src/HOL/Analysis/Measure_Space.thy
 author nipkow Sat Dec 29 15:43:53 2018 +0100 (9 months ago) changeset 69529 4ab9657b3257 parent 69517 dc20f278e8f3 child 69541 d466e0a639e4 permissions -rw-r--r--
capitalize proper names in lemma names
     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\<close>

     8

     9 theory Measure_Space

    10 imports

    11   Measurable "HOL-Library.Extended_Nonnegative_Real"

    12 begin

    13

    14 subsection%unimportant "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: sum.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 sum_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 sum.If_cases[OF \<open>finite P\<close>])

    55 qed

    56

    57 text \<open>

    58   The type for emeasure spaces is already defined in @{theory "HOL-Analysis.Sigma_Algebra"}, as it

    59   is also used to represent sigma algebras (with an arbitrary emeasure).

    60 \<close>

    61

    62 subsection%unimportant "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%unimportant \<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 "HOL-Analysis.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 sum.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>(A  UNIV) \<in> M \<longrightarrow> (\<Sum>i. f (A i)) = f (\<Union>(A  UNIV))"

   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[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%unimportant \<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 emeasure_Union:

   476   "A \<in> sets M \<Longrightarrow> B \<in> sets M \<Longrightarrow> emeasure M (A \<union> B) = emeasure M A + emeasure M (B - A)"

   477   using plus_emeasure[of A M "B - A"] by auto

   478

   479 lemma sum_emeasure:

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

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

   482   by (metis sets.additive_sum emeasure_positive emeasure_additive)

   483

   484 lemma emeasure_mono:

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

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

   487

   488 lemma emeasure_space:

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

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

   491

   492 lemma emeasure_Diff:

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

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

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

   496 proof -

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

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

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

   500     by (subst plus_emeasure[symmetric]) auto

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

   502     using finite by simp

   503 qed

   504

   505 lemma emeasure_compl:

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

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

   508

   509 lemma Lim_emeasure_incseq:

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

   511   using emeasure_countably_additive

   512   by (auto simp add: sets.countably_additive_iff_continuous_from_below emeasure_positive

   513     emeasure_additive)

   514

   515 lemma incseq_emeasure:

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

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

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

   519

   520 lemma SUP_emeasure_incseq:

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

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

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

   524   by (simp add: LIMSEQ_unique)

   525

   526 lemma decseq_emeasure:

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

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

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

   530

   531 lemma INF_emeasure_decseq:

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

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

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

   535 proof -

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

   537     using A by (auto intro!: emeasure_mono)

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

   539

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

   541     by (simp add: ennreal_INF_const_minus)

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

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

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

   545   proof (rule SUP_emeasure_incseq)

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

   547       using A by auto

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

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

   550   qed

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

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

   553   finally show ?thesis

   554     by (rule ennreal_minus_cancel[rotated 3])

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

   556 qed

   557

   558 lemma INF_emeasure_decseq':

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

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

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

   562 proof -

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

   564     by (auto simp: less_top)

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

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

   567

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

   569   proof (rule INF_eq)

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

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

   572   qed auto

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

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

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

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

   577   finally show ?thesis .

   578 qed

   579

   580 lemma emeasure_INT_decseq_subset:

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

   582   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"

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

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

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

   586 proof cases

   587   assume "finite I"

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

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

   590   moreover have "(INF i\<in>I. emeasure M (F i)) = emeasure M (F (Max I))"

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

   592   ultimately show ?thesis

   593     by simp

   594 next

   595   assume "infinite I"

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

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

   598     unfolding L_def

   599   proof (rule LeastI_ex)

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

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

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

   603   qed

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

   605     unfolding L_def by (intro Least_equality) auto

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

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

   608

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

   610   proof (intro INF_emeasure_decseq[symmetric])

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

   612       using L by (intro antimonoI F L_mono) auto

   613   qed (insert L fin, auto)

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

   615   proof (intro antisym INF_greatest)

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

   617       by (intro INF_lower2[of i]) auto

   618   qed (insert L, auto intro: INF_lower)

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

   620   proof (intro antisym INF_greatest)

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

   622       by (intro INF_lower2[of i]) auto

   623   qed (insert L, auto)

   624   finally show ?thesis .

   625 qed

   626

   627 lemma Lim_emeasure_decseq:

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

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

   630   using LIMSEQ_INF[OF decseq_emeasure, OF A]

   631   using INF_emeasure_decseq[OF A fin] by simp

   632

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

   634   assumes "P M"

   635   assumes cont: "sup_continuous F"

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

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

   638 proof -

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

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

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

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

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

   644   proof (rule incseq_SucI)

   645     fix i

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

   647     proof (induct i)

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

   649     next

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

   651     qed

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

   653       by auto

   654   qed

   655   ultimately show ?thesis

   656     by (subst SUP_emeasure_incseq) auto

   657 qed

   658

   659 lemma emeasure_lfp:

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

   661   assumes cont: "sup_continuous F" "sup_continuous f"

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

   663   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"

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

   665 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])

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

   667   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}))"

   668     unfolding SUP_apply[abs_def]

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

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

   671

   672 lemma emeasure_subadditive_finite:

   673   "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))"

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

   675

   676 lemma emeasure_subadditive:

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

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

   679

   680 lemma emeasure_subadditive_countably:

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

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

   683 proof -

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

   685     unfolding UN_disjointed_eq ..

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

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

   688     by (simp add:  disjoint_family_disjointed comp_def)

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

   690     using sets.range_disjointed_sets[OF assms] assms

   691     by (auto intro!: suminf_le emeasure_mono disjointed_subset)

   692   finally show ?thesis .

   693 qed

   694

   695 lemma emeasure_insert:

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

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

   698 proof -

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

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

   701 qed

   702

   703 lemma emeasure_insert_ne:

   704   "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"

   705   by (rule emeasure_insert)

   706

   707 lemma emeasure_eq_sum_singleton:

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

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

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

   711   by (auto simp: disjoint_family_on_def subset_eq)

   712

   713 lemma sum_emeasure_cover:

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

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

   716   assumes disj: "disjoint_family_on B S"

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

   718 proof -

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

   720   proof (rule sum_emeasure)

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

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

   723       unfolding disjoint_family_on_def by auto

   724   qed (insert assms, auto)

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

   726     using A by auto

   727   finally show ?thesis by simp

   728 qed

   729

   730 lemma emeasure_eq_0:

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

   732   by (metis emeasure_mono order_eq_iff zero_le)

   733

   734 lemma emeasure_UN_eq_0:

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

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

   737 proof -

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

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

   740   then show ?thesis

   741     by (auto intro: antisym zero_le)

   742 qed

   743

   744 lemma measure_eqI_finite:

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

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

   747   shows "M = N"

   748 proof (rule measure_eqI)

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

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

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

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

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

   754     using X eq by (auto intro!: sum.cong)

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

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

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

   758 qed simp

   759

   760 lemma measure_eqI_generator_eq:

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

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

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

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

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

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

   767   shows "M = N"

   768 proof -

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

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

   771   have "space M = \<Omega>"

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

   773     by blast

   774

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

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

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

   778     assume "D \<in> sets M"

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

   780       unfolding M

   781     proof (induct rule: sigma_sets_induct_disjoint)

   782       case (basic A)

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

   784       then show ?case using eq by auto

   785     next

   786       case empty then show ?case by simp

   787     next

   788       case (compl A)

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

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

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

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

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

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

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

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

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

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

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

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

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

   802       finally show ?case

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

   804     next

   805       case (union A)

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

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

   808       with A show ?case

   809         by auto

   810     qed }

   811   note * = this

   812   show "M = N"

   813   proof (rule measure_eqI)

   814     show "sets M = sets N"

   815       using M N by simp

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

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

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

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

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

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

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

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

   824       by (auto simp: subset_eq)

   825     have "disjoint_family ?D"

   826       by (auto simp: disjoint_family_disjointed)

   827     moreover

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

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

   830       fix i

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

   832         by (auto simp: disjointed_def)

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

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

   835     qed

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

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

   838   qed

   839 qed

   840

   841 lemma space_empty: "space M = {} \<Longrightarrow> M = count_space {}"

   842   by (rule measure_eqI) (simp_all add: space_empty_iff)

   843

   844 lemma measure_eqI_generator_eq_countable:

   845   fixes M N :: "'a measure" and E :: "'a set set" and A :: "'a set set"

   846   assumes E: "Int_stable E" "E \<subseteq> Pow \<Omega>" "\<And>X. X \<in> E \<Longrightarrow> emeasure M X = emeasure N X"

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

   848   and A: "A \<subseteq> E" "(\<Union>A) = \<Omega>" "countable A" "\<And>a. a \<in> A \<Longrightarrow> emeasure M a \<noteq> \<infinity>"

   849   shows "M = N"

   850 proof cases

   851   assume "\<Omega> = {}"

   852   have *: "sigma_sets \<Omega> E = sets (sigma \<Omega> E)"

   853     using E(2) by simp

   854   have "space M = \<Omega>" "space N = \<Omega>"

   855     using sets E(2) unfolding * by (auto dest: sets_eq_imp_space_eq simp del: sets_measure_of)

   856   then show "M = N"

   857     unfolding \<open>\<Omega> = {}\<close> by (auto dest: space_empty)

   858 next

   859   assume "\<Omega> \<noteq> {}" with \<open>\<Union>A = \<Omega>\<close> have "A \<noteq> {}" by auto

   860   from this \<open>countable A\<close> have rng: "range (from_nat_into A) = A"

   861     by (rule range_from_nat_into)

   862   show "M = N"

   863   proof (rule measure_eqI_generator_eq[OF E sets])

   864     show "range (from_nat_into A) \<subseteq> E"

   865       unfolding rng using \<open>A \<subseteq> E\<close> .

   866     show "(\<Union>i. from_nat_into A i) = \<Omega>"

   867       unfolding rng using \<open>\<Union>A = \<Omega>\<close> .

   868     show "emeasure M (from_nat_into A i) \<noteq> \<infinity>" for i

   869       using rng by (intro A) auto

   870   qed

   871 qed

   872

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

   874 proof (intro measure_eqI emeasure_measure_of_sigma)

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

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

   877     by (simp add: positive_def)

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

   879     by (simp add: emeasure_countably_additive)

   880 qed simp_all

   881

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

   883

   884 definition%important null_sets :: "'a measure \<Rightarrow> 'a set set" where

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

   886

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

   888   by (simp add: null_sets_def)

   889

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

   891   unfolding null_sets_def by simp

   892

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

   894   unfolding null_sets_def by simp

   895

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

   897 proof (rule ring_of_setsI)

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

   899     using sets.sets_into_space by auto

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

   901     by auto

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

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

   904     by auto

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

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

   907     by (auto intro!: emeasure_subadditive emeasure_mono)

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

   909     using null_sets by auto

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

   911     by (auto intro!: antisym zero_le)

   912 qed

   913

   914 lemma UN_from_nat_into:

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

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

   917 proof -

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

   919     using I by simp

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

   921     by simp

   922   finally show ?thesis by simp

   923 qed

   924

   925 lemma null_sets_UN':

   926   assumes "countable I"

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

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

   929 proof cases

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

   931 next

   932   assume "I \<noteq> {}"

   933   show ?thesis

   934   proof (intro conjI CollectI null_setsI)

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

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

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

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

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

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

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

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

   943       by (intro antisym zero_le) simp

   944   qed

   945 qed

   946

   947 lemma null_sets_UN[intro]:

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

   949   by (rule null_sets_UN') auto

   950

   951 lemma null_set_Int1:

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

   953 proof (intro CollectI conjI null_setsI)

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

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

   956 qed (insert assms, auto)

   957

   958 lemma null_set_Int2:

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

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

   961

   962 lemma emeasure_Diff_null_set:

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

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

   965 proof -

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

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

   968   then show ?thesis

   969     unfolding * using assms

   970     by (subst emeasure_Diff) auto

   971 qed

   972

   973 lemma null_set_Diff:

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

   975 proof (intro CollectI conjI null_setsI)

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

   977 qed (insert assms, auto)

   978

   979 lemma emeasure_Un_null_set:

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

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

   982 proof -

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

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

   985   then show ?thesis

   986     unfolding * using assms

   987     by (subst plus_emeasure[symmetric]) auto

   988 qed

   989

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

   991

   992 definition%important ae_filter :: "'a measure \<Rightarrow> 'a filter" where

   993   "ae_filter M = (INF N\<in>null_sets M. principal (space M - N))"

   994

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

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

   997

   998 syntax

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

  1000

  1001 translations

  1002   "AE x in M. P" \<rightleftharpoons> "CONST almost_everywhere M (\<lambda>x. P)"

  1003

  1004 abbreviation

  1005   "set_almost_everywhere A M P \<equiv> AE x in M. x \<in> A \<longrightarrow> P x"

  1006

  1007 syntax

  1008   "_set_almost_everywhere" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> bool \<Rightarrow> bool"

  1009   ("AE _\<in>_ in _./ _" [0,0,0,10] 10)

  1010

  1011 translations

  1012   "AE x\<in>A in M. P" \<rightleftharpoons> "CONST set_almost_everywhere A M (\<lambda>x. P)"

  1013

  1014 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)"

  1015   unfolding ae_filter_def by (subst eventually_INF_base) (auto simp: eventually_principal subset_eq)

  1016

  1017 lemma AE_I':

  1018   "N \<in> null_sets M \<Longrightarrow> {x\<in>space M. \<not> P x} \<subseteq> N \<Longrightarrow> (AE x in M. P x)"

  1019   unfolding eventually_ae_filter by auto

  1020

  1021 lemma AE_iff_null:

  1022   assumes "{x\<in>space M. \<not> P x} \<in> sets M" (is "?P \<in> sets M")

  1023   shows "(AE x in M. P x) \<longleftrightarrow> {x\<in>space M. \<not> P x} \<in> null_sets M"

  1024 proof

  1025   assume "AE x in M. P x" then obtain N where N: "N \<in> sets M" "?P \<subseteq> N" "emeasure M N = 0"

  1026     unfolding eventually_ae_filter by auto

  1027   have "emeasure M ?P \<le> emeasure M N"

  1028     using assms N(1,2) by (auto intro: emeasure_mono)

  1029   then have "emeasure M ?P = 0"

  1030     unfolding \<open>emeasure M N = 0\<close> by auto

  1031   then show "?P \<in> null_sets M" using assms by auto

  1032 next

  1033   assume "?P \<in> null_sets M" with assms show "AE x in M. P x" by (auto intro: AE_I')

  1034 qed

  1035

  1036 lemma AE_iff_null_sets:

  1037   "N \<in> sets M \<Longrightarrow> N \<in> null_sets M \<longleftrightarrow> (AE x in M. x \<notin> N)"

  1038   using Int_absorb1[OF sets.sets_into_space, of N M]

  1039   by (subst AE_iff_null) (auto simp: Int_def[symmetric])

  1040

  1041 lemma AE_not_in:

  1042   "N \<in> null_sets M \<Longrightarrow> AE x in M. x \<notin> N"

  1043   by (metis AE_iff_null_sets null_setsD2)

  1044

  1045 lemma AE_iff_measurable:

  1046   "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"

  1047   using AE_iff_null[of _ P] by auto

  1048

  1049 lemma AE_E[consumes 1]:

  1050   assumes "AE x in M. P x"

  1051   obtains N where "{x \<in> space M. \<not> P x} \<subseteq> N" "emeasure M N = 0" "N \<in> sets M"

  1052   using assms unfolding eventually_ae_filter by auto

  1053

  1054 lemma AE_E2:

  1055   assumes "AE x in M. P x" "{x\<in>space M. P x} \<in> sets M"

  1056   shows "emeasure M {x\<in>space M. \<not> P x} = 0" (is "emeasure M ?P = 0")

  1057 proof -

  1058   have "{x\<in>space M. \<not> P x} = space M - {x\<in>space M. P x}" by auto

  1059   with AE_iff_null[of M P] assms show ?thesis by auto

  1060 qed

  1061

  1062 lemma AE_E3:

  1063   assumes "AE x in M. P x"

  1064   obtains N where "\<And>x. x \<in> space M - N \<Longrightarrow> P x" "N \<in> null_sets M"

  1065 using assms unfolding eventually_ae_filter by auto

  1066

  1067 lemma AE_I:

  1068   assumes "{x \<in> space M. \<not> P x} \<subseteq> N" "emeasure M N = 0" "N \<in> sets M"

  1069   shows "AE x in M. P x"

  1070   using assms unfolding eventually_ae_filter by auto

  1071

  1072 lemma AE_mp[elim!]:

  1073   assumes AE_P: "AE x in M. P x" and AE_imp: "AE x in M. P x \<longrightarrow> Q x"

  1074   shows "AE x in M. Q x"

  1075 proof -

  1076   from AE_P obtain A where P: "{x\<in>space M. \<not> P x} \<subseteq> A"

  1077     and A: "A \<in> sets M" "emeasure M A = 0"

  1078     by (auto elim!: AE_E)

  1079

  1080   from AE_imp obtain B where imp: "{x\<in>space M. P x \<and> \<not> Q x} \<subseteq> B"

  1081     and B: "B \<in> sets M" "emeasure M B = 0"

  1082     by (auto elim!: AE_E)

  1083

  1084   show ?thesis

  1085   proof (intro AE_I)

  1086     have "emeasure M (A \<union> B) \<le> 0"

  1087       using emeasure_subadditive[of A M B] A B by auto

  1088     then show "A \<union> B \<in> sets M" "emeasure M (A \<union> B) = 0"

  1089       using A B by auto

  1090     show "{x\<in>space M. \<not> Q x} \<subseteq> A \<union> B"

  1091       using P imp by auto

  1092   qed

  1093 qed

  1094

  1095 text \<open>The next lemma is convenient to combine with a lemma whose conclusion is of the

  1096 form \<open>AE x in M. P x = Q x\<close>: for such a lemma, there is no \verb+[symmetric]+ variant,

  1097 but using \<open>AE_symmetric[OF...]\<close> will replace it.\<close>

  1098

  1099 (* depricated replace by laws about eventually *)

  1100 lemma

  1101   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"

  1102     and AE_disjI1: "AE x in M. P x \<Longrightarrow> AE x in M. P x \<or> Q x"

  1103     and AE_disjI2: "AE x in M. Q x \<Longrightarrow> AE x in M. P x \<or> Q x"

  1104     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"

  1105     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)"

  1106   by auto

  1107

  1108 lemma AE_symmetric:

  1109   assumes "AE x in M. P x = Q x"

  1110   shows "AE x in M. Q x = P x"

  1111   using assms by auto

  1112

  1113 lemma AE_impI:

  1114   "(P \<Longrightarrow> AE x in M. Q x) \<Longrightarrow> AE x in M. P \<longrightarrow> Q x"

  1115   by (cases P) auto

  1116

  1117 lemma AE_measure:

  1118   assumes AE: "AE x in M. P x" and sets: "{x\<in>space M. P x} \<in> sets M" (is "?P \<in> sets M")

  1119   shows "emeasure M {x\<in>space M. P x} = emeasure M (space M)"

  1120 proof -

  1121   from AE_E[OF AE] guess N . note N = this

  1122   with sets have "emeasure M (space M) \<le> emeasure M (?P \<union> N)"

  1123     by (intro emeasure_mono) auto

  1124   also have "\<dots> \<le> emeasure M ?P + emeasure M N"

  1125     using sets N by (intro emeasure_subadditive) auto

  1126   also have "\<dots> = emeasure M ?P" using N by simp

  1127   finally show "emeasure M ?P = emeasure M (space M)"

  1128     using emeasure_space[of M "?P"] by auto

  1129 qed

  1130

  1131 lemma AE_space: "AE x in M. x \<in> space M"

  1132   by (rule AE_I[where N="{}"]) auto

  1133

  1134 lemma AE_I2[simp, intro]:

  1135   "(\<And>x. x \<in> space M \<Longrightarrow> P x) \<Longrightarrow> AE x in M. P x"

  1136   using AE_space by force

  1137

  1138 lemma AE_Ball_mp:

  1139   "\<forall>x\<in>space M. P x \<Longrightarrow> AE x in M. P x \<longrightarrow> Q x \<Longrightarrow> AE x in M. Q x"

  1140   by auto

  1141

  1142 lemma AE_cong[cong]:

  1143   "(\<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)"

  1144   by auto

  1145

  1146 lemma AE_cong_strong: "M = N \<Longrightarrow> (\<And>x. x \<in> space N =simp=> P x = Q x) \<Longrightarrow> (AE x in M. P x) \<longleftrightarrow> (AE x in N. Q x)"

  1147   by (auto simp: simp_implies_def)

  1148

  1149 lemma AE_all_countable:

  1150   "(AE x in M. \<forall>i. P i x) \<longleftrightarrow> (\<forall>i::'i::countable. AE x in M. P i x)"

  1151 proof

  1152   assume "\<forall>i. AE x in M. P i x"

  1153   from this[unfolded eventually_ae_filter Bex_def, THEN choice]

  1154   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

  1155   have "{x\<in>space M. \<not> (\<forall>i. P i x)} \<subseteq> (\<Union>i. {x\<in>space M. \<not> P i x})" by auto

  1156   also have "\<dots> \<subseteq> (\<Union>i. N i)" using N by auto

  1157   finally have "{x\<in>space M. \<not> (\<forall>i. P i x)} \<subseteq> (\<Union>i. N i)" .

  1158   moreover from N have "(\<Union>i. N i) \<in> null_sets M"

  1159     by (intro null_sets_UN) auto

  1160   ultimately show "AE x in M. \<forall>i. P i x"

  1161     unfolding eventually_ae_filter by auto

  1162 qed auto

  1163

  1164 lemma AE_ball_countable:

  1165   assumes [intro]: "countable X"

  1166   shows "(AE x in M. \<forall>y\<in>X. P x y) \<longleftrightarrow> (\<forall>y\<in>X. AE x in M. P x y)"

  1167 proof

  1168   assume "\<forall>y\<in>X. AE x in M. P x y"

  1169   from this[unfolded eventually_ae_filter Bex_def, THEN bchoice]

  1170   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"

  1171     by auto

  1172   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})"

  1173     by auto

  1174   also have "\<dots> \<subseteq> (\<Union>y\<in>X. N y)"

  1175     using N by auto

  1176   finally have "{x\<in>space M. \<not> (\<forall>y\<in>X. P x y)} \<subseteq> (\<Union>y\<in>X. N y)" .

