src/HOL/Analysis/Measure_Space.thy
 author nipkow Mon Oct 17 11:46:22 2016 +0200 (2016-10-17) changeset 64267 b9a1486e79be parent 64008 17a20ca86d62 child 64283 979cdfdf7a79 permissions -rw-r--r--
setsum -> sum
     1 (*  Title:      HOL/Analysis/Measure_Space.thy

     2     Author:     Lawrence C Paulson

     3     Author:     Johannes Hölzl, TU München

     4     Author:     Armin Heller, TU München

     5 *)

     6

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

     8

     9 theory Measure_Space

    10 imports

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

    12 begin

    13

    14 subsection "Relate extended reals and the indicator function"

    15

    16 lemma suminf_cmult_indicator:

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

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

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

    20 proof -

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

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

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

    24     by (auto simp: 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 Sigma_Algebra}, as it is also used to

    59   represent sigma algebras (with an arbitrary emeasure).

    60 \<close>

    61

    62 subsection "Extend binary sets"

    63

    64 lemma LIMSEQ_binaryset:

    65   assumes f: "f {} = 0"

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

    67 proof -

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

    69     proof

    70       fix n

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

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

    73     qed

    74   moreover

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

    76   ultimately

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

    78     by metis

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

    80     by simp

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

    82 qed

    83

    84 lemma binaryset_sums:

    85   assumes f: "f {} = 0"

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

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

    88

    89 lemma suminf_binaryset_eq:

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

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

    92   by (metis binaryset_sums sums_unique)

    93

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

    95

    96 text \<open>

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

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

    99 \<close>

   100

   101 definition subadditive where

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

   103

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

   105   by (auto simp add: subadditive_def)

   106

   107 definition countably_subadditive where

   108   "countably_subadditive M f \<longleftrightarrow>

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

   110

   111 lemma (in ring_of_sets) countably_subadditive_subadditive:

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

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

   114   shows  "subadditive M f"

   115 proof (auto simp add: subadditive_def)

   116   fix x y

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

   118   hence "disjoint_family (binaryset x y)"

   119     by (auto simp add: disjoint_family_on_def binaryset_def)

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

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

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

   123     using cs by (auto simp add: countably_subadditive_def)

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

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

   126     by (simp add: range_binaryset_eq UN_binaryset_eq)

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

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

   129 qed

   130

   131 definition additive where

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

   133

   134 definition increasing where

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

   136

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

   138

   139 lemma positiveD_empty:

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

   141   by (auto simp add: positive_def)

   142

   143 lemma additiveD:

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

   145   by (auto simp add: additive_def)

   146

   147 lemma increasingD:

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

   149   by (auto simp add: increasing_def)

   150

   151 lemma countably_additiveI[case_names countably]:

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

   153   \<Longrightarrow> countably_additive M f"

   154   by (simp add: countably_additive_def)

   155

   156 lemma (in ring_of_sets) disjointed_additive:

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

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

   159 proof (induct n)

   160   case (Suc n)

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

   162     by simp

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

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

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

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

   167   finally show ?case .

   168 qed simp

   169

   170 lemma (in ring_of_sets) additive_sum:

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

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

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

   174       and disj: "disjoint_family_on A S"

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

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

   177 proof induct

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

   179 next

   180   case (insert s S)

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

   182     by (auto simp add: disjoint_family_on_def neq_iff)

   183   moreover

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

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

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

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

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

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

   190     by (auto simp add: additive_def subset_insertI)

   191 qed

   192

   193 lemma (in ring_of_sets) additive_increasing:

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

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

   196   shows "increasing M f"

   197 proof (auto simp add: increasing_def)

   198   fix x y

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

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

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

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

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

   204   also have "... = f y"

   205     by (metis Un_Diff_cancel Un_absorb1 xy(3))

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

   207 qed

   208

   209 lemma (in ring_of_sets) subadditive:

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

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

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

   213 using S A

   214 proof (induct S)

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

   216 next

   217   case (insert x F)

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

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

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

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

   222     by simp

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

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

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

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

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

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

   229 qed

   230

   231 lemma (in ring_of_sets) countably_additive_additive:

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

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

   234   shows "additive M f"

   235 proof (auto simp add: additive_def)

   236   fix x y

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

   238   hence "disjoint_family (binaryset x y)"

   239     by (auto simp add: disjoint_family_on_def binaryset_def)

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

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

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

   243     using ca

   244     by (simp add: countably_additive_def)

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

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

   247     by (simp add: range_binaryset_eq UN_binaryset_eq)

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

   249     by (auto simp add: Un suminf_binaryset_eq positive_def)

   250 qed

   251

   252 lemma (in algebra) increasing_additive_bound:

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

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

   255       and inc: "increasing M f"

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

   257       and disj: "disjoint_family A"

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

   259 proof (safe intro!: suminf_le_const)

   260   fix N

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

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

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

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

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

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

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

   268

   269 lemma (in ring_of_sets) countably_additiveI_finite:

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

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

   272   shows "countably_additive M \<mu>"

   273 proof (rule countably_additiveI)

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

   275

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

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

   278

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

   280   proof (rule inj_onI, simp)

