i))) \<longlonglongrightarrow> emeasure M (\<Union>i. A i)"

   507   using emeasure_countably_additive

   508   by (auto simp add: sets.countably_additive_iff_continuous_from_below emeasure_positive

   509     emeasure_additive)

   510

   511 lemma incseq_emeasure:

   512   assumes "range B \<subseteq> sets M" "incseq B"

   513   shows "incseq (\<lambda>i. emeasure M (B i))"

   514   using assms by (auto simp: incseq_def intro!: emeasure_mono)

   515

   516 lemma SUP_emeasure_incseq:

   517   assumes A: "range A \<subseteq> sets M" "incseq A"

   518   shows "(SUP n. emeasure M (A n)) = emeasure M (\<Union>i. A i)"

   519   using LIMSEQ_SUP[OF incseq_emeasure, OF A] Lim_emeasure_incseq[OF A]

   520   by (simp add: LIMSEQ_unique)

   521

   522 lemma decseq_emeasure:

   523   assumes "range B \<subseteq> sets M" "decseq B"

   524   shows "decseq (\<lambda>i. emeasure M (B i))"

   525   using assms by (auto simp: decseq_def intro!: emeasure_mono)

   526

   527 lemma INF_emeasure_decseq:

   528   assumes A: "range A \<subseteq> sets M" and "decseq A"

   529   and finite: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"

   530   shows "(INF n. emeasure M (A n)) = emeasure M (\<Inter>i. A i)"

   531 proof -

   532   have le_MI: "emeasure M (\<Inter>i. A i) \<le> emeasure M (A 0)"

   533     using A by (auto intro!: emeasure_mono)

   534   hence *: "emeasure M (\<Inter>i. A i) \<noteq> \<infinity>" using finite[of 0] by (auto simp: top_unique)

   535

   536   have "emeasure M (A 0) - (INF n. emeasure M (A n)) = (SUP n. emeasure M (A 0) - emeasure M (A n))"

   537     by (simp add: ennreal_INF_const_minus)

   538   also have "\<dots> = (SUP n. emeasure M (A 0 - A n))"

   539     using A finite \<open>decseq A\<close>[unfolded decseq_def] by (subst emeasure_Diff) auto

   540   also have "\<dots> = emeasure M (\<Union>i. A 0 - A i)"

   541   proof (rule SUP_emeasure_incseq)

   542     show "range (\<lambda>n. A 0 - A n) \<subseteq> sets M"

   543       using A by auto

   544     show "incseq (\<lambda>n. A 0 - A n)"

   545       using \<open>decseq A\<close> by (auto simp add: incseq_def decseq_def)

   546   qed

   547   also have "\<dots> = emeasure M (A 0) - emeasure M (\<Inter>i. A i)"

   548     using A finite * by (simp, subst emeasure_Diff) auto

   549   finally show ?thesis

   550     by (rule ennreal_minus_cancel[rotated 3])

   551        (insert finite A, auto intro: INF_lower emeasure_mono)

   552 qed

   553

   554 lemma INF_emeasure_decseq':

   555   assumes A: "\<And>i. A i \<in> sets M" and "decseq A"

   556   and finite: "\<exists>i. emeasure M (A i) \<noteq> \<infinity>"

   557   shows "(INF n. emeasure M (A n)) = emeasure M (\<Inter>i. A i)"

   558 proof -

   559   from finite obtain i where i: "emeasure M (A i) < \<infinity>"

   560     by (auto simp: less_top)

   561   have fin: "i \<le> j \<Longrightarrow> emeasure M (A j) < \<infinity>" for j

   562     by (rule le_less_trans[OF emeasure_mono i]) (auto intro!: decseqD[OF \<open>decseq A\<close>] A)

   563

   564   have "(INF n. emeasure M (A n)) = (INF n. emeasure M (A (n + i)))"

   565   proof (rule INF_eq)

   566     show "\<exists>j\<in>UNIV. emeasure M (A (j + i)) \<le> emeasure M (A i')" for i'

   567       by (intro bexI[of _ i'] emeasure_mono decseqD[OF \<open>decseq A\<close>] A) auto

   568   qed auto

   569   also have "\<dots> = emeasure M (INF n. (A (n + i)))"

   570     using A \<open>decseq A\<close> fin by (intro INF_emeasure_decseq) (auto simp: decseq_def less_top)

   571   also have "(INF n. (A (n + i))) = (INF n. A n)"

   572     by (meson INF_eq UNIV_I assms(2) decseqD le_add1)

   573   finally show ?thesis .

   574 qed

   575

   576 lemma emeasure_INT_decseq_subset:

   577   fixes F :: "nat \<Rightarrow> 'a set"

   578   assumes I: "I \<noteq> {}" and F: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> i \<le> j \<Longrightarrow> F j \<subseteq> F i"

   579   assumes F_sets[measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> sets M"

   580     and fin: "\<And>i. i \<in> I \<Longrightarrow> emeasure M (F i) \<noteq> \<infinity>"

   581   shows "emeasure M (\<Inter>i\<in>I. F i) = (INF i:I. emeasure M (F i))"

   582 proof cases

   583   assume "finite I"

   584   have "(\<Inter>i\<in>I. F i) = F (Max I)"

   585     using I \<open>finite I\<close> by (intro antisym INF_lower INF_greatest F) auto

   586   moreover have "(INF i:I. emeasure M (F i)) = emeasure M (F (Max I))"

   587     using I \<open>finite I\<close> by (intro antisym INF_lower INF_greatest F emeasure_mono) auto

   588   ultimately show ?thesis

   589     by simp

   590 next

   591   assume "infinite I"

   592   define L where "L n = (LEAST i. i \<in> I \<and> i \<ge> n)" for n

   593   have L: "L n \<in> I \<and> n \<le> L n" for n

   594     unfolding L_def

   595   proof (rule LeastI_ex)

   596     show "\<exists>x. x \<in> I \<and> n \<le> x"

   597       using \<open>infinite I\<close> finite_subset[of I "{..< n}"]

   598       by (rule_tac ccontr) (auto simp: not_le)

   599   qed

   600   have L_eq[simp]: "i \<in> I \<Longrightarrow> L i = i" for i

   601     unfolding L_def by (intro Least_equality) auto

   602   have L_mono: "i \<le> j \<Longrightarrow> L i \<le> L j" for i j

   603     using L[of j] unfolding L_def by (intro Least_le) (auto simp: L_def)

   604

   605   have "emeasure M (\<Inter>i. F (L i)) = (INF i. emeasure M (F (L i)))"

   606   proof (intro INF_emeasure_decseq[symmetric])

   607     show "decseq (\<lambda>i. F (L i))"

   608       using L by (intro antimonoI F L_mono) auto

   609   qed (insert L fin, auto)

   610   also have "\<dots> = (INF i:I. emeasure M (F i))"

   611   proof (intro antisym INF_greatest)

   612     show "i \<in> I \<Longrightarrow> (INF i. emeasure M (F (L i))) \<le> emeasure M (F i)" for i

   613       by (intro INF_lower2[of i]) auto

   614   qed (insert L, auto intro: INF_lower)

   615   also have "(\<Inter>i. F (L i)) = (\<Inter>i\<in>I. F i)"

   616   proof (intro antisym INF_greatest)

   617     show "i \<in> I \<Longrightarrow> (\<Inter>i. F (L i)) \<subseteq> F i" for i

   618       by (intro INF_lower2[of i]) auto

   619   qed (insert L, auto)

   620   finally show ?thesis .

   621 qed

   622

   623 lemma Lim_emeasure_decseq:

   624   assumes A: "range A \<subseteq> sets M" "decseq A" and fin: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"

   625   shows "(\<lambda>i. emeasure M (A i)) \<longlonglongrightarrow> emeasure M (\<Inter>i. A i)"

   626   using LIMSEQ_INF[OF decseq_emeasure, OF A]

   627   using INF_emeasure_decseq[OF A fin] by simp

   628

   629 lemma emeasure_lfp'[consumes 1, case_names cont measurable]:

   630   assumes "P M"

   631   assumes cont: "sup_continuous F"

   632   assumes *: "\<And>M A. P M \<Longrightarrow> (\<And>N. P N \<Longrightarrow> Measurable.pred N A) \<Longrightarrow> Measurable.pred M (F A)"

   633   shows "emeasure M {x\<in>space M. lfp F x} = (SUP i. emeasure M {x\<in>space M. (F ^^ i) (\<lambda>x. False) x})"

   634 proof -

   635   have "emeasure M {x\<in>space M. lfp F x} = emeasure M (\<Union>i. {x\<in>space M. (F ^^ i) (\<lambda>x. False) x})"

   636     using sup_continuous_lfp[OF cont] by (auto simp add: bot_fun_def intro!: arg_cong2[where f=emeasure])

   637   moreover { fix i from \<open>P M\<close> have "{x\<in>space M. (F ^^ i) (\<lambda>x. False) x} \<in> sets M"

   638     by (induct i arbitrary: M) (auto simp add: pred_def[symmetric] intro: *) }

   639   moreover have "incseq (\<lambda>i. {x\<in>space M. (F ^^ i) (\<lambda>x. False) x})"

   640   proof (rule incseq_SucI)

   641     fix i

   642     have "(F ^^ i) (\<lambda>x. False) \<le> (F ^^ (Suc i)) (\<lambda>x. False)"

   643     proof (induct i)

   644       case 0 show ?case by (simp add: le_fun_def)

   645     next

   646       case Suc thus ?case using monoD[OF sup_continuous_mono[OF cont] Suc] by auto

   647     qed

   648     then show "{x \<in> space M. (F ^^ i) (\<lambda>x. False) x} \<subseteq> {x \<in> space M. (F ^^ Suc i) (\<lambda>x. False) x}"

   649       by auto

   650   qed

   651   ultimately show ?thesis

   652     by (subst SUP_emeasure_incseq) auto

   653 qed

   654

   655 lemma emeasure_lfp:

   656   assumes [simp]: "\<And>s. sets (M s) = sets N"

   657   assumes cont: "sup_continuous F" "sup_continuous f"

   658   assumes meas: "\<And>P. Measurable.pred N P \<Longrightarrow> Measurable.pred N (F P)"

   659   assumes iter: "\<And>P s. Measurable.pred N P \<Longrightarrow> P \<le> lfp F \<Longrightarrow> emeasure (M s) {x\<in>space N. F P x} = f (\<lambda>s. emeasure (M s) {x\<in>space N. P x}) s"

   660   shows "emeasure (M s) {x\<in>space N. lfp F x} = lfp f s"

   661 proof (subst lfp_transfer_bounded[where \<alpha>="\<lambda>F s. emeasure (M s) {x\<in>space N. F x}" and g=f and f=F and P="Measurable.pred N", symmetric])

   662   fix C assume "incseq C" "\<And>i. Measurable.pred N (C i)"

   663   then show "(\<lambda>s. emeasure (M s) {x \<in> space N. (SUP i. C i) x}) = (SUP i. (\<lambda>s. emeasure (M s) {x \<in> space N. C i x}))"

   664     unfolding SUP_apply[abs_def]

   665     by (subst SUP_emeasure_incseq) (auto simp: mono_def fun_eq_iff intro!: arg_cong2[where f=emeasure])

   666 qed (auto simp add: iter le_fun_def SUP_apply[abs_def] intro!: meas cont)

   667

   668 lemma emeasure_subadditive_finite:

   669   "finite I \<Longrightarrow> A  I \<subseteq> sets M \<Longrightarrow> emeasure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. emeasure M (A i))"

   670   by (rule sets.subadditive[OF emeasure_positive emeasure_additive]) auto

   671

   672 lemma emeasure_subadditive:

   673   "A \<in> sets M \<Longrightarrow> B \<in> sets M \<Longrightarrow> emeasure M (A \<union> B) \<le> emeasure M A + emeasure M B"

   674   using emeasure_subadditive_finite[of "{True, False}" "\<lambda>True \<Rightarrow> A | False \<Rightarrow> B" M] by simp

   675

   676 lemma emeasure_subadditive_countably:

   677   assumes "range f \<subseteq> sets M"

   678   shows "emeasure M (\<Union>i. f i) \<le> (\<Sum>i. emeasure M (f i))"

   679 proof -

   680   have "emeasure M (\<Union>i. f i) = emeasure M (\<Union>i. disjointed f i)"

   681     unfolding UN_disjointed_eq ..

   682   also have "\<dots> = (\<Sum>i. emeasure M (disjointed f i))"

   683     using sets.range_disjointed_sets[OF assms] suminf_emeasure[of "disjointed f"]

   684     by (simp add:  disjoint_family_disjointed comp_def)

   685   also have "\<dots> \<le> (\<Sum>i. emeasure M (f i))"

   686     using sets.range_disjointed_sets[OF assms] assms

   687     by (auto intro!: suminf_le emeasure_mono disjointed_subset)

   688   finally show ?thesis .

   689 qed

   690

   691 lemma emeasure_insert:

   692   assumes sets: "{x} \<in> sets M" "A \<in> sets M" and "x \<notin> A"

   693   shows "emeasure M (insert x A) = emeasure M {x} + emeasure M A"

   694 proof -

   695   have "{x} \<inter> A = {}" using \<open>x \<notin> A\<close> by auto

   696   from plus_emeasure[OF sets this] show ?thesis by simp

   697 qed

   698

   699 lemma emeasure_insert_ne:

   700   "A \<noteq> {} \<Longrightarrow> {x} \<in> sets M \<Longrightarrow> A \<in> sets M \<Longrightarrow> x \<notin> A \<Longrightarrow> emeasure M (insert x A) = emeasure M {x} + emeasure M A"

   701   by (rule emeasure_insert)

   702

   703 lemma emeasure_eq_setsum_singleton:

   704   assumes "finite S" "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"

   705   shows "emeasure M S = (\<Sum>x\<in>S. emeasure M {x})"

   706   using setsum_emeasure[of "\<lambda>x. {x}" S M] assms

   707   by (auto simp: disjoint_family_on_def subset_eq)

   708

   709 lemma setsum_emeasure_cover:

   710   assumes "finite S" and "A \<in> sets M" and br_in_M: "B  S \<subseteq> sets M"

   711   assumes A: "A \<subseteq> (\<Union>i\<in>S. B i)"

   712   assumes disj: "disjoint_family_on B S"

   713   shows "emeasure M A = (\<Sum>i\<in>S. emeasure M (A \<inter> (B i)))"

   714 proof -

   715   have "(\<Sum>i\<in>S. emeasure M (A \<inter> (B i))) = emeasure M (\<Union>i\<in>S. A \<inter> (B i))"

   716   proof (rule setsum_emeasure)

   717     show "disjoint_family_on (\<lambda>i. A \<inter> B i) S"

   718       using \<open>disjoint_family_on B S\<close>

   719       unfolding disjoint_family_on_def by auto

   720   qed (insert assms, auto)

   721   also have "(\<Union>i\<in>S. A \<inter> (B i)) = A"

   722     using A by auto

   723   finally show ?thesis by simp

   724 qed

   725

   726 lemma emeasure_eq_0:

   727   "N \<in> sets M \<Longrightarrow> emeasure M N = 0 \<Longrightarrow> K \<subseteq> N \<Longrightarrow> emeasure M K = 0"

   728   by (metis emeasure_mono order_eq_iff zero_le)

   729

   730 lemma emeasure_UN_eq_0:

   731   assumes "\<And>i::nat. emeasure M (N i) = 0" and "range N \<subseteq> sets M"

   732   shows "emeasure M (\<Union>i. N i) = 0"

   733 proof -

   734   have "emeasure M (\<Union>i. N i) \<le> 0"

   735     using emeasure_subadditive_countably[OF assms(2)] assms(1) by simp

   736   then show ?thesis

   737     by (auto intro: antisym zero_le)

   738 qed

   739

   740 lemma measure_eqI_finite:

   741   assumes [simp]: "sets M = Pow A" "sets N = Pow A" and "finite A"

   742   assumes eq: "\<And>a. a \<in> A \<Longrightarrow> emeasure M {a} = emeasure N {a}"

   743   shows "M = N"

   744 proof (rule measure_eqI)

   745   fix X assume "X \<in> sets M"

   746   then have X: "X \<subseteq> A" by auto

   747   then have "emeasure M X = (\<Sum>a\<in>X. emeasure M {a})"

   748     using \<open>finite A\<close> by (subst emeasure_eq_setsum_singleton) (auto dest: finite_subset)

   749   also have "\<dots> = (\<Sum>a\<in>X. emeasure N {a})"

   750     using X eq by (auto intro!: setsum.cong)

   751   also have "\<dots> = emeasure N X"

   752     using X \<open>finite A\<close> by (subst emeasure_eq_setsum_singleton) (auto dest: finite_subset)

   753   finally show "emeasure M X = emeasure N X" .

   754 qed simp

   755

   756 lemma measure_eqI_generator_eq:

   757   fixes M N :: "'a measure" and E :: "'a set set" and A :: "nat \<Rightarrow> 'a set"

   758   assumes "Int_stable E" "E \<subseteq> Pow \<Omega>"

   759   and eq: "\<And>X. X \<in> E \<Longrightarrow> emeasure M X = emeasure N X"

   760   and M: "sets M = sigma_sets \<Omega> E"

   761   and N: "sets N = sigma_sets \<Omega> E"

   762   and A: "range A \<subseteq> E" "(\<Union>i. A i) = \<Omega>" "\<And>i. emeasure M (A i) \<noteq> \<infinity>"

   763   shows "M = N"

   764 proof -

   765   let ?\<mu>  = "emeasure M" and ?\<nu> = "emeasure N"

   766   interpret S: sigma_algebra \<Omega> "sigma_sets \<Omega> E" by (rule sigma_algebra_sigma_sets) fact

   767   have "space M = \<Omega>"

   768     using sets.top[of M] sets.space_closed[of M] S.top S.space_closed \<open>sets M = sigma_sets \<Omega> E\<close>

   769     by blast

   770

   771   { fix F D assume "F \<in> E" and "?\<mu> F \<noteq> \<infinity>"

   772     then have [intro]: "F \<in> sigma_sets \<Omega> E" by auto

   773     have "?\<nu> F \<noteq> \<infinity>" using \<open>?\<mu> F \<noteq> \<infinity>\<close> \<open>F \<in> E\<close> eq by simp

   774     assume "D \<in> sets M"

   775     with \<open>Int_stable E\<close> \<open>E \<subseteq> Pow \<Omega>\<close> have "emeasure M (F \<inter> D) = emeasure N (F \<inter> D)"

   776       unfolding M

   777     proof (induct rule: sigma_sets_induct_disjoint)

   778       case (basic A)

   779       then have "F \<inter> A \<in> E" using \<open>Int_stable E\<close> \<open>F \<in> E\<close> by (auto simp: Int_stable_def)

   780       then show ?case using eq by auto

   781     next

   782       case empty then show ?case by simp

   783     next

   784       case (compl A)

   785       then have **: "F \<inter> (\<Omega> - A) = F - (F \<inter> A)"

   786         and [intro]: "F \<inter> A \<in> sigma_sets \<Omega> E"

   787         using \<open>F \<in> E\<close> S.sets_into_space by (auto simp: M)

   788       have "?\<nu> (F \<inter> A) \<le> ?\<nu> F" by (auto intro!: emeasure_mono simp: M N)

   789       then have "?\<nu> (F \<inter> A) \<noteq> \<infinity>" using \<open>?\<nu> F \<noteq> \<infinity>\<close> by (auto simp: top_unique)

   790       have "?\<mu> (F \<inter> A) \<le> ?\<mu> F" by (auto intro!: emeasure_mono simp: M N)

   791       then have "?\<mu> (F \<inter> A) \<noteq> \<infinity>" using \<open>?\<mu> F \<noteq> \<infinity>\<close> by (auto simp: top_unique)

   792       then have "?\<mu> (F \<inter> (\<Omega> - A)) = ?\<mu> F - ?\<mu> (F \<inter> A)" unfolding **

   793         using \<open>F \<inter> A \<in> sigma_sets \<Omega> E\<close> by (auto intro!: emeasure_Diff simp: M N)

   794       also have "\<dots> = ?\<nu> F - ?\<nu> (F \<inter> A)" using eq \<open>F \<in> E\<close> compl by simp

   795       also have "\<dots> = ?\<nu> (F \<inter> (\<Omega> - A))" unfolding **

   796         using \<open>F \<inter> A \<in> sigma_sets \<Omega> E\<close> \<open>?\<nu> (F \<inter> A) \<noteq> \<infinity>\<close>

   797         by (auto intro!: emeasure_Diff[symmetric] simp: M N)

   798       finally show ?case

   799         using \<open>space M = \<Omega>\<close> by auto

   800     next

   801       case (union A)

   802       then have "?\<mu> (\<Union>x. F \<inter> A x) = ?\<nu> (\<Union>x. F \<inter> A x)"

   803         by (subst (1 2) suminf_emeasure[symmetric]) (auto simp: disjoint_family_on_def subset_eq M N)

   804       with A show ?case

   805         by auto

   806     qed }

   807   note * = this

   808   show "M = N"

   809   proof (rule measure_eqI)

   810     show "sets M = sets N"

   811       using M N by simp

   812     have [simp, intro]: "\<And>i. A i \<in> sets M"

   813       using A(1) by (auto simp: subset_eq M)

   814     fix F assume "F \<in> sets M"

   815     let ?D = "disjointed (\<lambda>i. F \<inter> A i)"

   816     from \<open>space M = \<Omega>\<close> have F_eq: "F = (\<Union>i. ?D i)"

   817       using \<open>F \<in> sets M\<close>[THEN sets.sets_into_space] A(2)[symmetric] by (auto simp: UN_disjointed_eq)

   818     have [simp, intro]: "\<And>i. ?D i \<in> sets M"

   819       using sets.range_disjointed_sets[of "\<lambda>i. F \<inter> A i" M] \<open>F \<in> sets M\<close>

   820       by (auto simp: subset_eq)

   821     have "disjoint_family ?D"

   822       by (auto simp: disjoint_family_disjointed)

   823     moreover

   824     have "(\<Sum>i. emeasure M (?D i)) = (\<Sum>i. emeasure N (?D i))"

   825     proof (intro arg_cong[where f=suminf] ext)

   826       fix i

   827       have "A i \<inter> ?D i = ?D i"

   828         by (auto simp: disjointed_def)

   829       then show "emeasure M (?D i) = emeasure N (?D i)"

   830         using *[of "A i" "?D i", OF _ A(3)] A(1) by auto

   831     qed

   832     ultimately show "emeasure M F = emeasure N F"

   833       by (simp add: image_subset_iff \<open>sets M = sets N\<close>[symmetric] F_eq[symmetric] suminf_emeasure)

   834   qed

   835 qed

   836

   837 lemma measure_of_of_measure: "measure_of (space M) (sets M) (emeasure M) = M"

   838 proof (intro measure_eqI emeasure_measure_of_sigma)

   839   show "sigma_algebra (space M) (sets M)" ..