  1177   moreover from N have "(\<Union>y\<in>X. N y) \<in> null_sets M"

  1178     by (intro null_sets_UN') auto

  1179   ultimately show "AE x in M. \<forall>y\<in>X. P x y"

  1180     unfolding eventually_ae_filter by auto

  1181 qed auto

  1182

  1183 lemma AE_ball_countable':

  1184   "(\<And>N. N \<in> I \<Longrightarrow> AE x in M. P N x) \<Longrightarrow> countable I \<Longrightarrow> AE x in M. \<forall>N \<in> I. P N x"

  1185   unfolding AE_ball_countable by simp

  1186

  1187 lemma pairwise_alt: "pairwise R S \<longleftrightarrow> (\<forall>x\<in>S. \<forall>y\<in>S-{x}. R x y)"

  1188   by (auto simp add: pairwise_def)

  1189

  1190 lemma AE_pairwise: "countable F \<Longrightarrow> pairwise (\<lambda>A B. AE x in M. R x A B) F \<longleftrightarrow> (AE x in M. pairwise (R x) F)"

  1191   unfolding pairwise_alt by (simp add: AE_ball_countable)

  1192

  1193 lemma AE_discrete_difference:

  1194   assumes X: "countable X"

  1195   assumes null: "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0"

  1196   assumes sets: "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"

  1197   shows "AE x in M. x \<notin> X"

  1198 proof -

  1199   have "(\<Union>x\<in>X. {x}) \<in> null_sets M"

  1200     using assms by (intro null_sets_UN') auto

  1201   from AE_not_in[OF this] show "AE x in M. x \<notin> X"

  1202     by auto

  1203 qed

  1204

  1205 lemma AE_finite_all:

  1206   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)"

  1207   using f by induct auto

  1208

  1209 lemma AE_finite_allI:

  1210   assumes "finite S"

  1211   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"

  1212   using AE_finite_all[OF \<open>finite S\<close>] by auto

  1213

  1214 lemma emeasure_mono_AE:

  1215   assumes imp: "AE x in M. x \<in> A \<longrightarrow> x \<in> B"

  1216     and B: "B \<in> sets M"

  1217   shows "emeasure M A \<le> emeasure M B"

  1218 proof cases

  1219   assume A: "A \<in> sets M"

  1220   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"

  1221     by (auto simp: eventually_ae_filter)

  1222   have "emeasure M A = emeasure M (A - N)"

  1223     using N A by (subst emeasure_Diff_null_set) auto

  1224   also have "emeasure M (A - N) \<le> emeasure M (B - N)"

  1225     using N A B sets.sets_into_space by (auto intro!: emeasure_mono)

  1226   also have "emeasure M (B - N) = emeasure M B"

  1227     using N B by (subst emeasure_Diff_null_set) auto

  1228   finally show ?thesis .

  1229 qed (simp add: emeasure_notin_sets)

  1230

  1231 lemma emeasure_eq_AE:

  1232   assumes iff: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B"

  1233   assumes A: "A \<in> sets M" and B: "B \<in> sets M"

  1234   shows "emeasure M A = emeasure M B"

  1235   using assms by (safe intro!: antisym emeasure_mono_AE) auto

  1236

  1237 lemma emeasure_Collect_eq_AE:

  1238   "AE x in M. P x \<longleftrightarrow> Q x \<Longrightarrow> Measurable.pred M Q \<Longrightarrow> Measurable.pred M P \<Longrightarrow>

  1239    emeasure M {x\<in>space M. P x} = emeasure M {x\<in>space M. Q x}"

  1240    by (intro emeasure_eq_AE) auto

  1241

  1242 lemma emeasure_eq_0_AE: "AE x in M. \<not> P x \<Longrightarrow> emeasure M {x\<in>space M. P x} = 0"

  1243   using AE_iff_measurable[OF _ refl, of M "\<lambda>x. \<not> P x"]

  1244   by (cases "{x\<in>space M. P x} \<in> sets M") (simp_all add: emeasure_notin_sets)

  1245

  1246 lemma emeasure_0_AE:

  1247   assumes "emeasure M (space M) = 0"

  1248   shows "AE x in M. P x"

  1249 using eventually_ae_filter assms by blast

  1250

  1251 lemma emeasure_add_AE:

  1252   assumes [measurable]: "A \<in> sets M" "B \<in> sets M" "C \<in> sets M"

  1253   assumes 1: "AE x in M. x \<in> C \<longleftrightarrow> x \<in> A \<or> x \<in> B"

  1254   assumes 2: "AE x in M. \<not> (x \<in> A \<and> x \<in> B)"

  1255   shows "emeasure M C = emeasure M A + emeasure M B"

  1256 proof -

  1257   have "emeasure M C = emeasure M (A \<union> B)"

  1258     by (rule emeasure_eq_AE) (insert 1, auto)

  1259   also have "\<dots> = emeasure M A + emeasure M (B - A)"

  1260     by (subst plus_emeasure) auto

  1261   also have "emeasure M (B - A) = emeasure M B"

  1262     by (rule emeasure_eq_AE) (insert 2, auto)

  1263   finally show ?thesis .

  1264 qed

  1265

  1266 subsection \<open>\<open>\<sigma>\<close>-finite Measures\<close>

  1267

  1268 locale%important sigma_finite_measure =

  1269   fixes M :: "'a measure"

  1270   assumes sigma_finite_countable:

  1271     "\<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>)"

  1272

  1273 lemma (in sigma_finite_measure) sigma_finite:

  1274   obtains A :: "nat \<Rightarrow> 'a set"

  1275   where "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>"

  1276 proof -

  1277   obtain A :: "'a set set" where

  1278     [simp]: "countable A" and

  1279     A: "A \<subseteq> sets M" "(\<Union>A) = space M" "\<And>a. a \<in> A \<Longrightarrow> emeasure M a \<noteq> \<infinity>"

  1280     using sigma_finite_countable by metis

  1281   show thesis

  1282   proof cases

  1283     assume "A = {}" with \<open>(\<Union>A) = space M\<close> show thesis

  1284       by (intro that[of "\<lambda>_. {}"]) auto

  1285   next

  1286     assume "A \<noteq> {}"

  1287     show thesis

  1288     proof

  1289       show "range (from_nat_into A) \<subseteq> sets M"

  1290         using \<open>A \<noteq> {}\<close> A by auto

  1291       have "(\<Union>i. from_nat_into A i) = \<Union>A"

  1292         using range_from_nat_into[OF \<open>A \<noteq> {}\<close> \<open>countable A\<close>] by auto

  1293       with A show "(\<Union>i. from_nat_into A i) = space M"

  1294         by auto

  1295     qed (intro A from_nat_into \<open>A \<noteq> {}\<close>)

  1296   qed

  1297 qed

  1298

  1299 lemma (in sigma_finite_measure) sigma_finite_disjoint:

  1300   obtains A :: "nat \<Rightarrow> 'a set"

  1301   where "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>" "disjoint_family A"

  1302 proof -

  1303   obtain A :: "nat \<Rightarrow> 'a set" where

  1304     range: "range A \<subseteq> sets M" and

  1305     space: "(\<Union>i. A i) = space M" and

  1306     measure: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"

  1307     using sigma_finite by blast

  1308   show thesis

  1309   proof (rule that[of "disjointed A"])

  1310     show "range (disjointed A) \<subseteq> sets M"

  1311       by (rule sets.range_disjointed_sets[OF range])

  1312     show "(\<Union>i. disjointed A i) = space M"

  1313       and "disjoint_family (disjointed A)"

  1314       using disjoint_family_disjointed UN_disjointed_eq[of A] space range

  1315       by auto

  1316     show "emeasure M (disjointed A i) \<noteq> \<infinity>" for i

  1317     proof -

  1318       have "emeasure M (disjointed A i) \<le> emeasure M (A i)"

  1319         using range disjointed_subset[of A i] by (auto intro!: emeasure_mono)

  1320       then show ?thesis using measure[of i] by (auto simp: top_unique)

  1321     qed

  1322   qed

  1323 qed

  1324

  1325 lemma (in sigma_finite_measure) sigma_finite_incseq:

  1326   obtains A :: "nat \<Rightarrow> 'a set"

  1327   where "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>" "incseq A"

  1328 proof -

  1329   obtain F :: "nat \<Rightarrow> 'a set" where

  1330     F: "range F \<subseteq> sets M" "(\<Union>i. F i) = space M" "\<And>i. emeasure M (F i) \<noteq> \<infinity>"

  1331     using sigma_finite by blast

  1332   show thesis

  1333   proof (rule that[of "\<lambda>n. \<Union>i\<le>n. F i"])

  1334     show "range (\<lambda>n. \<Union>i\<le>n. F i) \<subseteq> sets M"

  1335       using F by (force simp: incseq_def)

  1336     show "(\<Union>n. \<Union>i\<le>n. F i) = space M"

  1337     proof -

  1338       from F have "\<And>x. x \<in> space M \<Longrightarrow> \<exists>i. x \<in> F i" by auto

  1339       with F show ?thesis by fastforce

  1340     qed

  1341     show "emeasure M (\<Union>i\<le>n. F i) \<noteq> \<infinity>" for n

  1342     proof -

  1343       have "emeasure M (\<Union>i\<le>n. F i) \<le> (\<Sum>i\<le>n. emeasure M (F i))"

  1344         using F by (auto intro!: emeasure_subadditive_finite)

  1345       also have "\<dots> < \<infinity>"

  1346         using F by (auto simp: sum_Pinfty less_top)

  1347       finally show ?thesis by simp

  1348     qed

  1349     show "incseq (\<lambda>n. \<Union>i\<le>n. F i)"

  1350       by (force simp: incseq_def)

  1351   qed

  1352 qed

  1353

  1354 lemma (in sigma_finite_measure) approx_PInf_emeasure_with_finite:

  1355   fixes C::real

  1356   assumes W_meas: "W \<in> sets M"

  1357       and W_inf: "emeasure M W = \<infinity>"

  1358   obtains Z where "Z \<in> sets M" "Z \<subseteq> W" "emeasure M Z < \<infinity>" "emeasure M Z > C"

  1359 proof -

  1360   obtain A :: "nat \<Rightarrow> 'a set"

  1361     where A: "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>" "incseq A"

  1362     using sigma_finite_incseq by blast

  1363   define B where "B = (\<lambda>i. W \<inter> A i)"

  1364   have B_meas: "\<And>i. B i \<in> sets M" using W_meas \<open>range A \<subseteq> sets M\<close> B_def by blast

  1365   have b: "\<And>i. B i \<subseteq> W" using B_def by blast

  1366

  1367   { fix i

  1368     have "emeasure M (B i) \<le> emeasure M (A i)"

  1369       using A by (intro emeasure_mono) (auto simp: B_def)

  1370     also have "emeasure M (A i) < \<infinity>"

  1371       using \<open>\<And>i. emeasure M (A i) \<noteq> \<infinity>\<close> by (simp add: less_top)

  1372     finally have "emeasure M (B i) < \<infinity>" . }

  1373   note c = this

  1374

  1375   have "W = (\<Union>i. B i)" using B_def \<open>(\<Union>i. A i) = space M\<close> W_meas by auto

  1376   moreover have "incseq B" using B_def \<open>incseq A\<close> by (simp add: incseq_def subset_eq)

  1377   ultimately have "(\<lambda>i. emeasure M (B i)) \<longlonglongrightarrow> emeasure M W" using W_meas B_meas

  1378     by (simp add: B_meas Lim_emeasure_incseq image_subset_iff)

  1379   then have "(\<lambda>i. emeasure M (B i)) \<longlonglongrightarrow> \<infinity>" using W_inf by simp

  1380   from order_tendstoD(1)[OF this, of C]

  1381   obtain i where d: "emeasure M (B i) > C"

  1382     by (auto simp: eventually_sequentially)

  1383   have "B i \<in> sets M" "B i \<subseteq> W" "emeasure M (B i) < \<infinity>" "emeasure M (B i) > C"

  1384     using B_meas b c d by auto

  1385   then show ?thesis using that by blast

  1386 qed

  1387

  1388 subsection \<open>Measure space induced by distribution of @{const measurable}-functions\<close>

  1389

  1390 definition%important distr :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b measure" where

  1391   "distr M N f = measure_of (space N) (sets N) (\<lambda>A. emeasure M (f - A \<inter> space M))"

  1392

  1393 lemma

  1394   shows sets_distr[simp, measurable_cong]: "sets (distr M N f) = sets N"

  1395     and space_distr[simp]: "space (distr M N f) = space N"

  1396   by (auto simp: distr_def)

  1397

  1398 lemma

  1399   shows measurable_distr_eq1[simp]: "measurable (distr Mf Nf f) Mf' = measurable Nf Mf'"

  1400     and measurable_distr_eq2[simp]: "measurable Mg' (distr Mg Ng g) = measurable Mg' Ng"

  1401   by (auto simp: measurable_def)

  1402

  1403 lemma distr_cong:

  1404   "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"

  1405   using sets_eq_imp_space_eq[of N L] by (simp add: distr_def Int_def cong: rev_conj_cong)

  1406

  1407 lemma emeasure_distr:

  1408   fixes f :: "'a \<Rightarrow> 'b"

  1409   assumes f: "f \<in> measurable M N" and A: "A \<in> sets N"

  1410   shows "emeasure (distr M N f) A = emeasure M (f - A \<inter> space M)" (is "_ = ?\<mu> A")

  1411   unfolding distr_def

  1412 proof (rule emeasure_measure_of_sigma)

  1413   show "positive (sets N) ?\<mu>"

  1414     by (auto simp: positive_def)

  1415

  1416   show "countably_additive (sets N) ?\<mu>"

  1417   proof (intro countably_additiveI)

  1418     fix A :: "nat \<Rightarrow> 'b set" assume "range A \<subseteq> sets N" "disjoint_family A"

  1419     then have A: "\<And>i. A i \<in> sets N" "(\<Union>i. A i) \<in> sets N" by auto

  1420     then have *: "range (\<lambda>i. f - (A i) \<inter> space M) \<subseteq> sets M"

  1421       using f by (auto simp: measurable_def)

  1422     moreover have "(\<Union>i. f -  A i \<inter> space M) \<in> sets M"

  1423       using * by blast

  1424     moreover have **: "disjoint_family (\<lambda>i. f - A i \<inter> space M)"

  1425       using \<open>disjoint_family A\<close> by (auto simp: disjoint_family_on_def)

  1426     ultimately show "(\<Sum>i. ?\<mu> (A i)) = ?\<mu> (\<Union>i. A i)"

  1427       using suminf_emeasure[OF _ **] A f

  1428       by (auto simp: comp_def vimage_UN)

  1429   qed

  1430   show "sigma_algebra (space N) (sets N)" ..

  1431 qed fact

  1432

  1433 lemma emeasure_Collect_distr:

  1434   assumes X[measurable]: "X \<in> measurable M N" "Measurable.pred N P"

  1435   shows "emeasure (distr M N X) {x\<in>space N. P x} = emeasure M {x\<in>space M. P (X x)}"

  1436   by (subst emeasure_distr)

  1437      (auto intro!: arg_cong2[where f=emeasure] X(1)[THEN measurable_space])

  1438

  1439 lemma emeasure_lfp2[consumes 1, case_names cont f measurable]:

  1440   assumes "P M"

  1441   assumes cont: "sup_continuous F"

  1442   assumes f: "\<And>M. P M \<Longrightarrow> f \<in> measurable M' M"

  1443   assumes *: "\<And>M A. P M \<Longrightarrow> (\<And>N. P N \<Longrightarrow> Measurable.pred N A) \<Longrightarrow> Measurable.pred M (F A)"

  1444   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)})"

  1445 proof (subst (1 2) emeasure_Collect_distr[symmetric, where X=f])

  1446   show "f \<in> measurable M' M"  "f \<in> measurable M' M"

  1447     using f[OF \<open>P M\<close>] by auto

  1448   { fix i show "Measurable.pred M ((F ^^ i) (\<lambda>x. False))"

  1449     using \<open>P M\<close> by (induction i arbitrary: M) (auto intro!: *) }

  1450   show "Measurable.pred M (lfp F)"

  1451     using \<open>P M\<close> cont * by (rule measurable_lfp_coinduct[of P])

  1452

  1453   have "emeasure (distr M' M f) {x \<in> space (distr M' M f). lfp F x} =

  1454     (SUP i. emeasure (distr M' M f) {x \<in> space (distr M' M f). (F ^^ i) (\<lambda>x. False) x})"

  1455     using \<open>P M\<close>

  1456   proof (coinduction arbitrary: M rule: emeasure_lfp')

  1457     case (measurable A N) then have "\<And>N. P N \<Longrightarrow> Measurable.pred (distr M' N f) A"

  1458       by metis

  1459     then have "\<And>N. P N \<Longrightarrow> Measurable.pred N A"

  1460       by simp

  1461     with \<open>P N\<close>[THEN *] show ?case

  1462       by auto

  1463   qed fact

  1464   then show "emeasure (distr M' M f) {x \<in> space M. lfp F x} =

  1465     (SUP i. emeasure (distr M' M f) {x \<in> space M. (F ^^ i) (\<lambda>x. False) x})"

  1466    by simp

  1467 qed

  1468

  1469 lemma distr_id[simp]: "distr N N (\<lambda>x. x) = N"

  1470   by (rule measure_eqI) (auto simp: emeasure_distr)

  1471

  1472 lemma distr_id2: "sets M = sets N \<Longrightarrow> distr N M (\<lambda>x. x) = N"

  1473   by (rule measure_eqI) (auto simp: emeasure_distr)

  1474

  1475 lemma measure_distr:

  1476   "f \<in> measurable M N \<Longrightarrow> S \<in> sets N \<Longrightarrow> measure (distr M N f) S = measure M (f - S \<inter> space M)"

  1477   by (simp add: emeasure_distr measure_def)

  1478

  1479 lemma distr_cong_AE:

  1480   assumes 1: "M = K" "sets N = sets L" and

  1481     2: "(AE x in M. f x = g x)" and "f \<in> measurable M N" and "g \<in> measurable K L"

  1482   shows "distr M N f = distr K L g"

  1483 proof (rule measure_eqI)

  1484   fix A assume "A \<in> sets (distr M N f)"

  1485   with assms show "emeasure (distr M N f) A = emeasure (distr K L g) A"

  1486     by (auto simp add: emeasure_distr intro!: emeasure_eq_AE measurable_sets)

  1487 qed (insert 1, simp)

  1488

  1489 lemma AE_distrD:

  1490   assumes f: "f \<in> measurable M M'"

  1491     and AE: "AE x in distr M M' f. P x"

  1492   shows "AE x in M. P (f x)"

  1493 proof -

  1494   from AE[THEN AE_E] guess N .

  1495   with f show ?thesis

  1496     unfolding eventually_ae_filter

  1497     by (intro bexI[of _ "f - N \<inter> space M"])

  1498        (auto simp: emeasure_distr measurable_def)

  1499 qed

  1500

  1501 lemma AE_distr_iff:

  1502   assumes f[measurable]: "f \<in> measurable M N" and P[measurable]: "{x \<in> space N. P x} \<in> sets N"

  1503   shows "(AE x in distr M N f. P x) \<longleftrightarrow> (AE x in M. P (f x))"

  1504 proof (subst (1 2) AE_iff_measurable[OF _ refl])

  1505   have "f - {x\<in>space N. \<not> P x} \<inter> space M = {x \<in> space M. \<not> P (f x)}"

  1506     using f[THEN measurable_space] by auto

  1507   then show "(emeasure (distr M N f) {x \<in> space (distr M N f). \<not> P x} = 0) =

  1508     (emeasure M {x \<in> space M. \<not> P (f x)} = 0)"

  1509     by (simp add: emeasure_distr)

  1510 qed auto

  1511

  1512 lemma null_sets_distr_iff:

  1513   "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"

  1514   by (auto simp add: null_sets_def emeasure_distr)

  1515

  1516 proposition distr_distr:

  1517   "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)"

  1518   by (auto simp add: emeasure_distr measurable_space

  1519            intro!: arg_cong[where f="emeasure M"] measure_eqI)

  1520

  1521 subsection%unimportant \<open>Real measure values\<close>

  1522

  1523 lemma ring_of_finite_sets: "ring_of_sets (space M) {A\<in>sets M. emeasure M A \<noteq> top}"

  1524 proof (rule ring_of_setsI)

  1525   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>

  1526     a \<union> b \<in> {A \<in> sets M. emeasure M A \<noteq> top}" for a b

  1527     using emeasure_subadditive[of a M b] by (auto simp: top_unique)

  1528

  1529   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>

  1530     a - b \<in> {A \<in> sets M. emeasure M A \<noteq> top}" for a b

  1531     using emeasure_mono[of "a - b" a M] by (auto simp: top_unique)

  1532 qed (auto dest: sets.sets_into_space)

  1533

  1534 lemma measure_nonneg[simp]: "0 \<le> measure M A"

  1535   unfolding measure_def by auto

  1536

  1537 lemma measure_nonneg' [simp]: "\<not> measure M A < 0"

  1538   using measure_nonneg not_le by blast

  1539

  1540 lemma zero_less_measure_iff: "0 < measure M A \<longleftrightarrow> measure M A \<noteq> 0"

  1541   using measure_nonneg[of M A] by (auto simp add: le_less)

  1542

  1543 lemma measure_le_0_iff: "measure M X \<le> 0 \<longleftrightarrow> measure M X = 0"

  1544   using measure_nonneg[of M X] by linarith

  1545

  1546 lemma measure_empty[simp]: "measure M {} = 0"

  1547   unfolding measure_def by (simp add: zero_ennreal.rep_eq)

  1548

  1549 lemma emeasure_eq_ennreal_measure:

  1550   "emeasure M A \<noteq> top \<Longrightarrow> emeasure M A = ennreal (measure M A)"

  1551   by (cases "emeasure M A" rule: ennreal_cases) (auto simp: measure_def)

  1552

  1553 lemma measure_zero_top: "emeasure M A = top \<Longrightarrow> measure M A = 0"

  1554   by (simp add: measure_def enn2ereal_top)

  1555

  1556 lemma measure_eq_emeasure_eq_ennreal: "0 \<le> x \<Longrightarrow> emeasure M A = ennreal x \<Longrightarrow> measure M A = x"

  1557   using emeasure_eq_ennreal_measure[of M A]

  1558   by (cases "A \<in> M") (auto simp: measure_notin_sets emeasure_notin_sets)

  1559

  1560 lemma enn2real_plus:"a < top \<Longrightarrow> b < top \<Longrightarrow> enn2real (a + b) = enn2real a + enn2real b"

  1561   by (simp add: enn2real_def plus_ennreal.rep_eq real_of_ereal_add less_top

  1562            del: real_of_ereal_enn2ereal)

  1563

  1564 lemma measure_eq_AE:

  1565   assumes iff: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B"

  1566   assumes A: "A \<in> sets M" and B: "B \<in> sets M"

  1567   shows "measure M A = measure M B"

  1568   using assms emeasure_eq_AE[OF assms] by (simp add: measure_def)

  1569

  1570 lemma measure_Union:

  1571   "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>

  1572     measure M (A \<union> B) = measure M A + measure M B"

  1573   by (simp add: measure_def plus_emeasure[symmetric] enn2real_plus less_top)

  1574

  1575 lemma disjoint_family_on_insert:

  1576   "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"

  1577   by (fastforce simp: disjoint_family_on_def)

  1578

  1579 lemma measure_finite_Union:

  1580   "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>

  1581     measure M (\<Union>i\<in>S. A i) = (\<Sum>i\<in>S. measure M (A i))"

  1582   by (induction S rule: finite_induct)

  1583      (auto simp: disjoint_family_on_insert measure_Union sum_emeasure[symmetric] sets.countable_UN'[OF countable_finite])

  1584

  1585 lemma measure_Diff:

  1586   assumes finite: "emeasure M A \<noteq> \<infinity>"

  1587   and measurable: "A \<in> sets M" "B \<in> sets M" "B \<subseteq> A"

  1588   shows "measure M (A - B) = measure M A - measure M B"

  1589 proof -

  1590   have "emeasure M (A - B) \<le> emeasure M A" "emeasure M B \<le> emeasure M A"

  1591     using measurable by (auto intro!: emeasure_mono)

  1592   hence "measure M ((A - B) \<union> B) = measure M (A - B) + measure M B"

  1593     using measurable finite by (rule_tac measure_Union) (auto simp: top_unique)

  1594   thus ?thesis using \<open>B \<subseteq> A\<close> by (auto simp: Un_absorb2)

  1595 qed

  1596

  1597 lemma measure_UNION:

  1598   assumes measurable: "range A \<subseteq> sets M" "disjoint_family A"

  1599   assumes finite: "emeasure M (\<Union>i. A i) \<noteq> \<infinity>"

  1600   shows "(\<lambda>i. measure M (A i)) sums (measure M (\<Union>i. A i))"

  1601 proof -

  1602   have "(\<lambda>i. emeasure M (A i)) sums (emeasure M (\<Union>i. A i))"

  1603     unfolding suminf_emeasure[OF measurable, symmetric] by (simp add: summable_sums)

  1604   moreover

  1605   { fix i

  1606     have "emeasure M (A i) \<le> emeasure M (\<Union>i. A i)"

  1607       using measurable by (auto intro!: emeasure_mono)

  1608     then have "emeasure M (A i) = ennreal ((measure M (A i)))"

  1609       using finite by (intro emeasure_eq_ennreal_measure) (auto simp: top_unique) }

  1610   ultimately show ?thesis using finite

  1611     by (subst (asm) (2) emeasure_eq_ennreal_measure) simp_all

  1612 qed

  1613

  1614 lemma measure_subadditive:

  1615   assumes measurable: "A \<in> sets M" "B \<in> sets M"

  1616   and fin: "emeasure M A \<noteq> \<infinity>" "emeasure M B \<noteq> \<infinity>"

  1617   shows "measure M (A \<union> B) \<le> measure M A + measure M B"

  1618 proof -

  1619   have "emeasure M (A \<union> B) \<noteq> \<infinity>"

  1620     using emeasure_subadditive[OF measurable] fin by (auto simp: top_unique)

  1621   then show "(measure M (A \<union> B)) \<le> (measure M A) + (measure M B)"

  1622     using emeasure_subadditive[OF measurable] fin

  1623     apply simp

  1624     apply (subst (asm) (2 3 4) emeasure_eq_ennreal_measure)

  1625     apply (auto simp flip: ennreal_plus)

  1626     done

  1627 qed

  1628

  1629 lemma measure_subadditive_finite:

  1630   assumes A: "finite I" "AI \<subseteq> sets M" and fin: "\<And>i. i \<in> I \<Longrightarrow> emeasure M (A i) \<noteq> \<infinity>"

  1631   shows "measure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. measure M (A i))"

  1632 proof -

  1633   { have "emeasure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. emeasure M (A i))"

  1634       using emeasure_subadditive_finite[OF A] .