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

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

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

   284   qed

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

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

   287

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

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

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

   291   proof (rule finite_imageD)

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

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

   294       by (rule finite_subset) fact

   295   qed fact

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

   297     by (rule finite_subset)

   298

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

   300     using disj by (auto simp: disjoint_family_on_def)

   301

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

   303     by (rule suminf_finite) auto

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

   305     using fin_not_empty F_subset by (rule 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 UNIV A \<in> M \<longrightarrow> (\<Sum>i. f (A i)) = f (UNION UNIV A)"

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

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

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

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

   325     by (auto simp: UN_disjointed_eq disjoint_family_disjointed)

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

   327     using f(1)[unfolded positive_def] dA

   328     by (auto intro!: summable_LIMSEQ)

   329   from LIMSEQ_Suc[OF this]

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

   331     unfolding lessThan_Suc_atMost .

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

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

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

   335 next

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

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

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

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

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

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

   342       using A * by auto

   343   qed (force intro!: incseq_SucI)

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

   345     using A

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

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

   348   ultimately

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

   350     unfolding sums_def by simp

   351   from sums_unique[OF this]

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

   353 qed

   354

   355 lemma (in ring_of_sets) continuous_from_above_iff_empty_continuous:

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

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

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

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

   360 proof safe

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

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

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

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

   365 next

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

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

   368

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

   370     using additive_increasing[OF f] unfolding increasing_def by simp

   371

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

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

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

   375     using A by (auto simp: decseq_def)

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

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

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

   379     using A by (auto intro!: f_mono)

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

   381     using A by (auto simp: top_unique)

   382   { fix i

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

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

   385       using A by (auto simp: top_unique) }

   386   note f_fin = this

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

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

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

   390       using A by auto

   391   qed

   392   from INF_Lim_ereal[OF decseq_f this]

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

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

   395     by auto

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

   397     using A(4) f_fin f_Int_fin

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

   399   moreover {

   400     fix n

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

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

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

   404       by auto

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

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

   407     by simp

   408   with LIMSEQ_INF[OF decseq_fA]

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

   410 qed

   411

   412 lemma (in ring_of_sets) empty_continuous_imp_continuous_from_below:

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

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

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

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

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

   418 proof -

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

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

   421   moreover

   422   { fix i

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

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

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

   426       by auto

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

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

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

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

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

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

   433     by (auto intro: ennreal_tendsto_const_minus)

   434 qed

   435

   436 lemma (in ring_of_sets) empty_continuous_imp_countably_additive:

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

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

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

   440   shows "countably_additive M f"

   441   using countably_additive_iff_continuous_from_below[OF f]

   442   using empty_continuous_imp_continuous_from_below[OF f fin] cont

   443   by blast

   444

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

   446

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

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

   449

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

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

   452

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

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

   455

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

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

   458

   459 lemma suminf_emeasure:

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

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

   462   by (simp add: countably_additive_def)

   463

   464 lemma sums_emeasure:

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

   466   unfolding sums_iff by (intro conjI suminf_emeasure) auto

   467

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

   469   by (metis sets.countably_additive_additive emeasure_positive emeasure_countably_additive)

   470

   471 lemma plus_emeasure:

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

   473   using additiveD[OF emeasure_additive] ..

   474

   475 lemma 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: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: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: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 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 ae_filter :: "'a measure \<Rightarrow> 'a filter" where

   993   "ae_filter M = (INF N: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_I:

  1063   assumes "{x \<in> space M. \<not> P x} \<subseteq> N" "emeasure M N = 0" "N \<in> sets M"

  1064   shows "AE x in M. P x"

  1065   using assms unfolding eventually_ae_filter by auto

  1066

  1067 lemma AE_mp[elim!]:

  1068   assumes AE_P: "AE x in M. P x" and AE_imp: "AE x in M. P x \<longrightarrow> Q x"

  1069   shows "AE x in M. Q x"

  1070 proof -

  1071   from AE_P obtain A where P: "{x\<in>space M. \<not> P x} \<subseteq> A"

  1072     and A: "A \<in> sets M" "emeasure M A = 0"

  1073     by (auto elim!: AE_E)

  1074

  1075   from AE_imp obtain B where imp: "{x\<in>space M. P x \<and> \<not> Q x} \<subseteq> B"

  1076     and B: "B \<in> sets M" "emeasure M B = 0"

  1077     by (auto elim!: AE_E)

  1078

  1079   show ?thesis

  1080   proof (intro AE_I)

  1081     have "emeasure M (A \<union> B) \<le> 0"

  1082       using emeasure_subadditive[of A M B] A B by auto

  1083     then show "A \<union> B \<in> sets M" "emeasure M (A \<union> B) = 0"

  1084       using A B by auto

  1085     show "{x\<in>space M. \<not> Q x} \<subseteq> A \<union> B"

  1086       using P imp by auto

  1087   qed

  1088 qed

  1089

  1090 (* depricated replace by laws about eventually *)

  1091 lemma

  1092   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"

  1093     and AE_disjI1: "AE x in M. P x \<Longrightarrow> AE x in M. P x \<or> Q x"