   840   show "positive (sets M) (emeasure M)"

   841     by (simp add: positive_def)

   842   show "countably_additive (sets M) (emeasure M)"

   843     by (simp add: emeasure_countably_additive)

   844 qed simp_all

   845

   846 subsection \<open>\<open>\<mu>\<close>-null sets\<close>

   847

   848 definition null_sets :: "'a measure \<Rightarrow> 'a set set" where

   849   "null_sets M = {N\<in>sets M. emeasure M N = 0}"

   850

   851 lemma null_setsD1[dest]: "A \<in> null_sets M \<Longrightarrow> emeasure M A = 0"

   852   by (simp add: null_sets_def)

   853

   854 lemma null_setsD2[dest]: "A \<in> null_sets M \<Longrightarrow> A \<in> sets M"

   855   unfolding null_sets_def by simp

   856

   857 lemma null_setsI[intro]: "emeasure M A = 0 \<Longrightarrow> A \<in> sets M \<Longrightarrow> A \<in> null_sets M"

   858   unfolding null_sets_def by simp

   859

   860 interpretation null_sets: ring_of_sets "space M" "null_sets M" for M

   861 proof (rule ring_of_setsI)

   862   show "null_sets M \<subseteq> Pow (space M)"

   863     using sets.sets_into_space by auto

   864   show "{} \<in> null_sets M"

   865     by auto

   866   fix A B assume null_sets: "A \<in> null_sets M" "B \<in> null_sets M"

   867   then have sets: "A \<in> sets M" "B \<in> sets M"

   868     by auto

   869   then have *: "emeasure M (A \<union> B) \<le> emeasure M A + emeasure M B"

   870     "emeasure M (A - B) \<le> emeasure M A"

   871     by (auto intro!: emeasure_subadditive emeasure_mono)

   872   then have "emeasure M B = 0" "emeasure M A = 0"

   873     using null_sets by auto

   874   with sets * show "A - B \<in> null_sets M" "A \<union> B \<in> null_sets M"

   875     by (auto intro!: antisym zero_le)

   876 qed

   877

   878 lemma UN_from_nat_into:

   879   assumes I: "countable I" "I \<noteq> {}"

   880   shows "(\<Union>i\<in>I. N i) = (\<Union>i. N (from_nat_into I i))"

   881 proof -

   882   have "(\<Union>i\<in>I. N i) = \<Union>(N  range (from_nat_into I))"

   883     using I by simp

   884   also have "\<dots> = (\<Union>i. (N \<circ> from_nat_into I) i)"

   885     by simp

   886   finally show ?thesis by simp

   887 qed

   888

   889 lemma null_sets_UN':

   890   assumes "countable I"

   891   assumes "\<And>i. i \<in> I \<Longrightarrow> N i \<in> null_sets M"

   892   shows "(\<Union>i\<in>I. N i) \<in> null_sets M"

   893 proof cases

   894   assume "I = {}" then show ?thesis by simp

   895 next

   896   assume "I \<noteq> {}"

   897   show ?thesis

   898   proof (intro conjI CollectI null_setsI)

   899     show "(\<Union>i\<in>I. N i) \<in> sets M"

   900       using assms by (intro sets.countable_UN') auto

   901     have "emeasure M (\<Union>i\<in>I. N i) \<le> (\<Sum>n. emeasure M (N (from_nat_into I n)))"

   902       unfolding UN_from_nat_into[OF \<open>countable I\<close> \<open>I \<noteq> {}\<close>]

   903       using assms \<open>I \<noteq> {}\<close> by (intro emeasure_subadditive_countably) (auto intro: from_nat_into)

   904     also have "(\<lambda>n. emeasure M (N (from_nat_into I n))) = (\<lambda>_. 0)"

   905       using assms \<open>I \<noteq> {}\<close> by (auto intro: from_nat_into)

   906     finally show "emeasure M (\<Union>i\<in>I. N i) = 0"

   907       by (intro antisym zero_le) simp

   908   qed

   909 qed

   910

   911 lemma null_sets_UN[intro]:

   912   "(\<And>i::'i::countable. N i \<in> null_sets M) \<Longrightarrow> (\<Union>i. N i) \<in> null_sets M"

   913   by (rule null_sets_UN') auto

   914

   915 lemma null_set_Int1:

   916   assumes "B \<in> null_sets M" "A \<in> sets M" shows "A \<inter> B \<in> null_sets M"

   917 proof (intro CollectI conjI null_setsI)

   918   show "emeasure M (A \<inter> B) = 0" using assms

   919     by (intro emeasure_eq_0[of B _ "A \<inter> B"]) auto

   920 qed (insert assms, auto)

   921

   922 lemma null_set_Int2:

   923   assumes "B \<in> null_sets M" "A \<in> sets M" shows "B \<inter> A \<in> null_sets M"

   924   using assms by (subst Int_commute) (rule null_set_Int1)

   925

   926 lemma emeasure_Diff_null_set:

   927   assumes "B \<in> null_sets M" "A \<in> sets M"

   928   shows "emeasure M (A - B) = emeasure M A"

   929 proof -

   930   have *: "A - B = (A - (A \<inter> B))" by auto

   931   have "A \<inter> B \<in> null_sets M" using assms by (rule null_set_Int1)

   932   then show ?thesis

   933     unfolding * using assms

   934     by (subst emeasure_Diff) auto

   935 qed

   936

   937 lemma null_set_Diff:

   938   assumes "B \<in> null_sets M" "A \<in> sets M" shows "B - A \<in> null_sets M"

   939 proof (intro CollectI conjI null_setsI)

   940   show "emeasure M (B - A) = 0" using assms by (intro emeasure_eq_0[of B _ "B - A"]) auto

   941 qed (insert assms, auto)

   942

   943 lemma emeasure_Un_null_set:

   944   assumes "A \<in> sets M" "B \<in> null_sets M"

   945   shows "emeasure M (A \<union> B) = emeasure M A"

   946 proof -

   947   have *: "A \<union> B = A \<union> (B - A)" by auto

   948   have "B - A \<in> null_sets M" using assms(2,1) by (rule null_set_Diff)

   949   then show ?thesis

   950     unfolding * using assms

   951     by (subst plus_emeasure[symmetric]) auto

   952 qed

   953

   954 subsection \<open>The almost everywhere filter (i.e.\ quantifier)\<close>

   955

   956 definition ae_filter :: "'a measure \<Rightarrow> 'a filter" where

   957   "ae_filter M = (INF N:null_sets M. principal (space M - N))"

   958

   959 abbreviation almost_everywhere :: "'a measure \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" where

   960   "almost_everywhere M P \<equiv> eventually P (ae_filter M)"

   961

   962 syntax

   963   "_almost_everywhere" :: "pttrn \<Rightarrow> 'a \<Rightarrow> bool \<Rightarrow> bool" ("AE _ in _. _" [0,0,10] 10)

   964

   965 translations

   966   "AE x in M. P" \<rightleftharpoons> "CONST almost_everywhere M (\<lambda>x. P)"

   967

   968 lemma eventually_ae_filter: "eventually P (ae_filter M) \<longleftrightarrow> (\<exists>N\<in>null_sets M. {x \<in> space M. \<not> P x} \<subseteq> N)"

   969   unfolding ae_filter_def by (subst eventually_INF_base) (auto simp: eventually_principal subset_eq)

   970

   971 lemma AE_I':

   972   "N \<in> null_sets M \<Longrightarrow> {x\<in>space M. \<not> P x} \<subseteq> N \<Longrightarrow> (AE x in M. P x)"

   973   unfolding eventually_ae_filter by auto

   974

   975 lemma AE_iff_null:

   976   assumes "{x\<in>space M. \<not> P x} \<in> sets M" (is "?P \<in> sets M")

   977   shows "(AE x in M. P x) \<longleftrightarrow> {x\<in>space M. \<not> P x} \<in> null_sets M"

   978 proof

   979   assume "AE x in M. P x" then obtain N where N: "N \<in> sets M" "?P \<subseteq> N" "emeasure M N = 0"

   980     unfolding eventually_ae_filter by auto

   981   have "emeasure M ?P \<le> emeasure M N"

   982     using assms N(1,2) by (auto intro: emeasure_mono)

   983   then have "emeasure M ?P = 0"

   984     unfolding \<open>emeasure M N = 0\<close> by auto

   985   then show "?P \<in> null_sets M" using assms by auto

   986 next

   987   assume "?P \<in> null_sets M" with assms show "AE x in M. P x" by (auto intro: AE_I')

   988 qed

   989

   990 lemma AE_iff_null_sets:

   991   "N \<in> sets M \<Longrightarrow> N \<in> null_sets M \<longleftrightarrow> (AE x in M. x \<notin> N)"

   992   using Int_absorb1[OF sets.sets_into_space, of N M]

   993   by (subst AE_iff_null) (auto simp: Int_def[symmetric])

   994

   995 lemma AE_not_in:

   996   "N \<in> null_sets M \<Longrightarrow> AE x in M. x \<notin> N"

   997   by (metis AE_iff_null_sets null_setsD2)

   998

   999 lemma AE_iff_measurable:

  1000   "N \<in> sets M \<Longrightarrow> {x\<in>space M. \<not> P x} = N \<Longrightarrow> (AE x in M. P x) \<longleftrightarrow> emeasure M N = 0"

  1001   using AE_iff_null[of _ P] by auto

  1002

  1003 lemma AE_E[consumes 1]:

  1004   assumes "AE x in M. P x"

  1005   obtains N where "{x \<in> space M. \<not> P x} \<subseteq> N" "emeasure M N = 0" "N \<in> sets M"

  1006   using assms unfolding eventually_ae_filter by auto

  1007

  1008 lemma AE_E2:

  1009   assumes "AE x in M. P x" "{x\<in>space M. P x} \<in> sets M"

  1010   shows "emeasure M {x\<in>space M. \<not> P x} = 0" (is "emeasure M ?P = 0")

  1011 proof -

  1012   have "{x\<in>space M. \<not> P x} = space M - {x\<in>space M. P x}" by auto

  1013   with AE_iff_null[of M P] assms show ?thesis by auto

  1014 qed

  1015

  1016 lemma AE_I:

  1017   assumes "{x \<in> space M. \<not> P x} \<subseteq> N" "emeasure M N = 0" "N \<in> sets M"

  1018   shows "AE x in M. P x"

  1019   using assms unfolding eventually_ae_filter by auto

  1020

  1021 lemma AE_mp[elim!]:

  1022   assumes AE_P: "AE x in M. P x" and AE_imp: "AE x in M. P x \<longrightarrow> Q x"

  1023   shows "AE x in M. Q x"

  1024 proof -

  1025   from AE_P obtain A where P: "{x\<in>space M. \<not> P x} \<subseteq> A"

  1026     and A: "A \<in> sets M" "emeasure M A = 0"

  1027     by (auto elim!: AE_E)

  1028

  1029   from AE_imp obtain B where imp: "{x\<in>space M. P x \<and> \<not> Q x} \<subseteq> B"

  1030     and B: "B \<in> sets M" "emeasure M B = 0"

  1031     by (auto elim!: AE_E)

  1032

  1033   show ?thesis

  1034   proof (intro AE_I)

  1035     have "emeasure M (A \<union> B) \<le> 0"

  1036       using emeasure_subadditive[of A M B] A B by auto

  1037     then show "A \<union> B \<in> sets M" "emeasure M (A \<union> B) = 0"

  1038       using A B by auto

  1039     show "{x\<in>space M. \<not> Q x} \<subseteq> A \<union> B"

  1040       using P imp by auto

  1041   qed

  1042 qed

  1043

  1044 (* depricated replace by laws about eventually *)

  1045 lemma

  1046   shows AE_iffI: "AE x in M. P x \<Longrightarrow> AE x in M. P x \<longleftrightarrow> Q x \<Longrightarrow> AE x in M. Q x"

  1047     and AE_disjI1: "AE x in M. P x \<Longrightarrow> AE x in M. P x \<or> Q x"

  1048     and AE_disjI2: "AE x in M. Q x \<Longrightarrow> AE x in M. P x \<or> Q x"

  1049     and AE_conjI: "AE x in M. P x \<Longrightarrow> AE x in M. Q x \<Longrightarrow> AE x in M. P x \<and> Q x"

  1050     and AE_conj_iff[simp]: "(AE x in M. P x \<and> Q x) \<longleftrightarrow> (AE x in M. P x) \<and> (AE x in M. Q x)"

  1051   by auto

  1052

  1053 lemma AE_impI:

  1054   "(P \<Longrightarrow> AE x in M. Q x) \<Longrightarrow> AE x in M. P \<longrightarrow> Q x"

  1055   by (cases P) auto

  1056

  1057 lemma AE_measure:

  1058   assumes AE: "AE x in M. P x" and sets: "{x\<in>space M. P x} \<in> sets M" (is "?P \<in> sets M")

  1059   shows "emeasure M {x\<in>space M. P x} = emeasure M (space M)"

  1060 proof -

  1061   from AE_E[OF AE] guess N . note N = this

  1062   with sets have "emeasure M (space M) \<le> emeasure M (?P \<union> N)"

  1063     by (intro emeasure_mono) auto

  1064   also have "\<dots> \<le> emeasure M ?P + emeasure M N"

  1065     using sets N by (intro emeasure_subadditive) auto

  1066   also have "\<dots> = emeasure M ?P" using N by simp

  1067   finally show "emeasure M ?P = emeasure M (space M)"

  1068     using emeasure_space[of M "?P"] by auto

  1069 qed

  1070

  1071 lemma AE_space: "AE x in M. x \<in> space M"

  1072   by (rule AE_I[where N="{}"]) auto

  1073

  1074 lemma AE_I2[simp, intro]:

  1075   "(\<And>x. x \<in> space M \<Longrightarrow> P x) \<Longrightarrow> AE x in M. P x"

  1076   using AE_space by force

  1077

  1078 lemma AE_Ball_mp:

  1079   "\<forall>x\<in>space M. P x \<Longrightarrow> AE x in M. P x \<longrightarrow> Q x \<Longrightarrow> AE x in M. Q x"

  1080   by auto

  1081

  1082 lemma AE_cong[cong]:

  1083   "(\<And>x. x \<in> space M \<Longrightarrow> P x \<longleftrightarrow> Q x) \<Longrightarrow> (AE x in M. P x) \<longleftrightarrow> (AE x in M. Q x)"

  1084   by auto

  1085

  1086 lemma AE_all_countable:

  1087   "(AE x in M. \<forall>i. P i x) \<longleftrightarrow> (\<forall>i::'i::countable. AE x in M. P i x)"

  1088 proof

  1089   assume "\<forall>i. AE x in M. P i x"

  1090   from this[unfolded eventually_ae_filter Bex_def, THEN choice]

  1091   obtain N where N: "\<And>i. N i \<in> null_sets M" "\<And>i. {x\<in>space M. \<not> P i x} \<subseteq> N i" by auto

  1092   have "{x\<in>space M. \<not> (\<forall>i. P i x)} \<subseteq> (\<Union>i. {x\<in>space M. \<not> P i x})" by auto

  1093   also have "\<dots> \<subseteq> (\<Union>i. N i)" using N by auto

  1094   finally have "{x\<in>space M. \<not> (\<forall>i. P i x)} \<subseteq> (\<Union>i. N i)" .

  1095   moreover from N have "(\<Union>i. N i) \<in> null_sets M"

  1096     by (intro null_sets_UN) auto

  1097   ultimately show "AE x in M. \<forall>i. P i x"

  1098     unfolding eventually_ae_filter by auto

  1099 qed auto

  1100

  1101 lemma AE_ball_countable:

  1102   assumes [intro]: "countable X"

  1103   shows "(AE x in M. \<forall>y\<in>X. P x y) \<longleftrightarrow> (\<forall>y\<in>X. AE x in M. P x y)"

  1104 proof

  1105   assume "\<forall>y\<in>X. AE x in M. P x y"

  1106   from this[unfolded eventually_ae_filter Bex_def, THEN bchoice]

  1107   obtain N where N: "\<And>y. y \<in> X \<Longrightarrow> N y \<in> null_sets M" "\<And>y. y \<in> X \<Longrightarrow> {x\<in>space M. \<not> P x y} \<subseteq> N y"

  1108     by auto

  1109   have "{x\<in>space M. \<not> (\<forall>y\<in>X. P x y)} \<subseteq> (\<Union>y\<in>X. {x\<in>space M. \<not> P x y})"

  1110     by auto

  1111   also have "\<dots> \<subseteq> (\<Union>y\<in>X. N y)"

  1112     using N by auto

  1113   finally have "{x\<in>space M. \<not> (\<forall>y\<in>X. P x y)} \<subseteq> (\<Union>y\<in>X. N y)" .

  1114   moreover from N have "(\<Union>y\<in>X. N y) \<in> null_sets M"

  1115     by (intro null_sets_UN') auto

  1116   ultimately show "AE x in M. \<forall>y\<in>X. P x y"

  1117     unfolding eventually_ae_filter by auto

  1118 qed auto

  1119

  1120 lemma AE_discrete_difference:

  1121   assumes X: "countable X"

  1122   assumes null: "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0"

  1123   assumes sets: "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"

  1124   shows "AE x in M. x \<notin> X"

  1125 proof -

  1126   have "(\<Union>x\<in>X. {x}) \<in> null_sets M"

  1127     using assms by (intro null_sets_UN') auto

  1128   from AE_not_in[OF this] show "AE x in M. x \<notin> X"

  1129     by auto

  1130 qed

  1131

  1132 lemma AE_finite_all:

  1133   assumes f: "finite S" shows "(AE x in M. \<forall>i\<in>S. P i x) \<longleftrightarrow> (\<forall>i\<in>S. AE x in M. P i x)"

  1134   using f by induct auto

  1135

  1136 lemma AE_finite_allI:

  1137   assumes "finite S"

  1138   shows "(\<And>s. s \<in> S \<Longrightarrow> AE x in M. Q s x) \<Longrightarrow> AE x in M. \<forall>s\<in>S. Q s x"

  1139   using AE_finite_all[OF \<open>finite S\<close>] by auto

  1140

  1141 lemma emeasure_mono_AE:

  1142   assumes imp: "AE x in M. x \<in> A \<longrightarrow> x \<in> B"

  1143     and B: "B \<in> sets M"

  1144   shows "emeasure M A \<le> emeasure M B"

  1145 proof cases

  1146   assume A: "A \<in> sets M"

  1147   from imp obtain N where N: "{x\<in>space M. \<not> (x \<in> A \<longrightarrow> x \<in> B)} \<subseteq> N" "N \<in> null_sets M"

  1148     by (auto simp: eventually_ae_filter)

  1149   have "emeasure M A = emeasure M (A - N)"

  1150     using N A by (subst emeasure_Diff_null_set) auto

  1151   also have "emeasure M (A - N) \<le> emeasure M (B - N)"

  1152     using N A B sets.sets_into_space by (auto intro!: emeasure_mono)

  1153   also have "emeasure M (B - N) = emeasure M B"

  1154     using N B by (subst emeasure_Diff_null_set) auto

  1155   finally show ?thesis .

  1156 qed (simp add: emeasure_notin_sets)

  1157

  1158 lemma emeasure_eq_AE:

  1159   assumes iff: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B"

  1160   assumes A: "A \<in> sets M" and B: "B \<in> sets M"

  1161   shows "emeasure M A = emeasure M B"

  1162   using assms by (safe intro!: antisym emeasure_mono_AE) auto

  1163

  1164 lemma emeasure_Collect_eq_AE:

  1165   "AE x in M. P x \<longleftrightarrow> Q x \<Longrightarrow> Measurable.pred M Q \<Longrightarrow> Measurable.pred M P \<Longrightarrow>

  1166    emeasure M {x\<in>space M. P x} = emeasure M {x\<in>space M. Q x}"

  1167    by (intro emeasure_eq_AE) auto

  1168

  1169 lemma emeasure_eq_0_AE: "AE x in M. \<not> P x \<Longrightarrow> emeasure M {x\<in>space M. P x} = 0"

  1170   using AE_iff_measurable[OF _ refl, of M "\<lambda>x. \<not> P x"]

  1171   by (cases "{x\<in>space M. P x} \<in> sets M") (simp_all add: emeasure_notin_sets)

  1172

  1173 lemma emeasure_add_AE:

  1174   assumes [measurable]: "A \<in> sets M" "B \<in> sets M" "C \<in> sets M"

  1175   assumes 1: "AE x in M. x \<in> C \<longleftrightarrow> x \<in> A \<or> x \<in> B"

  1176   assumes 2: "AE x in M. \<not> (x \<in> A \<and> x \<in> B)"

  1177   shows "emeasure M C = emeasure M A + emeasure M B"

  1178 proof -

  1179   have "emeasure M C = emeasure M (A \<union> B)"

  1180     by (rule emeasure_eq_AE) (insert 1, auto)

  1181   also have "\<dots> = emeasure M A + emeasure M (B - A)"

  1182     by (subst plus_emeasure) auto

  1183   also have "emeasure M (B - A) = emeasure M B"

  1184     by (rule emeasure_eq_AE) (insert 2, auto)

  1185   finally show ?thesis .