  1635     also have "\<dots> < \<infinity>"

  1636       using fin by (simp add: less_top A)

  1637     finally have "emeasure M (\<Union>i\<in>I. A i) \<noteq> top" by simp }

  1638   note * = this

  1639   show ?thesis

  1640     using emeasure_subadditive_finite[OF A] fin

  1641     unfolding emeasure_eq_ennreal_measure[OF *]

  1642     by (simp_all add: sum_nonneg emeasure_eq_ennreal_measure)

  1643 qed

  1644

  1645 lemma measure_subadditive_countably:

  1646   assumes A: "range A \<subseteq> sets M" and fin: "(\<Sum>i. emeasure M (A i)) \<noteq> \<infinity>"

  1647   shows "measure M (\<Union>i. A i) \<le> (\<Sum>i. measure M (A i))"

  1648 proof -

  1649   from fin have **: "\<And>i. emeasure M (A i) \<noteq> top"

  1650     using ennreal_suminf_lessD[of "\<lambda>i. emeasure M (A i)"] by (simp add: less_top)

  1651   { have "emeasure M (\<Union>i. A i) \<le> (\<Sum>i. emeasure M (A i))"

  1652       using emeasure_subadditive_countably[OF A] .

  1653     also have "\<dots> < \<infinity>"

  1654       using fin by (simp add: less_top)

  1655     finally have "emeasure M (\<Union>i. A i) \<noteq> top" by simp }

  1656   then have "ennreal (measure M (\<Union>i. A i)) = emeasure M (\<Union>i. A i)"

  1657     by (rule emeasure_eq_ennreal_measure[symmetric])

  1658   also have "\<dots> \<le> (\<Sum>i. emeasure M (A i))"

  1659     using emeasure_subadditive_countably[OF A] .

  1660   also have "\<dots> = ennreal (\<Sum>i. measure M (A i))"

  1661     using fin unfolding emeasure_eq_ennreal_measure[OF **]

  1662     by (subst suminf_ennreal) (auto simp: **)

  1663   finally show ?thesis

  1664     apply (rule ennreal_le_iff[THEN iffD1, rotated])

  1665     apply (intro suminf_nonneg allI measure_nonneg summable_suminf_not_top)

  1666     using fin

  1667     apply (simp add: emeasure_eq_ennreal_measure[OF **])

  1668     done

  1669 qed

  1670

  1671 lemma measure_Un_null_set: "A \<in> sets M \<Longrightarrow> B \<in> null_sets M \<Longrightarrow> measure M (A \<union> B) = measure M A"

  1672   by (simp add: measure_def emeasure_Un_null_set)

  1673

  1674 lemma measure_Diff_null_set: "A \<in> sets M \<Longrightarrow> B \<in> null_sets M \<Longrightarrow> measure M (A - B) = measure M A"

  1675   by (simp add: measure_def emeasure_Diff_null_set)

  1676

  1677 lemma measure_eq_sum_singleton:

  1678   "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>

  1679     measure M S = (\<Sum>x\<in>S. measure M {x})"

  1680   using emeasure_eq_sum_singleton[of S M]

  1681   by (intro measure_eq_emeasure_eq_ennreal) (auto simp: sum_nonneg emeasure_eq_ennreal_measure)

  1682

  1683 lemma Lim_measure_incseq:

  1684   assumes A: "range A \<subseteq> sets M" "incseq A" and fin: "emeasure M (\<Union>i. A i) \<noteq> \<infinity>"

  1685   shows "(\<lambda>i. measure M (A i)) \<longlonglongrightarrow> measure M (\<Union>i. A i)"

  1686 proof (rule tendsto_ennrealD)

  1687   have "ennreal (measure M (\<Union>i. A i)) = emeasure M (\<Union>i. A i)"

  1688     using fin by (auto simp: emeasure_eq_ennreal_measure)

  1689   moreover have "ennreal (measure M (A i)) = emeasure M (A i)" for i

  1690     using assms emeasure_mono[of "A _" "\<Union>i. A i" M]

  1691     by (intro emeasure_eq_ennreal_measure[symmetric]) (auto simp: less_top UN_upper intro: le_less_trans)

  1692   ultimately show "(\<lambda>x. ennreal (measure M (A x))) \<longlonglongrightarrow> ennreal (measure M (\<Union>i. A i))"

  1693     using A by (auto intro!: Lim_emeasure_incseq)

  1694 qed auto

  1695

  1696 lemma Lim_measure_decseq:

  1697   assumes A: "range A \<subseteq> sets M" "decseq A" and fin: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"

  1698   shows "(\<lambda>n. measure M (A n)) \<longlonglongrightarrow> measure M (\<Inter>i. A i)"

  1699 proof (rule tendsto_ennrealD)

  1700   have "ennreal (measure M (\<Inter>i. A i)) = emeasure M (\<Inter>i. A i)"

  1701     using fin[of 0] A emeasure_mono[of "\<Inter>i. A i" "A 0" M]

  1702     by (auto intro!: emeasure_eq_ennreal_measure[symmetric] simp: INT_lower less_top intro: le_less_trans)

  1703   moreover have "ennreal (measure M (A i)) = emeasure M (A i)" for i

  1704     using A fin[of i] by (intro emeasure_eq_ennreal_measure[symmetric]) auto

  1705   ultimately show "(\<lambda>x. ennreal (measure M (A x))) \<longlonglongrightarrow> ennreal (measure M (\<Inter>i. A i))"

  1706     using fin A by (auto intro!: Lim_emeasure_decseq)

  1707 qed auto

  1708

  1709 subsection \<open>Set of measurable sets with finite measure\<close>

  1710

  1711 definition%important fmeasurable :: "'a measure \<Rightarrow> 'a set set"

  1712 where

  1713   "fmeasurable M = {A\<in>sets M. emeasure M A < \<infinity>}"

  1714

  1715 lemma fmeasurableD[dest, measurable_dest]: "A \<in> fmeasurable M \<Longrightarrow> A \<in> sets M"

  1716   by (auto simp: fmeasurable_def)

  1717

  1718 lemma fmeasurableD2: "A \<in> fmeasurable M \<Longrightarrow> emeasure M A \<noteq> top"

  1719   by (auto simp: fmeasurable_def)

  1720

  1721 lemma fmeasurableI: "A \<in> sets M \<Longrightarrow> emeasure M A < \<infinity> \<Longrightarrow> A \<in> fmeasurable M"

  1722   by (auto simp: fmeasurable_def)

  1723

  1724 lemma fmeasurableI_null_sets: "A \<in> null_sets M \<Longrightarrow> A \<in> fmeasurable M"

  1725   by (auto simp: fmeasurable_def)

  1726

  1727 lemma fmeasurableI2: "A \<in> fmeasurable M \<Longrightarrow> B \<subseteq> A \<Longrightarrow> B \<in> sets M \<Longrightarrow> B \<in> fmeasurable M"

  1728   using emeasure_mono[of B A M] by (auto simp: fmeasurable_def)

  1729

  1730 lemma measure_mono_fmeasurable:

  1731   "A \<subseteq> B \<Longrightarrow> A \<in> sets M \<Longrightarrow> B \<in> fmeasurable M \<Longrightarrow> measure M A \<le> measure M B"

  1732   by (auto simp: measure_def fmeasurable_def intro!: emeasure_mono enn2real_mono)

  1733

  1734 lemma emeasure_eq_measure2: "A \<in> fmeasurable M \<Longrightarrow> emeasure M A = measure M A"

  1735   by (simp add: emeasure_eq_ennreal_measure fmeasurable_def less_top)

  1736

  1737 interpretation fmeasurable: ring_of_sets "space M" "fmeasurable M"

  1738 proof (rule ring_of_setsI)

  1739   show "fmeasurable M \<subseteq> Pow (space M)" "{} \<in> fmeasurable M"

  1740     by (auto simp: fmeasurable_def dest: sets.sets_into_space)

  1741   fix a b assume *: "a \<in> fmeasurable M" "b \<in> fmeasurable M"

  1742   then have "emeasure M (a \<union> b) \<le> emeasure M a + emeasure M b"

  1743     by (intro emeasure_subadditive) auto

  1744   also have "\<dots> < top"

  1745     using * by (auto simp: fmeasurable_def)

  1746   finally show  "a \<union> b \<in> fmeasurable M"

  1747     using * by (auto intro: fmeasurableI)

  1748   show "a - b \<in> fmeasurable M"

  1749     using emeasure_mono[of "a - b" a M] * by (auto simp: fmeasurable_def)

  1750 qed

  1751

  1752 subsection\<open>Measurable sets formed by unions and intersections\<close>

  1753

  1754 lemma fmeasurable_Diff: "A \<in> fmeasurable M \<Longrightarrow> B \<in> sets M \<Longrightarrow> A - B \<in> fmeasurable M"

  1755   using fmeasurableI2[of A M "A - B"] by auto

  1756

  1757 lemma fmeasurable_Int_fmeasurable:

  1758    "\<lbrakk>S \<in> fmeasurable M; T \<in> sets M\<rbrakk> \<Longrightarrow> (S \<inter> T) \<in> fmeasurable M"

  1759   by (meson fmeasurableD fmeasurableI2 inf_le1 sets.Int)

  1760

  1761 lemma fmeasurable_UN:

  1762   assumes "countable I" "\<And>i. i \<in> I \<Longrightarrow> F i \<subseteq> A" "\<And>i. i \<in> I \<Longrightarrow> F i \<in> sets M" "A \<in> fmeasurable M"

  1763   shows "(\<Union>i\<in>I. F i) \<in> fmeasurable M"

  1764 proof (rule fmeasurableI2)

  1765   show "A \<in> fmeasurable M" "(\<Union>i\<in>I. F i) \<subseteq> A" using assms by auto

  1766   show "(\<Union>i\<in>I. F i) \<in> sets M"

  1767     using assms by (intro sets.countable_UN') auto

  1768 qed

  1769

  1770 lemma fmeasurable_INT:

  1771   assumes "countable I" "i \<in> I" "\<And>i. i \<in> I \<Longrightarrow> F i \<in> sets M" "F i \<in> fmeasurable M"

  1772   shows "(\<Inter>i\<in>I. F i) \<in> fmeasurable M"

  1773 proof (rule fmeasurableI2)

  1774   show "F i \<in> fmeasurable M" "(\<Inter>i\<in>I. F i) \<subseteq> F i"

  1775     using assms by auto

  1776   show "(\<Inter>i\<in>I. F i) \<in> sets M"

  1777     using assms by (intro sets.countable_INT') auto

  1778 qed

  1779

  1780 lemma measurable_measure_Diff:

  1781   assumes "A \<in> fmeasurable M" "B \<in> sets M" "B \<subseteq> A"

  1782   shows "measure M (A - B) = measure M A - measure M B"

  1783   by (simp add: assms fmeasurableD fmeasurableD2 measure_Diff)

  1784

  1785 lemma measurable_Un_null_set:

  1786   assumes "B \<in> null_sets M"

  1787   shows "(A \<union> B \<in> fmeasurable M \<and> A \<in> sets M) \<longleftrightarrow> A \<in> fmeasurable M"

  1788   using assms  by (fastforce simp add: fmeasurable.Un fmeasurableI_null_sets intro: fmeasurableI2)

  1789

  1790 lemma measurable_Diff_null_set:

  1791   assumes "B \<in> null_sets M"

  1792   shows "(A - B) \<in> fmeasurable M \<and> A \<in> sets M \<longleftrightarrow> A \<in> fmeasurable M"

  1793   using assms

  1794   by (metis Un_Diff_cancel2 fmeasurable.Diff fmeasurableD fmeasurableI_null_sets measurable_Un_null_set)

  1795

  1796 lemma fmeasurable_Diff_D:

  1797     assumes m: "T - S \<in> fmeasurable M" "S \<in> fmeasurable M" and sub: "S \<subseteq> T"

  1798     shows "T \<in> fmeasurable M"

  1799 proof -

  1800   have "T = S \<union> (T - S)"

  1801     using assms by blast

  1802   then show ?thesis

  1803     by (metis m fmeasurable.Un)

  1804 qed

  1805

  1806 lemma measure_Un2:

  1807   "A \<in> fmeasurable M \<Longrightarrow> B \<in> fmeasurable M \<Longrightarrow> measure M (A \<union> B) = measure M A + measure M (B - A)"

  1808   using measure_Union[of M A "B - A"] by (auto simp: fmeasurableD2 fmeasurable.Diff)

  1809

  1810 lemma measure_Un3:

  1811   assumes "A \<in> fmeasurable M" "B \<in> fmeasurable M"

  1812   shows "measure M (A \<union> B) = measure M A + measure M B - measure M (A \<inter> B)"

  1813 proof -

  1814   have "measure M (A \<union> B) = measure M A + measure M (B - A)"

  1815     using assms by (rule measure_Un2)

  1816   also have "B - A = B - (A \<inter> B)"

  1817     by auto

  1818   also have "measure M (B - (A \<inter> B)) = measure M B - measure M (A \<inter> B)"

  1819     using assms by (intro measure_Diff) (auto simp: fmeasurable_def)

  1820   finally show ?thesis

  1821     by simp

  1822 qed

  1823

  1824 lemma measure_Un_AE:

  1825   "AE x in M. x \<notin> A \<or> x \<notin> B \<Longrightarrow> A \<in> fmeasurable M \<Longrightarrow> B \<in> fmeasurable M \<Longrightarrow>

  1826   measure M (A \<union> B) = measure M A + measure M B"

  1827   by (subst measure_Un2) (auto intro!: measure_eq_AE)

  1828

  1829 lemma measure_UNION_AE:

  1830   assumes I: "finite I"

  1831   shows "(\<And>i. i \<in> I \<Longrightarrow> F i \<in> fmeasurable M) \<Longrightarrow> pairwise (\<lambda>i j. AE x in M. x \<notin> F i \<or> x \<notin> F j) I \<Longrightarrow>

  1832     measure M (\<Union>i\<in>I. F i) = (\<Sum>i\<in>I. measure M (F i))"

  1833   unfolding AE_pairwise[OF countable_finite, OF I]

  1834   using I

  1835 proof (induction I rule: finite_induct)

  1836   case (insert x I)

  1837   have "measure M (F x \<union> \<Union>(F  I)) = measure M (F x) + measure M (\<Union>(F  I))"

  1838     by (rule measure_Un_AE) (use insert in \<open>auto simp: pairwise_insert\<close>)

  1839     with insert show ?case

  1840       by (simp add: pairwise_insert )

  1841 qed simp

  1842

  1843 lemma measure_UNION':

  1844   "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> F i \<in> fmeasurable M) \<Longrightarrow> pairwise (\<lambda>i j. disjnt (F i) (F j)) I \<Longrightarrow>

  1845     measure M (\<Union>i\<in>I. F i) = (\<Sum>i\<in>I. measure M (F i))"

  1846   by (intro measure_UNION_AE) (auto simp: disjnt_def elim!: pairwise_mono intro!: always_eventually)

  1847

  1848 lemma measure_Union_AE:

  1849   "finite F \<Longrightarrow> (\<And>S. S \<in> F \<Longrightarrow> S \<in> fmeasurable M) \<Longrightarrow> pairwise (\<lambda>S T. AE x in M. x \<notin> S \<or> x \<notin> T) F \<Longrightarrow>

  1850     measure M (\<Union>F) = (\<Sum>S\<in>F. measure M S)"

  1851   using measure_UNION_AE[of F "\<lambda>x. x" M] by simp

  1852

  1853 lemma measure_Union':

  1854   "finite F \<Longrightarrow> (\<And>S. S \<in> F \<Longrightarrow> S \<in> fmeasurable M) \<Longrightarrow> pairwise disjnt F \<Longrightarrow> measure M (\<Union>F) = (\<Sum>S\<in>F. measure M S)"

  1855   using measure_UNION'[of F "\<lambda>x. x" M] by simp

  1856

  1857 lemma measure_Un_le:

  1858   assumes "A \<in> sets M" "B \<in> sets M" shows "measure M (A \<union> B) \<le> measure M A + measure M B"

  1859 proof cases

  1860   assume "A \<in> fmeasurable M \<and> B \<in> fmeasurable M"

  1861   with measure_subadditive[of A M B] assms show ?thesis

  1862     by (auto simp: fmeasurableD2)

  1863 next

  1864   assume "\<not> (A \<in> fmeasurable M \<and> B \<in> fmeasurable M)"

  1865   then have "A \<union> B \<notin> fmeasurable M"

  1866     using fmeasurableI2[of "A \<union> B" M A] fmeasurableI2[of "A \<union> B" M B] assms by auto

  1867   with assms show ?thesis

  1868     by (auto simp: fmeasurable_def measure_def less_top[symmetric])

  1869 qed

  1870

  1871 lemma measure_UNION_le:

  1872   "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> F i \<in> sets M) \<Longrightarrow> measure M (\<Union>i\<in>I. F i) \<le> (\<Sum>i\<in>I. measure M (F i))"

  1873 proof (induction I rule: finite_induct)

  1874   case (insert i I)

  1875   then have "measure M (\<Union>i\<in>insert i I. F i) \<le> measure M (F i) + measure M (\<Union>i\<in>I. F i)"

  1876     by (auto intro!: measure_Un_le)

  1877   also have "measure M (\<Union>i\<in>I. F i) \<le> (\<Sum>i\<in>I. measure M (F i))"

  1878     using insert by auto

  1879   finally show ?case

  1880     using insert by simp

  1881 qed simp

  1882

  1883 lemma measure_Union_le:

  1884   "finite F \<Longrightarrow> (\<And>S. S \<in> F \<Longrightarrow> S \<in> sets M) \<Longrightarrow> measure M (\<Union>F) \<le> (\<Sum>S\<in>F. measure M S)"

  1885   using measure_UNION_le[of F "\<lambda>x. x" M] by simp

  1886

  1887 text\<open>Version for indexed union over a countable set\<close>

  1888 lemma

  1889   assumes "countable I" and I: "\<And>i. i \<in> I \<Longrightarrow> A i \<in> fmeasurable M"

  1890     and bound: "\<And>I'. I' \<subseteq> I \<Longrightarrow> finite I' \<Longrightarrow> measure M (\<Union>i\<in>I'. A i) \<le> B"

  1891   shows fmeasurable_UN_bound: "(\<Union>i\<in>I. A i) \<in> fmeasurable M" (is ?fm)

  1892     and measure_UN_bound: "measure M (\<Union>i\<in>I. A i) \<le> B" (is ?m)

  1893 proof -

  1894   have "B \<ge> 0"

  1895     using bound by force

  1896   have "?fm \<and> ?m"

  1897   proof cases

  1898     assume "I = {}"

  1899     with \<open>B \<ge> 0\<close> show ?thesis

  1900       by simp

  1901   next

  1902     assume "I \<noteq> {}"

  1903     have "(\<Union>i\<in>I. A i) = (\<Union>i. (\<Union>n\<le>i. A (from_nat_into I n)))"

  1904       by (subst range_from_nat_into[symmetric, OF \<open>I \<noteq> {}\<close> \<open>countable I\<close>]) auto

  1905     then have "emeasure M (\<Union>i\<in>I. A i) = emeasure M (\<Union>i. (\<Union>n\<le>i. A (from_nat_into I n)))" by simp

  1906     also have "\<dots> = (SUP i. emeasure M (\<Union>n\<le>i. A (from_nat_into I n)))"

  1907       using I \<open>I \<noteq> {}\<close>[THEN from_nat_into] by (intro SUP_emeasure_incseq[symmetric]) (fastforce simp: incseq_Suc_iff)+

  1908     also have "\<dots> \<le> B"

  1909     proof (intro SUP_least)

  1910       fix i :: nat

  1911       have "emeasure M (\<Union>n\<le>i. A (from_nat_into I n)) = measure M (\<Union>n\<le>i. A (from_nat_into I n))"

  1912         using I \<open>I \<noteq> {}\<close>[THEN from_nat_into] by (intro emeasure_eq_measure2 fmeasurable.finite_UN) auto

  1913       also have "\<dots> = measure M (\<Union>n\<in>from_nat_into I  {..i}. A n)"

  1914         by simp

  1915       also have "\<dots> \<le> B"

  1916         by (intro ennreal_leI bound) (auto intro:  from_nat_into[OF \<open>I \<noteq> {}\<close>])

  1917       finally show "emeasure M (\<Union>n\<le>i. A (from_nat_into I n)) \<le> ennreal B" .

  1918     qed

  1919     finally have *: "emeasure M (\<Union>i\<in>I. A i) \<le> B" .