  1094     and AE_disjI2: "AE x in M. Q x \<Longrightarrow> AE x in M. P x \<or> Q x"

  1095     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"

  1096     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)"

  1097   by auto

  1098

  1099 lemma AE_impI:

  1100   "(P \<Longrightarrow> AE x in M. Q x) \<Longrightarrow> AE x in M. P \<longrightarrow> Q x"

  1101   by (cases P) auto

  1102

  1103 lemma AE_measure:

  1104   assumes AE: "AE x in M. P x" and sets: "{x\<in>space M. P x} \<in> sets M" (is "?P \<in> sets M")

  1105   shows "emeasure M {x\<in>space M. P x} = emeasure M (space M)"

  1106 proof -

  1107   from AE_E[OF AE] guess N . note N = this

  1108   with sets have "emeasure M (space M) \<le> emeasure M (?P \<union> N)"

  1109     by (intro emeasure_mono) auto

  1110   also have "\<dots> \<le> emeasure M ?P + emeasure M N"

  1111     using sets N by (intro emeasure_subadditive) auto

  1112   also have "\<dots> = emeasure M ?P" using N by simp

  1113   finally show "emeasure M ?P = emeasure M (space M)"

  1114     using emeasure_space[of M "?P"] by auto

  1115 qed

  1116

  1117 lemma AE_space: "AE x in M. x \<in> space M"

  1118   by (rule AE_I[where N="{}"]) auto

  1119

  1120 lemma AE_I2[simp, intro]:

  1121   "(\<And>x. x \<in> space M \<Longrightarrow> P x) \<Longrightarrow> AE x in M. P x"

  1122   using AE_space by force

  1123

  1124 lemma AE_Ball_mp:

  1125   "\<forall>x\<in>space M. P x \<Longrightarrow> AE x in M. P x \<longrightarrow> Q x \<Longrightarrow> AE x in M. Q x"

  1126   by auto

  1127

  1128 lemma AE_cong[cong]:

  1129   "(\<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)"

  1130   by auto

  1131

  1132 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)"

  1133   by (auto simp: simp_implies_def)

  1134

  1135 lemma AE_all_countable:

  1136   "(AE x in M. \<forall>i. P i x) \<longleftrightarrow> (\<forall>i::'i::countable. AE x in M. P i x)"

  1137 proof

  1138   assume "\<forall>i. AE x in M. P i x"

  1139   from this[unfolded eventually_ae_filter Bex_def, THEN choice]

  1140   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

  1141   have "{x\<in>space M. \<not> (\<forall>i. P i x)} \<subseteq> (\<Union>i. {x\<in>space M. \<not> P i x})" by auto

  1142   also have "\<dots> \<subseteq> (\<Union>i. N i)" using N by auto

  1143   finally have "{x\<in>space M. \<not> (\<forall>i. P i x)} \<subseteq> (\<Union>i. N i)" .

  1144   moreover from N have "(\<Union>i. N i) \<in> null_sets M"

  1145     by (intro null_sets_UN) auto

  1146   ultimately show "AE x in M. \<forall>i. P i x"

  1147     unfolding eventually_ae_filter by auto

  1148 qed auto

  1149

  1150 lemma AE_ball_countable:

  1151   assumes [intro]: "countable X"

  1152   shows "(AE x in M. \<forall>y\<in>X. P x y) \<longleftrightarrow> (\<forall>y\<in>X. AE x in M. P x y)"

  1153 proof

  1154   assume "\<forall>y\<in>X. AE x in M. P x y"

  1155   from this[unfolded eventually_ae_filter Bex_def, THEN bchoice]

  1156   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"

  1157     by auto

  1158   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})"

  1159     by auto

  1160   also have "\<dots> \<subseteq> (\<Union>y\<in>X. N y)"

  1161     using N by auto

  1162   finally have "{x\<in>space M. \<not> (\<forall>y\<in>X. P x y)} \<subseteq> (\<Union>y\<in>X. N y)" .

  1163   moreover from N have "(\<Union>y\<in>X. N y) \<in> null_sets M"

  1164     by (intro null_sets_UN') auto

  1165   ultimately show "AE x in M. \<forall>y\<in>X. P x y"

  1166     unfolding eventually_ae_filter by auto

  1167 qed auto

  1168

  1169 lemma pairwise_alt: "pairwise R S \<longleftrightarrow> (\<forall>x\<in>S. \<forall>y\<in>S-{x}. R x y)"

  1170   by (auto simp add: pairwise_def)

  1171

  1172 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)"

  1173   unfolding pairwise_alt by (simp add: AE_ball_countable)

  1174

  1175 lemma AE_discrete_difference:

  1176   assumes X: "countable X"

  1177   assumes null: "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0"

  1178   assumes sets: "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"

  1179   shows "AE x in M. x \<notin> X"

  1180 proof -

  1181   have "(\<Union>x\<in>X. {x}) \<in> null_sets M"

  1182     using assms by (intro null_sets_UN') auto

  1183   from AE_not_in[OF this] show "AE x in M. x \<notin> X"

  1184     by auto

  1185 qed

  1186

  1187 lemma AE_finite_all:

  1188   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)"