  1186 qed

  1187

  1188 subsection \<open>\<open>\<sigma>\<close>-finite Measures\<close>

  1189

  1190 locale sigma_finite_measure =

  1191   fixes M :: "'a measure"

  1192   assumes sigma_finite_countable:

  1193     "\<exists>A::'a set set. countable A \<and> A \<subseteq> sets M \<and> (\<Union>A) = space M \<and> (\<forall>a\<in>A. emeasure M a \<noteq> \<infinity>)"

  1194

  1195 lemma (in sigma_finite_measure) sigma_finite:

  1196   obtains A :: "nat \<Rightarrow> 'a set"

  1197   where "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>"

  1198 proof -

  1199   obtain A :: "'a set set" where

  1200     [simp]: "countable A" and

  1201     A: "A \<subseteq> sets M" "(\<Union>A) = space M" "\<And>a. a \<in> A \<Longrightarrow> emeasure M a \<noteq> \<infinity>"

  1202     using sigma_finite_countable by metis

  1203   show thesis

  1204   proof cases

  1205     assume "A = {}" with \<open>(\<Union>A) = space M\<close> show thesis

  1206       by (intro that[of "\<lambda>_. {}"]) auto

  1207   next

  1208     assume "A \<noteq> {}"

  1209     show thesis

  1210     proof

  1211       show "range (from_nat_into A) \<subseteq> sets M"

  1212         using \<open>A \<noteq> {}\<close> A by auto

  1213       have "(\<Union>i. from_nat_into A i) = \<Union>A"

  1214         using range_from_nat_into[OF \<open>A \<noteq> {}\<close> \<open>countable A\<close>] by auto

  1215       with A show "(\<Union>i. from_nat_into A i) = space M"

  1216         by auto

  1217     qed (intro A from_nat_into \<open>A \<noteq> {}\<close>)

  1218   qed

  1219 qed

  1220

  1221 lemma (in sigma_finite_measure) sigma_finite_disjoint:

  1222   obtains A :: "nat \<Rightarrow> 'a set"

  1223   where "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>" "disjoint_family A"

  1224 proof -

  1225   obtain A :: "nat \<Rightarrow> 'a set" where

  1226     range: "range A \<subseteq> sets M" and

  1227     space: "(\<Union>i. A i) = space M" and

  1228     measure: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"

  1229     using sigma_finite by blast

  1230   show thesis

  1231   proof (rule that[of "disjointed A"])

  1232     show "range (disjointed A) \<subseteq> sets M"

  1233       by (rule sets.range_disjointed_sets[OF range])

  1234     show "(\<Union>i. disjointed A i) = space M"

  1235       and "disjoint_family (disjointed A)"

  1236       using disjoint_family_disjointed UN_disjointed_eq[of A] space range

  1237       by auto

  1238     show "emeasure M (disjointed A i) \<noteq> \<infinity>" for i

  1239     proof -

  1240       have "emeasure M (disjointed A i) \<le> emeasure M (A i)"

  1241         using range disjointed_subset[of A i] by (auto intro!: emeasure_mono)

  1242       then show ?thesis using measure[of i] by (auto simp: top_unique)

  1243     qed

  1244   qed

  1245 qed

  1246

  1247 lemma (in sigma_finite_measure) sigma_finite_incseq:

  1248   obtains A :: "nat \<Rightarrow> 'a set"

  1249   where "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>" "incseq A"

  1250 proof -

  1251   obtain F :: "nat \<Rightarrow> 'a set" where

  1252     F: "range F \<subseteq> sets M" "(\<Union>i. F i) = space M" "\<And>i. emeasure M (F i) \<noteq> \<infinity>"

  1253     using sigma_finite by blast

  1254   show thesis

  1255   proof (rule that[of "\<lambda>n. \<Union>i\<le>n. F i"])

  1256     show "range (\<lambda>n. \<Union>i\<le>n. F i) \<subseteq> sets M"

  1257       using F by (force simp: incseq_def)

  1258     show "(\<Union>n. \<Union>i\<le>n. F i) = space M"

  1259     proof -

  1260       from F have "\<And>x. x \<in> space M \<Longrightarrow> \<exists>i. x \<in> F i" by auto

  1261       with F show ?thesis by fastforce

  1262     qed

  1263     show "emeasure M (\<Union>i\<le>n. F i) \<noteq> \<infinity>" for n

  1264     proof -

  1265       have "emeasure M (\<Union>i\<le>n. F i) \<le> (\<Sum>i\<le>n. emeasure M (F i))"

  1266         using F by (auto intro!: emeasure_subadditive_finite)

  1267       also have "\<dots> < \<infinity>"

  1268         using F by (auto simp: setsum_Pinfty less_top)

  1269       finally show ?thesis by simp

  1270     qed

  1271     show "incseq (\<lambda>n. \<Union>i\<le>n. F i)"

  1272       by (force simp: incseq_def)

  1273   qed

  1274 qed

  1275

  1276 subsection \<open>Measure space induced by distribution of @{const measurable}-functions\<close>

  1277

  1278 definition distr :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b measure" where

  1279   "distr M N f = measure_of (space N) (sets N) (\<lambda>A. emeasure M (f - A \<inter> space M))"

  1280

  1281 lemma

  1282   shows sets_distr[simp, measurable_cong]: "sets (distr M N f) = sets N"

  1283     and space_distr[simp]: "space (distr M N f) = space N"

  1284   by (auto simp: distr_def)

  1285

  1286 lemma

  1287   shows measurable_distr_eq1[simp]: "measurable (distr Mf Nf f) Mf' = measurable Nf Mf'"

  1288     and measurable_distr_eq2[simp]: "measurable Mg' (distr Mg Ng g) = measurable Mg' Ng"

  1289   by (auto simp: measurable_def)

  1290

  1291 lemma distr_cong:

  1292   "M = K \<Longrightarrow> sets N = sets L \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> distr M N f = distr K L g"

  1293   using sets_eq_imp_space_eq[of N L] by (simp add: distr_def Int_def cong: rev_conj_cong)

  1294

  1295 lemma emeasure_distr:

  1296   fixes f :: "'a \<Rightarrow> 'b"

  1297   assumes f: "f \<in> measurable M N" and A: "A \<in> sets N"

  1298   shows "emeasure (distr M N f) A = emeasure M (f - A \<inter> space M)" (is "_ = ?\<mu> A")

  1299   unfolding distr_def

  1300 proof (rule emeasure_measure_of_sigma)

  1301   show "positive (sets N) ?\<mu>"

  1302     by (auto simp: positive_def)

  1303

  1304   show "countably_additive (sets N) ?\<mu>"

  1305   proof (intro countably_additiveI)

  1306     fix A :: "nat \<Rightarrow> 'b set" assume "range A \<subseteq> sets N" "disjoint_family A"

  1307     then have A: "\<And>i. A i \<in> sets N" "(\<Union>i. A i) \<in> sets N" by auto

  1308     then have *: "range (\<lambda>i. f - (A i) \<inter> space M) \<subseteq> sets M"

  1309       using f by (auto simp: measurable_def)

  1310     moreover have "(\<Union>i. f -  A i \<inter> space M) \<in> sets M"

  1311       using * by blast

  1312     moreover have **: "disjoint_family (\<lambda>i. f - A i \<inter> space M)"

  1313       using \<open>disjoint_family A\<close> by (auto simp: disjoint_family_on_def)

  1314     ultimately show "(\<Sum>i. ?\<mu> (A i)) = ?\<mu> (\<Union>i. A i)"

  1315       using suminf_emeasure[OF _ **] A f

  1316       by (auto simp: comp_def vimage_UN)

  1317   qed

  1318   show "sigma_algebra (space N) (sets N)" ..

  1319 qed fact

  1320

  1321 lemma emeasure_Collect_distr:

  1322   assumes X[measurable]: "X \<in> measurable M N" "Measurable.pred N P"

  1323   shows "emeasure (distr M N X) {x\<in>space N. P x} = emeasure M {x\<in>space M. P (X x)}"

  1324   by (subst emeasure_distr)

  1325      (auto intro!: arg_cong2[where f=emeasure] X(1)[THEN measurable_space])

  1326

  1327 lemma emeasure_lfp2[consumes 1, case_names cont f measurable]:

  1328   assumes "P M"

  1329   assumes cont: "sup_continuous F"

  1330   assumes f: "\<And>M. P M \<Longrightarrow> f \<in> measurable M' M"

  1331   assumes *: "\<And>M A. P M \<Longrightarrow> (\<And>N. P N \<Longrightarrow> Measurable.pred N A) \<Longrightarrow> Measurable.pred M (F A)"

  1332   shows "emeasure M' {x\<in>space M'. lfp F (f x)} = (SUP i. emeasure M' {x\<in>space M'. (F ^^ i) (\<lambda>x. False) (f x)})"

  1333 proof (subst (1 2) emeasure_Collect_distr[symmetric, where X=f])

  1334   show "f \<in> measurable M' M"  "f \<in> measurable M' M"

  1335     using f[OF \<open>P M\<close>] by auto

  1336   { fix i show "Measurable.pred M ((F ^^ i) (\<lambda>x. False))"

  1337     using \<open>P M\<close> by (induction i arbitrary: M) (auto intro!: *) }

  1338   show "Measurable.pred M (lfp F)"

  1339     using \<open>P M\<close> cont * by (rule measurable_lfp_coinduct[of P])

  1340

  1341   have "emeasure (distr M' M f) {x \<in> space (distr M' M f). lfp F x} =

  1342     (SUP i. emeasure (distr M' M f) {x \<in> space (distr M' M f). (F ^^ i) (\<lambda>x. False) x})"

  1343     using \<open>P M\<close>

  1344   proof (coinduction arbitrary: M rule: emeasure_lfp')

  1345     case (measurable A N) then have "\<And>N. P N \<Longrightarrow> Measurable.pred (distr M' N f) A"

  1346       by metis

  1347     then have "\<And>N. P N \<Longrightarrow> Measurable.pred N A"

  1348       by simp

  1349     with \<open>P N\<close>[THEN *] show ?case

  1350       by auto

  1351   qed fact

  1352   then show "emeasure (distr M' M f) {x \<in> space M. lfp F x} =

  1353     (SUP i. emeasure (distr M' M f) {x \<in> space M. (F ^^ i) (\<lambda>x. False) x})"

  1354    by simp

  1355 qed

  1356

  1357 lemma distr_id[simp]: "distr N N (\<lambda>x. x) = N"

  1358   by (rule measure_eqI) (auto simp: emeasure_distr)

  1359

  1360 lemma measure_distr:

  1361   "f \<in> measurable M N \<Longrightarrow> S \<in> sets N \<Longrightarrow> measure (distr M N f) S = measure M (f - S \<inter> space M)"

  1362   by (simp add: emeasure_distr measure_def)

  1363

  1364 lemma distr_cong_AE:

  1365   assumes 1: "M = K" "sets N = sets L" and

  1366     2: "(AE x in M. f x = g x)" and "f \<in> measurable M N" and "g \<in> measurable K L"

  1367   shows "distr M N f = distr K L g"

  1368 proof (rule measure_eqI)

  1369   fix A assume "A \<in> sets (distr M N f)"

  1370   with assms show "emeasure (distr M N f) A = emeasure (distr K L g) A"

  1371     by (auto simp add: emeasure_distr intro!: emeasure_eq_AE measurable_sets)

  1372 qed (insert 1, simp)

  1373

  1374 lemma AE_distrD:

  1375   assumes f: "f \<in> measurable M M'"

  1376     and AE: "AE x in distr M M' f. P x"

  1377   shows "AE x in M. P (f x)"

  1378 proof -

  1379   from AE[THEN AE_E] guess N .

  1380   with f show ?thesis

  1381     unfolding eventually_ae_filter

  1382     by (intro bexI[of _ "f - N \<inter> space M"])

  1383        (auto simp: emeasure_distr measurable_def)

  1384 qed

  1385

  1386 lemma AE_distr_iff:

  1387   assumes f[measurable]: "f \<in> measurable M N" and P[measurable]: "{x \<in> space N. P x} \<in> sets N"

  1388   shows "(AE x in distr M N f. P x) \<longleftrightarrow> (AE x in M. P (f x))"

  1389 proof (subst (1 2) AE_iff_measurable[OF _ refl])

  1390   have "f - {x\<in>space N. \<not> P x} \<inter> space M = {x \<in> space M. \<not> P (f x)}"

  1391     using f[THEN measurable_space] by auto

  1392   then show "(emeasure (distr M N f) {x \<in> space (distr M N f). \<not> P x} = 0) =

  1393     (emeasure M {x \<in> space M. \<not> P (f x)} = 0)"

  1394     by (simp add: emeasure_distr)

  1395 qed auto

  1396

  1397 lemma null_sets_distr_iff:

  1398   "f \<in> measurable M N \<Longrightarrow> A \<in> null_sets (distr M N f) \<longleftrightarrow> f - A \<inter> space M \<in> null_sets M \<and> A \<in> sets N"

  1399   by (auto simp add: null_sets_def emeasure_distr)

  1400

  1401 lemma distr_distr:

  1402   "g \<in> measurable N L \<Longrightarrow> f \<in> measurable M N \<Longrightarrow> distr (distr M N f) L g = distr M L (g \<circ> f)"

  1403   by (auto simp add: emeasure_distr measurable_space

  1404            intro!: arg_cong[where f="emeasure M"] measure_eqI)

  1405

  1406 subsection \<open>Real measure values\<close>

  1407

  1408 lemma ring_of_finite_sets: "ring_of_sets (space M) {A\<in>sets M. emeasure M A \<noteq> top}"

  1409 proof (rule ring_of_setsI)

  1410   show "a \<in> {A \<in> sets M. emeasure M A \<noteq> top} \<Longrightarrow> b \<in> {A \<in> sets M. emeasure M A \<noteq> top} \<Longrightarrow>

  1411     a \<union> b \<in> {A \<in> sets M. emeasure M A \<noteq> top}" for a b

  1412     using emeasure_subadditive[of a M b] by (auto simp: top_unique)

  1413

  1414   show "a \<in> {A \<in> sets M. emeasure M A \<noteq> top} \<Longrightarrow> b \<in> {A \<in> sets M. emeasure M A \<noteq> top} \<Longrightarrow>

  1415     a - b \<in> {A \<in> sets M. emeasure M A \<noteq> top}" for a b

  1416     using emeasure_mono[of "a - b" a M] by (auto simp: Diff_subset top_unique)

  1417 qed (auto dest: sets.sets_into_space)

  1418

  1419 lemma measure_nonneg[simp]: "0 \<le> measure M A"

  1420   unfolding measure_def by auto

  1421

  1422 lemma zero_less_measure_iff: "0 < measure M A \<longleftrightarrow> measure M A \<noteq> 0"

  1423   using measure_nonneg[of M A] by (auto simp add: le_less)

  1424

  1425 lemma measure_le_0_iff: "measure M X \<le> 0 \<longleftrightarrow> measure M X = 0"

  1426   using measure_nonneg[of M X] by linarith

  1427

  1428 lemma measure_empty[simp]: "measure M {} = 0"

  1429   unfolding measure_def by (simp add: zero_ennreal.rep_eq)

  1430

  1431 lemma emeasure_eq_ennreal_measure:

  1432   "emeasure M A \<noteq> top \<Longrightarrow> emeasure M A = ennreal (measure M A)"

  1433   by (cases "emeasure M A" rule: ennreal_cases) (auto simp: measure_def)

  1434

  1435 lemma measure_zero_top: "emeasure M A = top \<Longrightarrow> measure M A = 0"

  1436   by (simp add: measure_def enn2ereal_top)

  1437

  1438 lemma measure_eq_emeasure_eq_ennreal: "0 \<le> x \<Longrightarrow> emeasure M A = ennreal x \<Longrightarrow> measure M A = x"

  1439   using emeasure_eq_ennreal_measure[of M A]

  1440   by (cases "A \<in> M") (auto simp: measure_notin_sets emeasure_notin_sets)

  1441

  1442 lemma enn2real_plus:"a < top \<Longrightarrow> b < top \<Longrightarrow> enn2real (a + b) = enn2real a + enn2real b"

  1443   by (simp add: enn2real_def plus_ennreal.rep_eq real_of_ereal_add less_top

  1444            del: real_of_ereal_enn2ereal)

  1445

  1446 lemma measure_Union:

  1447   "emeasure M A \<noteq> \<infinity> \<Longrightarrow> emeasure M B \<noteq> \<infinity> \<Longrightarrow> A \<in> sets M \<Longrightarrow> B \<in> sets M \<Longrightarrow> A \<inter> B = {} \<Longrightarrow>

  1448     measure M (A \<union> B) = measure M A + measure M B"

  1449   by (simp add: measure_def plus_emeasure[symmetric] enn2real_plus less_top)

  1450

  1451 lemma disjoint_family_on_insert:

  1452   "i \<notin> I \<Longrightarrow> disjoint_family_on A (insert i I) \<longleftrightarrow> A i \<inter> (\<Union>i\<in>I. A i) = {} \<and> disjoint_family_on A I"

  1453   by (fastforce simp: disjoint_family_on_def)

  1454

  1455 lemma measure_finite_Union:

  1456   "finite S \<Longrightarrow> AS \<subseteq> sets M \<Longrightarrow> disjoint_family_on A S \<Longrightarrow> (\<And>i. i \<in> S \<Longrightarrow> emeasure M (A i) \<noteq> \<infinity>) \<Longrightarrow>

  1457     measure M (\<Union>i\<in>S. A i) = (\<Sum>i\<in>S. measure M (A i))"

  1458   by (induction S rule: finite_induct)

  1459      (auto simp: disjoint_family_on_insert measure_Union setsum_emeasure[symmetric] sets.countable_UN'[OF countable_finite])

  1460

  1461 lemma measure_Diff:

  1462   assumes finite: "emeasure M A \<noteq> \<infinity>"

  1463   and measurable: "A \<in> sets M" "B \<in> sets M" "B \<subseteq> A"

  1464   shows "measure M (A - B) = measure M A - measure M B"

  1465 proof -

  1466   have "emeasure M (A - B) \<le> emeasure M A" "emeasure M B \<le> emeasure M A"

  1467     using measurable by (auto intro!: emeasure_mono)

  1468   hence "measure M ((A - B) \<union> B) = measure M (A - B) + measure M B"

  1469     using measurable finite by (rule_tac measure_Union) (auto simp: top_unique)

  1470   thus ?thesis using \<open>B \<subseteq> A\<close> by (auto simp: Un_absorb2)

  1471 qed

  1472

  1473 lemma measure_UNION:

  1474   assumes measurable: "range A \<subseteq> sets M" "disjoint_family A"

  1475   assumes finite: "emeasure M (\<Union>i. A i) \<noteq> \<infinity>"

  1476   shows "(\<lambda>i. measure M (A i)) sums (measure M (\<Union>i. A i))"

  1477 proof -

  1478   have "(\<lambda>i. emeasure M (A i)) sums (emeasure M (\<Union>i. A i))"

  1479     unfolding suminf_emeasure[OF measurable, symmetric] by (simp add: summable_sums)

  1480   moreover

  1481   { fix i

  1482     have "emeasure M (A i) \<le> emeasure M (\<Union>i. A i)"

  1483       using measurable by (auto intro!: emeasure_mono)

  1484     then have "emeasure M (A i) = ennreal ((measure M (A i)))"

  1485       using finite by (intro emeasure_eq_ennreal_measure) (auto simp: top_unique) }

  1486   ultimately show ?thesis using finite

  1487     by (subst (asm) (2) emeasure_eq_ennreal_measure) simp_all

  1488 qed

  1489

  1490 lemma measure_subadditive:

  1491   assumes measurable: "A \<in> sets M" "B \<in> sets M"

  1492   and fin: "emeasure M A \<noteq> \<infinity>" "emeasure M B \<noteq> \<infinity>"

  1493   shows "measure M (A \<union> B) \<le> measure M A + measure M B"

  1494 proof -

  1495   have "emeasure M (A \<union> B) \<noteq> \<infinity>"

  1496     using emeasure_subadditive[OF measurable] fin by (auto simp: top_unique)

  1497   then show "(measure M (A \<union> B)) \<le> (measure M A) + (measure M B)"

  1498     using emeasure_subadditive[OF measurable] fin

  1499     apply simp

  1500     apply (subst (asm) (2 3 4) emeasure_eq_ennreal_measure)

  1501     apply (auto simp add: ennreal_plus[symmetric] simp del: ennreal_plus)

  1502     done

  1503 qed

  1504

  1505 lemma measure_subadditive_finite:

  1506   assumes A: "finite I" "AI \<subseteq> sets M" and fin: "\<And>i. i \<in> I \<Longrightarrow> emeasure M (A i) \<noteq> \<infinity>"

  1507   shows "measure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. measure M (A i))"

  1508 proof -

  1509   { have "emeasure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. emeasure M (A i))"

  1510       using emeasure_subadditive_finite[OF A] .

  1511     also have "\<dots> < \<infinity>"

  1512       using fin by (simp add: less_top A)

  1513     finally have "emeasure M (\<Union>i\<in>I. A i) \<noteq> top" by simp }

  1514   note * = this

  1515   show ?thesis

  1516     using emeasure_subadditive_finite[OF A] fin

  1517     unfolding emeasure_eq_ennreal_measure[OF *]

  1518     by (simp_all add: setsum_nonneg emeasure_eq_ennreal_measure)

  1519 qed

  1520

  1521 lemma measure_subadditive_countably:

  1522   assumes A: "range A \<subseteq> sets M" and fin: "(\<Sum>i. emeasure M (A i)) \<noteq> \<infinity>"

  1523   shows "measure M (\<Union>i. A i) \<le> (\<Sum>i. measure M (A i))"

  1524 proof -

  1525   from fin have **: "\<And>i. emeasure M (A i) \<noteq> top"

  1526     using ennreal_suminf_lessD[of "\<lambda>i. emeasure M (A i)"] by (simp add: less_top)

  1527   { have "emeasure M (\<Union>i. A i) \<le> (\<Sum>i. emeasure M (A i))"

  1528       using emeasure_subadditive_countably[OF A] .

  1529     also have "\<dots> < \<infinity>"

  1530       using fin by (simp add: less_top)

  1531     finally have "emeasure M (\<Union>i. A i) \<noteq> top" by simp }

  1532   then have "ennreal (measure M (\<Union>i. A i)) = emeasure M (\<Union>i. A i)"

  1533     by (rule emeasure_eq_ennreal_measure[symmetric])

  1534   also have "\<dots> \<le> (\<Sum>i. emeasure M (A i))"

  1535     using emeasure_subadditive_countably[OF A] .