  1920     then have ?fm

  1921       using I \<open>countable I\<close> by (intro fmeasurableI conjI) (auto simp: less_top[symmetric] top_unique)

  1922     with * \<open>0\<le>B\<close> show ?thesis

  1923       by (simp add: emeasure_eq_measure2)

  1924   qed

  1925   then show ?fm ?m by auto

  1926 qed

  1927

  1928 text\<open>Version for big union of a countable set\<close>

  1929 lemma

  1930   assumes "countable \<D>"

  1931     and meas: "\<And>D. D \<in> \<D> \<Longrightarrow> D \<in> fmeasurable M"

  1932     and bound:  "\<And>\<E>. \<lbrakk>\<E> \<subseteq> \<D>; finite \<E>\<rbrakk> \<Longrightarrow> measure M (\<Union>\<E>) \<le> B"

  1933  shows fmeasurable_Union_bound: "\<Union>\<D> \<in> fmeasurable M"  (is ?fm)

  1934     and measure_Union_bound: "measure M (\<Union>\<D>) \<le> B"     (is ?m)

  1935 proof -

  1936   have "B \<ge> 0"

  1937     using bound by force

  1938   have "?fm \<and> ?m"

  1939   proof (cases "\<D> = {}")

  1940     case True

  1941     with \<open>B \<ge> 0\<close> show ?thesis

  1942       by auto

  1943   next

  1944     case False

  1945     then obtain D :: "nat \<Rightarrow> 'a set" where D: "\<D> = range D"

  1946       using \<open>countable \<D>\<close> uncountable_def by force

  1947       have 1: "\<And>i. D i \<in> fmeasurable M"

  1948         by (simp add: D meas)

  1949       have 2: "\<And>I'. finite I' \<Longrightarrow> measure M (\<Union>x\<in>I'. D x) \<le> B"

  1950         by (simp add: D bound image_subset_iff)

  1951       show ?thesis

  1952         unfolding D

  1953         by (intro conjI fmeasurable_UN_bound [OF _ 1 2] measure_UN_bound [OF _ 1 2]) auto

  1954     qed

  1955     then show ?fm ?m by auto

  1956 qed

  1957

  1958 text\<open>Version for indexed union over the type of naturals\<close>

  1959 lemma

  1960   fixes S :: "nat \<Rightarrow> 'a set"

  1961   assumes S: "\<And>i. S i \<in> fmeasurable M" and B: "\<And>n. measure M (\<Union>i\<le>n. S i) \<le> B"

  1962   shows fmeasurable_countable_Union: "(\<Union>i. S i) \<in> fmeasurable M"

  1963     and measure_countable_Union_le: "measure M (\<Union>i. S i) \<le> B"

  1964 proof -

  1965   have mB: "measure M (\<Union>i\<in>I. S i) \<le> B" if "finite I" for I

  1966   proof -

  1967     have "(\<Union>i\<in>I. S i) \<subseteq> (\<Union>i\<le>Max I. S i)"

  1968       using Max_ge that by force

  1969     then have "measure M (\<Union>i\<in>I. S i) \<le> measure M (\<Union>i \<le> Max I. S i)"

  1970       by (rule measure_mono_fmeasurable) (use S in \<open>blast+\<close>)

  1971     then show ?thesis

  1972       using B order_trans by blast

  1973   qed

  1974   show "(\<Union>i. S i) \<in> fmeasurable M"

  1975     by (auto intro: fmeasurable_UN_bound [OF _ S mB])

  1976   show "measure M (\<Union>n. S n) \<le> B"

  1977     by (auto intro: measure_UN_bound [OF _ S mB])

  1978 qed

  1979

  1980 lemma measure_diff_le_measure_setdiff:

  1981   assumes "S \<in> fmeasurable M" "T \<in> fmeasurable M"

  1982   shows "measure M S - measure M T \<le> measure M (S - T)"

  1983 proof -

  1984   have "measure M S \<le> measure M ((S - T) \<union> T)"

  1985     by (simp add: assms fmeasurable.Un fmeasurableD measure_mono_fmeasurable)

  1986   also have "\<dots> \<le> measure M (S - T) + measure M T"

  1987     using assms by (blast intro: measure_Un_le)

  1988   finally show ?thesis

  1989     by (simp add: algebra_simps)

  1990 qed

  1991

  1992 lemma suminf_exist_split2:

  1993   fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"

  1994   assumes "summable f"

  1995   shows "(\<lambda>n. (\<Sum>k. f(k+n))) \<longlonglongrightarrow> 0"

  1996 by (subst lim_sequentially, auto simp add: dist_norm suminf_exist_split[OF _ assms])

  1997

  1998 lemma emeasure_union_summable:

  1999   assumes [measurable]: "\<And>n. A n \<in> sets M"

  2000     and "\<And>n. emeasure M (A n) < \<infinity>" "summable (\<lambda>n. measure M (A n))"

  2001   shows "emeasure M (\<Union>n. A n) < \<infinity>" "emeasure M (\<Union>n. A n) \<le> (\<Sum>n. measure M (A n))"

  2002 proof -

  2003   define B where "B = (\<lambda>N. (\<Union>n\<in>{..<N}. A n))"

  2004   have [measurable]: "B N \<in> sets M" for N unfolding B_def by auto

  2005   have "(\<lambda>N. emeasure M (B N)) \<longlonglongrightarrow> emeasure M (\<Union>N. B N)"

  2006     apply (rule Lim_emeasure_incseq) unfolding B_def by (auto simp add: SUP_subset_mono incseq_def)

  2007   moreover have "emeasure M (B N) \<le> ennreal (\<Sum>n. measure M (A n))" for N

  2008   proof -

  2009     have *: "(\<Sum>n\<in>{..<N}. measure M (A n)) \<le> (\<Sum>n. measure M (A n))"

  2010       using assms(3) measure_nonneg sum_le_suminf by blast

  2011

  2012     have "emeasure M (B N) \<le> (\<Sum>n\<in>{..<N}. emeasure M (A n))"

  2013       unfolding B_def by (rule emeasure_subadditive_finite, auto)

  2014     also have "... = (\<Sum>n\<in>{..<N}. ennreal(measure M (A n)))"

  2015       using assms(2) by (simp add: emeasure_eq_ennreal_measure less_top)

  2016     also have "... = ennreal (\<Sum>n\<in>{..<N}. measure M (A n))"

  2017       by auto

  2018     also have "... \<le> ennreal (\<Sum>n. measure M (A n))"

  2019       using * by (auto simp: ennreal_leI)

  2020     finally show ?thesis by simp

  2021   qed

  2022   ultimately have "emeasure M (\<Union>N. B N) \<le> ennreal (\<Sum>n. measure M (A n))"

  2023     by (simp add: Lim_bounded)

  2024   then show "emeasure M (\<Union>n. A n) \<le> (\<Sum>n. measure M (A n))"

  2025     unfolding B_def by (metis UN_UN_flatten UN_lessThan_UNIV)

  2026   then show "emeasure M (\<Union>n. A n) < \<infinity>"

  2027     by (auto simp: less_top[symmetric] top_unique)

  2028 qed

  2029

  2030 lemma borel_cantelli_limsup1:

  2031   assumes [measurable]: "\<And>n. A n \<in> sets M"

  2032     and "\<And>n. emeasure M (A n) < \<infinity>" "summable (\<lambda>n. measure M (A n))"

  2033   shows "limsup A \<in> null_sets M"

  2034 proof -

  2035   have "emeasure M (limsup A) \<le> 0"

  2036   proof (rule LIMSEQ_le_const)

  2037     have "(\<lambda>n. (\<Sum>k. measure M (A (k+n)))) \<longlonglongrightarrow> 0" by (rule suminf_exist_split2[OF assms(3)])

  2038     then show "(\<lambda>n. ennreal (\<Sum>k. measure M (A (k+n)))) \<longlonglongrightarrow> 0"

  2039       unfolding ennreal_0[symmetric] by (intro tendsto_ennrealI)

  2040     have "emeasure M (limsup A) \<le> (\<Sum>k. measure M (A (k+n)))" for n

  2041     proof -

  2042       have I: "(\<Union>k\<in>{n..}. A k) = (\<Union>k. A (k+n))" by (auto, metis le_add_diff_inverse2, fastforce)

  2043       have "emeasure M (limsup A) \<le> emeasure M (\<Union>k\<in>{n..}. A k)"

  2044         by (rule emeasure_mono, auto simp add: limsup_INF_SUP)

  2045       also have "... = emeasure M (\<Union>k. A (k+n))"

  2046         using I by auto

  2047       also have "... \<le> (\<Sum>k. measure M (A (k+n)))"

  2048         apply (rule emeasure_union_summable)

  2049         using assms summable_ignore_initial_segment[OF assms(3), of n] by auto

  2050       finally show ?thesis by simp

  2051     qed

  2052     then show "\<exists>N. \<forall>n\<ge>N. emeasure M (limsup A) \<le> (\<Sum>k. measure M (A (k+n)))"

  2053       by auto

  2054   qed

  2055   then show ?thesis using assms(1) measurable_limsup by auto

  2056 qed

  2057

  2058 lemma borel_cantelli_AE1:

  2059   assumes [measurable]: "\<And>n. A n \<in> sets M"

  2060     and "\<And>n. emeasure M (A n) < \<infinity>" "summable (\<lambda>n. measure M (A n))"

  2061   shows "AE x in M. eventually (\<lambda>n. x \<in> space M - A n) sequentially"

  2062 proof -

  2063   have "AE x in M. x \<notin> limsup A"

  2064     using borel_cantelli_limsup1[OF assms] unfolding eventually_ae_filter by auto

  2065   moreover

  2066   {

  2067     fix x assume "x \<notin> limsup A"

  2068     then obtain N where "x \<notin> (\<Union>n\<in>{N..}. A n)" unfolding limsup_INF_SUP by blast

  2069     then have "eventually (\<lambda>n. x \<notin> A n) sequentially" using eventually_sequentially by auto

  2070   }

  2071   ultimately show ?thesis by auto

  2072 qed

  2073

  2074 subsection \<open>Measure spaces with @{term "emeasure M (space M) < \<infinity>"}\<close>

  2075

  2076 locale%important finite_measure = sigma_finite_measure M for M +

  2077   assumes finite_emeasure_space: "emeasure M (space M) \<noteq> top"

  2078

  2079 lemma finite_measureI[Pure.intro!]:

  2080   "emeasure M (space M) \<noteq> \<infinity> \<Longrightarrow> finite_measure M"

  2081   proof qed (auto intro!: exI[of _ "{space M}"])

  2082

  2083 lemma (in finite_measure) emeasure_finite[simp, intro]: "emeasure M A \<noteq> top"

  2084   using finite_emeasure_space emeasure_space[of M A] by (auto simp: top_unique)

  2085

  2086 lemma (in finite_measure) fmeasurable_eq_sets: "fmeasurable M = sets M"

  2087   by (auto simp: fmeasurable_def less_top[symmetric])

  2088

  2089 lemma (in finite_measure) emeasure_eq_measure: "emeasure M A = ennreal (measure M A)"

  2090   by (intro emeasure_eq_ennreal_measure) simp

  2091

  2092 lemma (in finite_measure) emeasure_real: "\<exists>r. 0 \<le> r \<and> emeasure M A = ennreal r"

  2093   using emeasure_finite[of A] by (cases "emeasure M A" rule: ennreal_cases) auto

  2094

  2095 lemma (in finite_measure) bounded_measure: "measure M A \<le> measure M (space M)"

  2096   using emeasure_space[of M A] emeasure_real[of A] emeasure_real[of "space M"] by (auto simp: measure_def)

  2097

  2098 lemma (in finite_measure) finite_measure_Diff:

  2099   assumes sets: "A \<in> sets M" "B \<in> sets M" and "B \<subseteq> A"

  2100   shows "measure M (A - B) = measure M A - measure M B"

  2101   using measure_Diff[OF _ assms] by simp

  2102

  2103 lemma (in finite_measure) finite_measure_Union:

  2104   assumes sets: "A \<in> sets M" "B \<in> sets M" and "A \<inter> B = {}"

  2105   shows "measure M (A \<union> B) = measure M A + measure M B"

  2106   using measure_Union[OF _ _ assms] by simp

  2107

  2108 lemma (in finite_measure) finite_measure_finite_Union:

  2109   assumes measurable: "finite S" "AS \<subseteq> sets M" "disjoint_family_on A S"

  2110   shows "measure M (\<Union>i\<in>S. A i) = (\<Sum>i\<in>S. measure M (A i))"

  2111   using measure_finite_Union[OF assms] by simp

  2112

  2113 lemma (in finite_measure) finite_measure_UNION:

  2114   assumes A: "range A \<subseteq> sets M" "disjoint_family A"

  2115   shows "(\<lambda>i. measure M (A i)) sums (measure M (\<Union>i. A i))"

  2116   using measure_UNION[OF A] by simp

  2117

  2118 lemma (in finite_measure) finite_measure_mono:

  2119   assumes "A \<subseteq> B" "B \<in> sets M" shows "measure M A \<le> measure M B"

  2120   using emeasure_mono[OF assms] emeasure_real[of A] emeasure_real[of B] by (auto simp: measure_def)

  2121

  2122 lemma (in finite_measure) finite_measure_subadditive:

  2123   assumes m: "A \<in> sets M" "B \<in> sets M"

  2124   shows "measure M (A \<union> B) \<le> measure M A + measure M B"

  2125   using measure_subadditive[OF m] by simp

  2126

  2127 lemma (in finite_measure) finite_measure_subadditive_finite:

  2128   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))"

  2129   using measure_subadditive_finite[OF assms] by simp

  2130

  2131 lemma (in finite_measure) finite_measure_subadditive_countably:

  2132   "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))"

  2133   by (rule measure_subadditive_countably)

  2134      (simp_all add: ennreal_suminf_neq_top emeasure_eq_measure)

  2135

  2136 lemma (in finite_measure) finite_measure_eq_sum_singleton:

  2137   assumes "finite S" and *: "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"

  2138   shows "measure M S = (\<Sum>x\<in>S. measure M {x})"

  2139   using measure_eq_sum_singleton[OF assms] by simp

  2140

  2141 lemma (in finite_measure) finite_Lim_measure_incseq:

  2142   assumes A: "range A \<subseteq> sets M" "incseq A"

  2143   shows "(\<lambda>i. measure M (A i)) \<longlonglongrightarrow> measure M (\<Union>i. A i)"

  2144   using Lim_measure_incseq[OF A] by simp

  2145

  2146 lemma (in finite_measure) finite_Lim_measure_decseq:

  2147   assumes A: "range A \<subseteq> sets M" "decseq A"

  2148   shows "(\<lambda>n. measure M (A n)) \<longlonglongrightarrow> measure M (\<Inter>i. A i)"

  2149   using Lim_measure_decseq[OF A] by simp

  2150

  2151 lemma (in finite_measure) finite_measure_compl:

  2152   assumes S: "S \<in> sets M"

  2153   shows "measure M (space M - S) = measure M (space M) - measure M S"

  2154   using measure_Diff[OF _ sets.top S sets.sets_into_space] S by simp

  2155

  2156 lemma (in finite_measure) finite_measure_mono_AE:

  2157   assumes imp: "AE x in M. x \<in> A \<longrightarrow> x \<in> B" and B: "B \<in> sets M"

  2158   shows "measure M A \<le> measure M B"

  2159   using assms emeasure_mono_AE[OF imp B]

  2160   by (simp add: emeasure_eq_measure)

  2161

  2162 lemma (in finite_measure) finite_measure_eq_AE:

  2163   assumes iff: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B"

  2164   assumes A: "A \<in> sets M" and B: "B \<in> sets M"

  2165   shows "measure M A = measure M B"

  2166   using assms emeasure_eq_AE[OF assms] by (simp add: emeasure_eq_measure)

  2167

  2168 lemma (in finite_measure) measure_increasing: "increasing M (measure M)"

  2169   by (auto intro!: finite_measure_mono simp: increasing_def)

  2170

  2171 lemma (in finite_measure) measure_zero_union:

  2172   assumes "s \<in> sets M" "t \<in> sets M" "measure M t = 0"

  2173   shows "measure M (s \<union> t) = measure M s"

  2174 using assms

  2175 proof -

  2176   have "measure M (s \<union> t) \<le> measure M s"

  2177     using finite_measure_subadditive[of s t] assms by auto

  2178   moreover have "measure M (s \<union> t) \<ge> measure M s"

  2179     using assms by (blast intro: finite_measure_mono)

  2180   ultimately show ?thesis by simp

  2181 qed

  2182

  2183 lemma (in finite_measure) measure_eq_compl:

  2184   assumes "s \<in> sets M" "t \<in> sets M"

  2185   assumes "measure M (space M - s) = measure M (space M - t)"

  2186   shows "measure M s = measure M t"

  2187   using assms finite_measure_compl by auto

  2188

  2189 lemma (in finite_measure) measure_eq_bigunion_image:

  2190   assumes "range f \<subseteq> sets M" "range g \<subseteq> sets M"

  2191   assumes "disjoint_family f" "disjoint_family g"

  2192   assumes "\<And> n :: nat. measure M (f n) = measure M (g n)"

  2193   shows "measure M (\<Union>i. f i) = measure M (\<Union>i. g i)"

  2194 using assms

  2195 proof -

  2196   have a: "(\<lambda> i. measure M (f i)) sums (measure M (\<Union>i. f i))"

  2197     by (rule finite_measure_UNION[OF assms(1,3)])

  2198   have b: "(\<lambda> i. measure M (g i)) sums (measure M (\<Union>i. g i))"

  2199     by (rule finite_measure_UNION[OF assms(2,4)])

  2200   show ?thesis using sums_unique[OF b] sums_unique[OF a] assms by simp

  2201 qed

  2202

  2203 lemma (in finite_measure) measure_countably_zero:

  2204   assumes "range c \<subseteq> sets M"

  2205   assumes "\<And> i. measure M (c i) = 0"

  2206   shows "measure M (\<Union>i :: nat. c i) = 0"

  2207 proof (rule antisym)

  2208   show "measure M (\<Union>i :: nat. c i) \<le> 0"

  2209     using finite_measure_subadditive_countably[OF assms(1)] by (simp add: assms(2))

  2210 qed simp

  2211

  2212 lemma (in finite_measure) measure_space_inter:

  2213   assumes events:"s \<in> sets M" "t \<in> sets M"

  2214   assumes "measure M t = measure M (space M)"

  2215   shows "measure M (s \<inter> t) = measure M s"

  2216 proof -

  2217   have "measure M ((space M - s) \<union> (space M - t)) = measure M (space M - s)"

  2218     using events assms finite_measure_compl[of "t"] by (auto intro!: measure_zero_union)

  2219   also have "(space M - s) \<union> (space M - t) = space M - (s \<inter> t)"

  2220     by blast

  2221   finally show "measure M (s \<inter> t) = measure M s"

  2222     using events by (auto intro!: measure_eq_compl[of "s \<inter> t" s])

  2223 qed

  2224

  2225 lemma (in finite_measure) measure_equiprobable_finite_unions:

  2226   assumes s: "finite s" "\<And>x. x \<in> s \<Longrightarrow> {x} \<in> sets M"

  2227   assumes "\<And> x y. \<lbrakk>x \<in> s; y \<in> s\<rbrakk> \<Longrightarrow> measure M {x} = measure M {y}"

  2228   shows "measure M s = real (card s) * measure M {SOME x. x \<in> s}"

  2229 proof cases

  2230   assume "s \<noteq> {}"

  2231   then have "\<exists> x. x \<in> s" by blast

  2232   from someI_ex[OF this] assms

  2233   have prob_some: "\<And> x. x \<in> s \<Longrightarrow> measure M {x} = measure M {SOME y. y \<in> s}" by blast

  2234   have "measure M s = (\<Sum> x \<in> s. measure M {x})"

  2235     using finite_measure_eq_sum_singleton[OF s] by simp

  2236   also have "\<dots> = (\<Sum> x \<in> s. measure M {SOME y. y \<in> s})" using prob_some by auto

  2237   also have "\<dots> = real (card s) * measure M {(SOME x. x \<in> s)}"

  2238     using sum_constant assms by simp

  2239   finally show ?thesis by simp

  2240 qed simp

  2241

  2242 lemma (in finite_measure) measure_real_sum_image_fn:

  2243   assumes "e \<in> sets M"

  2244   assumes "\<And> x. x \<in> s \<Longrightarrow> e \<inter> f x \<in> sets M"

  2245   assumes "finite s"

  2246   assumes disjoint: "\<And> x y. \<lbrakk>x \<in> s ; y \<in> s ; x \<noteq> y\<rbrakk> \<Longrightarrow> f x \<inter> f y = {}"

  2247   assumes upper: "space M \<subseteq> (\<Union>i \<in> s. f i)"

  2248   shows "measure M e = (\<Sum> x \<in> s. measure M (e \<inter> f x))"

  2249 proof -

  2250   have "e \<subseteq> (\<Union>i\<in>s. f i)"

  2251     using \<open>e \<in> sets M\<close> sets.sets_into_space upper by blast

  2252   then have e: "e = (\<Union>i \<in> s. e \<inter> f i)"

  2253     by auto

  2254   hence "measure M e = measure M (\<Union>i \<in> s. e \<inter> f i)" by simp

  2255   also have "\<dots> = (\<Sum> x \<in> s. measure M (e \<inter> f x))"

  2256   proof (rule finite_measure_finite_Union)

  2257     show "finite s" by fact

  2258     show "(\<lambda>i. e \<inter> f i)s \<subseteq> sets M" using assms(2) by auto

  2259     show "disjoint_family_on (\<lambda>i. e \<inter> f i) s"

  2260       using disjoint by (auto simp: disjoint_family_on_def)

  2261   qed

  2262   finally show ?thesis .