  1189   using f by induct auto

  1190

  1191 lemma AE_finite_allI:

  1192   assumes "finite S"

  1193   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"

  1194   using AE_finite_all[OF \<open>finite S\<close>] by auto

  1195

  1196 lemma emeasure_mono_AE:

  1197   assumes imp: "AE x in M. x \<in> A \<longrightarrow> x \<in> B"

  1198     and B: "B \<in> sets M"

  1199   shows "emeasure M A \<le> emeasure M B"

  1200 proof cases

  1201   assume A: "A \<in> sets M"

  1202   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"

  1203     by (auto simp: eventually_ae_filter)

  1204   have "emeasure M A = emeasure M (A - N)"

  1205     using N A by (subst emeasure_Diff_null_set) auto

  1206   also have "emeasure M (A - N) \<le> emeasure M (B - N)"

  1207     using N A B sets.sets_into_space by (auto intro!: emeasure_mono)

  1208   also have "emeasure M (B - N) = emeasure M B"

  1209     using N B by (subst emeasure_Diff_null_set) auto

  1210   finally show ?thesis .

  1211 qed (simp add: emeasure_notin_sets)

  1212

  1213 lemma emeasure_eq_AE:

  1214   assumes iff: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B"

  1215   assumes A: "A \<in> sets M" and B: "B \<in> sets M"

  1216   shows "emeasure M A = emeasure M B"

  1217   using assms by (safe intro!: antisym emeasure_mono_AE) auto

  1218

  1219 lemma emeasure_Collect_eq_AE:

  1220   "AE x in M. P x \<longleftrightarrow> Q x \<Longrightarrow> Measurable.pred M Q \<Longrightarrow> Measurable.pred M P \<Longrightarrow>

  1221    emeasure M {x\<in>space M. P x} = emeasure M {x\<in>space M. Q x}"

  1222    by (intro emeasure_eq_AE) auto

  1223

  1224 lemma emeasure_eq_0_AE: "AE x in M. \<not> P x \<Longrightarrow> emeasure M {x\<in>space M. P x} = 0"

  1225   using AE_iff_measurable[OF _ refl, of M "\<lambda>x. \<not> P x"]

  1226   by (cases "{x\<in>space M. P x} \<in> sets M") (simp_all add: emeasure_notin_sets)

  1227

  1228 lemma emeasure_add_AE:

  1229   assumes [measurable]: "A \<in> sets M" "B \<in> sets M" "C \<in> sets M"

  1230   assumes 1: "AE x in M. x \<in> C \<longleftrightarrow> x \<in> A \<or> x \<in> B"

  1231   assumes 2: "AE x in M. \<not> (x \<in> A \<and> x \<in> B)"

  1232   shows "emeasure M C = emeasure M A + emeasure M B"

  1233 proof -

  1234   have "emeasure M C = emeasure M (A \<union> B)"

  1235     by (rule emeasure_eq_AE) (insert 1, auto)

  1236   also have "\<dots> = emeasure M A + emeasure M (B - A)"

  1237     by (subst plus_emeasure) auto

  1238   also have "emeasure M (B - A) = emeasure M B"

  1239     by (rule emeasure_eq_AE) (insert 2, auto)

  1240   finally show ?thesis .

  1241 qed

  1242

  1243 subsection \<open>\<open>\<sigma>\<close>-finite Measures\<close>

  1244

  1245 locale sigma_finite_measure =

  1246   fixes M :: "'a measure"

  1247   assumes sigma_finite_countable:

  1248     "\<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>)"

  1249

  1250 lemma (in sigma_finite_measure) sigma_finite:

  1251   obtains A :: "nat \<Rightarrow> 'a set"

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

  1253 proof -

  1254   obtain A :: "'a set set" where

  1255     [simp]: "countable A" and

  1256     A: "A \<subseteq> sets M" "(\<Union>A) = space M" "\<And>a. a \<in> A \<Longrightarrow> emeasure M a \<noteq> \<infinity>"

  1257     using sigma_finite_countable by metis

  1258   show thesis

  1259   proof cases

  1260     assume "A = {}" with \<open>(\<Union>A) = space M\<close> show thesis

  1261       by (intro that[of "\<lambda>_. {}"]) auto

  1262   next

  1263     assume "A \<noteq> {}"

  1264     show thesis

  1265     proof

  1266       show "range (from_nat_into A) \<subseteq> sets M"

  1267         using \<open>A \<noteq> {}\<close> A by auto

  1268       have "(\<Union>i. from_nat_into A i) = \<Union>A"

  1269         using range_from_nat_into[OF \<open>A \<noteq> {}\<close> \<open>countable A\<close>] by auto

  1270       with A show "(\<Union>i. from_nat_into A i) = space M"

  1271         by auto

  1272     qed (intro A from_nat_into \<open>A \<noteq> {}\<close>)

  1273   qed

  1274 qed

  1275

  1276 lemma (in sigma_finite_measure) sigma_finite_disjoint:

  1277   obtains A :: "nat \<Rightarrow> 'a set"

  1278   where "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>" "disjoint_family A"