  1536   also have "\<dots> = ennreal (\<Sum>i. measure M (A i))"

  1537     using fin unfolding emeasure_eq_ennreal_measure[OF **]

  1538     by (subst suminf_ennreal) (auto simp: **)

  1539   finally show ?thesis

  1540     apply (rule ennreal_le_iff[THEN iffD1, rotated])

  1541     apply (intro suminf_nonneg allI measure_nonneg summable_suminf_not_top)

  1542     using fin

  1543     apply (simp add: emeasure_eq_ennreal_measure[OF **])

  1544     done

  1545 qed

  1546

  1547 lemma measure_eq_setsum_singleton:

  1548   "finite S \<Longrightarrow> (\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M) \<Longrightarrow> (\<And>x. x \<in> S \<Longrightarrow> emeasure M {x} \<noteq> \<infinity>) \<Longrightarrow>

  1549     measure M S = (\<Sum>x\<in>S. measure M {x})"

  1550   using emeasure_eq_setsum_singleton[of S M]

  1551   by (intro measure_eq_emeasure_eq_ennreal) (auto simp: setsum_nonneg emeasure_eq_ennreal_measure)

  1552

  1553 lemma Lim_measure_incseq:

  1554   assumes A: "range A \<subseteq> sets M" "incseq A" and fin: "emeasure M (\<Union>i. A i) \<noteq> \<infinity>"

  1555   shows "(\<lambda>i. measure M (A i)) \<longlonglongrightarrow> measure M (\<Union>i. A i)"

  1556 proof (rule tendsto_ennrealD)

  1557   have "ennreal (measure M (\<Union>i. A i)) = emeasure M (\<Union>i. A i)"

  1558     using fin by (auto simp: emeasure_eq_ennreal_measure)

  1559   moreover have "ennreal (measure M (A i)) = emeasure M (A i)" for i

  1560     using assms emeasure_mono[of "A _" "\<Union>i. A i" M]

  1561     by (intro emeasure_eq_ennreal_measure[symmetric]) (auto simp: less_top UN_upper intro: le_less_trans)

  1562   ultimately show "(\<lambda>x. ennreal (Sigma_Algebra.measure M (A x))) \<longlonglongrightarrow> ennreal (Sigma_Algebra.measure M (\<Union>i. A i))"

  1563     using A by (auto intro!: Lim_emeasure_incseq)

  1564 qed auto

  1565

  1566 lemma Lim_measure_decseq:

  1567   assumes A: "range A \<subseteq> sets M" "decseq A" and fin: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"

  1568   shows "(\<lambda>n. measure M (A n)) \<longlonglongrightarrow> measure M (\<Inter>i. A i)"

  1569 proof (rule tendsto_ennrealD)

  1570   have "ennreal (measure M (\<Inter>i. A i)) = emeasure M (\<Inter>i. A i)"

  1571     using fin[of 0] A emeasure_mono[of "\<Inter>i. A i" "A 0" M]

  1572     by (auto intro!: emeasure_eq_ennreal_measure[symmetric] simp: INT_lower less_top intro: le_less_trans)

  1573   moreover have "ennreal (measure M (A i)) = emeasure M (A i)" for i

  1574     using A fin[of i] by (intro emeasure_eq_ennreal_measure[symmetric]) auto

  1575   ultimately show "(\<lambda>x. ennreal (Sigma_Algebra.measure M (A x))) \<longlonglongrightarrow> ennreal (Sigma_Algebra.measure M (\<Inter>i. A i))"

  1576     using fin A by (auto intro!: Lim_emeasure_decseq)

  1577 qed auto

  1578

  1579 subsection \<open>Measure spaces with @{term "emeasure M (space M) < \<infinity>"}\<close>

  1580

  1581 locale finite_measure = sigma_finite_measure M for M +

  1582   assumes finite_emeasure_space: "emeasure M (space M) \<noteq> top"

  1583

  1584 lemma finite_measureI[Pure.intro!]:

  1585   "emeasure M (space M) \<noteq> \<infinity> \<Longrightarrow> finite_measure M"

  1586   proof qed (auto intro!: exI[of _ "{space M}"])

  1587

  1588 lemma (in finite_measure) emeasure_finite[simp, intro]: "emeasure M A \<noteq> top"

  1589   using finite_emeasure_space emeasure_space[of M A] by (auto simp: top_unique)

  1590

  1591 lemma (in finite_measure) emeasure_eq_measure: "emeasure M A = ennreal (measure M A)"

  1592   by (intro emeasure_eq_ennreal_measure) simp

  1593

  1594 lemma (in finite_measure) emeasure_real: "\<exists>r. 0 \<le> r \<and> emeasure M A = ennreal r"

  1595   using emeasure_finite[of A] by (cases "emeasure M A" rule: ennreal_cases) auto

  1596

  1597 lemma (in finite_measure) bounded_measure: "measure M A \<le> measure M (space M)"

  1598   using emeasure_space[of M A] emeasure_real[of A] emeasure_real[of "space M"] by (auto simp: measure_def)

  1599

  1600 lemma (in finite_measure) finite_measure_Diff:

  1601   assumes sets: "A \<in> sets M" "B \<in> sets M" and "B \<subseteq> A"

  1602   shows "measure M (A - B) = measure M A - measure M B"

  1603   using measure_Diff[OF _ assms] by simp

  1604

  1605 lemma (in finite_measure) finite_measure_Union:

  1606   assumes sets: "A \<in> sets M" "B \<in> sets M" and "A \<inter> B = {}"

  1607   shows "measure M (A \<union> B) = measure M A + measure M B"

  1608   using measure_Union[OF _ _ assms] by simp

  1609

  1610 lemma (in finite_measure) finite_measure_finite_Union:

  1611   assumes measurable: "finite S" "AS \<subseteq> sets M" "disjoint_family_on A S"

  1612   shows "measure M (\<Union>i\<in>S. A i) = (\<Sum>i\<in>S. measure M (A i))"

  1613   using measure_finite_Union[OF assms] by simp

  1614

  1615 lemma (in finite_measure) finite_measure_UNION:

  1616   assumes A: "range A \<subseteq> sets M" "disjoint_family A"

  1617   shows "(\<lambda>i. measure M (A i)) sums (measure M (\<Union>i. A i))"

  1618   using measure_UNION[OF A] by simp

  1619

  1620 lemma (in finite_measure) finite_measure_mono:

  1621   assumes "A \<subseteq> B" "B \<in> sets M" shows "measure M A \<le> measure M B"

  1622   using emeasure_mono[OF assms] emeasure_real[of A] emeasure_real[of B] by (auto simp: measure_def)

  1623

  1624 lemma (in finite_measure) finite_measure_subadditive:

  1625   assumes m: "A \<in> sets M" "B \<in> sets M"

  1626   shows "measure M (A \<union> B) \<le> measure M A + measure M B"

  1627   using measure_subadditive[OF m] by simp

  1628

  1629 lemma (in finite_measure) finite_measure_subadditive_finite:

  1630   assumes "finite I" "AI \<subseteq> sets M" shows "measure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. measure M (A i))"

  1631   using measure_subadditive_finite[OF assms] by simp

  1632

  1633 lemma (in finite_measure) finite_measure_subadditive_countably:

  1634   "range A \<subseteq> sets M \<Longrightarrow> summable (\<lambda>i. measure M (A i)) \<Longrightarrow> measure M (\<Union>i. A i) \<le> (\<Sum>i. measure M (A i))"

  1635   by (rule measure_subadditive_countably)

  1636      (simp_all add: ennreal_suminf_neq_top emeasure_eq_measure)

  1637

  1638 lemma (in finite_measure) finite_measure_eq_setsum_singleton:

  1639   assumes "finite S" and *: "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"

  1640   shows "measure M S = (\<Sum>x\<in>S. measure M {x})"

  1641   using measure_eq_setsum_singleton[OF assms] by simp

  1642

  1643 lemma (in finite_measure) finite_Lim_measure_incseq:

  1644   assumes A: "range A \<subseteq> sets M" "incseq A"

  1645   shows "(\<lambda>i. measure M (A i)) \<longlonglongrightarrow> measure M (\<Union>i. A i)"

  1646   using Lim_measure_incseq[OF A] by simp

  1647

  1648 lemma (in finite_measure) finite_Lim_measure_decseq:

  1649   assumes A: "range A \<subseteq> sets M" "decseq A"

  1650   shows "(\<lambda>n. measure M (A n)) \<longlonglongrightarrow> measure M (\<Inter>i. A i)"

  1651   using Lim_measure_decseq[OF A] by simp

  1652

  1653 lemma (in finite_measure) finite_measure_compl:

  1654   assumes S: "S \<in> sets M"

  1655   shows "measure M (space M - S) = measure M (space M) - measure M S"

  1656   using measure_Diff[OF _ sets.top S sets.sets_into_space] S by simp

  1657

  1658 lemma (in finite_measure) finite_measure_mono_AE:

  1659   assumes imp: "AE x in M. x \<in> A \<longrightarrow> x \<in> B" and B: "B \<in> sets M"

  1660   shows "measure M A \<le> measure M B"

  1661   using assms emeasure_mono_AE[OF imp B]

  1662   by (simp add: emeasure_eq_measure)

  1663

  1664 lemma (in finite_measure) finite_measure_eq_AE:

  1665   assumes iff: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B"

  1666   assumes A: "A \<in> sets M" and B: "B \<in> sets M"

  1667   shows "measure M A = measure M B"

  1668   using assms emeasure_eq_AE[OF assms] by (simp add: emeasure_eq_measure)

  1669

  1670 lemma (in finite_measure) measure_increasing: "increasing M (measure M)"

  1671   by (auto intro!: finite_measure_mono simp: increasing_def)

  1672

  1673 lemma (in finite_measure) measure_zero_union:

  1674   assumes "s \<in> sets M" "t \<in> sets M" "measure M t = 0"

  1675   shows "measure M (s \<union> t) = measure M s"

  1676 using assms

  1677 proof -

  1678   have "measure M (s \<union> t) \<le> measure M s"

  1679     using finite_measure_subadditive[of s t] assms by auto

  1680   moreover have "measure M (s \<union> t) \<ge> measure M s"

  1681     using assms by (blast intro: finite_measure_mono)

  1682   ultimately show ?thesis by simp

  1683 qed

  1684

  1685 lemma (in finite_measure) measure_eq_compl:

  1686   assumes "s \<in> sets M" "t \<in> sets M"

  1687   assumes "measure M (space M - s) = measure M (space M - t)"

  1688   shows "measure M s = measure M t"

  1689   using assms finite_measure_compl by auto

  1690

  1691 lemma (in finite_measure) measure_eq_bigunion_image:

  1692   assumes "range f \<subseteq> sets M" "range g \<subseteq> sets M"

  1693   assumes "disjoint_family f" "disjoint_family g"

  1694   assumes "\<And> n :: nat. measure M (f n) = measure M (g n)"

  1695   shows "measure M (\<Union>i. f i) = measure M (\<Union>i. g i)"

  1696 using assms

  1697 proof -

  1698   have a: "(\<lambda> i. measure M (f i)) sums (measure M (\<Union>i. f i))"

  1699     by (rule finite_measure_UNION[OF assms(1,3)])

  1700   have b: "(\<lambda> i. measure M (g i)) sums (measure M (\<Union>i. g i))"

  1701     by (rule finite_measure_UNION[OF assms(2,4)])

  1702   show ?thesis using sums_unique[OF b] sums_unique[OF a] assms by simp

  1703 qed

  1704

  1705 lemma (in finite_measure) measure_countably_zero:

  1706   assumes "range c \<subseteq> sets M"

  1707   assumes "\<And> i. measure M (c i) = 0"

  1708   shows "measure M (\<Union>i :: nat. c i) = 0"

  1709 proof (rule antisym)

  1710   show "measure M (\<Union>i :: nat. c i) \<le> 0"

  1711     using finite_measure_subadditive_countably[OF assms(1)] by (simp add: assms(2))

  1712 qed simp

  1713

  1714 lemma (in finite_measure) measure_space_inter:

  1715   assumes events:"s \<in> sets M" "t \<in> sets M"

  1716   assumes "measure M t = measure M (space M)"

  1717   shows "measure M (s \<inter> t) = measure M s"

  1718 proof -

  1719   have "measure M ((space M - s) \<union> (space M - t)) = measure M (space M - s)"

  1720     using events assms finite_measure_compl[of "t"] by (auto intro!: measure_zero_union)

  1721   also have "(space M - s) \<union> (space M - t) = space M - (s \<inter> t)"

  1722     by blast

  1723   finally show "measure M (s \<inter> t) = measure M s"

  1724     using events by (auto intro!: measure_eq_compl[of "s \<inter> t" s])

  1725 qed

  1726

  1727 lemma (in finite_measure) measure_equiprobable_finite_unions:

  1728   assumes s: "finite s" "\<And>x. x \<in> s \<Longrightarrow> {x} \<in> sets M"

  1729   assumes "\<And> x y. \<lbrakk>x \<in> s; y \<in> s\<rbrakk> \<Longrightarrow> measure M {x} = measure M {y}"

  1730   shows "measure M s = real (card s) * measure M {SOME x. x \<in> s}"

  1731 proof cases

  1732   assume "s \<noteq> {}"

  1733   then have "\<exists> x. x \<in> s" by blast

  1734   from someI_ex[OF this] assms

  1735   have prob_some: "\<And> x. x \<in> s \<Longrightarrow> measure M {x} = measure M {SOME y. y \<in> s}" by blast

  1736   have "measure M s = (\<Sum> x \<in> s. measure M {x})"

  1737     using finite_measure_eq_setsum_singleton[OF s] by simp

  1738   also have "\<dots> = (\<Sum> x \<in> s. measure M {SOME y. y \<in> s})" using prob_some by auto

  1739   also have "\<dots> = real (card s) * measure M {(SOME x. x \<in> s)}"

  1740     using setsum_constant assms by simp

  1741   finally show ?thesis by simp

  1742 qed simp

  1743

  1744 lemma (in finite_measure) measure_real_sum_image_fn:

  1745   assumes "e \<in> sets M"

  1746   assumes "\<And> x. x \<in> s \<Longrightarrow> e \<inter> f x \<in> sets M"

  1747   assumes "finite s"

  1748   assumes disjoint: "\<And> x y. \<lbrakk>x \<in> s ; y \<in> s ; x \<noteq> y\<rbrakk> \<Longrightarrow> f x \<inter> f y = {}"

  1749   assumes upper: "space M \<subseteq> (\<Union>i \<in> s. f i)"

  1750   shows "measure M e = (\<Sum> x \<in> s. measure M (e \<inter> f x))"

  1751 proof -

  1752   have "e \<subseteq> (\<Union>i\<in>s. f i)"

  1753     using \<open>e \<in> sets M\<close> sets.sets_into_space upper by blast

  1754   then have e: "e = (\<Union>i \<in> s. e \<inter> f i)"

  1755     by auto

  1756   hence "measure M e = measure M (\<Union>i \<in> s. e \<inter> f i)" by simp

  1757   also have "\<dots> = (\<Sum> x \<in> s. measure M (e \<inter> f x))"

  1758   proof (rule finite_measure_finite_Union)

  1759     show "finite s" by fact

  1760     show "(\<lambda>i. e \<inter> f i)s \<subseteq> sets M" using assms(2) by auto

  1761     show "disjoint_family_on (\<lambda>i. e \<inter> f i) s"

  1762       using disjoint by (auto simp: disjoint_family_on_def)

  1763   qed

  1764   finally show ?thesis .

  1765 qed

  1766

  1767 lemma (in finite_measure) measure_exclude:

  1768   assumes "A \<in> sets M" "B \<in> sets M"

  1769   assumes "measure M A = measure M (space M)" "A \<inter> B = {}"

  1770   shows "measure M B = 0"

  1771   using measure_space_inter[of B A] assms by (auto simp: ac_simps)

  1772 lemma (in finite_measure) finite_measure_distr:

  1773   assumes f: "f \<in> measurable M M'"

  1774   shows "finite_measure (distr M M' f)"

  1775 proof (rule finite_measureI)

  1776   have "f - space M' \<inter> space M = space M" using f by (auto dest: measurable_space)

  1777   with f show "emeasure (distr M M' f) (space (distr M M' f)) \<noteq> \<infinity>" by (auto simp: emeasure_distr)

  1778 qed

  1779

  1780 lemma emeasure_gfp[consumes 1, case_names cont measurable]:

  1781   assumes sets[simp]: "\<And>s. sets (M s) = sets N"

  1782   assumes "\<And>s. finite_measure (M s)"

  1783   assumes cont: "inf_continuous F" "inf_continuous f"

  1784   assumes meas: "\<And>P. Measurable.pred N P \<Longrightarrow> Measurable.pred N (F P)"

  1785   assumes iter: "\<And>P s. Measurable.pred N P \<Longrightarrow> emeasure (M s) {x\<in>space N. F P x} = f (\<lambda>s. emeasure (M s) {x\<in>space N. P x}) s"

  1786   assumes bound: "\<And>P. f P \<le> f (\<lambda>s. emeasure (M s) (space (M s)))"

  1787   shows "emeasure (M s) {x\<in>space N. gfp F x} = gfp f s"

  1788 proof (subst gfp_transfer_bounded[where \<alpha>="\<lambda>F s. emeasure (M s) {x\<in>space N. F x}" and g=f and f=F and

  1789     P="Measurable.pred N", symmetric])

  1790   interpret finite_measure "M s" for s by fact

  1791   fix C assume "decseq C" "\<And>i. Measurable.pred N (C i)"

  1792   then show "(\<lambda>s. emeasure (M s) {x \<in> space N. (INF i. C i) x}) = (INF i. (\<lambda>s. emeasure (M s) {x \<in> space N. C i x}))"

  1793     unfolding INF_apply[abs_def]

  1794     by (subst INF_emeasure_decseq) (auto simp: antimono_def fun_eq_iff intro!: arg_cong2[where f=emeasure])

  1795 next

  1796   show "f x \<le> (\<lambda>s. emeasure (M s) {x \<in> space N. F top x})" for x

  1797     using bound[of x] sets_eq_imp_space_eq[OF sets] by (simp add: iter)

  1798 qed (auto simp add: iter le_fun_def INF_apply[abs_def] intro!: meas cont)

  1799

  1800 subsection \<open>Counting space\<close>

  1801

  1802 lemma strict_monoI_Suc:

  1803   assumes ord [simp]: "(\<And>n. f n < f (Suc n))" shows "strict_mono f"

  1804   unfolding strict_mono_def

  1805 proof safe

  1806   fix n m :: nat assume "n < m" then show "f n < f m"

  1807     by (induct m) (auto simp: less_Suc_eq intro: less_trans ord)

  1808 qed

  1809

  1810 lemma emeasure_count_space:

  1811   assumes "X \<subseteq> A" shows "emeasure (count_space A) X = (if finite X then of_nat (card X) else \<infinity>)"

  1812     (is "_ = ?M X")

  1813   unfolding count_space_def

  1814 proof (rule emeasure_measure_of_sigma)

  1815   show "X \<in> Pow A" using \<open>X \<subseteq> A\<close> by auto

  1816   show "sigma_algebra A (Pow A)" by (rule sigma_algebra_Pow)

  1817   show positive: "positive (Pow A) ?M"

  1818     by (auto simp: positive_def)

  1819   have additive: "additive (Pow A) ?M"

  1820     by (auto simp: card_Un_disjoint additive_def)

  1821

  1822   interpret ring_of_sets A "Pow A"

  1823     by (rule ring_of_setsI) auto

  1824   show "countably_additive (Pow A) ?M"

  1825     unfolding countably_additive_iff_continuous_from_below[OF positive additive]

  1826   proof safe

  1827     fix F :: "nat \<Rightarrow> 'a set" assume "incseq F"

  1828     show "(\<lambda>i. ?M (F i)) \<longlonglongrightarrow> ?M (\<Union>i. F i)"

  1829     proof cases

  1830       assume "\<exists>i. \<forall>j\<ge>i. F i = F j"

  1831       then guess i .. note i = this

  1832       { fix j from i \<open>incseq F\<close> have "F j \<subseteq> F i"

  1833           by (cases "i \<le> j") (auto simp: incseq_def) }

  1834       then have eq: "(\<Union>i. F i) = F i"

  1835         by auto

  1836       with i show ?thesis

  1837         by (auto intro!: Lim_transform_eventually[OF _ tendsto_const] eventually_sequentiallyI[where c=i])

  1838     next

  1839       assume "\<not> (\<exists>i. \<forall>j\<ge>i. F i = F j)"

  1840       then obtain f where f: "\<And>i. i \<le> f i" "\<And>i. F i \<noteq> F (f i)" by metis

  1841       then have "\<And>i. F i \<subseteq> F (f i)" using \<open>incseq F\<close> by (auto simp: incseq_def)

  1842       with f have *: "\<And>i. F i \<subset> F (f i)" by auto

  1843

  1844       have "incseq (\<lambda>i. ?M (F i))"

  1845         using \<open>incseq F\<close> unfolding incseq_def by (auto simp: card_mono dest: finite_subset)

  1846       then have "(\<lambda>i. ?M (F i)) \<longlonglongrightarrow> (SUP n. ?M (F n))"

  1847         by (rule LIMSEQ_SUP)

  1848

  1849       moreover have "(SUP n. ?M (F n)) = top"

  1850       proof (rule ennreal_SUP_eq_top)

  1851         fix n :: nat show "\<exists>k::nat\<in>UNIV. of_nat n \<le> ?M (F k)"

  1852         proof (induct n)

  1853           case (Suc n)

  1854           then guess k .. note k = this

  1855           moreover have "finite (F k) \<Longrightarrow> finite (F (f k)) \<Longrightarrow> card (F k) < card (F (f k))"

  1856             using \<open>F k \<subset> F (f k)\<close> by (simp add: psubset_card_mono)

  1857           moreover have "finite (F (f k)) \<Longrightarrow> finite (F k)"

  1858             using \<open>k \<le> f k\<close> \<open>incseq F\<close> by (auto simp: incseq_def dest: finite_subset)

  1859           ultimately show ?case

  1860             by (auto intro!: exI[of _ "f k"] simp del: of_nat_Suc)

  1861         qed auto

  1862       qed

  1863

  1864       moreover

  1865       have "inj (\<lambda>n. F ((f ^^ n) 0))"

  1866         by (intro strict_mono_imp_inj_on strict_monoI_Suc) (simp add: *)

  1867       then have 1: "infinite (range (\<lambda>i. F ((f ^^ i) 0)))"

  1868         by (rule range_inj_infinite)

  1869       have "infinite (Pow (\<Union>i. F i))"

  1870         by (rule infinite_super[OF _ 1]) auto

  1871       then have "infinite (\<Union>i. F i)"

  1872         by auto

  1873

  1874       ultimately show ?thesis by auto

  1875     qed

  1876   qed

  1877 qed

  1878

  1879 lemma distr_bij_count_space:

  1880   assumes f: "bij_betw f A B"

  1881   shows "distr (count_space A) (count_space B) f = count_space B"

  1882 proof (rule measure_eqI)

  1883   have f': "f \<in> measurable (count_space A) (count_space B)"

  1884     using f unfolding Pi_def bij_betw_def by auto

  1885   fix X assume "X \<in> sets (distr (count_space A) (count_space B) f)"

  1886   then have X: "X \<in> sets (count_space B)" by auto

  1887   moreover from X have "f - X \<inter> A = the_inv_into A f  X"

  1888     using f by (auto simp: bij_betw_def subset_image_iff image_iff the_inv_into_f_f intro: the_inv_into_f_f[symmetric])

  1889   moreover have "inj_on (the_inv_into A f) B"

  1890     using X f by (auto simp: bij_betw_def inj_on_the_inv_into)

  1891   with X have "inj_on (the_inv_into A f) X"

  1892     by (auto intro: subset_inj_on)

  1893   ultimately show "emeasure (distr (count_space A) (count_space B) f) X = emeasure (count_space B) X"

  1894     using f unfolding emeasure_distr[OF f' X]

  1895     by (subst (1 2) emeasure_count_space) (auto simp: card_image dest: finite_imageD)

  1896 qed simp

  1897

  1898 lemma emeasure_count_space_finite[simp]:

  1899   "X \<subseteq> A \<Longrightarrow> finite X \<Longrightarrow> emeasure (count_space A) X = of_nat (card X)"

  1900   using emeasure_count_space[of X A] by simp

  1901

  1902 lemma emeasure_count_space_infinite[simp]:

  1903   "X \<subseteq> A \<Longrightarrow> infinite X \<Longrightarrow> emeasure (count_space A) X = \<infinity>"

  1904   using emeasure_count_space[of X A] by simp

  1905

  1906 lemma measure_count_space: "measure (count_space A) X = (if X \<subseteq> A then of_nat (card X) else 0)"

  1907   by (cases "finite X") (auto simp: measure_notin_sets ennreal_of_nat_eq_real_of_nat

  1908                                     measure_zero_top measure_eq_emeasure_eq_ennreal)

  1909

  1910 lemma emeasure_count_space_eq_0:

  1911   "emeasure (count_space A) X = 0 \<longleftrightarrow> (X \<subseteq> A \<longrightarrow> X = {})"

  1912 proof cases

  1913   assume X: "X \<subseteq> A"

  1914   then show ?thesis

  1915   proof (intro iffI impI)

  1916     assume "emeasure (count_space A) X = 0"

  1917     with X show "X = {}"

  1918       by (subst (asm) emeasure_count_space) (auto split: if_split_asm)

  1919   qed simp

  1920 qed (simp add: emeasure_notin_sets)

  1921

  1922 lemma space_empty: "space M = {} \<Longrightarrow> M = count_space {}"

  1923   by (rule measure_eqI) (simp_all add: space_empty_iff)

  1924

  1925 lemma null_sets_count_space: "null_sets (count_space A) = { {} }"

  1926   unfolding null_sets_def by (auto simp add: emeasure_count_space_eq_0)

  1927

  1928 lemma AE_count_space: "(AE x in count_space A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x)"

  1929   unfolding eventually_ae_filter by (auto simp add: null_sets_count_space)

  1930

  1931 lemma sigma_finite_measure_count_space_countable:

  1932   assumes A: "countable A"

  1933   shows "sigma_finite_measure (count_space A)"

  1934   proof qed (insert A, auto intro!: exI[of _ "(\<lambda>a. {a})  A"])

  1935

  1936 lemma sigma_finite_measure_count_space:

  1937   fixes A :: "'a::countable set" shows "sigma_finite_measure (count_space A)"

  1938   by (rule sigma_finite_measure_count_space_countable) auto

  1939

  1940 lemma finite_measure_count_space:

  1941   assumes [simp]: "finite A"

  1942   shows "finite_measure (count_space A)"

  1943   by rule simp

  1944

  1945 lemma sigma_finite_measure_count_space_finite:

  1946   assumes A: "finite A" shows "sigma_finite_measure (count_space A)"

  1947 proof -

  1948   interpret finite_measure "count_space A" using A by (rule finite_measure_count_space)

  1949   show "sigma_finite_measure (count_space A)" ..