  2263 qed

  2264

  2265 lemma (in finite_measure) measure_exclude:

  2266   assumes "A \<in> sets M" "B \<in> sets M"

  2267   assumes "measure M A = measure M (space M)" "A \<inter> B = {}"

  2268   shows "measure M B = 0"

  2269   using measure_space_inter[of B A] assms by (auto simp: ac_simps)

  2270 lemma (in finite_measure) finite_measure_distr:

  2271   assumes f: "f \<in> measurable M M'"

  2272   shows "finite_measure (distr M M' f)"

  2273 proof (rule finite_measureI)

  2274   have "f - space M' \<inter> space M = space M" using f by (auto dest: measurable_space)

  2275   with f show "emeasure (distr M M' f) (space (distr M M' f)) \<noteq> \<infinity>" by (auto simp: emeasure_distr)

  2276 qed

  2277

  2278 lemma emeasure_gfp[consumes 1, case_names cont measurable]:

  2279   assumes sets[simp]: "\<And>s. sets (M s) = sets N"

  2280   assumes "\<And>s. finite_measure (M s)"

  2281   assumes cont: "inf_continuous F" "inf_continuous f"

  2282   assumes meas: "\<And>P. Measurable.pred N P \<Longrightarrow> Measurable.pred N (F P)"

  2283   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"

  2284   assumes bound: "\<And>P. f P \<le> f (\<lambda>s. emeasure (M s) (space (M s)))"

  2285   shows "emeasure (M s) {x\<in>space N. gfp F x} = gfp f s"

  2286 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

  2287     P="Measurable.pred N", symmetric])

  2288   interpret finite_measure "M s" for s by fact

  2289   fix C assume "decseq C" "\<And>i. Measurable.pred N (C i)"

  2290   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}))"

  2291     unfolding INF_apply[abs_def]

  2292     by (subst INF_emeasure_decseq) (auto simp: antimono_def fun_eq_iff intro!: arg_cong2[where f=emeasure])

  2293 next

  2294   show "f x \<le> (\<lambda>s. emeasure (M s) {x \<in> space N. F top x})" for x

  2295     using bound[of x] sets_eq_imp_space_eq[OF sets] by (simp add: iter)

  2296 qed (auto simp add: iter le_fun_def INF_apply[abs_def] intro!: meas cont)

  2297

  2298 subsection%unimportant \<open>Counting space\<close>

  2299

  2300 lemma strict_monoI_Suc:

  2301   assumes ord [simp]: "(\<And>n. f n < f (Suc n))" shows "strict_mono f"

  2302   unfolding strict_mono_def

  2303 proof safe

  2304   fix n m :: nat assume "n < m" then show "f n < f m"

  2305     by (induct m) (auto simp: less_Suc_eq intro: less_trans ord)

  2306 qed

  2307

  2308 lemma emeasure_count_space:

  2309   assumes "X \<subseteq> A" shows "emeasure (count_space A) X = (if finite X then of_nat (card X) else \<infinity>)"

  2310     (is "_ = ?M X")

  2311   unfolding count_space_def

  2312 proof (rule emeasure_measure_of_sigma)

  2313   show "X \<in> Pow A" using \<open>X \<subseteq> A\<close> by auto

  2314   show "sigma_algebra A (Pow A)" by (rule sigma_algebra_Pow)

  2315   show positive: "positive (Pow A) ?M"

  2316     by (auto simp: positive_def)

  2317   have additive: "additive (Pow A) ?M"

  2318     by (auto simp: card_Un_disjoint additive_def)

  2319

  2320   interpret ring_of_sets A "Pow A"

  2321     by (rule ring_of_setsI) auto

  2322   show "countably_additive (Pow A) ?M"

  2323     unfolding countably_additive_iff_continuous_from_below[OF positive additive]

  2324   proof safe

  2325     fix F :: "nat \<Rightarrow> 'a set" assume "incseq F"

  2326     show "(\<lambda>i. ?M (F i)) \<longlonglongrightarrow> ?M (\<Union>i. F i)"

  2327     proof cases

  2328       assume "\<exists>i. \<forall>j\<ge>i. F i = F j"

  2329       then guess i .. note i = this

  2330       { fix j from i \<open>incseq F\<close> have "F j \<subseteq> F i"

  2331           by (cases "i \<le> j") (auto simp: incseq_def) }

  2332       then have eq: "(\<Union>i. F i) = F i"

  2333         by auto

  2334       with i show ?thesis

  2335         by (auto intro!: Lim_transform_eventually[OF _ tendsto_const] eventually_sequentiallyI[where c=i])

  2336     next

  2337       assume "\<not> (\<exists>i. \<forall>j\<ge>i. F i = F j)"

  2338       then obtain f where f: "\<And>i. i \<le> f i" "\<And>i. F i \<noteq> F (f i)" by metis

  2339       then have "\<And>i. F i \<subseteq> F (f i)" using \<open>incseq F\<close> by (auto simp: incseq_def)

  2340       with f have *: "\<And>i. F i \<subset> F (f i)" by auto

  2341

  2342       have "incseq (\<lambda>i. ?M (F i))"

  2343         using \<open>incseq F\<close> unfolding incseq_def by (auto simp: card_mono dest: finite_subset)

  2344       then have "(\<lambda>i. ?M (F i)) \<longlonglongrightarrow> (SUP n. ?M (F n))"

  2345         by (rule LIMSEQ_SUP)

  2346

  2347       moreover have "(SUP n. ?M (F n)) = top"

  2348       proof (rule ennreal_SUP_eq_top)

  2349         fix n :: nat show "\<exists>k::nat\<in>UNIV. of_nat n \<le> ?M (F k)"

  2350         proof (induct n)

  2351           case (Suc n)

  2352           then guess k .. note k = this

  2353           moreover have "finite (F k) \<Longrightarrow> finite (F (f k)) \<Longrightarrow> card (F k) < card (F (f k))"

  2354             using \<open>F k \<subset> F (f k)\<close> by (simp add: psubset_card_mono)

  2355           moreover have "finite (F (f k)) \<Longrightarrow> finite (F k)"

  2356             using \<open>k \<le> f k\<close> \<open>incseq F\<close> by (auto simp: incseq_def dest: finite_subset)

  2357           ultimately show ?case

  2358             by (auto intro!: exI[of _ "f k"] simp del: of_nat_Suc)

  2359         qed auto

  2360       qed

  2361

  2362       moreover

  2363       have "inj (\<lambda>n. F ((f ^^ n) 0))"

  2364         by (intro strict_mono_imp_inj_on strict_monoI_Suc) (simp add: *)

  2365       then have 1: "infinite (range (\<lambda>i. F ((f ^^ i) 0)))"

  2366         by (rule range_inj_infinite)

  2367       have "infinite (Pow (\<Union>i. F i))"

  2368         by (rule infinite_super[OF _ 1]) auto

  2369       then have "infinite (\<Union>i. F i)"

  2370         by auto

  2371

  2372       ultimately show ?thesis by auto

  2373     qed

  2374   qed

  2375 qed

  2376

  2377 lemma distr_bij_count_space:

  2378   assumes f: "bij_betw f A B"

  2379   shows "distr (count_space A) (count_space B) f = count_space B"

  2380 proof (rule measure_eqI)

  2381   have f': "f \<in> measurable (count_space A) (count_space B)"

  2382     using f unfolding Pi_def bij_betw_def by auto

  2383   fix X assume "X \<in> sets (distr (count_space A) (count_space B) f)"

  2384   then have X: "X \<in> sets (count_space B)" by auto

  2385   moreover from X have "f - X \<inter> A = the_inv_into A f  X"

  2386     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])

  2387   moreover have "inj_on (the_inv_into A f) B"

  2388     using X f by (auto simp: bij_betw_def inj_on_the_inv_into)

  2389   with X have "inj_on (the_inv_into A f) X"

  2390     by (auto intro: subset_inj_on)

  2391   ultimately show "emeasure (distr (count_space A) (count_space B) f) X = emeasure (count_space B) X"

  2392     using f unfolding emeasure_distr[OF f' X]

  2393     by (subst (1 2) emeasure_count_space) (auto simp: card_image dest: finite_imageD)

  2394 qed simp

  2395

  2396 lemma emeasure_count_space_finite[simp]:

  2397   "X \<subseteq> A \<Longrightarrow> finite X \<Longrightarrow> emeasure (count_space A) X = of_nat (card X)"

  2398   using emeasure_count_space[of X A] by simp

  2399

  2400 lemma emeasure_count_space_infinite[simp]:

  2401   "X \<subseteq> A \<Longrightarrow> infinite X \<Longrightarrow> emeasure (count_space A) X = \<infinity>"

  2402   using emeasure_count_space[of X A] by simp

  2403

  2404 lemma measure_count_space: "measure (count_space A) X = (if X \<subseteq> A then of_nat (card X) else 0)"

  2405   by (cases "finite X") (auto simp: measure_notin_sets ennreal_of_nat_eq_real_of_nat

  2406                                     measure_zero_top measure_eq_emeasure_eq_ennreal)

  2407

  2408 lemma emeasure_count_space_eq_0:

  2409   "emeasure (count_space A) X = 0 \<longleftrightarrow> (X \<subseteq> A \<longrightarrow> X = {})"

  2410 proof cases

  2411   assume X: "X \<subseteq> A"

  2412   then show ?thesis

  2413   proof (intro iffI impI)

  2414     assume "emeasure (count_space A) X = 0"

  2415     with X show "X = {}"

  2416       by (subst (asm) emeasure_count_space) (auto split: if_split_asm)

  2417   qed simp

  2418 qed (simp add: emeasure_notin_sets)

  2419

  2420 lemma null_sets_count_space: "null_sets (count_space A) = { {} }"

  2421   unfolding null_sets_def by (auto simp add: emeasure_count_space_eq_0)

  2422

  2423 lemma AE_count_space: "(AE x in count_space A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x)"

  2424   unfolding eventually_ae_filter by (auto simp add: null_sets_count_space)

  2425

  2426 lemma sigma_finite_measure_count_space_countable:

  2427   assumes A: "countable A"

  2428   shows "sigma_finite_measure (count_space A)"

  2429   proof qed (insert A, auto intro!: exI[of _ "(\<lambda>a. {a})  A"])

  2430

  2431 lemma sigma_finite_measure_count_space:

  2432   fixes A :: "'a::countable set" shows "sigma_finite_measure (count_space A)"

  2433   by (rule sigma_finite_measure_count_space_countable) auto

  2434

  2435 lemma finite_measure_count_space:

  2436   assumes [simp]: "finite A"

  2437   shows "finite_measure (count_space A)"

  2438   by rule simp

  2439

  2440 lemma sigma_finite_measure_count_space_finite:

  2441   assumes A: "finite A" shows "sigma_finite_measure (count_space A)"

  2442 proof -

  2443   interpret finite_measure "count_space A" using A by (rule finite_measure_count_space)

  2444   show "sigma_finite_measure (count_space A)" ..

  2445 qed

  2446

  2447 subsection%unimportant \<open>Measure restricted to space\<close>

  2448

  2449 lemma emeasure_restrict_space:

  2450   assumes "\<Omega> \<inter> space M \<in> sets M" "A \<subseteq> \<Omega>"

  2451   shows "emeasure (restrict_space M \<Omega>) A = emeasure M A"

  2452 proof (cases "A \<in> sets M")

  2453   case True

  2454   show ?thesis

  2455   proof (rule emeasure_measure_of[OF restrict_space_def])

  2456     show "(\<inter>) \<Omega>  sets M \<subseteq> Pow (\<Omega> \<inter> space M)" "A \<in> sets (restrict_space M \<Omega>)"

  2457       using \<open>A \<subseteq> \<Omega>\<close> \<open>A \<in> sets M\<close> sets.space_closed by (auto simp: sets_restrict_space)

  2458     show "positive (sets (restrict_space M \<Omega>)) (emeasure M)"

  2459       by (auto simp: positive_def)

  2460     show "countably_additive (sets (restrict_space M \<Omega>)) (emeasure M)"

  2461     proof (rule countably_additiveI)

  2462       fix A :: "nat \<Rightarrow> _" assume "range A \<subseteq> sets (restrict_space M \<Omega>)" "disjoint_family A"

  2463       with assms have "\<And>i. A i \<in> sets M" "\<And>i. A i \<subseteq> space M" "disjoint_family A"

  2464         by (fastforce simp: sets_restrict_space_iff[OF assms(1)] image_subset_iff

  2465                       dest: sets.sets_into_space)+

  2466       then show "(\<Sum>i. emeasure M (A i)) = emeasure M (\<Union>i. A i)"

  2467         by (subst suminf_emeasure) (auto simp: disjoint_family_subset)

  2468     qed

  2469   qed

  2470 next

  2471   case False

  2472   with assms have "A \<notin> sets (restrict_space M \<Omega>)"

  2473     by (simp add: sets_restrict_space_iff)

  2474   with False show ?thesis

  2475     by (simp add: emeasure_notin_sets)

  2476 qed

  2477

  2478 lemma measure_restrict_space:

  2479   assumes "\<Omega> \<inter> space M \<in> sets M" "A \<subseteq> \<Omega>"

  2480   shows "measure (restrict_space M \<Omega>) A = measure M A"

  2481   using emeasure_restrict_space[OF assms] by (simp add: measure_def)

  2482

  2483 lemma AE_restrict_space_iff:

  2484   assumes "\<Omega> \<inter> space M \<in> sets M"

  2485   shows "(AE x in restrict_space M \<Omega>. P x) \<longleftrightarrow> (AE x in M. x \<in> \<Omega> \<longrightarrow> P x)"

  2486 proof -

  2487   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)"

  2488     by auto

  2489   { fix X assume X: "X \<in> sets M" "emeasure M X = 0"

  2490     then have "emeasure M (\<Omega> \<inter> space M \<inter> X) \<le> emeasure M X"

  2491       by (intro emeasure_mono) auto

  2492     then have "emeasure M (\<Omega> \<inter> space M \<inter> X) = 0"

  2493       using X by (auto intro!: antisym) }

  2494   with assms show ?thesis

  2495     unfolding eventually_ae_filter

  2496     by (auto simp add: space_restrict_space null_sets_def sets_restrict_space_iff

  2497                        emeasure_restrict_space cong: conj_cong

  2498              intro!: ex_cong[where f="\<lambda>X. (\<Omega> \<inter> space M) \<inter> X"])

  2499 qed

  2500

  2501 lemma restrict_restrict_space:

  2502   assumes "A \<inter> space M \<in> sets M" "B \<inter> space M \<in> sets M"

  2503   shows "restrict_space (restrict_space M A) B = restrict_space M (A \<inter> B)" (is "?l = ?r")

  2504 proof (rule measure_eqI[symmetric])

  2505   show "sets ?r = sets ?l"

  2506     unfolding sets_restrict_space image_comp by (intro image_cong) auto

  2507 next

  2508   fix X assume "X \<in> sets (restrict_space M (A \<inter> B))"

  2509   then obtain Y where "Y \<in> sets M" "X = Y \<inter> A \<inter> B"

  2510     by (auto simp: sets_restrict_space)

  2511   with assms sets.Int[OF assms] show "emeasure ?r X = emeasure ?l X"

  2512     by (subst (1 2) emeasure_restrict_space)

  2513        (auto simp: space_restrict_space sets_restrict_space_iff emeasure_restrict_space ac_simps)

  2514 qed

  2515

  2516 lemma restrict_count_space: "restrict_space (count_space B) A = count_space (A \<inter> B)"

  2517 proof (rule measure_eqI)

  2518   show "sets (restrict_space (count_space B) A) = sets (count_space (A \<inter> B))"

  2519     by (subst sets_restrict_space) auto

  2520   moreover fix X assume "X \<in> sets (restrict_space (count_space B) A)"

  2521   ultimately have "X \<subseteq> A \<inter> B" by auto

  2522   then show "emeasure (restrict_space (count_space B) A) X = emeasure (count_space (A \<inter> B)) X"

  2523     by (cases "finite X") (auto simp add: emeasure_restrict_space)

  2524 qed

  2525

  2526 lemma sigma_finite_measure_restrict_space:

  2527   assumes "sigma_finite_measure M"

  2528   and A: "A \<in> sets M"

  2529   shows "sigma_finite_measure (restrict_space M A)"

  2530 proof -

  2531   interpret sigma_finite_measure M by fact

  2532   from sigma_finite_countable obtain C

  2533     where C: "countable C" "C \<subseteq> sets M" "(\<Union>C) = space M" "\<forall>a\<in>C. emeasure M a \<noteq> \<infinity>"

  2534     by blast

  2535   let ?C = "(\<inter>) A  C"

  2536   from C have "countable ?C" "?C \<subseteq> sets (restrict_space M A)" "(\<Union>?C) = space (restrict_space M A)"

  2537     by(auto simp add: sets_restrict_space space_restrict_space)

  2538   moreover {

  2539     fix a

  2540     assume "a \<in> ?C"

  2541     then obtain a' where "a = A \<inter> a'" "a' \<in> C" ..

  2542     then have "emeasure (restrict_space M A) a \<le> emeasure M a'"

  2543       using A C by(auto simp add: emeasure_restrict_space intro: emeasure_mono)

  2544     also have "\<dots> < \<infinity>" using C(4)[rule_format, of a'] \<open>a' \<in> C\<close> by (simp add: less_top)

  2545     finally have "emeasure (restrict_space M A) a \<noteq> \<infinity>" by simp }

  2546   ultimately show ?thesis

  2547     by unfold_locales (rule exI conjI|assumption|blast)+

  2548 qed

  2549

  2550 lemma finite_measure_restrict_space:

  2551   assumes "finite_measure M"

  2552   and A: "A \<in> sets M"

  2553   shows "finite_measure (restrict_space M A)"

  2554 proof -

  2555   interpret finite_measure M by fact

  2556   show ?thesis

  2557     by(rule finite_measureI)(simp add: emeasure_restrict_space A space_restrict_space)

  2558 qed

  2559

  2560 lemma restrict_distr:

  2561   assumes [measurable]: "f \<in> measurable M N"

  2562   assumes [simp]: "\<Omega> \<inter> space N \<in> sets N" and restrict: "f \<in> space M \<rightarrow> \<Omega>"

  2563   shows "restrict_space (distr M N f) \<Omega> = distr M (restrict_space N \<Omega>) f"

  2564   (is "?l = ?r")

  2565 proof (rule measure_eqI)

  2566   fix A assume "A \<in> sets (restrict_space (distr M N f) \<Omega>)"

  2567   with restrict show "emeasure ?l A = emeasure ?r A"

  2568     by (subst emeasure_distr)

  2569        (auto simp: sets_restrict_space_iff emeasure_restrict_space emeasure_distr

  2570              intro!: measurable_restrict_space2)

  2571 qed (simp add: sets_restrict_space)

  2572

  2573 lemma measure_eqI_restrict_generator:

  2574   assumes E: "Int_stable E" "E \<subseteq> Pow \<Omega>" "\<And>X. X \<in> E \<Longrightarrow> emeasure M X = emeasure N X"

  2575   assumes sets_eq: "sets M = sets N" and \<Omega>: "\<Omega> \<in> sets M"

  2576   assumes "sets (restrict_space M \<Omega>) = sigma_sets \<Omega> E"

  2577   assumes "sets (restrict_space N \<Omega>) = sigma_sets \<Omega> E"

  2578   assumes ae: "AE x in M. x \<in> \<Omega>" "AE x in N. x \<in> \<Omega>"

  2579   assumes A: "countable A" "A \<noteq> {}" "A \<subseteq> E" "\<Union>A = \<Omega>" "\<And>a. a \<in> A \<Longrightarrow> emeasure M a \<noteq> \<infinity>"

  2580   shows "M = N"

  2581 proof (rule measure_eqI)

  2582   fix X assume X: "X \<in> sets M"

  2583   then have "emeasure M X = emeasure (restrict_space M \<Omega>) (X \<inter> \<Omega>)"

  2584     using ae \<Omega> by (auto simp add: emeasure_restrict_space intro!: emeasure_eq_AE)

  2585   also have "restrict_space M \<Omega> = restrict_space N \<Omega>"

  2586   proof (rule measure_eqI_generator_eq)

  2587     fix X assume "X \<in> E"

  2588     then show "emeasure (restrict_space M \<Omega>) X = emeasure (restrict_space N \<Omega>) X"

  2589       using E \<Omega> by (subst (1 2) emeasure_restrict_space) (auto simp: sets_eq sets_eq[THEN sets_eq_imp_space_eq])

  2590   next

  2591     show "range (from_nat_into A) \<subseteq> E" "(\<Union>i. from_nat_into A i) = \<Omega>"

  2592       using A by (auto cong del: SUP_cong_strong)

  2593   next

  2594     fix i

  2595     have "emeasure (restrict_space M \<Omega>) (from_nat_into A i) = emeasure M (from_nat_into A i)"

  2596       using A \<Omega> by (subst emeasure_restrict_space)

  2597                    (auto simp: sets_eq sets_eq[THEN sets_eq_imp_space_eq] intro: from_nat_into)

  2598     with A show "emeasure (restrict_space M \<Omega>) (from_nat_into A i) \<noteq> \<infinity>"

  2599       by (auto intro: from_nat_into)

  2600   qed fact+

  2601   also have "emeasure (restrict_space N \<Omega>) (X \<inter> \<Omega>) = emeasure N X"

  2602     using X ae \<Omega> by (auto simp add: emeasure_restrict_space sets_eq intro!: emeasure_eq_AE)

  2603   finally show "emeasure M X = emeasure N X" .

  2604 qed fact

  2605

  2606 subsection%unimportant \<open>Null measure\<close>

  2607

  2608 definition "null_measure M = sigma (space M) (sets M)"

  2609

  2610 lemma space_null_measure[simp]: "space (null_measure M) = space M"

  2611   by (simp add: null_measure_def)

  2612

  2613 lemma sets_null_measure[simp, measurable_cong]: "sets (null_measure M) = sets M"

  2614   by (simp add: null_measure_def)

  2615

  2616 lemma emeasure_null_measure[simp]: "emeasure (null_measure M) X = 0"

  2617   by (cases "X \<in> sets M", rule emeasure_measure_of)

  2618      (auto simp: positive_def countably_additive_def emeasure_notin_sets null_measure_def

  2619            dest: sets.sets_into_space)

  2620

  2621 lemma measure_null_measure[simp]: "measure (null_measure M) X = 0"

  2622   by (intro measure_eq_emeasure_eq_ennreal) auto

  2623

  2624 lemma null_measure_idem [simp]: "null_measure (null_measure M) = null_measure M"

  2625   by(rule measure_eqI) simp_all

  2626

  2627 subsection \<open>Scaling a measure\<close>

  2628

  2629 definition%important scale_measure :: "ennreal \<Rightarrow> 'a measure \<Rightarrow> 'a measure"

  2630 where

  2631   "scale_measure r M = measure_of (space M) (sets M) (\<lambda>A. r * emeasure M A)"

  2632

  2633 lemma space_scale_measure: "space (scale_measure r M) = space M"

  2634   by (simp add: scale_measure_def)

  2635

  2636 lemma sets_scale_measure [simp, measurable_cong]: "sets (scale_measure r M) = sets M"

  2637   by (simp add: scale_measure_def)

  2638

  2639 lemma emeasure_scale_measure [simp]:

  2640   "emeasure (scale_measure r M) A = r * emeasure M A"

  2641   (is "_ = ?\<mu> A")

  2642 proof(cases "A \<in> sets M")

  2643   case True

  2644   show ?thesis unfolding scale_measure_def

  2645   proof(rule emeasure_measure_of_sigma)

  2646     show "sigma_algebra (space M) (sets M)" ..

  2647     show "positive (sets M) ?\<mu>" by (simp add: positive_def)

  2648     show "countably_additive (sets M) ?\<mu>"

  2649     proof (rule countably_additiveI)

  2650       fix A :: "nat \<Rightarrow> _"  assume *: "range A \<subseteq> sets M" "disjoint_family A"

  2651       have "(\<Sum>i. ?\<mu> (A i)) = r * (\<Sum>i. emeasure M (A i))"

  2652         by simp

  2653       also have "\<dots> = ?\<mu> (\<Union>i. A i)" using * by(simp add: suminf_emeasure)

  2654       finally show "(\<Sum>i. ?\<mu> (A i)) = ?\<mu> (\<Union>i. A i)" .