  1279 proof -

  1280   obtain A :: "nat \<Rightarrow> 'a set" where

  1281     range: "range A \<subseteq> sets M" and

  1282     space: "(\<Union>i. A i) = space M" and

  1283     measure: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"

  1284     using sigma_finite by blast

  1285   show thesis

  1286   proof (rule that[of "disjointed A"])

  1287     show "range (disjointed A) \<subseteq> sets M"

  1288       by (rule sets.range_disjointed_sets[OF range])

  1289     show "(\<Union>i. disjointed A i) = space M"

  1290       and "disjoint_family (disjointed A)"

  1291       using disjoint_family_disjointed UN_disjointed_eq[of A] space range

  1292       by auto

  1293     show "emeasure M (disjointed A i) \<noteq> \<infinity>" for i

  1294     proof -

  1295       have "emeasure M (disjointed A i) \<le> emeasure M (A i)"

  1296         using range disjointed_subset[of A i] by (auto intro!: emeasure_mono)

  1297       then show ?thesis using measure[of i] by (auto simp: top_unique)

  1298     qed

  1299   qed

  1300 qed

  1301

  1302 lemma (in sigma_finite_measure) sigma_finite_incseq:

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

  1304   where "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>" "incseq A"

  1305 proof -

  1306   obtain F :: "nat \<Rightarrow> 'a set" where

  1307     F: "range F \<subseteq> sets M" "(\<Union>i. F i) = space M" "\<And>i. emeasure M (F i) \<noteq> \<infinity>"

  1308     using sigma_finite by blast

  1309   show thesis

  1310   proof (rule that[of "\<lambda>n. \<Union>i\<le>n. F i"])

  1311     show "range (\<lambda>n. \<Union>i\<le>n. F i) \<subseteq> sets M"

  1312       using F by (force simp: incseq_def)

  1313     show "(\<Union>n. \<Union>i\<le>n. F i) = space M"

  1314     proof -

  1315       from F have "\<And>x. x \<in> space M \<Longrightarrow> \<exists>i. x \<in> F i" by auto

  1316       with F show ?thesis by fastforce

  1317     qed

  1318     show "emeasure M (\<Union>i\<le>n. F i) \<noteq> \<infinity>" for n

  1319     proof -

  1320       have "emeasure M (\<Union>i\<le>n. F i) \<le> (\<Sum>i\<le>n. emeasure M (F i))"

  1321         using F by (auto intro!: emeasure_subadditive_finite)

  1322       also have "\<dots> < \<infinity>"

  1323         using F by (auto simp: sum_Pinfty less_top)

  1324       finally show ?thesis by simp

  1325     qed

  1326     show "incseq (\<lambda>n. \<Union>i\<le>n. F i)"

  1327       by (force simp: incseq_def)

  1328   qed

  1329 qed

  1330

  1331 subsection \<open>Measure space induced by distribution of @{const measurable}-functions\<close>

  1332

  1333 definition distr :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b measure" where

  1334   "distr M N f = measure_of (space N) (sets N) (\<lambda>A. emeasure M (f - A \<inter> space M))"

  1335

  1336 lemma

  1337   shows sets_distr[simp, measurable_cong]: "sets (distr M N f) = sets N"

  1338     and space_distr[simp]: "space (distr M N f) = space N"

  1339   by (auto simp: distr_def)

  1340

  1341 lemma

  1342   shows measurable_distr_eq1[simp]: "measurable (distr Mf Nf f) Mf' = measurable Nf Mf'"

  1343     and measurable_distr_eq2[simp]: "measurable Mg' (distr Mg Ng g) = measurable Mg' Ng"

  1344   by (auto simp: measurable_def)

  1345

  1346 lemma distr_cong:

  1347   "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"

  1348   using sets_eq_imp_space_eq[of N L] by (simp add: distr_def Int_def cong: rev_conj_cong)

  1349

  1350 lemma emeasure_distr:

  1351   fixes f :: "'a \<Rightarrow> 'b"

  1352   assumes f: "f \<in> measurable M N" and A: "A \<in> sets N"

  1353   shows "emeasure (distr M N f) A = emeasure M (f - A \<inter> space M)" (is "_ = ?\<mu> A")

  1354   unfolding distr_def

  1355 proof (rule emeasure_measure_of_sigma)

  1356   show "positive (sets N) ?\<mu>"

  1357     by (auto simp: positive_def)

  1358

  1359   show "countably_additive (sets N) ?\<mu>"

  1360   proof (intro countably_additiveI)

  1361     fix A :: "nat \<Rightarrow> 'b set" assume "range A \<subseteq> sets N" "disjoint_family A"

  1362     then have A: "\<And>i. A i \<in> sets N" "(\<Union>i. A i) \<in> sets N" by auto

  1363     then have *: "range (\<lambda>i. f - (A i) \<inter> space M) \<subseteq> sets M"

  1364       using f by (auto simp: measurable_def)

  1365     moreover have "(\<Union>i. f -  A i \<inter> space M) \<in> sets M"

  1366       using * by blast

  1367     moreover have **: "disjoint_family (\<lambda>i. f - A i \<inter> space M)"

  1368       using \<open>disjoint_family A\<close> by (auto simp: disjoint_family_on_def)

  1369     ultimately show "(\<Sum>i. ?\<mu> (A i)) = ?\<mu> (\<Union>i. A i)"

  1370       using suminf_emeasure[OF _ **] A f

  1371       by (auto simp: comp_def vimage_UN)

  1372   qed

  1373   show "sigma_algebra (space N) (sets N)" ..