  1950 qed

  1951

  1952 subsection \<open>Measure restricted to space\<close>

  1953

  1954 lemma emeasure_restrict_space:

  1955   assumes "\<Omega> \<inter> space M \<in> sets M" "A \<subseteq> \<Omega>"

  1956   shows "emeasure (restrict_space M \<Omega>) A = emeasure M A"

  1957 proof (cases "A \<in> sets M")

  1958   case True

  1959   show ?thesis

  1960   proof (rule emeasure_measure_of[OF restrict_space_def])

  1961     show "op \<inter> \<Omega>  sets M \<subseteq> Pow (\<Omega> \<inter> space M)" "A \<in> sets (restrict_space M \<Omega>)"

  1962       using \<open>A \<subseteq> \<Omega>\<close> \<open>A \<in> sets M\<close> sets.space_closed by (auto simp: sets_restrict_space)

  1963     show "positive (sets (restrict_space M \<Omega>)) (emeasure M)"

  1964       by (auto simp: positive_def)

  1965     show "countably_additive (sets (restrict_space M \<Omega>)) (emeasure M)"

  1966     proof (rule countably_additiveI)

  1967       fix A :: "nat \<Rightarrow> _" assume "range A \<subseteq> sets (restrict_space M \<Omega>)" "disjoint_family A"

  1968       with assms have "\<And>i. A i \<in> sets M" "\<And>i. A i \<subseteq> space M" "disjoint_family A"

  1969         by (fastforce simp: sets_restrict_space_iff[OF assms(1)] image_subset_iff

  1970                       dest: sets.sets_into_space)+

  1971       then show "(\<Sum>i. emeasure M (A i)) = emeasure M (\<Union>i. A i)"

  1972         by (subst suminf_emeasure) (auto simp: disjoint_family_subset)

  1973     qed

  1974   qed

  1975 next

  1976   case False

  1977   with assms have "A \<notin> sets (restrict_space M \<Omega>)"

  1978     by (simp add: sets_restrict_space_iff)

  1979   with False show ?thesis

  1980     by (simp add: emeasure_notin_sets)

  1981 qed

  1982

  1983 lemma measure_restrict_space:

  1984   assumes "\<Omega> \<inter> space M \<in> sets M" "A \<subseteq> \<Omega>"

  1985   shows "measure (restrict_space M \<Omega>) A = measure M A"

  1986   using emeasure_restrict_space[OF assms] by (simp add: measure_def)

  1987

  1988 lemma AE_restrict_space_iff:

  1989   assumes "\<Omega> \<inter> space M \<in> sets M"

  1990   shows "(AE x in restrict_space M \<Omega>. P x) \<longleftrightarrow> (AE x in M. x \<in> \<Omega> \<longrightarrow> P x)"

  1991 proof -

  1992   have ex_cong: "\<And>P Q f. (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> (\<And>x. Q x \<Longrightarrow> P (f x)) \<Longrightarrow> (\<exists>x. P x) \<longleftrightarrow> (\<exists>x. Q x)"

  1993     by auto

  1994   { fix X assume X: "X \<in> sets M" "emeasure M X = 0"

  1995     then have "emeasure M (\<Omega> \<inter> space M \<inter> X) \<le> emeasure M X"

  1996       by (intro emeasure_mono) auto

  1997     then have "emeasure M (\<Omega> \<inter> space M \<inter> X) = 0"

  1998       using X by (auto intro!: antisym) }

  1999   with assms show ?thesis

  2000     unfolding eventually_ae_filter

  2001     by (auto simp add: space_restrict_space null_sets_def sets_restrict_space_iff

  2002                        emeasure_restrict_space cong: conj_cong

  2003              intro!: ex_cong[where f="\<lambda>X. (\<Omega> \<inter> space M) \<inter> X"])

  2004 qed

  2005

  2006 lemma restrict_restrict_space:

  2007   assumes "A \<inter> space M \<in> sets M" "B \<inter> space M \<in> sets M"

  2008   shows "restrict_space (restrict_space M A) B = restrict_space M (A \<inter> B)" (is "?l = ?r")

  2009 proof (rule measure_eqI[symmetric])

  2010   show "sets ?r = sets ?l"

  2011     unfolding sets_restrict_space image_comp by (intro image_cong) auto

  2012 next

  2013   fix X assume "X \<in> sets (restrict_space M (A \<inter> B))"

  2014   then obtain Y where "Y \<in> sets M" "X = Y \<inter> A \<inter> B"

  2015     by (auto simp: sets_restrict_space)

  2016   with assms sets.Int[OF assms] show "emeasure ?r X = emeasure ?l X"

  2017     by (subst (1 2) emeasure_restrict_space)

  2018        (auto simp: space_restrict_space sets_restrict_space_iff emeasure_restrict_space ac_simps)

  2019 qed

  2020

  2021 lemma restrict_count_space: "restrict_space (count_space B) A = count_space (A \<inter> B)"

  2022 proof (rule measure_eqI)

  2023   show "sets (restrict_space (count_space B) A) = sets (count_space (A \<inter> B))"

  2024     by (subst sets_restrict_space) auto

  2025   moreover fix X assume "X \<in> sets (restrict_space (count_space B) A)"

  2026   ultimately have "X \<subseteq> A \<inter> B" by auto

  2027   then show "emeasure (restrict_space (count_space B) A) X = emeasure (count_space (A \<inter> B)) X"

  2028     by (cases "finite X") (auto simp add: emeasure_restrict_space)

  2029 qed

  2030

  2031 lemma sigma_finite_measure_restrict_space:

  2032   assumes "sigma_finite_measure M"

  2033   and A: "A \<in> sets M"

  2034   shows "sigma_finite_measure (restrict_space M A)"

  2035 proof -

  2036   interpret sigma_finite_measure M by fact

  2037   from sigma_finite_countable obtain C

  2038     where C: "countable C" "C \<subseteq> sets M" "(\<Union>C) = space M" "\<forall>a\<in>C. emeasure M a \<noteq> \<infinity>"

  2039     by blast

  2040   let ?C = "op \<inter> A  C"

  2041   from C have "countable ?C" "?C \<subseteq> sets (restrict_space M A)" "(\<Union>?C) = space (restrict_space M A)"

  2042     by(auto simp add: sets_restrict_space space_restrict_space)

  2043   moreover {

  2044     fix a

  2045     assume "a \<in> ?C"

  2046     then obtain a' where "a = A \<inter> a'" "a' \<in> C" ..

  2047     then have "emeasure (restrict_space M A) a \<le> emeasure M a'"

  2048       using A C by(auto simp add: emeasure_restrict_space intro: emeasure_mono)

  2049     also have "\<dots> < \<infinity>" using C(4)[rule_format, of a'] \<open>a' \<in> C\<close> by (simp add: less_top)

  2050     finally have "emeasure (restrict_space M A) a \<noteq> \<infinity>" by simp }

  2051   ultimately show ?thesis

  2052     by unfold_locales (rule exI conjI|assumption|blast)+

  2053 qed

  2054

  2055 lemma finite_measure_restrict_space:

  2056   assumes "finite_measure M"

  2057   and A: "A \<in> sets M"

  2058   shows "finite_measure (restrict_space M A)"

  2059 proof -

  2060   interpret finite_measure M by fact

  2061   show ?thesis

  2062     by(rule finite_measureI)(simp add: emeasure_restrict_space A space_restrict_space)

  2063 qed

  2064

  2065 lemma restrict_distr:

  2066   assumes [measurable]: "f \<in> measurable M N"

  2067   assumes [simp]: "\<Omega> \<inter> space N \<in> sets N" and restrict: "f \<in> space M \<rightarrow> \<Omega>"

  2068   shows "restrict_space (distr M N f) \<Omega> = distr M (restrict_space N \<Omega>) f"

  2069   (is "?l = ?r")

  2070 proof (rule measure_eqI)

  2071   fix A assume "A \<in> sets (restrict_space (distr M N f) \<Omega>)"

  2072   with restrict show "emeasure ?l A = emeasure ?r A"

  2073     by (subst emeasure_distr)

  2074        (auto simp: sets_restrict_space_iff emeasure_restrict_space emeasure_distr

  2075              intro!: measurable_restrict_space2)

  2076 qed (simp add: sets_restrict_space)

  2077

  2078 lemma measure_eqI_restrict_generator:

  2079   assumes E: "Int_stable E" "E \<subseteq> Pow \<Omega>" "\<And>X. X \<in> E \<Longrightarrow> emeasure M X = emeasure N X"

  2080   assumes sets_eq: "sets M = sets N" and \<Omega>: "\<Omega> \<in> sets M"

  2081   assumes "sets (restrict_space M \<Omega>) = sigma_sets \<Omega> E"

  2082   assumes "sets (restrict_space N \<Omega>) = sigma_sets \<Omega> E"

  2083   assumes ae: "AE x in M. x \<in> \<Omega>" "AE x in N. x \<in> \<Omega>"

  2084   assumes A: "countable A" "A \<noteq> {}" "A \<subseteq> E" "\<Union>A = \<Omega>" "\<And>a. a \<in> A \<Longrightarrow> emeasure M a \<noteq> \<infinity>"

  2085   shows "M = N"

  2086 proof (rule measure_eqI)

  2087   fix X assume X: "X \<in> sets M"

  2088   then have "emeasure M X = emeasure (restrict_space M \<Omega>) (X \<inter> \<Omega>)"

  2089     using ae \<Omega> by (auto simp add: emeasure_restrict_space intro!: emeasure_eq_AE)

  2090   also have "restrict_space M \<Omega> = restrict_space N \<Omega>"

  2091   proof (rule measure_eqI_generator_eq)

  2092     fix X assume "X \<in> E"

  2093     then show "emeasure (restrict_space M \<Omega>) X = emeasure (restrict_space N \<Omega>) X"

  2094       using E \<Omega> by (subst (1 2) emeasure_restrict_space) (auto simp: sets_eq sets_eq[THEN sets_eq_imp_space_eq])

  2095   next

  2096     show "range (from_nat_into A) \<subseteq> E" "(\<Union>i. from_nat_into A i) = \<Omega>"

  2097       using A by (auto cong del: strong_SUP_cong)

  2098   next

  2099     fix i

  2100     have "emeasure (restrict_space M \<Omega>) (from_nat_into A i) = emeasure M (from_nat_into A i)"

  2101       using A \<Omega> by (subst emeasure_restrict_space)

  2102                    (auto simp: sets_eq sets_eq[THEN sets_eq_imp_space_eq] intro: from_nat_into)

  2103     with A show "emeasure (restrict_space M \<Omega>) (from_nat_into A i) \<noteq> \<infinity>"

  2104       by (auto intro: from_nat_into)

  2105   qed fact+

  2106   also have "emeasure (restrict_space N \<Omega>) (X \<inter> \<Omega>) = emeasure N X"

  2107     using X ae \<Omega> by (auto simp add: emeasure_restrict_space sets_eq intro!: emeasure_eq_AE)

  2108   finally show "emeasure M X = emeasure N X" .

  2109 qed fact

  2110

  2111 subsection \<open>Null measure\<close>

  2112

  2113 definition "null_measure M = sigma (space M) (sets M)"

  2114

  2115 lemma space_null_measure[simp]: "space (null_measure M) = space M"

  2116   by (simp add: null_measure_def)

  2117

  2118 lemma sets_null_measure[simp, measurable_cong]: "sets (null_measure M) = sets M"

  2119   by (simp add: null_measure_def)

  2120

  2121 lemma emeasure_null_measure[simp]: "emeasure (null_measure M) X = 0"

  2122   by (cases "X \<in> sets M", rule emeasure_measure_of)

  2123      (auto simp: positive_def countably_additive_def emeasure_notin_sets null_measure_def

  2124            dest: sets.sets_into_space)

  2125

  2126 lemma measure_null_measure[simp]: "measure (null_measure M) X = 0"

  2127   by (intro measure_eq_emeasure_eq_ennreal) auto

  2128

  2129 lemma null_measure_idem [simp]: "null_measure (null_measure M) = null_measure M"

  2130   by(rule measure_eqI) simp_all

  2131

  2132 subsection \<open>Scaling a measure\<close>

  2133

  2134 definition scale_measure :: "ennreal \<Rightarrow> 'a measure \<Rightarrow> 'a measure"

  2135 where

  2136   "scale_measure r M = measure_of (space M) (sets M) (\<lambda>A. r * emeasure M A)"

  2137

  2138 lemma space_scale_measure: "space (scale_measure r M) = space M"

  2139   by (simp add: scale_measure_def)

  2140

  2141 lemma sets_scale_measure [simp, measurable_cong]: "sets (scale_measure r M) = sets M"

  2142   by (simp add: scale_measure_def)

  2143

  2144 lemma emeasure_scale_measure [simp]:

  2145   "emeasure (scale_measure r M) A = r * emeasure M A"

  2146   (is "_ = ?\<mu> A")

  2147 proof(cases "A \<in> sets M")

  2148   case True

  2149   show ?thesis unfolding scale_measure_def

  2150   proof(rule emeasure_measure_of_sigma)

  2151     show "sigma_algebra (space M) (sets M)" ..

  2152     show "positive (sets M) ?\<mu>" by (simp add: positive_def)

  2153     show "countably_additive (sets M) ?\<mu>"

  2154     proof (rule countably_additiveI)

  2155       fix A :: "nat \<Rightarrow> _"  assume *: "range A \<subseteq> sets M" "disjoint_family A"

  2156       have "(\<Sum>i. ?\<mu> (A i)) = r * (\<Sum>i. emeasure M (A i))"

  2157         by simp

  2158       also have "\<dots> = ?\<mu> (\<Union>i. A i)" using * by(simp add: suminf_emeasure)

  2159       finally show "(\<Sum>i. ?\<mu> (A i)) = ?\<mu> (\<Union>i. A i)" .

  2160     qed

  2161   qed(fact True)

  2162 qed(simp add: emeasure_notin_sets)

  2163

  2164 lemma scale_measure_1 [simp]: "scale_measure 1 M = M"

  2165   by(rule measure_eqI) simp_all

  2166

  2167 lemma scale_measure_0[simp]: "scale_measure 0 M = null_measure M"

  2168   by(rule measure_eqI) simp_all

  2169

  2170 lemma measure_scale_measure [simp]: "0 \<le> r \<Longrightarrow> measure (scale_measure r M) A = r * measure M A"

  2171   using emeasure_scale_measure[of r M A]

  2172     emeasure_eq_ennreal_measure[of M A]

  2173     measure_eq_emeasure_eq_ennreal[of _ "scale_measure r M" A]

  2174   by (cases "emeasure (scale_measure r M) A = top")

  2175      (auto simp del: emeasure_scale_measure

  2176            simp: ennreal_top_eq_mult_iff ennreal_mult_eq_top_iff measure_zero_top ennreal_mult[symmetric])

  2177

  2178 lemma scale_scale_measure [simp]:

  2179   "scale_measure r (scale_measure r' M) = scale_measure (r * r') M"

  2180   by (rule measure_eqI) (simp_all add: max_def mult.assoc)

  2181

  2182 lemma scale_null_measure [simp]: "scale_measure r (null_measure M) = null_measure M"

  2183   by (rule measure_eqI) simp_all

  2184

  2185

  2186 subsection \<open>Complete lattice structure on measures\<close>

  2187

  2188 lemma (in finite_measure) finite_measure_Diff':

  2189   "A \<in> sets M \<Longrightarrow> B \<in> sets M \<Longrightarrow> measure M (A - B) = measure M A - measure M (A \<inter> B)"

  2190   using finite_measure_Diff[of A "A \<inter> B"] by (auto simp: Diff_Int)

  2191

  2192 lemma (in finite_measure) finite_measure_Union':

  2193   "A \<in> sets M \<Longrightarrow> B \<in> sets M \<Longrightarrow> measure M (A \<union> B) = measure M A + measure M (B - A)"

  2194   using finite_measure_Union[of A "B - A"] by auto

  2195

  2196 lemma finite_unsigned_Hahn_decomposition:

  2197   assumes "finite_measure M" "finite_measure N" and [simp]: "sets N = sets M"

  2198   shows "\<exists>Y\<in>sets M. (\<forall>X\<in>sets M. X \<subseteq> Y \<longrightarrow> N X \<le> M X) \<and> (\<forall>X\<in>sets M. X \<inter> Y = {} \<longrightarrow> M X \<le> N X)"

  2199 proof -

  2200   interpret M: finite_measure M by fact

  2201   interpret N: finite_measure N by fact

  2202

  2203   define d where "d X = measure M X - measure N X" for X

  2204

  2205   have [intro]: "bdd_above (dsets M)"

  2206     using sets.sets_into_space[of _ M]

  2207     by (intro bdd_aboveI[where M="measure M (space M)"])

  2208        (auto simp: d_def field_simps subset_eq intro!: add_increasing M.finite_measure_mono)

  2209

  2210   define \<gamma> where "\<gamma> = (SUP X:sets M. d X)"

  2211   have le_\<gamma>[intro]: "X \<in> sets M \<Longrightarrow> d X \<le> \<gamma>" for X

  2212     by (auto simp: \<gamma>_def intro!: cSUP_upper)

  2213

  2214   have "\<exists>f. \<forall>n. f n \<in> sets M \<and> d (f n) > \<gamma> - 1 / 2^n"

  2215   proof (intro choice_iff[THEN iffD1] allI)

  2216     fix n

  2217     have "\<exists>X\<in>sets M. \<gamma> - 1 / 2^n < d X"

  2218       unfolding \<gamma>_def by (intro less_cSUP_iff[THEN iffD1]) auto

  2219     then show "\<exists>y. y \<in> sets M \<and> \<gamma> - 1 / 2 ^ n < d y"

  2220       by auto

  2221   qed

  2222   then obtain E where [measurable]: "E n \<in> sets M" and E: "d (E n) > \<gamma> - 1 / 2^n" for n

  2223     by auto

  2224

  2225   define F where "F m n = (if m \<le> n then \<Inter>i\<in>{m..n}. E i else space M)" for m n

  2226

  2227   have [measurable]: "m \<le> n \<Longrightarrow> F m n \<in> sets M" for m n

  2228     by (auto simp: F_def)

  2229

  2230   have 1: "\<gamma> - 2 / 2 ^ m + 1 / 2 ^ n \<le> d (F m n)" if "m \<le> n" for m n

  2231     using that

  2232   proof (induct rule: dec_induct)

  2233     case base with E[of m] show ?case

  2234       by (simp add: F_def field_simps)

  2235   next

  2236     case (step i)

  2237     have F_Suc: "F m (Suc i) = F m i \<inter> E (Suc i)"

  2238       using \<open>m \<le> i\<close> by (auto simp: F_def le_Suc_eq)

  2239

  2240     have "\<gamma> + (\<gamma> - 2 / 2^m + 1 / 2 ^ Suc i) \<le> (\<gamma> - 1 / 2^Suc i) + (\<gamma> - 2 / 2^m + 1 / 2^i)"

  2241       by (simp add: field_simps)

  2242     also have "\<dots> \<le> d (E (Suc i)) + d (F m i)"

  2243       using E[of "Suc i"] by (intro add_mono step) auto

  2244     also have "\<dots> = d (E (Suc i)) + d (F m i - E (Suc i)) + d (F m (Suc i))"

  2245       using \<open>m \<le> i\<close> by (simp add: d_def field_simps F_Suc M.finite_measure_Diff' N.finite_measure_Diff')

  2246     also have "\<dots> = d (E (Suc i) \<union> F m i) + d (F m (Suc i))"

  2247       using \<open>m \<le> i\<close> by (simp add: d_def field_simps M.finite_measure_Union' N.finite_measure_Union')

  2248     also have "\<dots> \<le> \<gamma> + d (F m (Suc i))"

  2249       using \<open>m \<le> i\<close> by auto

  2250     finally show ?case

  2251       by auto

  2252   qed

  2253

  2254   define F' where "F' m = (\<Inter>i\<in>{m..}. E i)" for m

  2255   have F'_eq: "F' m = (\<Inter>i. F m (i + m))" for m

  2256     by (fastforce simp: le_iff_add[of m] F'_def F_def)

  2257

  2258   have [measurable]: "F' m \<in> sets M" for m

  2259     by (auto simp: F'_def)

  2260

  2261   have \<gamma>_le: "\<gamma> - 0 \<le> d (\<Union>m. F' m)"

  2262   proof (rule LIMSEQ_le)

  2263     show "(\<lambda>n. \<gamma> - 2 / 2 ^ n) \<longlonglongrightarrow> \<gamma> - 0"

  2264       by (intro tendsto_intros LIMSEQ_divide_realpow_zero) auto

  2265     have "incseq F'"

  2266       by (auto simp: incseq_def F'_def)

  2267     then show "(\<lambda>m. d (F' m)) \<longlonglongrightarrow> d (\<Union>m. F' m)"

  2268       unfolding d_def

  2269       by (intro tendsto_diff M.finite_Lim_measure_incseq N.finite_Lim_measure_incseq) auto

  2270

  2271     have "\<gamma> - 2 / 2 ^ m + 0 \<le> d (F' m)" for m

  2272     proof (rule LIMSEQ_le)

  2273       have *: "decseq (\<lambda>n. F m (n + m))"

  2274         by (auto simp: decseq_def F_def)

  2275       show "(\<lambda>n. d (F m n)) \<longlonglongrightarrow> d (F' m)"

  2276         unfolding d_def F'_eq

  2277         by (rule LIMSEQ_offset[where k=m])

  2278            (auto intro!: tendsto_diff M.finite_Lim_measure_decseq N.finite_Lim_measure_decseq *)

  2279       show "(\<lambda>n. \<gamma> - 2 / 2 ^ m + 1 / 2 ^ n) \<longlonglongrightarrow> \<gamma> - 2 / 2 ^ m + 0"

  2280         by (intro tendsto_add LIMSEQ_divide_realpow_zero tendsto_const) auto

  2281       show "\<exists>N. \<forall>n\<ge>N. \<gamma> - 2 / 2 ^ m + 1 / 2 ^ n \<le> d (F m n)"

  2282         using 1[of m] by (intro exI[of _ m]) auto

  2283     qed

  2284     then show "\<exists>N. \<forall>n\<ge>N. \<gamma> - 2 / 2 ^ n \<le> d (F' n)"

  2285       by auto

  2286   qed

  2287

  2288   show ?thesis

  2289   proof (safe intro!: bexI[of _ "\<Union>m. F' m"])

  2290     fix X assume [measurable]: "X \<in> sets M" and X: "X \<subseteq> (\<Union>m. F' m)"

  2291     have "d (\<Union>m. F' m) - d X = d ((\<Union>m. F' m) - X)"

  2292       using X by (auto simp: d_def M.finite_measure_Diff N.finite_measure_Diff)

  2293     also have "\<dots> \<le> \<gamma>"

  2294       by auto

  2295     finally have "0 \<le> d X"

  2296       using \<gamma>_le by auto

  2297     then show "emeasure N X \<le> emeasure M X"

  2298       by (auto simp: d_def M.emeasure_eq_measure N.emeasure_eq_measure)

  2299   next

  2300     fix X assume [measurable]: "X \<in> sets M" and X: "X \<inter> (\<Union>m. F' m) = {}"

  2301     then have "d (\<Union>m. F' m) + d X = d (X \<union> (\<Union>m. F' m))"

  2302       by (auto simp: d_def M.finite_measure_Union N.finite_measure_Union)

  2303     also have "\<dots> \<le> \<gamma>"

  2304       by auto

  2305     finally have "d X \<le> 0"

  2306       using \<gamma>_le by auto

  2307     then show "emeasure M X \<le> emeasure N X"

  2308       by (auto simp: d_def M.emeasure_eq_measure N.emeasure_eq_measure)

  2309   qed auto

  2310 qed

  2311

  2312 lemma unsigned_Hahn_decomposition:

  2313   assumes [simp]: "sets N = sets M" and [measurable]: "A \<in> sets M"

  2314     and [simp]: "emeasure M A \<noteq> top" "emeasure N A \<noteq> top"

  2315   shows "\<exists>Y\<in>sets M. Y \<subseteq> A \<and> (\<forall>X\<in>sets M. X \<subseteq> Y \<longrightarrow> N X \<le> M X) \<and> (\<forall>X\<in>sets M. X \<subseteq> A \<longrightarrow> X \<inter> Y = {} \<longrightarrow> M X \<le> N X)"

  2316 proof -

  2317   have "\<exists>Y\<in>sets (restrict_space M A).