  2655     qed

  2656   qed(fact True)

  2657 qed(simp add: emeasure_notin_sets)

  2658

  2659 lemma scale_measure_1 [simp]: "scale_measure 1 M = M"

  2660   by(rule measure_eqI) simp_all

  2661

  2662 lemma scale_measure_0[simp]: "scale_measure 0 M = null_measure M"

  2663   by(rule measure_eqI) simp_all

  2664

  2665 lemma measure_scale_measure [simp]: "0 \<le> r \<Longrightarrow> measure (scale_measure r M) A = r * measure M A"

  2666   using emeasure_scale_measure[of r M A]

  2667     emeasure_eq_ennreal_measure[of M A]

  2668     measure_eq_emeasure_eq_ennreal[of _ "scale_measure r M" A]

  2669   by (cases "emeasure (scale_measure r M) A = top")

  2670      (auto simp del: emeasure_scale_measure

  2671            simp: ennreal_top_eq_mult_iff ennreal_mult_eq_top_iff measure_zero_top ennreal_mult[symmetric])

  2672

  2673 lemma scale_scale_measure [simp]:

  2674   "scale_measure r (scale_measure r' M) = scale_measure (r * r') M"

  2675   by (rule measure_eqI) (simp_all add: max_def mult.assoc)

  2676

  2677 lemma scale_null_measure [simp]: "scale_measure r (null_measure M) = null_measure M"

  2678   by (rule measure_eqI) simp_all

  2679

  2680

  2681 subsection \<open>Complete lattice structure on measures\<close>

  2682

  2683 lemma (in finite_measure) finite_measure_Diff':

  2684   "A \<in> sets M \<Longrightarrow> B \<in> sets M \<Longrightarrow> measure M (A - B) = measure M A - measure M (A \<inter> B)"

  2685   using finite_measure_Diff[of A "A \<inter> B"] by (auto simp: Diff_Int)

  2686

  2687 lemma (in finite_measure) finite_measure_Union':

  2688   "A \<in> sets M \<Longrightarrow> B \<in> sets M \<Longrightarrow> measure M (A \<union> B) = measure M A + measure M (B - A)"

  2689   using finite_measure_Union[of A "B - A"] by auto

  2690

  2691 lemma finite_unsigned_Hahn_decomposition:

  2692   assumes "finite_measure M" "finite_measure N" and [simp]: "sets N = sets M"

  2693   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)"

  2694 proof -

  2695   interpret M: finite_measure M by fact

  2696   interpret N: finite_measure N by fact

  2697

  2698   define d where "d X = measure M X - measure N X" for X

  2699

  2700   have [intro]: "bdd_above (dsets M)"

  2701     using sets.sets_into_space[of _ M]

  2702     by (intro bdd_aboveI[where M="measure M (space M)"])

  2703        (auto simp: d_def field_simps subset_eq intro!: add_increasing M.finite_measure_mono)

  2704

  2705   define \<gamma> where "\<gamma> = (SUP X\<in>sets M. d X)"

  2706   have le_\<gamma>[intro]: "X \<in> sets M \<Longrightarrow> d X \<le> \<gamma>" for X

  2707     by (auto simp: \<gamma>_def intro!: cSUP_upper)

  2708

  2709   have "\<exists>f. \<forall>n. f n \<in> sets M \<and> d (f n) > \<gamma> - 1 / 2^n"

  2710   proof (intro choice_iff[THEN iffD1] allI)

  2711     fix n

  2712     have "\<exists>X\<in>sets M. \<gamma> - 1 / 2^n < d X"

  2713       unfolding \<gamma>_def by (intro less_cSUP_iff[THEN iffD1]) auto

  2714     then show "\<exists>y. y \<in> sets M \<and> \<gamma> - 1 / 2 ^ n < d y"

  2715       by auto

  2716   qed

  2717   then obtain E where [measurable]: "E n \<in> sets M" and E: "d (E n) > \<gamma> - 1 / 2^n" for n

  2718     by auto

  2719

  2720   define F where "F m n = (if m \<le> n then \<Inter>i\<in>{m..n}. E i else space M)" for m n

  2721

  2722   have [measurable]: "m \<le> n \<Longrightarrow> F m n \<in> sets M" for m n

  2723     by (auto simp: F_def)

  2724

  2725   have 1: "\<gamma> - 2 / 2 ^ m + 1 / 2 ^ n \<le> d (F m n)" if "m \<le> n" for m n

  2726     using that

  2727   proof (induct rule: dec_induct)

  2728     case base with E[of m] show ?case

  2729       by (simp add: F_def field_simps)

  2730   next

  2731     case (step i)

  2732     have F_Suc: "F m (Suc i) = F m i \<inter> E (Suc i)"

  2733       using \<open>m \<le> i\<close> by (auto simp: F_def le_Suc_eq)

  2734

  2735     have "\<gamma> + (\<gamma> - 2 / 2^m + 1 / 2 ^ Suc i) \<le> (\<gamma> - 1 / 2^Suc i) + (\<gamma> - 2 / 2^m + 1 / 2^i)"

  2736       by (simp add: field_simps)

  2737     also have "\<dots> \<le> d (E (Suc i)) + d (F m i)"

  2738       using E[of "Suc i"] by (intro add_mono step) auto

  2739     also have "\<dots> = d (E (Suc i)) + d (F m i - E (Suc i)) + d (F m (Suc i))"

  2740       using \<open>m \<le> i\<close> by (simp add: d_def field_simps F_Suc M.finite_measure_Diff' N.finite_measure_Diff')

  2741     also have "\<dots> = d (E (Suc i) \<union> F m i) + d (F m (Suc i))"

  2742       using \<open>m \<le> i\<close> by (simp add: d_def field_simps M.finite_measure_Union' N.finite_measure_Union')

  2743     also have "\<dots> \<le> \<gamma> + d (F m (Suc i))"

  2744       using \<open>m \<le> i\<close> by auto

  2745     finally show ?case

  2746       by auto

  2747   qed

  2748

  2749   define F' where "F' m = (\<Inter>i\<in>{m..}. E i)" for m

  2750   have F'_eq: "F' m = (\<Inter>i. F m (i + m))" for m

  2751     by (fastforce simp: le_iff_add[of m] F'_def F_def)

  2752

  2753   have [measurable]: "F' m \<in> sets M" for m

  2754     by (auto simp: F'_def)

  2755

  2756   have \<gamma>_le: "\<gamma> - 0 \<le> d (\<Union>m. F' m)"

  2757   proof (rule LIMSEQ_le)

  2758     show "(\<lambda>n. \<gamma> - 2 / 2 ^ n) \<longlonglongrightarrow> \<gamma> - 0"

  2759       by (intro tendsto_intros LIMSEQ_divide_realpow_zero) auto

  2760     have "incseq F'"

  2761       by (auto simp: incseq_def F'_def)

  2762     then show "(\<lambda>m. d (F' m)) \<longlonglongrightarrow> d (\<Union>m. F' m)"

  2763       unfolding d_def

  2764       by (intro tendsto_diff M.finite_Lim_measure_incseq N.finite_Lim_measure_incseq) auto

  2765

  2766     have "\<gamma> - 2 / 2 ^ m + 0 \<le> d (F' m)" for m

  2767     proof (rule LIMSEQ_le)

  2768       have *: "decseq (\<lambda>n. F m (n + m))"

  2769         by (auto simp: decseq_def F_def)

  2770       show "(\<lambda>n. d (F m n)) \<longlonglongrightarrow> d (F' m)"

  2771         unfolding d_def F'_eq

  2772         by (rule LIMSEQ_offset[where k=m])

  2773            (auto intro!: tendsto_diff M.finite_Lim_measure_decseq N.finite_Lim_measure_decseq *)

  2774       show "(\<lambda>n. \<gamma> - 2 / 2 ^ m + 1 / 2 ^ n) \<longlonglongrightarrow> \<gamma> - 2 / 2 ^ m + 0"

  2775         by (intro tendsto_add LIMSEQ_divide_realpow_zero tendsto_const) auto

  2776       show "\<exists>N. \<forall>n\<ge>N. \<gamma> - 2 / 2 ^ m + 1 / 2 ^ n \<le> d (F m n)"

  2777         using 1[of m] by (intro exI[of _ m]) auto

  2778     qed

  2779     then show "\<exists>N. \<forall>n\<ge>N. \<gamma> - 2 / 2 ^ n \<le> d (F' n)"

  2780       by auto

  2781   qed

  2782

  2783   show ?thesis

  2784   proof (safe intro!: bexI[of _ "\<Union>m. F' m"])

  2785     fix X assume [measurable]: "X \<in> sets M" and X: "X \<subseteq> (\<Union>m. F' m)"

  2786     have "d (\<Union>m. F' m) - d X = d ((\<Union>m. F' m) - X)"

  2787       using X by (auto simp: d_def M.finite_measure_Diff N.finite_measure_Diff)

  2788     also have "\<dots> \<le> \<gamma>"

  2789       by auto

  2790     finally have "0 \<le> d X"

  2791       using \<gamma>_le by auto

  2792     then show "emeasure N X \<le> emeasure M X"

  2793       by (auto simp: d_def M.emeasure_eq_measure N.emeasure_eq_measure)

  2794   next

  2795     fix X assume [measurable]: "X \<in> sets M" and X: "X \<inter> (\<Union>m. F' m) = {}"

  2796     then have "d (\<Union>m. F' m) + d X = d (X \<union> (\<Union>m. F' m))"

  2797       by (auto simp: d_def M.finite_measure_Union N.finite_measure_Union)

  2798     also have "\<dots> \<le> \<gamma>"

  2799       by auto

  2800     finally have "d X \<le> 0"

  2801       using \<gamma>_le by auto

  2802     then show "emeasure M X \<le> emeasure N X"

  2803       by (auto simp: d_def M.emeasure_eq_measure N.emeasure_eq_measure)

  2804   qed auto

  2805 qed

  2806

  2807 proposition unsigned_Hahn_decomposition:

  2808   assumes [simp]: "sets N = sets M" and [measurable]: "A \<in> sets M"

  2809     and [simp]: "emeasure M A \<noteq> top" "emeasure N A \<noteq> top"

  2810   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)"

  2811 proof -

  2812   have "\<exists>Y\<in>sets (restrict_space M A).

  2813     (\<forall>X\<in>sets (restrict_space M A). X \<subseteq> Y \<longrightarrow> (restrict_space N A) X \<le> (restrict_space M A) X) \<and>

  2814     (\<forall>X\<in>sets (restrict_space M A). X \<inter> Y = {} \<longrightarrow> (restrict_space M A) X \<le> (restrict_space N A) X)"

  2815   proof (rule finite_unsigned_Hahn_decomposition)

  2816     show "finite_measure (restrict_space M A)" "finite_measure (restrict_space N A)"

  2817       by (auto simp: space_restrict_space emeasure_restrict_space less_top intro!: finite_measureI)

  2818   qed (simp add: sets_restrict_space)

  2819   then guess Y ..

  2820   then show ?thesis

  2821     apply (intro bexI[of _ Y] conjI ballI conjI)

  2822     apply (simp_all add: sets_restrict_space emeasure_restrict_space)

  2823     apply safe

  2824     subgoal for X Z

  2825       by (erule ballE[of _ _ X]) (auto simp add: Int_absorb1)

  2826     subgoal for X Z

  2827       by (erule ballE[of _ _ X])  (auto simp add: Int_absorb1 ac_simps)

  2828     apply auto

  2829     done

  2830 qed

  2831

  2832 text%important \<open>

  2833   Define a lexicographical order on @{type measure}, in the order space, sets and measure. The parts

  2834   of the lexicographical order are point-wise ordered.

  2835 \<close>

  2836

  2837 instantiation measure :: (type) order_bot

  2838 begin

  2839

  2840 inductive less_eq_measure :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> bool" where

  2841   "space M \<subset> space N \<Longrightarrow> less_eq_measure M N"

  2842 | "space M = space N \<Longrightarrow> sets M \<subset> sets N \<Longrightarrow> less_eq_measure M N"

  2843 | "space M = space N \<Longrightarrow> sets M = sets N \<Longrightarrow> emeasure M \<le> emeasure N \<Longrightarrow> less_eq_measure M N"

  2844

  2845 lemma le_measure_iff:

  2846   "M \<le> N \<longleftrightarrow> (if space M = space N then

  2847     if sets M = sets N then emeasure M \<le> emeasure N else sets M \<subseteq> sets N else space M \<subseteq> space N)"

  2848   by (auto elim: less_eq_measure.cases intro: less_eq_measure.intros)

  2849

  2850 definition%important less_measure :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> bool" where

  2851   "less_measure M N \<longleftrightarrow> (M \<le> N \<and> \<not> N \<le> M)"

  2852

  2853 definition%important bot_measure :: "'a measure" where

  2854   "bot_measure = sigma {} {}"

  2855

  2856 lemma

  2857   shows space_bot[simp]: "space bot = {}"

  2858     and sets_bot[simp]: "sets bot = {{}}"

  2859     and emeasure_bot[simp]: "emeasure bot X = 0"

  2860   by (auto simp: bot_measure_def sigma_sets_empty_eq emeasure_sigma)

  2861

  2862 instance

  2863 proof standard

  2864   show "bot \<le> a" for a :: "'a measure"

  2865     by (simp add: le_measure_iff bot_measure_def sigma_sets_empty_eq emeasure_sigma le_fun_def)

  2866 qed (auto simp: le_measure_iff less_measure_def split: if_split_asm intro: measure_eqI)

  2867

  2868 end

  2869

  2870 proposition le_measure: "sets M = sets N \<Longrightarrow> M \<le> N \<longleftrightarrow> (\<forall>A\<in>sets M. emeasure M A \<le> emeasure N A)"

  2871   apply -

  2872   apply (auto simp: le_measure_iff le_fun_def dest: sets_eq_imp_space_eq)

  2873   subgoal for X

  2874     by (cases "X \<in> sets M") (auto simp: emeasure_notin_sets)

  2875   done

  2876

  2877 definition%important sup_measure' :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> 'a measure"

  2878 where

  2879   "sup_measure' A B = measure_of (space A) (sets A) (\<lambda>X. SUP Y\<in>sets A. emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y))"

  2880

  2881 lemma assumes [simp]: "sets B = sets A"

  2882   shows space_sup_measure'[simp]: "space (sup_measure' A B) = space A"

  2883     and sets_sup_measure'[simp]: "sets (sup_measure' A B) = sets A"

  2884   using sets_eq_imp_space_eq[OF assms] by (simp_all add: sup_measure'_def)

  2885

  2886 lemma emeasure_sup_measure':

  2887   assumes sets_eq[simp]: "sets B = sets A" and [simp, intro]: "X \<in> sets A"

  2888   shows "emeasure (sup_measure' A B) X = (SUP Y\<in>sets A. emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y))"

  2889     (is "_ = ?S X")

  2890 proof -

  2891   note sets_eq_imp_space_eq[OF sets_eq, simp]

  2892   show ?thesis

  2893     using sup_measure'_def

  2894   proof (rule emeasure_measure_of)

  2895     let ?d = "\<lambda>X Y. emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y)"

  2896     show "countably_additive (sets (sup_measure' A B)) (\<lambda>X. SUP Y \<in> sets A. emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y))"

  2897     proof (rule countably_additiveI, goal_cases)

  2898       case (1 X)

  2899       then have [measurable]: "\<And>i. X i \<in> sets A" and "disjoint_family X"

  2900         by auto

  2901       have "(\<Sum>i. ?S (X i)) = (SUP Y\<in>sets A. \<Sum>i. ?d (X i) Y)"

  2902       proof (rule ennreal_suminf_SUP_eq_directed)

  2903         fix J :: "nat set" and a b assume "finite J" and [measurable]: "a \<in> sets A" "b \<in> sets A"

  2904         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

  2905         proof cases

  2906           assume "emeasure A (X i) = top \<or> emeasure B (X i) = top"

  2907           then show ?thesis

  2908           proof

  2909             assume "emeasure A (X i) = top" then show ?thesis

  2910               by (intro bexI[of _ "X i"]) auto

  2911           next

  2912             assume "emeasure B (X i) = top" then show ?thesis

  2913               by (intro bexI[of _ "{}"]) auto

  2914           qed

  2915         next

  2916           assume finite: "\<not> (emeasure A (X i) = top \<or> emeasure B (X i) = top)"

  2917           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)"

  2918             using unsigned_Hahn_decomposition[of B A "X i"] by simp

  2919           then obtain Y where [measurable]: "Y \<in> sets A" and [simp]: "Y \<subseteq> X i"

  2920             and B_le_A: "\<And>C. C \<in> sets A \<Longrightarrow> C \<subseteq> Y \<Longrightarrow> B C \<le> A C"

  2921             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"

  2922             by auto

  2923

  2924           show ?thesis

  2925           proof (intro bexI[of _ Y] ballI conjI)

  2926             fix a assume [measurable]: "a \<in> sets A"

  2927             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"

  2928               for a Y by auto

  2929             then have "?d (X i) a =

  2930               (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))"

  2931               by (subst (1 2) plus_emeasure) (auto simp: Diff_eq[symmetric])

  2932             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))"

  2933               by (intro add_mono order_refl B_le_A A_le_B) (auto simp: Diff_eq[symmetric])

  2934             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))"

  2935               by (simp add: ac_simps)

  2936             also have "\<dots> \<le> A (X i \<inter> Y) + B (X i \<inter> - Y)"

  2937               by (subst (1 2) plus_emeasure) (auto simp: Diff_eq[symmetric] *)

  2938             finally show "?d (X i) a \<le> ?d (X i) Y" .

  2939           qed auto

  2940         qed

  2941         then obtain C where [measurable]: "C i \<in> sets A" and "C i \<subseteq> X i"

  2942           and C: "\<And>a. a \<in> sets A \<Longrightarrow> ?d (X i) a \<le> ?d (X i) (C i)" for i

  2943           by metis

  2944         have *: "X i \<inter> (\<Union>i. C i) = X i \<inter> C i" for i

  2945         proof safe

  2946           fix x j assume "x \<in> X i" "x \<in> C j"

  2947           moreover have "i = j \<or> X i \<inter> X j = {}"

  2948             using \<open>disjoint_family X\<close> by (auto simp: disjoint_family_on_def)

  2949           ultimately show "x \<in> C i"

  2950             using \<open>C i \<subseteq> X i\<close> \<open>C j \<subseteq> X j\<close> by auto

  2951         qed auto

  2952         have **: "X i \<inter> - (\<Union>i. C i) = X i \<inter> - C i" for i

  2953         proof safe

  2954           fix x j assume "x \<in> X i" "x \<notin> C i" "x \<in> C j"

  2955           moreover have "i = j \<or> X i \<inter> X j = {}"

  2956             using \<open>disjoint_family X\<close> by (auto simp: disjoint_family_on_def)

  2957           ultimately show False

  2958             using \<open>C i \<subseteq> X i\<close> \<open>C j \<subseteq> X j\<close> by auto

  2959         qed auto

  2960         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"

  2961           apply (intro bexI[of _ "\<Union>i. C i"])

  2962           unfolding * **

  2963           apply (intro C ballI conjI)

  2964           apply auto

  2965           done

  2966       qed

  2967       also have "\<dots> = ?S (\<Union>i. X i)"

  2968         unfolding UN_extend_simps(4)

  2969         by (auto simp add: suminf_add[symmetric] Diff_eq[symmetric] simp del: UN_simps

  2970                  intro!: SUP_cong arg_cong2[where f="(+)"] suminf_emeasure

  2971                          disjoint_family_on_bisimulation[OF \<open>disjoint_family X\<close>])

  2972       finally show "(\<Sum>i. ?S (X i)) = ?S (\<Union>i. X i)" .

  2973     qed

  2974   qed (auto dest: sets.sets_into_space simp: positive_def intro!: SUP_const)

  2975 qed

  2976

  2977 lemma le_emeasure_sup_measure'1:

  2978   assumes "sets B = sets A" "X \<in> sets A" shows "emeasure A X \<le> emeasure (sup_measure' A B) X"

  2979   by (subst emeasure_sup_measure'[OF assms]) (auto intro!: SUP_upper2[of "X"] assms)

  2980

  2981 lemma le_emeasure_sup_measure'2:

  2982   assumes "sets B = sets A" "X \<in> sets A" shows "emeasure B X \<le> emeasure (sup_measure' A B) X"

  2983   by (subst emeasure_sup_measure'[OF assms]) (auto intro!: SUP_upper2[of "{}"] assms)

  2984

  2985 lemma emeasure_sup_measure'_le2:

  2986   assumes [simp]: "sets B = sets C" "sets A = sets C" and [measurable]: "X \<in> sets C"

  2987   assumes A: "\<And>Y. Y \<subseteq> X \<Longrightarrow> Y \<in> sets A \<Longrightarrow> emeasure A Y \<le> emeasure C Y"

  2988   assumes B: "\<And>Y. Y \<subseteq> X \<Longrightarrow> Y \<in> sets A \<Longrightarrow> emeasure B Y \<le> emeasure C Y"

  2989   shows "emeasure (sup_measure' A B) X \<le> emeasure C X"

  2990 proof (subst emeasure_sup_measure')

  2991   show "(SUP Y\<in>sets A. emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y)) \<le> emeasure C X"

  2992     unfolding \<open>sets A = sets C\<close>

  2993   proof (intro SUP_least)

  2994     fix Y assume [measurable]: "Y \<in> sets C"

  2995     have [simp]: "X \<inter> Y \<union> (X - Y) = X"

  2996       by auto

  2997     have "emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y) \<le> emeasure C (X \<inter> Y) + emeasure C (X \<inter> - Y)"

  2998       by (intro add_mono A B) (auto simp: Diff_eq[symmetric])

  2999     also have "\<dots> = emeasure C X"

  3000       by (subst plus_emeasure) (auto simp: Diff_eq[symmetric])

  3001     finally show "emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y) \<le> emeasure C X" .

  3002   qed

  3003 qed simp_all

  3004

  3005 definition%important sup_lexord :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b::order) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a"

  3006 where

  3007   "sup_lexord A B k s c =

  3008     (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)"

  3009

  3010 lemma sup_lexord:

  3011   "(k A < k B \<Longrightarrow> P B) \<Longrightarrow> (k B < k A \<Longrightarrow> P A) \<Longrightarrow> (k A = k B \<Longrightarrow> P c) \<Longrightarrow>

  3012     (\<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)"

  3013   by (auto simp: sup_lexord_def)

  3014

  3015 lemmas le_sup_lexord = sup_lexord[where P="\<lambda>a. c \<le> a" for c]

  3016

  3017 lemma sup_lexord1: "k A = k B \<Longrightarrow> sup_lexord A B k s c = c"

  3018   by (simp add: sup_lexord_def)

  3019

  3020 lemma sup_lexord_commute: "sup_lexord A B k s c = sup_lexord B A k s c"

  3021   by (auto simp: sup_lexord_def)

  3022

  3023 lemma sigma_sets_le_sets_iff: "(sigma_sets (space x) \<A> \<subseteq> sets x) = (\<A> \<subseteq> sets x)"

  3024   using sets.sigma_sets_subset[of \<A> x] by auto

  3025

  3026 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))"

  3027   by (cases "\<Omega> = space x")

  3028      (simp_all add: eq_commute[of _ "sets x"] le_measure_iff emeasure_sigma le_fun_def

  3029                     sigma_sets_superset_generator sigma_sets_le_sets_iff)

  3030

  3031 instantiation measure :: (type) semilattice_sup

  3032 begin

  3033

  3034 definition%important sup_measure :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> 'a measure"

  3035 where

  3036   "sup_measure A B =

  3037     sup_lexord A B space (sigma (space A \<union> space B) {})

  3038       (sup_lexord A B sets (sigma (space A) (sets A \<union> sets B)) (sup_measure' A B))"

  3039

  3040 instance

  3041 proof

  3042   fix x y z :: "'a measure"

  3043   show "x \<le> sup x y"

  3044     unfolding sup_measure_def

  3045   proof (intro le_sup_lexord)

  3046     assume "space x = space y"

  3047     then have *: "sets x \<union> sets y \<subseteq> Pow (space x)"

  3048       using sets.space_closed by auto

  3049     assume "\<not> sets y \<subseteq> sets x" "\<not> sets x \<subseteq> sets y"

  3050     then have "sets x \<subset> sets x \<union> sets y"

  3051       by auto

  3052     also have "\<dots> \<le> sigma (space x) (sets x \<union> sets y)"

  3053       by (subst sets_measure_of[OF *]) (rule sigma_sets_superset_generator)

  3054     finally show "x \<le> sigma (space x) (sets x \<union> sets y)"

  3055       by (simp add: space_measure_of[OF *] less_eq_measure.intros(2))

  3056   next

  3057     assume "\<not> space y \<subseteq> space x" "\<not> space x \<subseteq> space y"

  3058     then show "x \<le> sigma (space x \<union> space y) {}"

  3059       by (intro less_eq_measure.intros) auto

  3060   next

  3061     assume "sets x = sets y" then show "x \<le> sup_measure' x y"

  3062       by (simp add: le_measure le_emeasure_sup_measure'1)

  3063   qed (auto intro: less_eq_measure.intros)

  3064   show "y \<le> sup x y"

  3065     unfolding sup_measure_def

  3066   proof (intro le_sup_lexord)

  3067     assume **: "space x = space y"

  3068     then have *: "sets x \<union> sets y \<subseteq> Pow (space y)"

  3069       using sets.space_closed by auto

  3070     assume "\<not> sets y \<subseteq> sets x" "\<not> sets x \<subseteq> sets y"

  3071     then have "sets y \<subset> sets x \<union> sets y"

  3072       by auto

  3073     also have "\<dots> \<le> sigma (space y) (sets x \<union> sets y)"

  3074       by (subst sets_measure_of[OF *]) (rule sigma_sets_superset_generator)

  3075     finally show "y \<le> sigma (space x) (sets x \<union> sets y)"

  3076       by (simp add: ** space_measure_of[OF *] less_eq_measure.intros(2))

  3077   next

  3078     assume "\<not> space y \<subseteq> space x" "\<not> space x \<subseteq> space y"

  3079     then show "y \<le> sigma (space x \<union> space y) {}"

  3080       by (intro less_eq_measure.intros) auto

  3081   next

  3082     assume "sets x = sets y" then show "y \<le> sup_measure' x y"

  3083       by (simp add: le_measure le_emeasure_sup_measure'2)

  3084   qed (auto intro: less_eq_measure.intros)

  3085   show "x \<le> y \<Longrightarrow> z \<le> y \<Longrightarrow> sup x z \<le> y"