  1374 qed fact

  1375

  1376 lemma emeasure_Collect_distr:

  1377   assumes X[measurable]: "X \<in> measurable M N" "Measurable.pred N P"

  1378   shows "emeasure (distr M N X) {x\<in>space N. P x} = emeasure M {x\<in>space M. P (X x)}"

  1379   by (subst emeasure_distr)

  1380      (auto intro!: arg_cong2[where f=emeasure] X(1)[THEN measurable_space])

  1381

  1382 lemma emeasure_lfp2[consumes 1, case_names cont f measurable]:

  1383   assumes "P M"

  1384   assumes cont: "sup_continuous F"

  1385   assumes f: "\<And>M. P M \<Longrightarrow> f \<in> measurable M' M"

  1386   assumes *: "\<And>M A. P M \<Longrightarrow> (\<And>N. P N \<Longrightarrow> Measurable.pred N A) \<Longrightarrow> Measurable.pred M (F A)"

  1387   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)})"

  1388 proof (subst (1 2) emeasure_Collect_distr[symmetric, where X=f])

  1389   show "f \<in> measurable M' M"  "f \<in> measurable M' M"

  1390     using f[OF \<open>P M\<close>] by auto

  1391   { fix i show "Measurable.pred M ((F ^^ i) (\<lambda>x. False))"

  1392     using \<open>P M\<close> by (induction i arbitrary: M) (auto intro!: *) }

  1393   show "Measurable.pred M (lfp F)"

  1394     using \<open>P M\<close> cont * by (rule measurable_lfp_coinduct[of P])

  1395

  1396   have "emeasure (distr M' M f) {x \<in> space (distr M' M f). lfp F x} =

  1397     (SUP i. emeasure (distr M' M f) {x \<in> space (distr M' M f). (F ^^ i) (\<lambda>x. False) x})"

  1398     using \<open>P M\<close>

  1399   proof (coinduction arbitrary: M rule: emeasure_lfp')

  1400     case (measurable A N) then have "\<And>N. P N \<Longrightarrow> Measurable.pred (distr M' N f) A"

  1401       by metis

  1402     then have "\<And>N. P N \<Longrightarrow> Measurable.pred N A"

  1403       by simp

  1404     with \<open>P N\<close>[THEN *] show ?case

  1405       by auto

  1406   qed fact

  1407   then show "emeasure (distr M' M f) {x \<in> space M. lfp F x} =

  1408     (SUP i. emeasure (distr M' M f) {x \<in> space M. (F ^^ i) (\<lambda>x. False) x})"

  1409    by simp

  1410 qed

  1411

  1412 lemma distr_id[simp]: "distr N N (\<lambda>x. x) = N"

  1413   by (rule measure_eqI) (auto simp: emeasure_distr)

  1414

  1415 lemma measure_distr:

  1416   "f \<in> measurable M N \<Longrightarrow> S \<in> sets N \<Longrightarrow> measure (distr M N f) S = measure M (f - S \<inter> space M)"

  1417   by (simp add: emeasure_distr measure_def)

  1418

  1419 lemma distr_cong_AE:

  1420   assumes 1: "M = K" "sets N = sets L" and

  1421     2: "(AE x in M. f x = g x)" and "f \<in> measurable M N" and "g \<in> measurable K L"

  1422   shows "distr M N f = distr K L g"

  1423 proof (rule measure_eqI)

  1424   fix A assume "A \<in> sets (distr M N f)"

  1425   with assms show "emeasure (distr M N f) A = emeasure (distr K L g) A"

  1426     by (auto simp add: emeasure_distr intro!: emeasure_eq_AE measurable_sets)

  1427 qed (insert 1, simp)

  1428

  1429 lemma AE_distrD:

  1430   assumes f: "f \<in> measurable M M'"

  1431     and AE: "AE x in distr M M' f. P x"

  1432   shows "AE x in M. P (f x)"

  1433 proof -

  1434   from AE[THEN AE_E] guess N .

  1435   with f show ?thesis

  1436     unfolding eventually_ae_filter

  1437     by (intro bexI[of _ "f - N \<inter> space M"])

  1438        (auto simp: emeasure_distr measurable_def)

  1439 qed

  1440

  1441 lemma AE_distr_iff:

  1442   assumes f[measurable]: "f \<in> measurable M N" and P[measurable]: "{x \<in> space N. P x} \<in> sets N"

  1443   shows "(AE x in distr M N f. P x) \<longleftrightarrow> (AE x in M. P (f x))"

  1444 proof (subst (1 2) AE_iff_measurable[OF _ refl])

  1445   have "f - {x\<in>space N. \<not> P x} \<inter> space M = {x \<in> space M. \<not> P (f x)}"

  1446     using f[THEN measurable_space] by auto

  1447   then show "(emeasure (distr M N f) {x \<in> space (distr M N f). \<not> P x} = 0) =