  2318     (\<forall>X\<in>sets (restrict_space M A). X \<subseteq> Y \<longrightarrow> (restrict_space N A) X \<le> (restrict_space M A) X) \<and>

  2319     (\<forall>X\<in>sets (restrict_space M A). X \<inter> Y = {} \<longrightarrow> (restrict_space M A) X \<le> (restrict_space N A) X)"

  2320   proof (rule finite_unsigned_Hahn_decomposition)

  2321     show "finite_measure (restrict_space M A)" "finite_measure (restrict_space N A)"

  2322       by (auto simp: space_restrict_space emeasure_restrict_space less_top intro!: finite_measureI)

  2323   qed (simp add: sets_restrict_space)

  2324   then guess Y ..

  2325   then show ?thesis

  2326     apply (intro bexI[of _ Y] conjI ballI conjI)

  2327     apply (simp_all add: sets_restrict_space emeasure_restrict_space)

  2328     apply safe

  2329     subgoal for X Z

  2330       by (erule ballE[of _ _ X]) (auto simp add: Int_absorb1)

  2331     subgoal for X Z

  2332       by (erule ballE[of _ _ X])  (auto simp add: Int_absorb1 ac_simps)

  2333     apply auto

  2334     done

  2335 qed

  2336

  2337 text \<open>

  2338   Define a lexicographical order on @{type measure}, in the order space, sets and measure. The parts

  2339   of the lexicographical order are point-wise ordered.

  2340 \<close>

  2341

  2342 instantiation measure :: (type) order_bot

  2343 begin

  2344

  2345 inductive less_eq_measure :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> bool" where

  2346   "space M \<subset> space N \<Longrightarrow> less_eq_measure M N"

  2347 | "space M = space N \<Longrightarrow> sets M \<subset> sets N \<Longrightarrow> less_eq_measure M N"

  2348 | "space M = space N \<Longrightarrow> sets M = sets N \<Longrightarrow> emeasure M \<le> emeasure N \<Longrightarrow> less_eq_measure M N"

  2349

  2350 lemma le_measure_iff:

  2351   "M \<le> N \<longleftrightarrow> (if space M = space N then

  2352     if sets M = sets N then emeasure M \<le> emeasure N else sets M \<subseteq> sets N else space M \<subseteq> space N)"

  2353   by (auto elim: less_eq_measure.cases intro: less_eq_measure.intros)

  2354

  2355 definition less_measure :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> bool" where

  2356   "less_measure M N \<longleftrightarrow> (M \<le> N \<and> \<not> N \<le> M)"

  2357

  2358 definition bot_measure :: "'a measure" where

  2359   "bot_measure = sigma {} {}"

  2360

  2361 lemma

  2362   shows space_bot[simp]: "space bot = {}"

  2363     and sets_bot[simp]: "sets bot = {{}}"

  2364     and emeasure_bot[simp]: "emeasure bot X = 0"

  2365   by (auto simp: bot_measure_def sigma_sets_empty_eq emeasure_sigma)

  2366

  2367 instance

  2368 proof standard

  2369   show "bot \<le> a" for a :: "'a measure"

  2370     by (simp add: le_measure_iff bot_measure_def sigma_sets_empty_eq emeasure_sigma le_fun_def)

  2371 qed (auto simp: le_measure_iff less_measure_def split: if_split_asm intro: measure_eqI)

  2372

  2373 end

  2374

  2375 lemma le_measure: "sets M = sets N \<Longrightarrow> M \<le> N \<longleftrightarrow> (\<forall>A\<in>sets M. emeasure M A \<le> emeasure N A)"

  2376   apply (auto simp: le_measure_iff le_fun_def dest: sets_eq_imp_space_eq)

  2377   subgoal for X

  2378     by (cases "X \<in> sets M") (auto simp: emeasure_notin_sets)

  2379   done

  2380

  2381 definition sup_measure' :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> 'a measure"

  2382 where

  2383   "sup_measure' A B = measure_of (space A) (sets A) (\<lambda>X. SUP Y:sets A. emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y))"

  2384

  2385 lemma assumes [simp]: "sets B = sets A"

  2386   shows space_sup_measure'[simp]: "space (sup_measure' A B) = space A"

  2387     and sets_sup_measure'[simp]: "sets (sup_measure' A B) = sets A"

  2388   using sets_eq_imp_space_eq[OF assms] by (simp_all add: sup_measure'_def)

  2389

  2390 lemma emeasure_sup_measure':

  2391   assumes sets_eq[simp]: "sets B = sets A" and [simp, intro]: "X \<in> sets A"

  2392   shows "emeasure (sup_measure' A B) X = (SUP Y:sets A. emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y))"

  2393     (is "_ = ?S X")

  2394 proof -

  2395   note sets_eq_imp_space_eq[OF sets_eq, simp]

  2396   show ?thesis

  2397     using sup_measure'_def

  2398   proof (rule emeasure_measure_of)

  2399     let ?d = "\<lambda>X Y. emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y)"

  2400     show "countably_additive (sets (sup_measure' A B)) (\<lambda>X. SUP Y : sets A. emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y))"

  2401     proof (rule countably_additiveI, goal_cases)

  2402       case (1 X)

  2403       then have [measurable]: "\<And>i. X i \<in> sets A" and "disjoint_family X"

  2404         by auto

  2405       have "(\<Sum>i. ?S (X i)) = (SUP Y:sets A. \<Sum>i. ?d (X i) Y)"

  2406       proof (rule ennreal_suminf_SUP_eq_directed)

  2407         fix J :: "nat set" and a b assume "finite J" and [measurable]: "a \<in> sets A" "b \<in> sets A"

  2408         have "\<exists>c\<in>sets A. c \<subseteq> X i \<and> (\<forall>a\<in>sets A. ?d (X i) a \<le> ?d (X i) c)" for i

  2409         proof cases

  2410           assume "emeasure A (X i) = top \<or> emeasure B (X i) = top"

  2411           then show ?thesis

  2412           proof

  2413             assume "emeasure A (X i) = top" then show ?thesis

  2414               by (intro bexI[of _ "X i"]) auto

  2415           next

  2416             assume "emeasure B (X i) = top" then show ?thesis

  2417               by (intro bexI[of _ "{}"]) auto

  2418           qed

  2419         next

  2420           assume finite: "\<not> (emeasure A (X i) = top \<or> emeasure B (X i) = top)"

  2421           then have "\<exists>Y\<in>sets A. Y \<subseteq> X i \<and> (\<forall>C\<in>sets A. C \<subseteq> Y \<longrightarrow> B C \<le> A C) \<and> (\<forall>C\<in>sets A. C \<subseteq> X i \<longrightarrow> C \<inter> Y = {} \<longrightarrow> A C \<le> B C)"

  2422             using unsigned_Hahn_decomposition[of B A "X i"] by simp

  2423           then obtain Y where [measurable]: "Y \<in> sets A" and [simp]: "Y \<subseteq> X i"

  2424             and B_le_A: "\<And>C. C \<in> sets A \<Longrightarrow> C \<subseteq> Y \<Longrightarrow> B C \<le> A C"

  2425             and A_le_B: "\<And>C. C \<in> sets A \<Longrightarrow> C \<subseteq> X i \<Longrightarrow> C \<inter> Y = {} \<Longrightarrow> A C \<le> B C"

  2426             by auto

  2427

  2428           show ?thesis

  2429           proof (intro bexI[of _ Y] ballI conjI)

  2430             fix a assume [measurable]: "a \<in> sets A"

  2431             have *: "(X i \<inter> a \<inter> Y \<union> (X i \<inter> a - Y)) = X i \<inter> a" "(X i - a) \<inter> Y \<union> (X i - a - Y) = X i \<inter> - a"

  2432               for a Y by auto

  2433             then have "?d (X i) a =

  2434               (A (X i \<inter> a \<inter> Y) + A (X i \<inter> a \<inter> - Y)) + (B (X i \<inter> - a \<inter> Y) + B (X i \<inter> - a \<inter> - Y))"

  2435               by (subst (1 2) plus_emeasure) (auto simp: Diff_eq[symmetric])

  2436             also have "\<dots> \<le> (A (X i \<inter> a \<inter> Y) + B (X i \<inter> a \<inter> - Y)) + (A (X i \<inter> - a \<inter> Y) + B (X i \<inter> - a \<inter> - Y))"

  2437               by (intro add_mono order_refl B_le_A A_le_B) (auto simp: Diff_eq[symmetric])

  2438             also have "\<dots> \<le> (A (X i \<inter> Y \<inter> a) + A (X i \<inter> Y \<inter> - a)) + (B (X i \<inter> - Y \<inter> a) + B (X i \<inter> - Y \<inter> - a))"

  2439               by (simp add: ac_simps)

  2440             also have "\<dots> \<le> A (X i \<inter> Y) + B (X i \<inter> - Y)"

  2441               by (subst (1 2) plus_emeasure) (auto simp: Diff_eq[symmetric] *)

  2442             finally show "?d (X i) a \<le> ?d (X i) Y" .

  2443           qed auto

  2444         qed

  2445         then obtain C where [measurable]: "C i \<in> sets A" and "C i \<subseteq> X i"

  2446           and C: "\<And>a. a \<in> sets A \<Longrightarrow> ?d (X i) a \<le> ?d (X i) (C i)" for i

  2447           by metis

  2448         have *: "X i \<inter> (\<Union>i. C i) = X i \<inter> C i" for i

  2449         proof safe

  2450           fix x j assume "x \<in> X i" "x \<in> C j"

  2451           moreover have "i = j \<or> X i \<inter> X j = {}"

  2452             using \<open>disjoint_family X\<close> by (auto simp: disjoint_family_on_def)

  2453           ultimately show "x \<in> C i"

  2454             using \<open>C i \<subseteq> X i\<close> \<open>C j \<subseteq> X j\<close> by auto

  2455         qed auto

  2456         have **: "X i \<inter> - (\<Union>i. C i) = X i \<inter> - C i" for i

  2457         proof safe

  2458           fix x j assume "x \<in> X i" "x \<notin> C i" "x \<in> C j"

  2459           moreover have "i = j \<or> X i \<inter> X j = {}"

  2460             using \<open>disjoint_family X\<close> by (auto simp: disjoint_family_on_def)

  2461           ultimately show False

  2462             using \<open>C i \<subseteq> X i\<close> \<open>C j \<subseteq> X j\<close> by auto

  2463         qed auto

  2464         show "\<exists>c\<in>sets A. \<forall>i\<in>J. ?d (X i) a \<le> ?d (X i) c \<and> ?d (X i) b \<le> ?d (X i) c"

  2465           apply (intro bexI[of _ "\<Union>i. C i"])

  2466           unfolding * **

  2467           apply (intro C ballI conjI)

  2468           apply auto

  2469           done

  2470       qed

  2471       also have "\<dots> = ?S (\<Union>i. X i)"

  2472         unfolding UN_extend_simps(4)

  2473         by (auto simp add: suminf_add[symmetric] Diff_eq[symmetric] simp del: UN_simps

  2474                  intro!: SUP_cong arg_cong2[where f="op +"] suminf_emeasure

  2475                          disjoint_family_on_bisimulation[OF \<open>disjoint_family X\<close>])

  2476       finally show "(\<Sum>i. ?S (X i)) = ?S (\<Union>i. X i)" .

  2477     qed

  2478   qed (auto dest: sets.sets_into_space simp: positive_def intro!: SUP_const)

  2479 qed

  2480

  2481 lemma le_emeasure_sup_measure'1:

  2482   assumes "sets B = sets A" "X \<in> sets A" shows "emeasure A X \<le> emeasure (sup_measure' A B) X"

  2483   by (subst emeasure_sup_measure'[OF assms]) (auto intro!: SUP_upper2[of "X"] assms)

  2484

  2485 lemma le_emeasure_sup_measure'2:

  2486   assumes "sets B = sets A" "X \<in> sets A" shows "emeasure B X \<le> emeasure (sup_measure' A B) X"

  2487   by (subst emeasure_sup_measure'[OF assms]) (auto intro!: SUP_upper2[of "{}"] assms)

  2488

  2489 lemma emeasure_sup_measure'_le2:

  2490   assumes [simp]: "sets B = sets C" "sets A = sets C" and [measurable]: "X \<in> sets C"

  2491   assumes A: "\<And>Y. Y \<subseteq> X \<Longrightarrow> Y \<in> sets A \<Longrightarrow> emeasure A Y \<le> emeasure C Y"

  2492   assumes B: "\<And>Y. Y \<subseteq> X \<Longrightarrow> Y \<in> sets A \<Longrightarrow> emeasure B Y \<le> emeasure C Y"

  2493   shows "emeasure (sup_measure' A B) X \<le> emeasure C X"

  2494 proof (subst emeasure_sup_measure')

  2495   show "(SUP Y:sets A. emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y)) \<le> emeasure C X"

  2496     unfolding \<open>sets A = sets C\<close>

  2497   proof (intro SUP_least)

  2498     fix Y assume [measurable]: "Y \<in> sets C"

  2499     have [simp]: "X \<inter> Y \<union> (X - Y) = X"

  2500       by auto

  2501     have "emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y) \<le> emeasure C (X \<inter> Y) + emeasure C (X \<inter> - Y)"

  2502       by (intro add_mono A B) (auto simp: Diff_eq[symmetric])

  2503     also have "\<dots> = emeasure C X"

  2504       by (subst plus_emeasure) (auto simp: Diff_eq[symmetric])

  2505     finally show "emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y) \<le> emeasure C X" .

  2506   qed

  2507 qed simp_all

  2508

  2509 definition sup_lexord :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b::order) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a"

  2510 where

  2511   "sup_lexord A B k s c =

  2512     (if k A = k B then c else if \<not> k A \<le> k B \<and> \<not> k B \<le> k A then s else if k B \<le> k A then A else B)"

  2513

  2514 lemma sup_lexord:

  2515   "(k A < k B \<Longrightarrow> P B) \<Longrightarrow> (k B < k A \<Longrightarrow> P A) \<Longrightarrow> (k A = k B \<Longrightarrow> P c) \<Longrightarrow>

  2516     (\<not> k B \<le> k A \<Longrightarrow> \<not> k A \<le> k B \<Longrightarrow> P s) \<Longrightarrow> P (sup_lexord A B k s c)"

  2517   by (auto simp: sup_lexord_def)

  2518

  2519 lemmas le_sup_lexord = sup_lexord[where P="\<lambda>a. c \<le> a" for c]

  2520

  2521 lemma sup_lexord1: "k A = k B \<Longrightarrow> sup_lexord A B k s c = c"

  2522   by (simp add: sup_lexord_def)

  2523

  2524 lemma sup_lexord_commute: "sup_lexord A B k s c = sup_lexord B A k s c"

  2525   by (auto simp: sup_lexord_def)

  2526

  2527 lemma sigma_sets_le_sets_iff: "(sigma_sets (space x) \<A> \<subseteq> sets x) = (\<A> \<subseteq> sets x)"

  2528   using sets.sigma_sets_subset[of \<A> x] by auto

  2529

  2530 lemma sigma_le_iff: "\<A> \<subseteq> Pow \<Omega> \<Longrightarrow> sigma \<Omega> \<A> \<le> x \<longleftrightarrow> (\<Omega> \<subseteq> space x \<and> (space x = \<Omega> \<longrightarrow> \<A> \<subseteq> sets x))"

  2531   by (cases "\<Omega> = space x")

  2532      (simp_all add: eq_commute[of _ "sets x"] le_measure_iff emeasure_sigma le_fun_def

  2533                     sigma_sets_superset_generator sigma_sets_le_sets_iff)

  2534

  2535 instantiation measure :: (type) semilattice_sup

  2536 begin

  2537

  2538 definition sup_measure :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> 'a measure"

  2539 where

  2540   "sup_measure A B =

  2541     sup_lexord A B space (sigma (space A \<union> space B) {})

  2542       (sup_lexord A B sets (sigma (space A) (sets A \<union> sets B)) (sup_measure' A B))"

  2543

  2544 instance

  2545 proof

  2546   fix x y z :: "'a measure"

  2547   show "x \<le> sup x y"

  2548     unfolding sup_measure_def

  2549   proof (intro le_sup_lexord)

  2550     assume "space x = space y"

  2551     then have *: "sets x \<union> sets y \<subseteq> Pow (space x)"

  2552       using sets.space_closed by auto

  2553     assume "\<not> sets y \<subseteq> sets x" "\<not> sets x \<subseteq> sets y"

  2554     then have "sets x \<subset> sets x \<union> sets y"

  2555       by auto

  2556     also have "\<dots> \<le> sigma (space x) (sets x \<union> sets y)"

  2557       by (subst sets_measure_of[OF *]) (rule sigma_sets_superset_generator)

  2558     finally show "x \<le> sigma (space x) (sets x \<union> sets y)"

  2559       by (simp add: space_measure_of[OF *] less_eq_measure.intros(2))

  2560   next

  2561     assume "\<not> space y \<subseteq> space x" "\<not> space x \<subseteq> space y"

  2562     then show "x \<le> sigma (space x \<union> space y) {}"

  2563       by (intro less_eq_measure.intros) auto

  2564   next

  2565     assume "sets x = sets y" then show "x \<le> sup_measure' x y"

  2566       by (simp add: le_measure le_emeasure_sup_measure'1)

  2567   qed (auto intro: less_eq_measure.intros)

  2568   show "y \<le> sup x y"

  2569     unfolding sup_measure_def

  2570   proof (intro le_sup_lexord)

  2571     assume **: "space x = space y"

  2572     then have *: "sets x \<union> sets y \<subseteq> Pow (space y)"

  2573       using sets.space_closed by auto

  2574     assume "\<not> sets y \<subseteq> sets x" "\<not> sets x \<subseteq> sets y"

  2575     then have "sets y \<subset> sets x \<union> sets y"

  2576       by auto

  2577     also have "\<dots> \<le> sigma (space y) (sets x \<union> sets y)"

  2578       by (subst sets_measure_of[OF *]) (rule sigma_sets_superset_generator)

  2579     finally show "y \<le> sigma (space x) (sets x \<union> sets y)"

  2580       by (simp add: ** space_measure_of[OF *] less_eq_measure.intros(2))

  2581   next

  2582     assume "\<not> space y \<subseteq> space x" "\<not> space x \<subseteq> space y"

  2583     then show "y \<le> sigma (space x \<union> space y) {}"

  2584       by (intro less_eq_measure.intros) auto

  2585   next

  2586     assume "sets x = sets y" then show "y \<le> sup_measure' x y"

  2587       by (simp add: le_measure le_emeasure_sup_measure'2)

  2588   qed (auto intro: less_eq_measure.intros)

  2589   show "x \<le> y \<Longrightarrow> z \<le> y \<Longrightarrow> sup x z \<le> y"

  2590     unfolding sup_measure_def

  2591   proof (intro sup_lexord[where P="\<lambda>x. x \<le> y"])

  2592     assume "x \<le> y" "z \<le> y" and [simp]: "space x = space z" "sets x = sets z"

  2593     from \<open>x \<le> y\<close> show "sup_measure' x z \<le> y"

  2594     proof cases

  2595       case 1 then show ?thesis

  2596         by (intro less_eq_measure.intros(1)) simp

  2597     next

  2598       case 2 then show ?thesis

  2599         by (intro less_eq_measure.intros(2)) simp_all

  2600     next

  2601       case 3 with \<open>z \<le> y\<close> \<open>x \<le> y\<close> show ?thesis

  2602         by (auto simp add: le_measure intro!: emeasure_sup_measure'_le2)

  2603     qed

  2604   next

  2605     assume **: "x \<le> y" "z \<le> y" "space x = space z" "\<not> sets z \<subseteq> sets x" "\<not> sets x \<subseteq> sets z"

  2606     then have *: "sets x \<union> sets z \<subseteq> Pow (space x)"

  2607       using sets.space_closed by auto

  2608     show "sigma (space x) (sets x \<union> sets z) \<le> y"

  2609       unfolding sigma_le_iff[OF *] using ** by (auto simp: le_measure_iff split: if_split_asm)

  2610   next

  2611     assume "x \<le> y" "z \<le> y" "\<not> space z \<subseteq> space x" "\<not> space x \<subseteq> space z"

  2612     then have "space x \<subseteq> space y" "space z \<subseteq> space y"

  2613       by (auto simp: le_measure_iff split: if_split_asm)

  2614     then show "sigma (space x \<union> space z) {} \<le> y"

  2615       by (simp add: sigma_le_iff)

  2616   qed

  2617 qed

  2618

  2619 end

  2620

  2621 lemma space_empty_eq_bot: "space a = {} \<longleftrightarrow> a = bot"

  2622   using space_empty[of a] by (auto intro!: measure_eqI)

  2623

  2624 lemma sets_eq_iff_bounded: "A \<le> B \<Longrightarrow> B \<le> C \<Longrightarrow> sets A = sets C \<Longrightarrow> sets B = sets A"

  2625   by (auto dest: sets_eq_imp_space_eq simp add: le_measure_iff split: if_split_asm)

  2626

  2627 lemma sets_sup: "sets A = sets M \<Longrightarrow> sets B = sets M \<Longrightarrow> sets (sup A B) = sets M"

  2628   by (auto simp add: sup_measure_def sup_lexord_def dest: sets_eq_imp_space_eq)

  2629

  2630 lemma le_measureD1: "A \<le> B \<Longrightarrow> space A \<le> space B"