  3086     unfolding sup_measure_def

  3087   proof (intro sup_lexord[where P="\<lambda>x. x \<le> y"])

  3088     assume "x \<le> y" "z \<le> y" and [simp]: "space x = space z" "sets x = sets z"

  3089     from \<open>x \<le> y\<close> show "sup_measure' x z \<le> y"

  3090     proof cases

  3091       case 1 then show ?thesis

  3092         by (intro less_eq_measure.intros(1)) simp

  3093     next

  3094       case 2 then show ?thesis

  3095         by (intro less_eq_measure.intros(2)) simp_all

  3096     next

  3097       case 3 with \<open>z \<le> y\<close> \<open>x \<le> y\<close> show ?thesis

  3098         by (auto simp add: le_measure intro!: emeasure_sup_measure'_le2)

  3099     qed

  3100   next

  3101     assume **: "x \<le> y" "z \<le> y" "space x = space z" "\<not> sets z \<subseteq> sets x" "\<not> sets x \<subseteq> sets z"

  3102     then have *: "sets x \<union> sets z \<subseteq> Pow (space x)"

  3103       using sets.space_closed by auto

  3104     show "sigma (space x) (sets x \<union> sets z) \<le> y"

  3105       unfolding sigma_le_iff[OF *] using ** by (auto simp: le_measure_iff split: if_split_asm)

  3106   next

  3107     assume "x \<le> y" "z \<le> y" "\<not> space z \<subseteq> space x" "\<not> space x \<subseteq> space z"

  3108     then have "space x \<subseteq> space y" "space z \<subseteq> space y"

  3109       by (auto simp: le_measure_iff split: if_split_asm)

  3110     then show "sigma (space x \<union> space z) {} \<le> y"

  3111       by (simp add: sigma_le_iff)

  3112   qed

  3113 qed

  3114

  3115 end

  3116

  3117 lemma space_empty_eq_bot: "space a = {} \<longleftrightarrow> a = bot"

  3118   using space_empty[of a] by (auto intro!: measure_eqI)

  3119

  3120 lemma sets_eq_iff_bounded: "A \<le> B \<Longrightarrow> B \<le> C \<Longrightarrow> sets A = sets C \<Longrightarrow> sets B = sets A"

  3121   by (auto dest: sets_eq_imp_space_eq simp add: le_measure_iff split: if_split_asm)

  3122

  3123 lemma sets_sup: "sets A = sets M \<Longrightarrow> sets B = sets M \<Longrightarrow> sets (sup A B) = sets M"

  3124   by (auto simp add: sup_measure_def sup_lexord_def dest: sets_eq_imp_space_eq)

  3125

  3126 lemma le_measureD1: "A \<le> B \<Longrightarrow> space A \<le> space B"

  3127   by (auto simp: le_measure_iff split: if_split_asm)

  3128

  3129 lemma le_measureD2: "A \<le> B \<Longrightarrow> space A = space B \<Longrightarrow> sets A \<le> sets B"

  3130   by (auto simp: le_measure_iff split: if_split_asm)

  3131

  3132 lemma le_measureD3: "A \<le> B \<Longrightarrow> sets A = sets B \<Longrightarrow> emeasure A X \<le> emeasure B X"

  3133   by (auto simp: le_measure_iff le_fun_def dest: sets_eq_imp_space_eq split: if_split_asm)

  3134

  3135 lemma UN_space_closed: "\<Union>(sets  S) \<subseteq> Pow (\<Union>(space  S))"

  3136   using sets.space_closed by auto

  3137

  3138 definition Sup_lexord :: "('a \<Rightarrow> 'b::complete_lattice) \<Rightarrow> ('a set \<Rightarrow> 'a) \<Rightarrow> ('a set \<Rightarrow> 'a) \<Rightarrow> 'a set \<Rightarrow> 'a"

  3139 where

  3140   "Sup_lexord k c s A = (let U = (SUP a\<in>A. k a) in if \<exists>a\<in>A. k a = U then c {a\<in>A. k a = U} else s A)"

  3141

  3142 lemma Sup_lexord:

  3143   "(\<And>a S. a \<in> A \<Longrightarrow> k a = (SUP a\<in>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\<in>A. k a)) \<Longrightarrow> P (s A)) \<Longrightarrow>

  3144     P (Sup_lexord k c s A)"

  3145   by (auto simp: Sup_lexord_def Let_def)

  3146

  3147 lemma Sup_lexord1:

  3148   assumes A: "A \<noteq> {}" "(\<And>a. a \<in> A \<Longrightarrow> k a = (\<Union>a\<in>A. k a))" "P (c A)"

  3149   shows "P (Sup_lexord k c s A)"

  3150   unfolding Sup_lexord_def Let_def

  3151 proof (clarsimp, safe)

  3152   show "\<forall>a\<in>A. k a \<noteq> (\<Union>x\<in>A. k x) \<Longrightarrow> P (s A)"

  3153     by (metis assms(1,2) ex_in_conv)

  3154 next

  3155   fix a assume "a \<in> A" "k a = (\<Union>x\<in>A. k x)"

  3156   then have "{a \<in> A. k a = (\<Union>x\<in>A. k x)} = {a \<in> A. k a = k a}"

  3157     by (metis A(2)[symmetric])

  3158   then show "P (c {a \<in> A. k a = (\<Union>x\<in>A. k x)})"

  3159     by (simp add: A(3))

  3160 qed

  3161

  3162 instantiation measure :: (type) complete_lattice

  3163 begin

  3164

  3165 interpretation sup_measure: comm_monoid_set sup "bot :: 'a measure"

  3166   by standard (auto intro!: antisym)

  3167

  3168 lemma sup_measure_F_mono':

  3169   "finite J \<Longrightarrow> finite I \<Longrightarrow> sup_measure.F id I \<le> sup_measure.F id (I \<union> J)"

  3170 proof (induction J rule: finite_induct)

  3171   case empty then show ?case

  3172     by simp

  3173 next

  3174   case (insert i J)

  3175   show ?case

  3176   proof cases

  3177     assume "i \<in> I" with insert show ?thesis

  3178       by (auto simp: insert_absorb)

  3179   next

  3180     assume "i \<notin> I"

  3181     have "sup_measure.F id I \<le> sup_measure.F id (I \<union> J)"

  3182       by (intro insert)

  3183     also have "\<dots> \<le> sup_measure.F id (insert i (I \<union> J))"

  3184       using insert \<open>i \<notin> I\<close> by (subst sup_measure.insert) auto

  3185     finally show ?thesis

  3186       by auto

  3187   qed

  3188 qed

  3189

  3190 lemma sup_measure_F_mono: "finite I \<Longrightarrow> J \<subseteq> I \<Longrightarrow> sup_measure.F id J \<le> sup_measure.F id I"

  3191   using sup_measure_F_mono'[of I J] by (auto simp: finite_subset Un_absorb1)

  3192

  3193 lemma sets_sup_measure_F:

  3194   "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"

  3195   by (induction I rule: finite_ne_induct) (simp_all add: sets_sup)

  3196

  3197 definition%important Sup_measure' :: "'a measure set \<Rightarrow> 'a measure"

  3198 where

  3199   "Sup_measure' M = measure_of (\<Union>a\<in>M. space a) (\<Union>a\<in>M. sets a)

  3200     (\<lambda>X. (SUP P\<in>{P. finite P \<and> P \<subseteq> M }. sup_measure.F id P X))"

  3201

  3202 lemma space_Sup_measure'2: "space (Sup_measure' M) = (\<Union>m\<in>M. space m)"

  3203   unfolding Sup_measure'_def by (intro space_measure_of[OF UN_space_closed])

  3204

  3205 lemma sets_Sup_measure'2: "sets (Sup_measure' M) = sigma_sets (\<Union>m\<in>M. space m) (\<Union>m\<in>M. sets m)"

  3206   unfolding Sup_measure'_def by (intro sets_measure_of[OF UN_space_closed])

  3207

  3208 lemma sets_Sup_measure':

  3209   assumes sets_eq[simp]: "\<And>m. m \<in> M \<Longrightarrow> sets m = sets A" and "M \<noteq> {}"

  3210   shows "sets (Sup_measure' M) = sets A"

  3211   using sets_eq[THEN sets_eq_imp_space_eq, simp] \<open>M \<noteq> {}\<close> by (simp add: Sup_measure'_def)

  3212

  3213 lemma space_Sup_measure':

  3214   assumes sets_eq[simp]: "\<And>m. m \<in> M \<Longrightarrow> sets m = sets A" and "M \<noteq> {}"

  3215   shows "space (Sup_measure' M) = space A"

  3216   using sets_eq[THEN sets_eq_imp_space_eq, simp] \<open>M \<noteq> {}\<close>

  3217   by (simp add: Sup_measure'_def )

  3218

  3219 lemma emeasure_Sup_measure':

  3220   assumes sets_eq[simp]: "\<And>m. m \<in> M \<Longrightarrow> sets m = sets A" and "X \<in> sets A" "M \<noteq> {}"

  3221   shows "emeasure (Sup_measure' M) X = (SUP P\<in>{P. finite P \<and> P \<subseteq> M}. sup_measure.F id P X)"

  3222     (is "_ = ?S X")

  3223   using Sup_measure'_def

  3224 proof (rule emeasure_measure_of)

  3225   note sets_eq[THEN sets_eq_imp_space_eq, simp]

  3226   have *: "sets (Sup_measure' M) = sets A" "space (Sup_measure' M) = space A"

  3227     using \<open>M \<noteq> {}\<close> by (simp_all add: Sup_measure'_def)

  3228   let ?\<mu> = "sup_measure.F id"

  3229   show "countably_additive (sets (Sup_measure' M)) ?S"

  3230   proof (rule countably_additiveI, goal_cases)

  3231     case (1 F)

  3232     then have **: "range F \<subseteq> sets A"

  3233       by (auto simp: *)

  3234     show "(\<Sum>i. ?S (F i)) = ?S (\<Union>i. F i)"

  3235     proof (subst ennreal_suminf_SUP_eq_directed)

  3236       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}"

  3237       have "(i \<noteq> {} \<longrightarrow> sets (?\<mu> i) = sets A) \<and> (j \<noteq> {} \<longrightarrow> sets (?\<mu> j) = sets A) \<and>

  3238         (i \<noteq> {} \<or> j \<noteq> {} \<longrightarrow> sets (?\<mu> (i \<union> j)) = sets A)"

  3239         using ij by (intro impI sets_sup_measure_F conjI) auto

  3240       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

  3241         using ij

  3242         by (cases "i = {}"; cases "j = {}")

  3243            (auto intro!: le_measureD3 sup_measure_F_mono simp: sets_sup_measure_F

  3244                  simp del: id_apply)

  3245       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)"

  3246         by (safe intro!: bexI[of _ "i \<union> j"]) auto

  3247     next

  3248       show "(SUP P \<in> {P. finite P \<and> P \<subseteq> M}. \<Sum>n. ?\<mu> P (F n)) = (SUP P \<in> {P. finite P \<and> P \<subseteq> M}. ?\<mu> P (\<Union>(F  UNIV)))"

  3249       proof (intro SUP_cong refl)

  3250         fix i assume i: "i \<in> {P. finite P \<and> P \<subseteq> M}"

  3251         show "(\<Sum>n. ?\<mu> i (F n)) = ?\<mu> i (\<Union>(F  UNIV))"

  3252         proof cases

  3253           assume "i \<noteq> {}" with i ** show ?thesis

  3254             apply (intro suminf_emeasure \<open>disjoint_family F\<close>)

  3255             apply (subst sets_sup_measure_F[OF _ _ sets_eq])

  3256             apply auto

  3257             done

  3258         qed simp

  3259       qed

  3260     qed

  3261   qed

  3262   show "positive (sets (Sup_measure' M)) ?S"

  3263     by (auto simp: positive_def bot_ennreal[symmetric])

  3264   show "X \<in> sets (Sup_measure' M)"

  3265     using assms * by auto

  3266 qed (rule UN_space_closed)

  3267

  3268 definition%important Sup_measure :: "'a measure set \<Rightarrow> 'a measure"

  3269 where

  3270   "Sup_measure = Sup_lexord space (Sup_lexord sets Sup_measure'

  3271     (\<lambda>U. sigma (\<Union>u\<in>U. space u) (\<Union>u\<in>U. sets u))) (\<lambda>U. sigma (\<Union>u\<in>U. space u) {})"

  3272

  3273 definition%important Inf_measure :: "'a measure set \<Rightarrow> 'a measure"

  3274 where

  3275   "Inf_measure A = Sup {x. \<forall>a\<in>A. x \<le> a}"

  3276

  3277 definition%important inf_measure :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> 'a measure"

  3278 where

  3279   "inf_measure a b = Inf {a, b}"

  3280

  3281 definition%important top_measure :: "'a measure"

  3282 where

  3283   "top_measure = Inf {}"

  3284

  3285 instance

  3286 proof

  3287   note UN_space_closed [simp]

  3288   show upper: "x \<le> Sup A" if x: "x \<in> A" for x :: "'a measure" and A

  3289     unfolding Sup_measure_def

  3290   proof (intro Sup_lexord[where P="\<lambda>y. x \<le> y"])

  3291     assume "\<And>a. a \<in> A \<Longrightarrow> space a \<noteq> (\<Union>a\<in>A. space a)"

  3292     from this[OF \<open>x \<in> A\<close>] \<open>x \<in> A\<close> show "x \<le> sigma (\<Union>a\<in>A. space a) {}"

  3293       by (intro less_eq_measure.intros) auto

  3294   next

  3295     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}"

  3296       and neq: "\<And>aa. aa \<in> S \<Longrightarrow> sets aa \<noteq> (\<Union>a\<in>S. sets a)"

  3297     have sp_a: "space a = (\<Union>(space  S))"

  3298       using \<open>a\<in>A\<close> by (auto simp: S)

  3299     show "x \<le> sigma (\<Union>(space  S)) (\<Union>(sets  S))"

  3300     proof cases

  3301       assume [simp]: "space x = space a"

  3302       have "sets x \<subset> (\<Union>a\<in>S. sets a)"

  3303         using \<open>x\<in>A\<close> neq[of x] by (auto simp: S)

  3304       also have "\<dots> \<subseteq> sigma_sets (\<Union>x\<in>S. space x) (\<Union>x\<in>S. sets x)"

  3305         by (rule sigma_sets_superset_generator)

  3306       finally show ?thesis

  3307         by (intro less_eq_measure.intros(2)) (simp_all add: sp_a)

  3308     next

  3309       assume "space x \<noteq> space a"

  3310       moreover have "space x \<le> space a"

  3311         unfolding a using \<open>x\<in>A\<close> by auto

  3312       ultimately show ?thesis

  3313         by (intro less_eq_measure.intros) (simp add: less_le sp_a)

  3314     qed

  3315   next

  3316     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}"

  3317       and "b \<in> S" and b: "sets b = (\<Union>a\<in>S. sets a)" and S': "S' = {a' \<in> S. sets a' = sets b}"

  3318     then have "S' \<noteq> {}" "space b = space a"

  3319       by auto

  3320     have sets_eq: "\<And>x. x \<in> S' \<Longrightarrow> sets x = sets b"

  3321       by (auto simp: S')

  3322     note sets_eq[THEN sets_eq_imp_space_eq, simp]

  3323     have *: "sets (Sup_measure' S') = sets b" "space (Sup_measure' S') = space b"

  3324       using \<open>S' \<noteq> {}\<close> by (simp_all add: Sup_measure'_def sets_eq)

  3325     show "x \<le> Sup_measure' S'"

  3326     proof cases

  3327       assume "x \<in> S"

  3328       with \<open>b \<in> S\<close> have "space x = space b"

  3329         by (simp add: S)

  3330       show ?thesis

  3331       proof cases

  3332         assume "x \<in> S'"

  3333         show "x \<le> Sup_measure' S'"

  3334         proof (intro le_measure[THEN iffD2] ballI)

  3335           show "sets x = sets (Sup_measure' S')"

  3336             using \<open>x\<in>S'\<close> * by (simp add: S')

  3337           fix X assume "X \<in> sets x"

  3338           show "emeasure x X \<le> emeasure (Sup_measure' S') X"

  3339           proof (subst emeasure_Sup_measure'[OF _ \<open>X \<in> sets x\<close>])

  3340             show "emeasure x X \<le> (SUP P \<in> {P. finite P \<and> P \<subseteq> S'}. emeasure (sup_measure.F id P) X)"

  3341               using \<open>x\<in>S'\<close> by (intro SUP_upper2[where i="{x}"]) auto

  3342           qed (insert \<open>x\<in>S'\<close> S', auto)

  3343         qed

  3344       next

  3345         assume "x \<notin> S'"

  3346         then have "sets x \<noteq> sets b"

  3347           using \<open>x\<in>S\<close> by (auto simp: S')

  3348         moreover have "sets x \<le> sets b"

  3349           using \<open>x\<in>S\<close> unfolding b by auto

  3350         ultimately show ?thesis

  3351           using * \<open>x \<in> S\<close>

  3352           by (intro less_eq_measure.intros(2))

  3353              (simp_all add: * \<open>space x = space b\<close> less_le)

  3354       qed

  3355     next

  3356       assume "x \<notin> S"

  3357       with \<open>x\<in>A\<close> \<open>x \<notin> S\<close> \<open>space b = space a\<close> show ?thesis

  3358         by (intro less_eq_measure.intros)

  3359            (simp_all add: * less_le a SUP_upper S)

  3360     qed

  3361   qed

  3362   show least: "Sup A \<le> x" if x: "\<And>z. z \<in> A \<Longrightarrow> z \<le> x" for x :: "'a measure" and A

  3363     unfolding Sup_measure_def

  3364   proof (intro Sup_lexord[where P="\<lambda>y. y \<le> x"])

  3365     assume "\<And>a. a \<in> A \<Longrightarrow> space a \<noteq> (\<Union>a\<in>A. space a)"

  3366     show "sigma (\<Union>(space  A)) {} \<le> x"

  3367       using x[THEN le_measureD1] by (subst sigma_le_iff) auto

  3368   next

  3369     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}"

  3370       "\<And>a. a \<in> S \<Longrightarrow> sets a \<noteq> (\<Union>a\<in>S. sets a)"

  3371     have "\<Union>(space  S) \<subseteq> space x"

  3372       using S le_measureD1[OF x] by auto

  3373     moreover

  3374     have "\<Union>(space  S) = space a"

  3375       using \<open>a\<in>A\<close> S by auto

  3376     then have "space x = \<Union>(space  S) \<Longrightarrow> \<Union>(sets  S) \<subseteq> sets x"

  3377       using \<open>a \<in> A\<close> le_measureD2[OF x] by (auto simp: S)

  3378     ultimately show "sigma (\<Union>(space  S)) (\<Union>(sets  S)) \<le> x"

  3379       by (subst sigma_le_iff) simp_all

  3380   next

  3381     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}"

  3382       and "b \<in> S" and b: "sets b = (\<Union>a\<in>S. sets a)" and S': "S' = {a' \<in> S. sets a' = sets b}"

  3383     then have "S' \<noteq> {}" "space b = space a"

  3384       by auto

  3385     have sets_eq: "\<And>x. x \<in> S' \<Longrightarrow> sets x = sets b"

  3386       by (auto simp: S')

  3387     note sets_eq[THEN sets_eq_imp_space_eq, simp]

  3388     have *: "sets (Sup_measure' S') = sets b" "space (Sup_measure' S') = space b"

  3389       using \<open>S' \<noteq> {}\<close> by (simp_all add: Sup_measure'_def sets_eq)

  3390     show "Sup_measure' S' \<le> x"

  3391     proof cases

  3392       assume "space x = space a"

  3393       show ?thesis

  3394       proof cases

  3395         assume **: "sets x = sets b"

  3396         show ?thesis

  3397         proof (intro le_measure[THEN iffD2] ballI)

  3398           show ***: "sets (Sup_measure' S') = sets x"

  3399             by (simp add: * **)

  3400           fix X assume "X \<in> sets (Sup_measure' S')"

  3401           show "emeasure (Sup_measure' S') X \<le> emeasure x X"

  3402             unfolding ***

  3403           proof (subst emeasure_Sup_measure'[OF _ \<open>X \<in> sets (Sup_measure' S')\<close>])

  3404             show "(SUP P \<in> {P. finite P \<and> P \<subseteq> S'}. emeasure (sup_measure.F id P) X) \<le> emeasure x X"

  3405             proof (safe intro!: SUP_least)

  3406               fix P assume P: "finite P" "P \<subseteq> S'"

  3407               show "emeasure (sup_measure.F id P) X \<le> emeasure x X"

  3408               proof cases

  3409                 assume "P = {}" then show ?thesis

  3410                   by auto

  3411               next

  3412                 assume "P \<noteq> {}"

  3413                 from P have "finite P" "P \<subseteq> A"

  3414                   unfolding S' S by (simp_all add: subset_eq)

  3415                 then have "sup_measure.F id P \<le> x"

  3416                   by (induction P) (auto simp: x)

  3417                 moreover have "sets (sup_measure.F id P) = sets x"

  3418                   using \<open>finite P\<close> \<open>P \<noteq> {}\<close> \<open>P \<subseteq> S'\<close> \<open>sets x = sets b\<close>

  3419                   by (intro sets_sup_measure_F) (auto simp: S')

  3420                 ultimately show "emeasure (sup_measure.F id P) X \<le> emeasure x X"

  3421                   by (rule le_measureD3)

  3422               qed

  3423             qed

  3424             show "m \<in> S' \<Longrightarrow> sets m = sets (Sup_measure' S')" for m

  3425               unfolding * by (simp add: S')

  3426           qed fact

  3427         qed

  3428       next

  3429         assume "sets x \<noteq> sets b"

  3430         moreover have "sets b \<le> sets x"

  3431           unfolding b S using x[THEN le_measureD2] \<open>space x = space a\<close> by auto

  3432         ultimately show "Sup_measure' S' \<le> x"

  3433           using \<open>space x = space a\<close> \<open>b \<in> S\<close>

  3434           by (intro less_eq_measure.intros(2)) (simp_all add: * S)

  3435       qed

  3436     next

  3437       assume "space x \<noteq> space a"

  3438       then have "space a < space x"

  3439         using le_measureD1[OF x[OF \<open>a\<in>A\<close>]] by auto

  3440       then show "Sup_measure' S' \<le> x"

  3441         by (intro less_eq_measure.intros) (simp add: * \<open>space b = space a\<close>)

  3442     qed

  3443   qed

  3444   show "Sup {} = (bot::'a measure)" "Inf {} = (top::'a measure)"

  3445     by (auto intro!: antisym least simp: top_measure_def)

  3446   show lower: "x \<in> A \<Longrightarrow> Inf A \<le> x" for x :: "'a measure" and A

  3447     unfolding Inf_measure_def by (intro least) auto

  3448   show greatest: "(\<And>z. z \<in> A \<Longrightarrow> x \<le> z) \<Longrightarrow> x \<le> Inf A" for x :: "'a measure" and A

  3449     unfolding Inf_measure_def by (intro upper) auto

  3450   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"

  3451     by (auto simp: inf_measure_def intro!: lower greatest)

  3452 qed

  3453

  3454 end

  3455

  3456 lemma sets_SUP:

  3457   assumes "\<And>x. x \<in> I \<Longrightarrow> sets (M x) = sets N"

  3458   shows "I \<noteq> {} \<Longrightarrow> sets (SUP i\<in>I. M i) = sets N"

  3459   unfolding Sup_measure_def

  3460   using assms assms[THEN sets_eq_imp_space_eq]

  3461     sets_Sup_measure'[where A=N and M="MI"]

  3462   by (intro Sup_lexord1[where P="\<lambda>x. sets x = sets N"]) auto

  3463

  3464 lemma emeasure_SUP:

  3465   assumes sets: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets N" "X \<in> sets N" "I \<noteq> {}"

  3466   shows "emeasure (SUP i\<in>I. M i) X = (SUP J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emeasure (SUP i\<in>J. M i) X)"

  3467 proof -

  3468   interpret sup_measure: comm_monoid_set sup "bot :: 'b measure"

  3469     by standard (auto intro!: antisym)

  3470   have eq: "finite J \<Longrightarrow> sup_measure.F id J = (SUP i\<in>J. i)" for J :: "'b measure set"

  3471     by (induction J rule: finite_induct) auto

  3472   have 1: "J \<noteq> {} \<Longrightarrow> J \<subseteq> I \<Longrightarrow> sets (SUP x\<in>J. M x) = sets N" for J

  3473     by (intro sets_SUP sets) (auto )

  3474   from \<open>I \<noteq> {}\<close> obtain i where "i\<in>I" by auto

  3475   have "Sup_measure' (MI) X = (SUP P\<in>{P. finite P \<and> P \<subseteq> MI}. sup_measure.F id P X)"

  3476     using sets by (intro emeasure_Sup_measure') auto

  3477   also have "Sup_measure' (MI) = (SUP i\<in>I. M i)"

  3478     unfolding Sup_measure_def using \<open>I \<noteq> {}\<close> sets sets(1)[THEN sets_eq_imp_space_eq]

  3479     by (intro Sup_lexord1[where P="\<lambda>x. _ = x"]) auto

  3480   also have "(SUP P\<in>{P. finite P \<and> P \<subseteq> MI}. sup_measure.F id P X) =

  3481     (SUP J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. (SUP i\<in>J. M i) X)"

  3482   proof (intro SUP_eq)

  3483     fix J assume "J \<in> {P. finite P \<and> P \<subseteq> MI}"

  3484     then obtain J' where J': "J' \<subseteq> I" "finite J'" and J: "J = MJ'" and "finite J"

  3485       using finite_subset_image[of J M I] by auto

  3486     show "\<exists>j\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. sup_measure.F id J X \<le> (SUP i\<in>j. M i) X"

  3487     proof cases

  3488       assume "J' = {}" with \<open>i \<in> I\<close> show ?thesis

  3489         by (auto simp add: J)

  3490     next

  3491       assume "J' \<noteq> {}" with J J' show ?thesis

  3492         by (intro bexI[of _ "J'"]) (auto simp add: eq simp del: id_apply)

  3493     qed

  3494   next

  3495     fix J assume J: "J \<in> {P. P \<noteq> {} \<and> finite P \<and> P \<subseteq> I}"

  3496     show "\<exists>J'\<in>{J. finite J \<and> J \<subseteq> MI}. (SUP i\<in>J. M i) X \<le> sup_measure.F id J' X"

  3497       using J by (intro bexI[of _ "MJ"]) (auto simp add: eq simp del: id_apply)

  3498   qed

  3499   finally show ?thesis .