  2631   by (auto simp: le_measure_iff split: if_split_asm)

  2632

  2633 lemma le_measureD2: "A \<le> B \<Longrightarrow> space A = space B \<Longrightarrow> sets A \<le> sets B"

  2634   by (auto simp: le_measure_iff split: if_split_asm)

  2635

  2636 lemma le_measureD3: "A \<le> B \<Longrightarrow> sets A = sets B \<Longrightarrow> emeasure A X \<le> emeasure B X"

  2637   by (auto simp: le_measure_iff le_fun_def dest: sets_eq_imp_space_eq split: if_split_asm)

  2638

  2639 lemma UN_space_closed: "UNION S sets \<subseteq> Pow (UNION S space)"

  2640   using sets.space_closed by auto

  2641

  2642 definition Sup_lexord :: "('a \<Rightarrow> 'b::complete_lattice) \<Rightarrow> ('a set \<Rightarrow> 'a) \<Rightarrow> ('a set \<Rightarrow> 'a) \<Rightarrow> 'a set \<Rightarrow> 'a"

  2643 where

  2644   "Sup_lexord k c s A = (let U = (SUP a:A. k a) in if \<exists>a\<in>A. k a = U then c {a\<in>A. k a = U} else s A)"

  2645

  2646 lemma Sup_lexord:

  2647   "(\<And>a S. a \<in> A \<Longrightarrow> k a = (SUP a:A. k a) \<Longrightarrow> S = {a'\<in>A. k a' = k a} \<Longrightarrow> P (c S)) \<Longrightarrow> ((\<And>a. a \<in> A \<Longrightarrow> k a \<noteq> (SUP a:A. k a)) \<Longrightarrow> P (s A)) \<Longrightarrow>

  2648     P (Sup_lexord k c s A)"

  2649   by (auto simp: Sup_lexord_def Let_def)

  2650

  2651 lemma Sup_lexord1:

  2652   assumes A: "A \<noteq> {}" "(\<And>a. a \<in> A \<Longrightarrow> k a = (\<Union>a\<in>A. k a))" "P (c A)"

  2653   shows "P (Sup_lexord k c s A)"

  2654   unfolding Sup_lexord_def Let_def

  2655 proof (clarsimp, safe)

  2656   show "\<forall>a\<in>A. k a \<noteq> (\<Union>x\<in>A. k x) \<Longrightarrow> P (s A)"

  2657     by (metis assms(1,2) ex_in_conv)

  2658 next

  2659   fix a assume "a \<in> A" "k a = (\<Union>x\<in>A. k x)"

  2660   then have "{a \<in> A. k a = (\<Union>x\<in>A. k x)} = {a \<in> A. k a = k a}"

  2661     by (metis A(2)[symmetric])

  2662   then show "P (c {a \<in> A. k a = (\<Union>x\<in>A. k x)})"

  2663     by (simp add: A(3))

  2664 qed

  2665

  2666 instantiation measure :: (type) complete_lattice

  2667 begin

  2668

  2669 interpretation sup_measure: comm_monoid_set sup "bot :: 'a measure"

  2670   by standard (auto intro!: antisym)

  2671

  2672 lemma sup_measure_F_mono':

  2673   "finite J \<Longrightarrow> finite I \<Longrightarrow> sup_measure.F id I \<le> sup_measure.F id (I \<union> J)"

  2674 proof (induction J rule: finite_induct)

  2675   case empty then show ?case

  2676     by simp

  2677 next

  2678   case (insert i J)

  2679   show ?case

  2680   proof cases

  2681     assume "i \<in> I" with insert show ?thesis

  2682       by (auto simp: insert_absorb)

  2683   next

  2684     assume "i \<notin> I"

  2685     have "sup_measure.F id I \<le> sup_measure.F id (I \<union> J)"

  2686       by (intro insert)

  2687     also have "\<dots> \<le> sup_measure.F id (insert i (I \<union> J))"

  2688       using insert \<open>i \<notin> I\<close> by (subst sup_measure.insert) auto

  2689     finally show ?thesis

  2690       by auto

  2691   qed

  2692 qed

  2693

  2694 lemma sup_measure_F_mono: "finite I \<Longrightarrow> J \<subseteq> I \<Longrightarrow> sup_measure.F id J \<le> sup_measure.F id I"

  2695   using sup_measure_F_mono'[of I J] by (auto simp: finite_subset Un_absorb1)

  2696

  2697 lemma sets_sup_measure_F:

  2698   "finite I \<Longrightarrow> I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> sets i = sets M) \<Longrightarrow> sets (sup_measure.F id I) = sets M"

  2699   by (induction I rule: finite_ne_induct) (simp_all add: sets_sup)

  2700

  2701 definition Sup_measure' :: "'a measure set \<Rightarrow> 'a measure"

  2702 where

  2703   "Sup_measure' M = measure_of (\<Union>a\<in>M. space a) (\<Union>a\<in>M. sets a)

  2704     (\<lambda>X. (SUP P:{P. finite P \<and> P \<subseteq> M }. sup_measure.F id P X))"

  2705

  2706 lemma space_Sup_measure'2: "space (Sup_measure' M) = (\<Union>m\<in>M. space m)"

  2707   unfolding Sup_measure'_def by (intro space_measure_of[OF UN_space_closed])

  2708

  2709 lemma sets_Sup_measure'2: "sets (Sup_measure' M) = sigma_sets (\<Union>m\<in>M. space m) (\<Union>m\<in>M. sets m)"

  2710   unfolding Sup_measure'_def by (intro sets_measure_of[OF UN_space_closed])

  2711

  2712 lemma sets_Sup_measure':

  2713   assumes sets_eq[simp]: "\<And>m. m \<in> M \<Longrightarrow> sets m = sets A" and "M \<noteq> {}"

  2714   shows "sets (Sup_measure' M) = sets A"

  2715   using sets_eq[THEN sets_eq_imp_space_eq, simp] \<open>M \<noteq> {}\<close> by (simp add: Sup_measure'_def)

  2716

  2717 lemma space_Sup_measure':

  2718   assumes sets_eq[simp]: "\<And>m. m \<in> M \<Longrightarrow> sets m = sets A" and "M \<noteq> {}"

  2719   shows "space (Sup_measure' M) = space A"

  2720   using sets_eq[THEN sets_eq_imp_space_eq, simp] \<open>M \<noteq> {}\<close>

  2721   by (simp add: Sup_measure'_def )

  2722

  2723 lemma emeasure_Sup_measure':

  2724   assumes sets_eq[simp]: "\<And>m. m \<in> M \<Longrightarrow> sets m = sets A" and "X \<in> sets A" "M \<noteq> {}"

  2725   shows "emeasure (Sup_measure' M) X = (SUP P:{P. finite P \<and> P \<subseteq> M}. sup_measure.F id P X)"

  2726     (is "_ = ?S X")

  2727   using Sup_measure'_def

  2728 proof (rule emeasure_measure_of)

  2729   note sets_eq[THEN sets_eq_imp_space_eq, simp]

  2730   have *: "sets (Sup_measure' M) = sets A" "space (Sup_measure' M) = space A"

  2731     using \<open>M \<noteq> {}\<close> by (simp_all add: Sup_measure'_def)

  2732   let ?\<mu> = "sup_measure.F id"

  2733   show "countably_additive (sets (Sup_measure' M)) ?S"

  2734   proof (rule countably_additiveI, goal_cases)

  2735     case (1 F)

  2736     then have **: "range F \<subseteq> sets A"

  2737       by (auto simp: *)

  2738     show "(\<Sum>i. ?S (F i)) = ?S (\<Union>i. F i)"

  2739     proof (subst ennreal_suminf_SUP_eq_directed)

  2740       fix i j and N :: "nat set" assume ij: "i \<in> {P. finite P \<and> P \<subseteq> M}" "j \<in> {P. finite P \<and> P \<subseteq> M}"

  2741       have "(i \<noteq> {} \<longrightarrow> sets (?\<mu> i) = sets A) \<and> (j \<noteq> {} \<longrightarrow> sets (?\<mu> j) = sets A) \<and>

  2742         (i \<noteq> {} \<or> j \<noteq> {} \<longrightarrow> sets (?\<mu> (i \<union> j)) = sets A)"

  2743         using ij by (intro impI sets_sup_measure_F conjI) auto

  2744       then have "?\<mu> j (F n) \<le> ?\<mu> (i \<union> j) (F n) \<and> ?\<mu> i (F n) \<le> ?\<mu> (i \<union> j) (F n)" for n

  2745         using ij

  2746         by (cases "i = {}"; cases "j = {}")

  2747            (auto intro!: le_measureD3 sup_measure_F_mono simp: sets_sup_measure_F

  2748                  simp del: id_apply)

  2749       with ij show "\<exists>k\<in>{P. finite P \<and> P \<subseteq> M}. \<forall>n\<in>N. ?\<mu> i (F n) \<le> ?\<mu> k (F n) \<and> ?\<mu> j (F n) \<le> ?\<mu> k (F n)"

  2750         by (safe intro!: bexI[of _ "i \<union> j"]) auto

  2751     next

  2752       show "(SUP P : {P. finite P \<and> P \<subseteq> M}. \<Sum>n. ?\<mu> P (F n)) = (SUP P : {P. finite P \<and> P \<subseteq> M}. ?\<mu> P (UNION UNIV F))"

  2753       proof (intro SUP_cong refl)

  2754         fix i assume i: "i \<in> {P. finite P \<and> P \<subseteq> M}"

  2755         show "(\<Sum>n. ?\<mu> i (F n)) = ?\<mu> i (UNION UNIV F)"

  2756         proof cases

  2757           assume "i \<noteq> {}" with i ** show ?thesis

  2758             apply (intro suminf_emeasure \<open>disjoint_family F\<close>)

  2759             apply (subst sets_sup_measure_F[OF _ _ sets_eq])

  2760             apply auto

  2761             done

  2762         qed simp

  2763       qed

  2764     qed

  2765   qed

  2766   show "positive (sets (Sup_measure' M)) ?S"

  2767     by (auto simp: positive_def bot_ennreal[symmetric])

  2768   show "X \<in> sets (Sup_measure' M)"

  2769     using assms * by auto

  2770 qed (rule UN_space_closed)

  2771

  2772 definition Sup_measure :: "'a measure set \<Rightarrow> 'a measure"

  2773 where

  2774   "Sup_measure = Sup_lexord space (Sup_lexord sets Sup_measure'

  2775     (\<lambda>U. sigma (\<Union>u\<in>U. space u) (\<Union>u\<in>U. sets u))) (\<lambda>U. sigma (\<Union>u\<in>U. space u) {})"

  2776

  2777 definition Inf_measure :: "'a measure set \<Rightarrow> 'a measure"

  2778 where

  2779   "Inf_measure A = Sup {x. \<forall>a\<in>A. x \<le> a}"

  2780

  2781 definition inf_measure :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> 'a measure"

  2782 where

  2783   "inf_measure a b = Inf {a, b}"

  2784

  2785 definition top_measure :: "'a measure"

  2786 where

  2787   "top_measure = Inf {}"

  2788

  2789 instance

  2790 proof

  2791   note UN_space_closed [simp]

  2792   show upper: "x \<le> Sup A" if x: "x \<in> A" for x :: "'a measure" and A

  2793     unfolding Sup_measure_def

  2794   proof (intro Sup_lexord[where P="\<lambda>y. x \<le> y"])

  2795     assume "\<And>a. a \<in> A \<Longrightarrow> space a \<noteq> (\<Union>a\<in>A. space a)"

  2796     from this[OF \<open>x \<in> A\<close>] \<open>x \<in> A\<close> show "x \<le> sigma (\<Union>a\<in>A. space a) {}"

  2797       by (intro less_eq_measure.intros) auto

  2798   next

  2799     fix a S assume "a \<in> A" and a: "space a = (\<Union>a\<in>A. space a)" and S: "S = {a' \<in> A. space a' = space a}"

  2800       and neq: "\<And>aa. aa \<in> S \<Longrightarrow> sets aa \<noteq> (\<Union>a\<in>S. sets a)"

  2801     have sp_a: "space a = (UNION S space)"

  2802       using \<open>a\<in>A\<close> by (auto simp: S)

  2803     show "x \<le> sigma (UNION S space) (UNION S sets)"

  2804     proof cases

  2805       assume [simp]: "space x = space a"

  2806       have "sets x \<subset> (\<Union>a\<in>S. sets a)"

  2807         using \<open>x\<in>A\<close> neq[of x] by (auto simp: S)

  2808       also have "\<dots> \<subseteq> sigma_sets (\<Union>x\<in>S. space x) (\<Union>x\<in>S. sets x)"

  2809         by (rule sigma_sets_superset_generator)

  2810       finally show ?thesis

  2811         by (intro less_eq_measure.intros(2)) (simp_all add: sp_a)

  2812     next

  2813       assume "space x \<noteq> space a"

  2814       moreover have "space x \<le> space a"

  2815         unfolding a using \<open>x\<in>A\<close> by auto

  2816       ultimately show ?thesis

  2817         by (intro less_eq_measure.intros) (simp add: less_le sp_a)

  2818     qed

  2819   next

  2820     fix a b S S' assume "a \<in> A" and a: "space a = (\<Union>a\<in>A. space a)" and S: "S = {a' \<in> A. space a' = space a}"

  2821       and "b \<in> S" and b: "sets b = (\<Union>a\<in>S. sets a)" and S': "S' = {a' \<in> S. sets a' = sets b}"

  2822     then have "S' \<noteq> {}" "space b = space a"

  2823       by auto

  2824     have sets_eq: "\<And>x. x \<in> S' \<Longrightarrow> sets x = sets b"

  2825       by (auto simp: S')

  2826     note sets_eq[THEN sets_eq_imp_space_eq, simp]

  2827     have *: "sets (Sup_measure' S') = sets b" "space (Sup_measure' S') = space b"

  2828       using \<open>S' \<noteq> {}\<close> by (simp_all add: Sup_measure'_def sets_eq)

  2829     show "x \<le> Sup_measure' S'"

  2830     proof cases

  2831       assume "x \<in> S"

  2832       with \<open>b \<in> S\<close> have "space x = space b"

  2833         by (simp add: S)

  2834       show ?thesis

  2835       proof cases

  2836         assume "x \<in> S'"

  2837         show "x \<le> Sup_measure' S'"

  2838         proof (intro le_measure[THEN iffD2] ballI)

  2839           show "sets x = sets (Sup_measure' S')"

  2840             using \<open>x\<in>S'\<close> * by (simp add: S')

  2841           fix X assume "X \<in> sets x"

  2842           show "emeasure x X \<le> emeasure (Sup_measure' S') X"

  2843           proof (subst emeasure_Sup_measure'[OF _ \<open>X \<in> sets x\<close>])

  2844             show "emeasure x X \<le> (SUP P : {P. finite P \<and> P \<subseteq> S'}. emeasure (sup_measure.F id P) X)"

  2845               using \<open>x\<in>S'\<close> by (intro SUP_upper2[where i="{x}"]) auto

  2846           qed (insert \<open>x\<in>S'\<close> S', auto)

  2847         qed

  2848       next

  2849         assume "x \<notin> S'"

  2850         then have "sets x \<noteq> sets b"

  2851           using \<open>x\<in>S\<close> by (auto simp: S')

  2852         moreover have "sets x \<le> sets b"

  2853           using \<open>x\<in>S\<close> unfolding b by auto

  2854         ultimately show ?thesis

  2855           using * \<open>x \<in> S\<close>

  2856           by (intro less_eq_measure.intros(2))

  2857              (simp_all add: * \<open>space x = space b\<close> less_le)

  2858       qed

  2859     next

  2860       assume "x \<notin> S"

  2861       with \<open>x\<in>A\<close> \<open>x \<notin> S\<close> \<open>space b = space a\<close> show ?thesis

  2862         by (intro less_eq_measure.intros)

  2863            (simp_all add: * less_le a SUP_upper S)

  2864     qed

  2865   qed

  2866   show least: "Sup A \<le> x" if x: "\<And>z. z \<in> A \<Longrightarrow> z \<le> x" for x :: "'a measure" and A

  2867     unfolding Sup_measure_def

  2868   proof (intro Sup_lexord[where P="\<lambda>y. y \<le> x"])

  2869     assume "\<And>a. a \<in> A \<Longrightarrow> space a \<noteq> (\<Union>a\<in>A. space a)"

  2870     show "sigma (UNION A space) {} \<le> x"

  2871       using x[THEN le_measureD1] by (subst sigma_le_iff) auto

  2872   next

  2873     fix a S assume "a \<in> A" "space a = (\<Union>a\<in>A. space a)" and S: "S = {a' \<in> A. space a' = space a}"

  2874       "\<And>a. a \<in> S \<Longrightarrow> sets a \<noteq> (\<Union>a\<in>S. sets a)"

  2875     have "UNION S space \<subseteq> space x"

  2876       using S le_measureD1[OF x] by auto

  2877     moreover

  2878     have "UNION S space = space a"

  2879       using \<open>a\<in>A\<close> S by auto

  2880     then have "space x = UNION S space \<Longrightarrow> UNION S sets \<subseteq> sets x"

  2881       using \<open>a \<in> A\<close> le_measureD2[OF x] by (auto simp: S)

  2882     ultimately show "sigma (UNION S space) (UNION S sets) \<le> x"

  2883       by (subst sigma_le_iff) simp_all

  2884   next

  2885     fix a b S S' assume "a \<in> A" and a: "space a = (\<Union>a\<in>A. space a)" and S: "S = {a' \<in> A. space a' = space a}"

  2886       and "b \<in> S" and b: "sets b = (\<Union>a\<in>S. sets a)" and S': "S' = {a' \<in> S. sets a' = sets b}"

  2887     then have "S' \<noteq> {}" "space b = space a"

  2888       by auto

  2889     have sets_eq: "\<And>x. x \<in> S' \<Longrightarrow> sets x = sets b"

  2890       by (auto simp: S')

  2891     note sets_eq[THEN sets_eq_imp_space_eq, simp]

  2892     have *: "sets (Sup_measure' S') = sets b" "space (Sup_measure' S') = space b"

  2893       using \<open>S' \<noteq> {}\<close> by (simp_all add: Sup_measure'_def sets_eq)

  2894     show "Sup_measure' S' \<le> x"

  2895     proof cases

  2896       assume "space x = space a"

  2897       show ?thesis

  2898       proof cases

  2899         assume **: "sets x = sets b"

  2900         show ?thesis

  2901         proof (intro le_measure[THEN iffD2] ballI)

  2902           show ***: "sets (Sup_measure' S') = sets x"

  2903             by (simp add: * **)

  2904           fix X assume "X \<in> sets (Sup_measure' S')"

  2905           show "emeasure (Sup_measure' S') X \<le> emeasure x X"

  2906             unfolding ***

  2907           proof (subst emeasure_Sup_measure'[OF _ \<open>X \<in> sets (Sup_measure' S')\<close>])

  2908             show "(SUP P : {P. finite P \<and> P \<subseteq> S'}. emeasure (sup_measure.F id P) X) \<le> emeasure x X"

  2909             proof (safe intro!: SUP_least)

  2910               fix P assume P: "finite P" "P \<subseteq> S'"

  2911               show "emeasure (sup_measure.F id P) X \<le> emeasure x X"

  2912               proof cases

  2913                 assume "P = {}" then show ?thesis

  2914                   by auto

  2915               next

  2916                 assume "P \<noteq> {}"

  2917                 from P have "finite P" "P \<subseteq> A"

  2918                   unfolding S' S by (simp_all add: subset_eq)

  2919                 then have "sup_measure.F id P \<le> x"

  2920                   by (induction P) (auto simp: x)

  2921                 moreover have "sets (sup_measure.F id P) = sets x"

  2922                   using \<open>finite P\<close> \<open>P \<noteq> {}\<close> \<open>P \<subseteq> S'\<close> \<open>sets x = sets b\<close>

  2923                   by (intro sets_sup_measure_F) (auto simp: S')

  2924                 ultimately show "emeasure (sup_measure.F id P) X \<le> emeasure x X"

  2925                   by (rule le_measureD3)

  2926               qed

  2927             qed

  2928             show "m \<in> S' \<Longrightarrow> sets m = sets (Sup_measure' S')" for m

  2929               unfolding * by (simp add: S')

  2930           qed fact

  2931         qed

  2932       next

  2933         assume "sets x \<noteq> sets b"

  2934         moreover have "sets b \<le> sets x"

  2935           unfolding b S using x[THEN le_measureD2] \<open>space x = space a\<close> by auto

  2936         ultimately show "Sup_measure' S' \<le> x"

  2937           using \<open>space x = space a\<close> \<open>b \<in> S\<close>

  2938           by (intro less_eq_measure.intros(2)) (simp_all add: * S)

  2939       qed

  2940     next

  2941       assume "space x \<noteq> space a"

  2942       then have "space a < space x"

  2943         using le_measureD1[OF x[OF \<open>a\<in>A\<close>]] by auto

  2944       then show "Sup_measure' S' \<le> x"

  2945         by (intro less_eq_measure.intros) (simp add: * \<open>space b = space a\<close>)

  2946     qed

  2947   qed

  2948   show "Sup {} = (bot::'a measure)" "Inf {} = (top::'a measure)"

  2949     by (auto intro!: antisym least simp: top_measure_def)

  2950   show lower: "x \<in> A \<Longrightarrow> Inf A \<le> x" for x :: "'a measure" and A

  2951     unfolding Inf_measure_def by (intro least) auto

  2952   show greatest: "(\<And>z. z \<in> A \<Longrightarrow> x \<le> z) \<Longrightarrow> x \<le> Inf A" for x :: "'a measure" and A

  2953     unfolding Inf_measure_def by (intro upper) auto

  2954   show "inf x y \<le> x" "inf x y \<le> y" "x \<le> y \<Longrightarrow> x \<le> z \<Longrightarrow> x \<le> inf y z" for x y z :: "'a measure"

  2955     by (auto simp: inf_measure_def intro!: lower greatest)

  2956 qed

  2957

  2958 end

  2959

  2960 lemma sets_SUP:

  2961   assumes "\<And>x. x \<in> I \<Longrightarrow> sets (M x) = sets N"

  2962   shows "I \<noteq> {} \<Longrightarrow> sets (SUP i:I. M i) = sets N"

  2963   unfolding Sup_measure_def

  2964   using assms assms[THEN sets_eq_imp_space_eq]

  2965     sets_Sup_measure'[where A=N and M="MI"]

  2966   by (intro Sup_lexord1[where P="\<lambda>x. sets x = sets N"]) auto

  2967

  2968 lemma emeasure_SUP:

  2969   assumes sets: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets N" "X \<in> sets N" "I \<noteq> {}"

  2970   shows "emeasure (SUP i:I. M i) X = (SUP J:{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emeasure (SUP i:J. M i) X)"

  2971 proof -

  2972   interpret sup_measure: comm_monoid_set sup "bot :: 'b measure"

  2973     by standard (auto intro!: antisym)

  2974   have eq: "finite J \<Longrightarrow> sup_measure.F id J = (SUP i:J. i)" for J :: "'b measure set"

  2975     by (induction J rule: finite_induct) auto

  2976   have 1: "J \<noteq> {} \<Longrightarrow> J \<subseteq> I \<Longrightarrow> sets (SUP x:J. M x) = sets N" for J

  2977     by (intro sets_SUP sets) (auto )

  2978   from \<open>I \<noteq> {}\<close> obtain i where "i\<in>I" by auto

  2979   have "Sup_measure' (MI) X = (SUP P:{P. finite P \<and> P \<subseteq> MI}. sup_measure.F id P X)"

  2980     using sets by (intro emeasure_Sup_measure') auto

  2981   also have "Sup_measure' (MI) = (SUP i:I. M i)"

  2982     unfolding Sup_measure_def using \<open>I \<noteq> {}\<close> sets sets(1)[THEN sets_eq_imp_space_eq]

  2983     by (intro Sup_lexord1[where P="\<lambda>x. _ = x"]) auto

  2984   also have "(SUP P:{P. finite P \<and> P \<subseteq> MI}. sup_measure.F id P X) =

  2985     (SUP J:{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. (SUP i:J. M i) X)"

  2986   proof (intro SUP_eq)

  2987     fix J assume "J \<in> {P. finite P \<and> P \<subseteq> MI}"

  2988     then obtain J' where J': "J' \<subseteq> I" "finite J'" and J: "J = MJ'" and "finite J"

  2989       using finite_subset_image[of J M I] by auto

  2990     show "\<exists>j\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. sup_measure.F id J X \<le> (SUP i:j. M i) X"

  2991     proof cases

  2992       assume "J' = {}" with \<open>i \<in> I\<close> show ?thesis

  2993         by (auto simp add: J)

  2994     next

  2995       assume "J' \<noteq> {}" with J J' show ?thesis

  2996         by (intro bexI[of _ "J'"]) (auto simp add: eq simp del: id_apply)

  2997     qed

  2998   next

  2999     fix J assume J: "J \<in> {P. P \<noteq> {} \<and> finite P \<and> P \<subseteq> I}"

  3000     show "\<exists>J'\<in>{J. finite J \<and> J \<subseteq> MI}. (SUP i:J. M i) X \<le> sup_measure.F id J' X"

  3001       using J by (intro bexI[of _ "MJ"]) (auto simp add: eq simp del: id_apply)

  3002   qed

  3003   finally show ?thesis .