  3500 qed

  3501

  3502 lemma emeasure_SUP_chain:

  3503   assumes sets: "\<And>i. i \<in> A \<Longrightarrow> sets (M i) = sets N" "X \<in> sets N"

  3504   assumes ch: "Complete_Partial_Order.chain (\<le>) (M  A)" and "A \<noteq> {}"

  3505   shows "emeasure (SUP i\<in>A. M i) X = (SUP i\<in>A. emeasure (M i) X)"

  3506 proof (subst emeasure_SUP[OF sets \<open>A \<noteq> {}\<close>])

  3507   show "(SUP J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> A}. emeasure (Sup (M  J)) X) = (SUP i\<in>A. emeasure (M i) X)"

  3508   proof (rule SUP_eq)

  3509     fix J assume "J \<in> {J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> A}"

  3510     then have J: "Complete_Partial_Order.chain (\<le>) (M  J)" "finite J" "J \<noteq> {}" and "J \<subseteq> A"

  3511       using ch[THEN chain_subset, of "MJ"] by auto

  3512     with in_chain_finite[OF J(1)] obtain j where "j \<in> J" "(SUP j\<in>J. M j) = M j"

  3513       by auto

  3514     with \<open>J \<subseteq> A\<close> show "\<exists>j\<in>A. emeasure (Sup (M  J)) X \<le> emeasure (M j) X"

  3515       by auto

  3516   next

  3517     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 (Sup (M  i)) X"

  3518       by (intro bexI[of _ "{j}"]) auto

  3519   qed

  3520 qed

  3521

  3522 subsubsection%unimportant \<open>Supremum of a set of $\sigma$-algebras\<close>

  3523

  3524 lemma space_Sup_eq_UN: "space (Sup M) = (\<Union>x\<in>M. space x)"

  3525   unfolding Sup_measure_def

  3526   apply (intro Sup_lexord[where P="\<lambda>x. space x = _"])

  3527   apply (subst space_Sup_measure'2)

  3528   apply auto []

  3529   apply (subst space_measure_of[OF UN_space_closed])

  3530   apply auto

  3531   done

  3532

  3533 lemma sets_Sup_eq:

  3534   assumes *: "\<And>m. m \<in> M \<Longrightarrow> space m = X" and "M \<noteq> {}"

  3535   shows "sets (Sup M) = sigma_sets X (\<Union>x\<in>M. sets x)"

  3536   unfolding Sup_measure_def

  3537   apply (rule Sup_lexord1)

  3538   apply fact

  3539   apply (simp add: assms)

  3540   apply (rule Sup_lexord)

  3541   subgoal premises that for a S

  3542     unfolding that(3) that(2)[symmetric]

  3543     using that(1)

  3544     apply (subst sets_Sup_measure'2)

  3545     apply (intro arg_cong2[where f=sigma_sets])

  3546     apply (auto simp: *)

  3547     done

  3548   apply (subst sets_measure_of[OF UN_space_closed])

  3549   apply (simp add:  assms)

  3550   done

  3551

  3552 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)"

  3553   by (subst sets_Sup_eq[where X=X]) auto

  3554

  3555 lemma Sup_lexord_rel:

  3556   assumes "\<And>i. i \<in> I \<Longrightarrow> k (A i) = k (B i)"

  3557     "R (c (A  {a \<in> I. k (B a) = (SUP x\<in>I. k (B x))})) (c (B  {a \<in> I. k (B a) = (SUP x\<in>I. k (B x))}))"

  3558     "R (s (AI)) (s (BI))"

  3559   shows "R (Sup_lexord k c s (AI)) (Sup_lexord k c s (BI))"

  3560 proof -

  3561   have "A  {a \<in> I. k (B a) = (SUP x\<in>I. k (B x))} =  {a \<in> A  I. k a = (SUP x\<in>I. k (B x))}"

  3562     using assms(1) by auto

  3563   moreover have "B  {a \<in> I. k (B a) = (SUP x\<in>I. k (B x))} =  {a \<in> B  I. k a = (SUP x\<in>I. k (B x))}"

  3564     by auto

  3565   ultimately show ?thesis

  3566     using assms by (auto simp: Sup_lexord_def Let_def)

  3567 qed

  3568

  3569 lemma sets_SUP_cong:

  3570   assumes eq: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets (N i)" shows "sets (SUP i\<in>I. M i) = sets (SUP i\<in>I. N i)"

  3571   unfolding Sup_measure_def

  3572   using eq eq[THEN sets_eq_imp_space_eq]

  3573   apply (intro Sup_lexord_rel[where R="\<lambda>x y. sets x = sets y"])

  3574   apply simp

  3575   apply simp

  3576   apply (simp add: sets_Sup_measure'2)

  3577   apply (intro arg_cong2[where f="\<lambda>x y. sets (sigma x y)"])

  3578   apply auto

  3579   done

  3580

  3581 lemma sets_Sup_in_sets:

  3582   assumes "M \<noteq> {}"

  3583   assumes "\<And>m. m \<in> M \<Longrightarrow> space m = space N"

  3584   assumes "\<And>m. m \<in> M \<Longrightarrow> sets m \<subseteq> sets N"

  3585   shows "sets (Sup M) \<subseteq> sets N"

  3586 proof -

  3587   have *: "\<Union>(space  M) = space N"

  3588     using assms by auto

  3589   show ?thesis

  3590     unfolding * using assms by (subst sets_Sup_eq[of M "space N"]) (auto intro!: sets.sigma_sets_subset)

  3591 qed

  3592

  3593 lemma measurable_Sup1:

  3594   assumes m: "m \<in> M" and f: "f \<in> measurable m N"

  3595     and const_space: "\<And>m n. m \<in> M \<Longrightarrow> n \<in> M \<Longrightarrow> space m = space n"

  3596   shows "f \<in> measurable (Sup M) N"

  3597 proof -

  3598   have "space (Sup M) = space m"

  3599     using m by (auto simp add: space_Sup_eq_UN dest: const_space)

  3600   then show ?thesis

  3601     using m f unfolding measurable_def by (auto intro: in_sets_Sup[OF const_space])

  3602 qed

  3603

  3604 lemma measurable_Sup2:

  3605   assumes M: "M \<noteq> {}"

  3606   assumes f: "\<And>m. m \<in> M \<Longrightarrow> f \<in> measurable N m"

  3607     and const_space: "\<And>m n. m \<in> M \<Longrightarrow> n \<in> M \<Longrightarrow> space m = space n"

  3608   shows "f \<in> measurable N (Sup M)"

  3609 proof -

  3610   from M obtain m where "m \<in> M" by auto

  3611   have space_eq: "\<And>n. n \<in> M \<Longrightarrow> space n = space m"

  3612     by (intro const_space \<open>m \<in> M\<close>)

  3613   have "f \<in> measurable N (sigma (\<Union>m\<in>M. space m) (\<Union>m\<in>M. sets m))"

  3614   proof (rule measurable_measure_of)

  3615     show "f \<in> space N \<rightarrow> \<Union>(space  M)"

  3616       using measurable_space[OF f] M by auto

  3617   qed (auto intro: measurable_sets f dest: sets.sets_into_space)

  3618   also have "measurable N (sigma (\<Union>m\<in>M. space m) (\<Union>m\<in>M. sets m)) = measurable N (Sup M)"

  3619     apply (intro measurable_cong_sets refl)

  3620     apply (subst sets_Sup_eq[OF space_eq M])

  3621     apply simp

  3622     apply (subst sets_measure_of[OF UN_space_closed])

  3623     apply (simp add: space_eq M)

  3624     done

  3625   finally show ?thesis .

  3626 qed

  3627

  3628 lemma measurable_SUP2:

  3629   "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f \<in> measurable N (M i)) \<Longrightarrow>

  3630     (\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> space (M i) = space (M j)) \<Longrightarrow> f \<in> measurable N (SUP i\<in>I. M i)"

  3631   by (auto intro!: measurable_Sup2)

  3632

  3633 lemma sets_Sup_sigma:

  3634   assumes [simp]: "M \<noteq> {}" and M: "\<And>m. m \<in> M \<Longrightarrow> m \<subseteq> Pow \<Omega>"

  3635   shows "sets (SUP m\<in>M. sigma \<Omega> m) = sets (sigma \<Omega> (\<Union>M))"

  3636 proof -

  3637   { fix a m assume "a \<in> sigma_sets \<Omega> m" "m \<in> M"

  3638     then have "a \<in> sigma_sets \<Omega> (\<Union>M)"

  3639      by induction (auto intro: sigma_sets.intros(2-)) }

  3640   then show "sets (SUP m\<in>M. sigma \<Omega> m) = sets (sigma \<Omega> (\<Union>M))"

  3641     apply (subst sets_Sup_eq[where X="\<Omega>"])

  3642     apply (auto simp add: M) []

  3643     apply auto []

  3644     apply (simp add: space_measure_of_conv M Union_least)

  3645     apply (rule sigma_sets_eqI)

  3646     apply auto

  3647     done

  3648 qed

  3649

  3650 lemma Sup_sigma:

  3651   assumes [simp]: "M \<noteq> {}" and M: "\<And>m. m \<in> M \<Longrightarrow> m \<subseteq> Pow \<Omega>"

  3652   shows "(SUP m\<in>M. sigma \<Omega> m) = (sigma \<Omega> (\<Union>M))"

  3653 proof (intro antisym SUP_least)

  3654   have *: "\<Union>M \<subseteq> Pow \<Omega>"

  3655     using M by auto

  3656   show "sigma \<Omega> (\<Union>M) \<le> (SUP m\<in>M. sigma \<Omega> m)"

  3657   proof (intro less_eq_measure.intros(3))

  3658     show "space (sigma \<Omega> (\<Union>M)) = space (SUP m\<in>M. sigma \<Omega> m)"

  3659       "sets (sigma \<Omega> (\<Union>M)) = sets (SUP m\<in>M. sigma \<Omega> m)"

  3660       using sets_Sup_sigma[OF assms] sets_Sup_sigma[OF assms, THEN sets_eq_imp_space_eq]

  3661       by auto

  3662   qed (simp add: emeasure_sigma le_fun_def)

  3663   fix m assume "m \<in> M" then show "sigma \<Omega> m \<le> sigma \<Omega> (\<Union>M)"

  3664     by (subst sigma_le_iff) (auto simp add: M *)

  3665 qed

  3666

  3667 lemma SUP_sigma_sigma:

  3668   "M \<noteq> {} \<Longrightarrow> (\<And>m. m \<in> M \<Longrightarrow> f m \<subseteq> Pow \<Omega>) \<Longrightarrow> (SUP m\<in>M. sigma \<Omega> (f m)) = sigma \<Omega> (\<Union>m\<in>M. f m)"

  3669   using Sup_sigma[of "fM" \<Omega>] by auto

  3670

  3671 lemma sets_vimage_Sup_eq:

  3672   assumes *: "M \<noteq> {}" "f \<in> X \<rightarrow> Y" "\<And>m. m \<in> M \<Longrightarrow> space m = Y"

  3673   shows "sets (vimage_algebra X f (Sup M)) = sets (SUP m \<in> M. vimage_algebra X f m)"

  3674   (is "?IS = ?SI")

  3675 proof

  3676   show "?IS \<subseteq> ?SI"

  3677     apply (intro sets_image_in_sets measurable_Sup2)

  3678     apply (simp add: space_Sup_eq_UN *)

  3679     apply (simp add: *)

  3680     apply (intro measurable_Sup1)

  3681     apply (rule imageI)

  3682     apply assumption

  3683     apply (rule measurable_vimage_algebra1)

  3684     apply (auto simp: *)

  3685     done

  3686   show "?SI \<subseteq> ?IS"

  3687     apply (intro sets_Sup_in_sets)

  3688     apply (auto simp: *) []

  3689     apply (auto simp: *) []

  3690     apply (elim imageE)

  3691     apply simp

  3692     apply (rule sets_image_in_sets)

  3693     apply simp

  3694     apply (simp add: measurable_def)

  3695     apply (simp add: * space_Sup_eq_UN sets_vimage_algebra2)

  3696     apply (auto intro: in_sets_Sup[OF *(3)])

  3697     done

  3698 qed

  3699

  3700 lemma restrict_space_eq_vimage_algebra':

  3701   "sets (restrict_space M \<Omega>) = sets (vimage_algebra (\<Omega> \<inter> space M) (\<lambda>x. x) M)"

  3702 proof -

  3703   have *: "{A \<inter> (\<Omega> \<inter> space M) |A. A \<in> sets M} = {A \<inter> \<Omega> |A. A \<in> sets M}"

  3704     using sets.sets_into_space[of _ M] by blast

  3705

  3706   show ?thesis

  3707     unfolding restrict_space_def

  3708     by (subst sets_measure_of)

  3709        (auto simp add: image_subset_iff sets_vimage_algebra * dest: sets.sets_into_space intro!: arg_cong2[where f=sigma_sets])

  3710 qed

  3711

  3712 lemma sigma_le_sets:

  3713   assumes [simp]: "A \<subseteq> Pow X" shows "sets (sigma X A) \<subseteq> sets N \<longleftrightarrow> X \<in> sets N \<and> A \<subseteq> sets N"

  3714 proof

  3715   have "X \<in> sigma_sets X A" "A \<subseteq> sigma_sets X A"

  3716     by (auto intro: sigma_sets_top)

  3717   moreover assume "sets (sigma X A) \<subseteq> sets N"

  3718   ultimately show "X \<in> sets N \<and> A \<subseteq> sets N"

  3719     by auto

  3720 next

  3721   assume *: "X \<in> sets N \<and> A \<subseteq> sets N"

  3722   { fix Y assume "Y \<in> sigma_sets X A" from this * have "Y \<in> sets N"

  3723       by induction auto }

  3724   then show "sets (sigma X A) \<subseteq> sets N"

  3725     by auto

  3726 qed

  3727

  3728 lemma measurable_iff_sets:

  3729   "f \<in> measurable M N \<longleftrightarrow> (f \<in> space M \<rightarrow> space N \<and> sets (vimage_algebra (space M) f N) \<subseteq> sets M)"

  3730 proof -

  3731   have *: "{f - A \<inter> space M |A. A \<in> sets N} \<subseteq> Pow (space M)"

  3732     by auto

  3733   show ?thesis

  3734     unfolding measurable_def

  3735     by (auto simp add: vimage_algebra_def sigma_le_sets[OF *])

  3736 qed

  3737

  3738 lemma sets_vimage_algebra_space: "X \<in> sets (vimage_algebra X f M)"

  3739   using sets.top[of "vimage_algebra X f M"] by simp

  3740

  3741 lemma measurable_mono:

  3742   assumes N: "sets N' \<le> sets N" "space N = space N'"

  3743   assumes M: "sets M \<le> sets M'" "space M = space M'"

  3744   shows "measurable M N \<subseteq> measurable M' N'"

  3745   unfolding measurable_def

  3746 proof safe

  3747   fix f A assume "f \<in> space M \<rightarrow> space N" "A \<in> sets N'"

  3748   moreover assume "\<forall>y\<in>sets N. f - y \<inter> space M \<in> sets M" note this[THEN bspec, of A]

  3749   ultimately show "f - A \<inter> space M' \<in> sets M'"

  3750     using assms by auto

  3751 qed (insert N M, auto)

  3752

  3753 lemma measurable_Sup_measurable:

  3754   assumes f: "f \<in> space N \<rightarrow> A"

  3755   shows "f \<in> measurable N (Sup {M. space M = A \<and> f \<in> measurable N M})"

  3756 proof (rule measurable_Sup2)

  3757   show "{M. space M = A \<and> f \<in> measurable N M} \<noteq> {}"

  3758     using f unfolding ex_in_conv[symmetric]

  3759     by (intro exI[of _ "sigma A {}"]) (auto intro!: measurable_measure_of)

  3760 qed auto

  3761

  3762 lemma (in sigma_algebra) sigma_sets_subset':

  3763   assumes a: "a \<subseteq> M" "\<Omega>' \<in> M"

  3764   shows "sigma_sets \<Omega>' a \<subseteq> M"

  3765 proof

  3766   show "x \<in> M" if x: "x \<in> sigma_sets \<Omega>' a" for x

  3767     using x by (induct rule: sigma_sets.induct) (insert a, auto)

  3768 qed

  3769

  3770 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\<in>I. M i)"

  3771   by (intro in_sets_Sup[where X=Y]) auto

  3772

  3773 lemma measurable_SUP1:

  3774   "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>

  3775     f \<in> measurable (SUP i\<in>I. M i) N"

  3776   by (auto intro: measurable_Sup1)

  3777

  3778 lemma sets_image_in_sets':

  3779   assumes X: "X \<in> sets N"

  3780   assumes f: "\<And>A. A \<in> sets M \<Longrightarrow> f - A \<inter> X \<in> sets N"

  3781   shows "sets (vimage_algebra X f M) \<subseteq> sets N"

  3782   unfolding sets_vimage_algebra

  3783   by (rule sets.sigma_sets_subset') (auto intro!: measurable_sets X f)

  3784

  3785 lemma mono_vimage_algebra:

  3786   "sets M \<le> sets N \<Longrightarrow> sets (vimage_algebra X f M) \<subseteq> sets (vimage_algebra X f N)"

  3787   using sets.top[of "sigma X {f - A \<inter> X |A. A \<in> sets N}"]

  3788   unfolding vimage_algebra_def

  3789   apply (subst (asm) space_measure_of)

  3790   apply auto []

  3791   apply (subst sigma_le_sets)

  3792   apply auto

  3793   done

  3794

  3795 lemma mono_restrict_space: "sets M \<le> sets N \<Longrightarrow> sets (restrict_space M X) \<subseteq> sets (restrict_space N X)"

  3796   unfolding sets_restrict_space by (rule image_mono)

  3797

  3798 lemma sets_eq_bot: "sets M = {{}} \<longleftrightarrow> M = bot"

  3799   apply safe

  3800   apply (intro measure_eqI)

  3801   apply auto

  3802   done

  3803

  3804 lemma sets_eq_bot2: "{{}} = sets M \<longleftrightarrow> M = bot"

  3805   using sets_eq_bot[of M] by blast

  3806

  3807

  3808 lemma (in finite_measure) countable_support:

  3809   "countable {x. measure M {x} \<noteq> 0}"

  3810 proof cases

  3811   assume "measure M (space M) = 0"

  3812   with bounded_measure measure_le_0_iff have "{x. measure M {x} \<noteq> 0} = {}"

  3813     by auto

  3814   then show ?thesis

  3815     by simp

  3816 next

  3817   let ?M = "measure M (space M)" and ?m = "\<lambda>x. measure M {x}"

  3818   assume "?M \<noteq> 0"

  3819   then have *: "{x. ?m x \<noteq> 0} = (\<Union>n. {x. ?M / Suc n < ?m x})"

  3820     using reals_Archimedean[of "?m x / ?M" for x]

  3821     by (auto simp: field_simps not_le[symmetric] divide_le_0_iff measure_le_0_iff)

  3822   have **: "\<And>n. finite {x. ?M / Suc n < ?m x}"

  3823   proof (rule ccontr)

  3824     fix n assume "infinite {x. ?M / Suc n < ?m x}" (is "infinite ?X")

  3825     then obtain X where "finite X" "card X = Suc (Suc n)" "X \<subseteq> ?X"

  3826       by (metis infinite_arbitrarily_large)

  3827     from this(3) have *: "\<And>x. x \<in> X \<Longrightarrow> ?M / Suc n \<le> ?m x"

  3828       by auto

  3829     { fix x assume "x \<in> X"

  3830       from \<open>?M \<noteq> 0\<close> *[OF this] have "?m x \<noteq> 0" by (auto simp: field_simps measure_le_0_iff)

  3831       then have "{x} \<in> sets M" by (auto dest: measure_notin_sets) }

  3832     note singleton_sets = this

  3833     have "?M < (\<Sum>x\<in>X. ?M / Suc n)"

  3834       using \<open>?M \<noteq> 0\<close>

  3835       by (simp add: \<open>card X = Suc (Suc n)\<close> field_simps less_le)

  3836     also have "\<dots> \<le> (\<Sum>x\<in>X. ?m x)"

  3837       by (rule sum_mono) fact

  3838     also have "\<dots> = measure M (\<Union>x\<in>X. {x})"

  3839       using singleton_sets \<open>finite X\<close>

  3840       by (intro finite_measure_finite_Union[symmetric]) (auto simp: disjoint_family_on_def)

  3841     finally have "?M < measure M (\<Union>x\<in>X. {x})" .

  3842     moreover have "measure M (\<Union>x\<in>X. {x}) \<le> ?M"

  3843       using singleton_sets[THEN sets.sets_into_space] by (intro finite_measure_mono) auto

  3844     ultimately show False by simp

  3845   qed

  3846   show ?thesis

  3847     unfolding * by (intro countable_UN countableI_type countable_finite[OF **])

  3848 qed

  3849

  3850 end
`