  3004 qed

  3005

  3006 lemma emeasure_SUP_chain:

  3007   assumes sets: "\<And>i. i \<in> A \<Longrightarrow> sets (M i) = sets N" "X \<in> sets N"

  3008   assumes ch: "Complete_Partial_Order.chain op \<le> (M  A)" and "A \<noteq> {}"

  3009   shows "emeasure (SUP i:A. M i) X = (SUP i:A. emeasure (M i) X)"

  3010 proof (subst emeasure_SUP[OF sets \<open>A \<noteq> {}\<close>])

  3011   show "(SUP J:{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> A}. emeasure (SUPREMUM J M) X) = (SUP i:A. emeasure (M i) X)"

  3012   proof (rule SUP_eq)

  3013     fix J assume "J \<in> {J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> A}"

  3014     then have J: "Complete_Partial_Order.chain op \<le> (M  J)" "finite J" "J \<noteq> {}" and "J \<subseteq> A"

  3015       using ch[THEN chain_subset, of "MJ"] by auto

  3016     with in_chain_finite[OF J(1)] obtain j where "j \<in> J" "(SUP j:J. M j) = M j"

  3017       by auto

  3018     with \<open>J \<subseteq> A\<close> show "\<exists>j\<in>A. emeasure (SUPREMUM J M) X \<le> emeasure (M j) X"

  3019       by auto

  3020   next

  3021     fix j assume "j\<in>A" then show "\<exists>i\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> A}. emeasure (M j) X \<le> emeasure (SUPREMUM i M) X"

  3022       by (intro bexI[of _ "{j}"]) auto

  3023   qed

  3024 qed

  3025

  3026 subsubsection \<open>Supremum of a set of $\sigma$-algebras\<close>

  3027

  3028 lemma space_Sup_eq_UN: "space (Sup M) = (\<Union>x\<in>M. space x)"

  3029   unfolding Sup_measure_def

  3030   apply (intro Sup_lexord[where P="\<lambda>x. space x = _"])

  3031   apply (subst space_Sup_measure'2)

  3032   apply auto []

  3033   apply (subst space_measure_of[OF UN_space_closed])

  3034   apply auto

  3035   done

  3036

  3037 lemma sets_Sup_eq:

  3038   assumes *: "\<And>m. m \<in> M \<Longrightarrow> space m = X" and "M \<noteq> {}"

  3039   shows "sets (Sup M) = sigma_sets X (\<Union>x\<in>M. sets x)"

  3040   unfolding Sup_measure_def

  3041   apply (rule Sup_lexord1)

  3042   apply fact

  3043   apply (simp add: assms)

  3044   apply (rule Sup_lexord)

  3045   subgoal premises that for a S

  3046     unfolding that(3) that(2)[symmetric]

  3047     using that(1)

  3048     apply (subst sets_Sup_measure'2)

  3049     apply (intro arg_cong2[where f=sigma_sets])

  3050     apply (auto simp: *)

  3051     done

  3052   apply (subst sets_measure_of[OF UN_space_closed])

  3053   apply (simp add:  assms)

  3054   done

  3055

  3056 lemma in_sets_Sup: "(\<And>m. m \<in> M \<Longrightarrow> space m = X) \<Longrightarrow> m \<in> M \<Longrightarrow> A \<in> sets m \<Longrightarrow> A \<in> sets (Sup M)"

  3057   by (subst sets_Sup_eq[where X=X]) auto

  3058

  3059 lemma Sup_lexord_rel:

  3060   assumes "\<And>i. i \<in> I \<Longrightarrow> k (A i) = k (B i)"

  3061     "R (c (A  {a \<in> I. k (B a) = (SUP x:I. k (B x))})) (c (B  {a \<in> I. k (B a) = (SUP x:I. k (B x))}))"

  3062     "R (s (AI)) (s (BI))"

  3063   shows "R (Sup_lexord k c s (AI)) (Sup_lexord k c s (BI))"

  3064 proof -

  3065   have "A  {a \<in> I. k (B a) = (SUP x:I. k (B x))} =  {a \<in> A  I. k a = (SUP x:I. k (B x))}"

  3066     using assms(1) by auto

  3067   moreover have "B  {a \<in> I. k (B a) = (SUP x:I. k (B x))} =  {a \<in> B  I. k a = (SUP x:I. k (B x))}"

  3068     by auto

  3069   ultimately show ?thesis

  3070     using assms by (auto simp: Sup_lexord_def Let_def)

  3071 qed

  3072

  3073 lemma sets_SUP_cong:

  3074   assumes eq: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets (N i)" shows "sets (SUP i:I. M i) = sets (SUP i:I. N i)"

  3075   unfolding Sup_measure_def

  3076   using eq eq[THEN sets_eq_imp_space_eq]

  3077   apply (intro Sup_lexord_rel[where R="\<lambda>x y. sets x = sets y"])

  3078   apply simp

  3079   apply simp

  3080   apply (simp add: sets_Sup_measure'2)

  3081   apply (intro arg_cong2[where f="\<lambda>x y. sets (sigma x y)"])

  3082   apply auto

  3083   done

  3084

  3085 lemma sets_Sup_in_sets:

  3086   assumes "M \<noteq> {}"

  3087   assumes "\<And>m. m \<in> M \<Longrightarrow> space m = space N"

  3088   assumes "\<And>m. m \<in> M \<Longrightarrow> sets m \<subseteq> sets N"

  3089   shows "sets (Sup M) \<subseteq> sets N"

  3090 proof -

  3091   have *: "UNION M space = space N"

  3092     using assms by auto

  3093   show ?thesis

  3094     unfolding * using assms by (subst sets_Sup_eq[of M "space N"]) (auto intro!: sets.sigma_sets_subset)

  3095 qed

  3096

  3097 lemma measurable_Sup1:

  3098   assumes m: "m \<in> M" and f: "f \<in> measurable m N"

  3099     and const_space: "\<And>m n. m \<in> M \<Longrightarrow> n \<in> M \<Longrightarrow> space m = space n"

  3100   shows "f \<in> measurable (Sup M) N"

  3101 proof -

  3102   have "space (Sup M) = space m"

  3103     using m by (auto simp add: space_Sup_eq_UN dest: const_space)

  3104   then show ?thesis

  3105     using m f unfolding measurable_def by (auto intro: in_sets_Sup[OF const_space])

  3106 qed

  3107

  3108 lemma measurable_Sup2:

  3109   assumes M: "M \<noteq> {}"

  3110   assumes f: "\<And>m. m \<in> M \<Longrightarrow> f \<in> measurable N m"

  3111     and const_space: "\<And>m n. m \<in> M \<Longrightarrow> n \<in> M \<Longrightarrow> space m = space n"

  3112   shows "f \<in> measurable N (Sup M)"

  3113 proof -

  3114   from M obtain m where "m \<in> M" by auto

  3115   have space_eq: "\<And>n. n \<in> M \<Longrightarrow> space n = space m"

  3116     by (intro const_space \<open>m \<in> M\<close>)

  3117   have "f \<in> measurable N (sigma (\<Union>m\<in>M. space m) (\<Union>m\<in>M. sets m))"

  3118   proof (rule measurable_measure_of)

  3119     show "f \<in> space N \<rightarrow> UNION M space"

  3120       using measurable_space[OF f] M by auto

  3121   qed (auto intro: measurable_sets f dest: sets.sets_into_space)

  3122   also have "measurable N (sigma (\<Union>m\<in>M. space m) (\<Union>m\<in>M. sets m)) = measurable N (Sup M)"

  3123     apply (intro measurable_cong_sets refl)

  3124     apply (subst sets_Sup_eq[OF space_eq M])

  3125     apply simp

  3126     apply (subst sets_measure_of[OF UN_space_closed])

  3127     apply (simp add: space_eq M)

  3128     done

  3129   finally show ?thesis .

  3130 qed

  3131

  3132 lemma sets_Sup_sigma:

  3133   assumes [simp]: "M \<noteq> {}" and M: "\<And>m. m \<in> M \<Longrightarrow> m \<subseteq> Pow \<Omega>"

  3134   shows "sets (SUP m:M. sigma \<Omega> m) = sets (sigma \<Omega> (\<Union>M))"

  3135 proof -

  3136   { fix a m assume "a \<in> sigma_sets \<Omega> m" "m \<in> M"

  3137     then have "a \<in> sigma_sets \<Omega> (\<Union>M)"

  3138      by induction (auto intro: sigma_sets.intros) }

  3139   then show "sets (SUP m:M. sigma \<Omega> m) = sets (sigma \<Omega> (\<Union>M))"

  3140     apply (subst sets_Sup_eq[where X="\<Omega>"])

  3141     apply (auto simp add: M) []

  3142     apply auto []

  3143     apply (simp add: space_measure_of_conv M Union_least)

  3144     apply (rule sigma_sets_eqI)

  3145     apply auto

  3146     done

  3147 qed

  3148

  3149 lemma Sup_sigma:

  3150   assumes [simp]: "M \<noteq> {}" and M: "\<And>m. m \<in> M \<Longrightarrow> m \<subseteq> Pow \<Omega>"

  3151   shows "(SUP m:M. sigma \<Omega> m) = (sigma \<Omega> (\<Union>M))"

  3152 proof (intro antisym SUP_least)

  3153   have *: "\<Union>M \<subseteq> Pow \<Omega>"

  3154     using M by auto

  3155   show "sigma \<Omega> (\<Union>M) \<le> (SUP m:M. sigma \<Omega> m)"

  3156   proof (intro less_eq_measure.intros(3))

  3157     show "space (sigma \<Omega> (\<Union>M)) = space (SUP m:M. sigma \<Omega> m)"

  3158       "sets (sigma \<Omega> (\<Union>M)) = sets (SUP m:M. sigma \<Omega> m)"

  3159       using sets_Sup_sigma[OF assms] sets_Sup_sigma[OF assms, THEN sets_eq_imp_space_eq]

  3160       by auto

  3161   qed (simp add: emeasure_sigma le_fun_def)

  3162   fix m assume "m \<in> M" then show "sigma \<Omega> m \<le> sigma \<Omega> (\<Union>M)"

  3163     by (subst sigma_le_iff) (auto simp add: M *)

  3164 qed

  3165

  3166 lemma SUP_sigma_sigma:

  3167   "M \<noteq> {} \<Longrightarrow> (\<And>m. m \<in> M \<Longrightarrow> f m \<subseteq> Pow \<Omega>) \<Longrightarrow> (SUP m:M. sigma \<Omega> (f m)) = sigma \<Omega> (\<Union>m\<in>M. f m)"

  3168   using Sup_sigma[of "fM" \<Omega>] by auto

  3169

  3170 lemma sets_vimage_Sup_eq:

  3171   assumes *: "M \<noteq> {}" "f \<in> X \<rightarrow> Y" "\<And>m. m \<in> M \<Longrightarrow> space m = Y"

  3172   shows "sets (vimage_algebra X f (Sup M)) = sets (SUP m : M. vimage_algebra X f m)"

  3173   (is "?IS = ?SI")

  3174 proof

  3175   show "?IS \<subseteq> ?SI"

  3176     apply (intro sets_image_in_sets measurable_Sup2)

  3177     apply (simp add: space_Sup_eq_UN *)

  3178     apply (simp add: *)

  3179     apply (intro measurable_Sup1)

  3180     apply (rule imageI)

  3181     apply assumption

  3182     apply (rule measurable_vimage_algebra1)

  3183     apply (auto simp: *)

  3184     done

  3185   show "?SI \<subseteq> ?IS"

  3186     apply (intro sets_Sup_in_sets)

  3187     apply (auto simp: *) []

  3188     apply (auto simp: *) []

  3189     apply (elim imageE)

  3190     apply simp

  3191     apply (rule sets_image_in_sets)

  3192     apply simp

  3193     apply (simp add: measurable_def)

  3194     apply (simp add: * space_Sup_eq_UN sets_vimage_algebra2)

  3195     apply (auto intro: in_sets_Sup[OF *(3)])

  3196     done

  3197 qed

  3198

  3199 lemma restrict_space_eq_vimage_algebra':

  3200   "sets (restrict_space M \<Omega>) = sets (vimage_algebra (\<Omega> \<inter> space M) (\<lambda>x. x) M)"

  3201 proof -

  3202   have *: "{A \<inter> (\<Omega> \<inter> space M) |A. A \<in> sets M} = {A \<inter> \<Omega> |A. A \<in> sets M}"

  3203     using sets.sets_into_space[of _ M] by blast

  3204

  3205   show ?thesis

  3206     unfolding restrict_space_def

  3207     by (subst sets_measure_of)

  3208        (auto simp add: image_subset_iff sets_vimage_algebra * dest: sets.sets_into_space intro!: arg_cong2[where f=sigma_sets])

  3209 qed

  3210

  3211 lemma sigma_le_sets:

  3212   assumes [simp]: "A \<subseteq> Pow X" shows "sets (sigma X A) \<subseteq> sets N \<longleftrightarrow> X \<in> sets N \<and> A \<subseteq> sets N"

  3213 proof

  3214   have "X \<in> sigma_sets X A" "A \<subseteq> sigma_sets X A"

  3215     by (auto intro: sigma_sets_top)

  3216   moreover assume "sets (sigma X A) \<subseteq> sets N"

  3217   ultimately show "X \<in> sets N \<and> A \<subseteq> sets N"

  3218     by auto

  3219 next

  3220   assume *: "X \<in> sets N \<and> A \<subseteq> sets N"

  3221   { fix Y assume "Y \<in> sigma_sets X A" from this * have "Y \<in> sets N"

  3222       by induction auto }

  3223   then show "sets (sigma X A) \<subseteq> sets N"

  3224     by auto

  3225 qed

  3226

  3227 lemma measurable_iff_sets:

  3228   "f \<in> measurable M N \<longleftrightarrow> (f \<in> space M \<rightarrow> space N \<and> sets (vimage_algebra (space M) f N) \<subseteq> sets M)"

  3229 proof -

  3230   have *: "{f - A \<inter> space M |A. A \<in> sets N} \<subseteq> Pow (space M)"

  3231     by auto

  3232   show ?thesis

  3233     unfolding measurable_def

  3234     by (auto simp add: vimage_algebra_def sigma_le_sets[OF *])

  3235 qed

  3236

  3237 lemma sets_vimage_algebra_space: "X \<in> sets (vimage_algebra X f M)"

  3238   using sets.top[of "vimage_algebra X f M"] by simp

  3239

  3240 lemma measurable_mono:

  3241   assumes N: "sets N' \<le> sets N" "space N = space N'"

  3242   assumes M: "sets M \<le> sets M'" "space M = space M'"

  3243   shows "measurable M N \<subseteq> measurable M' N'"

  3244   unfolding measurable_def

  3245 proof safe

  3246   fix f A assume "f \<in> space M \<rightarrow> space N" "A \<in> sets N'"

  3247   moreover assume "\<forall>y\<in>sets N. f - y \<inter> space M \<in> sets M" note this[THEN bspec, of A]

  3248   ultimately show "f - A \<inter> space M' \<in> sets M'"

  3249     using assms by auto

  3250 qed (insert N M, auto)

  3251

  3252 lemma measurable_Sup_measurable:

  3253   assumes f: "f \<in> space N \<rightarrow> A"

  3254   shows "f \<in> measurable N (Sup {M. space M = A \<and> f \<in> measurable N M})"

  3255 proof (rule measurable_Sup2)

  3256   show "{M. space M = A \<and> f \<in> measurable N M} \<noteq> {}"

  3257     using f unfolding ex_in_conv[symmetric]

  3258     by (intro exI[of _ "sigma A {}"]) (auto intro!: measurable_measure_of)

  3259 qed auto

  3260

  3261 lemma (in sigma_algebra) sigma_sets_subset':

  3262   assumes a: "a \<subseteq> M" "\<Omega>' \<in> M"

  3263   shows "sigma_sets \<Omega>' a \<subseteq> M"

  3264 proof

  3265   show "x \<in> M" if x: "x \<in> sigma_sets \<Omega>' a" for x

  3266     using x by (induct rule: sigma_sets.induct) (insert a, auto)

  3267 qed

  3268

  3269 lemma in_sets_SUP: "i \<in> I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> space (M i) = Y) \<Longrightarrow> X \<in> sets (M i) \<Longrightarrow> X \<in> sets (SUP i:I. M i)"

  3270   by (intro in_sets_Sup[where X=Y]) auto

  3271

  3272 lemma measurable_SUP1:

  3273   "i \<in> I \<Longrightarrow> f \<in> measurable (M i) N \<Longrightarrow> (\<And>m n. m \<in> I \<Longrightarrow> n \<in> I \<Longrightarrow> space (M m) = space (M n)) \<Longrightarrow>

  3274     f \<in> measurable (SUP i:I. M i) N"

  3275   by (auto intro: measurable_Sup1)

  3276

  3277 lemma sets_image_in_sets':

  3278   assumes X: "X \<in> sets N"

  3279   assumes f: "\<And>A. A \<in> sets M \<Longrightarrow> f - A \<inter> X \<in> sets N"

  3280   shows "sets (vimage_algebra X f M) \<subseteq> sets N"

  3281   unfolding sets_vimage_algebra

  3282   by (rule sets.sigma_sets_subset') (auto intro!: measurable_sets X f)

  3283

  3284 lemma mono_vimage_algebra:

  3285   "sets M \<le> sets N \<Longrightarrow> sets (vimage_algebra X f M) \<subseteq> sets (vimage_algebra X f N)"

  3286   using sets.top[of "sigma X {f - A \<inter> X |A. A \<in> sets N}"]

  3287   unfolding vimage_algebra_def

  3288   apply (subst (asm) space_measure_of)

  3289   apply auto []

  3290   apply (subst sigma_le_sets)

  3291   apply auto

  3292   done

  3293

  3294 lemma mono_restrict_space: "sets M \<le> sets N \<Longrightarrow> sets (restrict_space M X) \<subseteq> sets (restrict_space N X)"

  3295   unfolding sets_restrict_space by (rule image_mono)

  3296

  3297 lemma sets_eq_bot: "sets M = {{}} \<longleftrightarrow> M = bot"

  3298   apply safe

  3299   apply (intro measure_eqI)

  3300   apply auto

  3301   done

  3302

  3303 lemma sets_eq_bot2: "{{}} = sets M \<longleftrightarrow> M = bot"

  3304   using sets_eq_bot[of M] by blast

  3305

  3306

  3307 lemma (in finite_measure) countable_support:

  3308   "countable {x. measure M {x} \<noteq> 0}"

  3309 proof cases

  3310   assume "measure M (space M) = 0"

  3311   with bounded_measure measure_le_0_iff have "{x. measure M {x} \<noteq> 0} = {}"

  3312     by auto

  3313   then show ?thesis

  3314     by simp

  3315 next

  3316   let ?M = "measure M (space M)" and ?m = "\<lambda>x. measure M {x}"

  3317   assume "?M \<noteq> 0"

  3318   then have *: "{x. ?m x \<noteq> 0} = (\<Union>n. {x. ?M / Suc n < ?m x})"

  3319     using reals_Archimedean[of "?m x / ?M" for x]

  3320     by (auto simp: field_simps not_le[symmetric] measure_nonneg divide_le_0_iff measure_le_0_iff)

  3321   have **: "\<And>n. finite {x. ?M / Suc n < ?m x}"

  3322   proof (rule ccontr)

  3323     fix n assume "infinite {x. ?M / Suc n < ?m x}" (is "infinite ?X")

  3324     then obtain X where "finite X" "card X = Suc (Suc n)" "X \<subseteq> ?X"

  3325       by (metis infinite_arbitrarily_large)

  3326     from this(3) have *: "\<And>x. x \<in> X \<Longrightarrow> ?M / Suc n \<le> ?m x"

  3327       by auto

  3328     { fix x assume "x \<in> X"

  3329       from \<open>?M \<noteq> 0\<close> *[OF this] have "?m x \<noteq> 0" by (auto simp: field_simps measure_le_0_iff)

  3330       then have "{x} \<in> sets M" by (auto dest: measure_notin_sets) }

  3331     note singleton_sets = this

  3332     have "?M < (\<Sum>x\<in>X. ?M / Suc n)"

  3333       using \<open>?M \<noteq> 0\<close>

  3334       by (simp add: \<open>card X = Suc (Suc n)\<close> field_simps less_le)

  3335     also have "\<dots> \<le> (\<Sum>x\<in>X. ?m x)"

  3336       by (rule setsum_mono) fact

  3337     also have "\<dots> = measure M (\<Union>x\<in>X. {x})"

  3338       using singleton_sets \<open>finite X\<close>

  3339       by (intro finite_measure_finite_Union[symmetric]) (auto simp: disjoint_family_on_def)

  3340     finally have "?M < measure M (\<Union>x\<in>X. {x})" .

  3341     moreover have "measure M (\<Union>x\<in>X. {x}) \<le> ?M"

  3342       using singleton_sets[THEN sets.sets_into_space] by (intro finite_measure_mono) auto

  3343     ultimately show False by simp

  3344   qed

  3345   show ?thesis

  3346     unfolding * by (intro countable_UN countableI_type countable_finite[OF **])

  3347 qed

  3348

  3349 end
`