  1448     (emeasure M {x \<in> space M. \<not> P (f x)} = 0)"

  1449     by (simp add: emeasure_distr)

  1450 qed auto

  1451

  1452 lemma null_sets_distr_iff:

  1453   "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"

  1454   by (auto simp add: null_sets_def emeasure_distr)

  1455

  1456 lemma distr_distr:

  1457   "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)"

  1458   by (auto simp add: emeasure_distr measurable_space

  1459            intro!: arg_cong[where f="emeasure M"] measure_eqI)

  1460

  1461 subsection \<open>Real measure values\<close>

  1462

  1463 lemma ring_of_finite_sets: "ring_of_sets (space M) {A\<in>sets M. emeasure M A \<noteq> top}"

  1464 proof (rule ring_of_setsI)

  1465   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>

  1466     a \<union> b \<in> {A \<in> sets M. emeasure M A \<noteq> top}" for a b

  1467     using emeasure_subadditive[of a M b] by (auto simp: top_unique)

  1468

  1469   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>

  1470     a - b \<in> {A \<in> sets M. emeasure M A \<noteq> top}" for a b

  1471     using emeasure_mono[of "a - b" a M] by (auto simp: Diff_subset top_unique)

  1472 qed (auto dest: sets.sets_into_space)

  1473

  1474 lemma measure_nonneg[simp]: "0 \<le> measure M A"

  1475   unfolding measure_def by auto

  1476

  1477 lemma zero_less_measure_iff: "0 < measure M A \<longleftrightarrow> measure M A \<noteq> 0"

  1478   using measure_nonneg[of M A] by (auto simp add: le_less)

  1479

  1480 lemma measure_le_0_iff: "measure M X \<le> 0 \<longleftrightarrow> measure M X = 0"

  1481   using measure_nonneg[of M X] by linarith

  1482

  1483 lemma measure_empty[simp]: "measure M {} = 0"

  1484   unfolding measure_def by (simp add: zero_ennreal.rep_eq)

  1485

  1486 lemma emeasure_eq_ennreal_measure:

  1487   "emeasure M A \<noteq> top \<Longrightarrow> emeasure M A = ennreal (measure M A)"

  1488   by (cases "emeasure M A" rule: ennreal_cases) (auto simp: measure_def)

  1489

  1490 lemma measure_zero_top: "emeasure M A = top \<Longrightarrow> measure M A = 0"

  1491   by (simp add: measure_def enn2ereal_top)

  1492

  1493 lemma measure_eq_emeasure_eq_ennreal: "0 \<le> x \<Longrightarrow> emeasure M A = ennreal x \<Longrightarrow> measure M A = x"

  1494   using emeasure_eq_ennreal_measure[of M A]

  1495   by (cases "A \<in> M") (auto simp: measure_notin_sets emeasure_notin_sets)

  1496

  1497 lemma enn2real_plus:"a < top \<Longrightarrow> b < top \<Longrightarrow> enn2real (a + b) = enn2real a + enn2real b"

  1498   by (simp add: enn2real_def plus_ennreal.rep_eq real_of_ereal_add less_top

  1499            del: real_of_ereal_enn2ereal)

  1500

  1501 lemma measure_eq_AE:

  1502   assumes iff: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B"

  1503   assumes A: "A \<in> sets M" and B: "B \<in> sets M"

  1504   shows "measure M A = measure M B"

  1505   using assms emeasure_eq_AE[OF assms] by (simp add: measure_def)

  1506

  1507 lemma measure_Union:

  1508   "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>

  1509     measure M (A \<union> B) = measure M A + measure M B"

  1510   by (simp add: measure_def plus_emeasure[symmetric] enn2real_plus less_top)

  1511

  1512 lemma disjoint_family_on_insert:

  1513   "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"

  1514   by (fastforce simp: disjoint_family_on_def)

  1515

  1516 lemma measure_finite_Union:

  1517   "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>

  1518     measure M (\<Union>i\<in>S. A i) = (\<Sum>i\<in>S. measure M (A i))"

  1519   by (induction S rule: finite_induct)

  1520      (auto simp: disjoint_family_on_insert measure_Union sum_emeasure[symmetric] sets.countable_UN'[OF countable_finite])

  1521

  1522 lemma measure_Diff:

  1523   assumes finite: "emeasure M A \<noteq> \<infinity>"

  1524   and measurable: "A \<in> sets M" "B \<in> sets M" "B \<subseteq> A"

  1525   shows "measure M (A - B) = measure M A - measure M B"

  1526 proof -

  1527   have "emeasure M (A - B) \<le> emeasure M A" "emeasure M B \<le> emeasure M A"

  1528     using measurable by (auto intro!: emeasure_mono)

  1529   hence "measure M ((A - B) \<union> B) = measure M (A - B) + measure M B"

  1530     using measurable finite by (rule_tac measure_Union) (auto simp: top_unique)

  1531   thus ?thesis using \<open>B \<subseteq> A\<close> by (auto simp: Un_absorb2)

  1532 qed

  1533

  1534 lemma measure_UNION:

  1535   assumes measurable: "range A \<subseteq> sets M" "disjoint_family A"

  1536   assumes finite: "emeasure M (\<Union>i. A i) \<noteq> \<infinity>"

  1537   shows "(\<lambda>i. measure M (A i)) sums (measure M (\<Union>i. A i))"

  1538 proof -

  1539   have "(\<lambda>i. emeasure M (A i)) sums (emeasure M (\<Union>i. A i))"

  1540     unfolding suminf_emeasure[OF measurable, symmetric] by (simp add: summable_sums)

  1541   moreover

  1542   { fix i

  1543     have "emeasure M (A i) \<le> emeasure M (\<Union>i. A i)"

  1544       using measurable by (auto intro!: emeasure_mono)

  1545     then have "emeasure M (A i) = ennreal ((measure M (A i)))"

  1546       using finite by (intro emeasure_eq_ennreal_measure) (auto simp: top_unique) }

  1547   ultimately show ?thesis using finite

  1548     by (subst (asm) (2) emeasure_eq_ennreal_measure) simp_all

  1549 qed

  1550

  1551 lemma measure_subadditive:

  1552   assumes measurable: "A \<in> sets M" "B \<in> sets M"

  1553   and fin: "emeasure M A \<noteq> \<infinity>" "emeasure M B \<noteq> \<infinity>"

  1554   shows "measure M (A \<union> B) \<le> measure M A + measure M B"

  1555 proof -

  1556   have "emeasure M (A \<union> B) \<noteq> \<infinity>"

  1557     using emeasure_subadditive[OF measurable] fin by (auto simp: top_unique)

  1558   then show "(measure M (A \<union> B)) \<le> (measure M A) + (measure M B)"

  1559     using emeasure_subadditive[OF measurable] fin

  1560     apply simp

  1561     apply (subst (asm) (2 3 4) emeasure_eq_ennreal_measure)

  1562     apply (auto simp add: ennreal_plus[symmetric] simp del: ennreal_plus)

  1563     done

  1564 qed

  1565

  1566 lemma measure_subadditive_finite:

  1567   assumes A: "finite I" "AI \<subseteq> sets M" and fin: "\<And>i. i \<in> I \<Longrightarrow> emeasure M (A i) \<noteq> \<infinity>"

  1568   shows "measure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. measure M (A i))"

  1569 proof -

  1570   { have "emeasure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. emeasure M (A i))"

  1571       using emeasure_subadditive_finite[OF A] .

  1572     also have "\<dots> < \<infinity>"

  1573       using fin by (simp add: less_top A)

  1574     finally have "emeasure M (\<Union>i\<in>I. A i) \<noteq> top" by simp }

  1575   note * = this

  1576   show ?thesis

  1577     using emeasure_subadditive_finite[OF A] fin

  1578     unfolding emeasure_eq_ennreal_measure[OF *]

  1579     by (simp_all add: sum_nonneg emeasure_eq_ennreal_measure)

  1580 qed

  1581

  1582 lemma measure_subadditive_countably:

  1583   assumes A: "range A \<subseteq> sets M" and fin: "(\<Sum>i. emeasure M (A i)) \<noteq> \<infinity>"

  1584   shows "measure M (\<Union>i. A i) \<le> (\<Sum>i. measure M (A i))"

  1585 proof -

  1586   from fin have **: "\<And>i. emeasure M (A i) \<noteq> top"

  1587     using ennreal_suminf_lessD[of "\<lambda>i. emeasure M (A i)"] by (simp add: less_top)

  1588   { have "emeasure M (\<Union>i. A i) \<le> (\<Sum>i. emeasure M (A i))"

  1589       using emeasure_subadditive_countably[OF A] .

  1590     also have "\<dots> < \<infinity>"

  1591       using fin by (simp add: less_top)

  1592     finally have "emeasure M (\<Union>i. A i) \<noteq> top" by simp }

  1593   then have "ennreal (measure M (\<Union>i. A i)) = emeasure M (\<Union>i. A i)"

  1594     by (rule emeasure_eq_ennreal_measure[symmetric])

  1595   also have "\<dots> \<le> (\<Sum>i. emeasure M (A i))"

  1596     using emeasure_subadditive_countably[OF A] .

  1597   also have "\<dots> = ennreal (\<Sum>i. measure M (A i))"

  1598     using fin unfolding emeasure_eq_ennreal_measure[OF **]

  1599     by (subst suminf_ennreal) (auto simp: **)

  1600   finally show ?thesis

  1601     apply (rule ennreal_le_iff[THEN iffD1, rotated])

  1602     apply (intro suminf_nonneg allI measure_nonneg summable_suminf_not_top)

  1603     using fin

  1604     apply (simp add: emeasure_eq_ennreal_measure[OF **])

  1605     done

  1606 qed

  1607

  1608 lemma measure_Un_null_set: "A \<in> sets M \<Longrightarrow> B \<in> null_sets M \<Longrightarrow> measure M (A \<union> B) = measure M A"

  1609   by (simp add: measure_def emeasure_Un_null_set)

  1610

  1611 lemma measure_Diff_null_set: "A \<in> sets M \<Longrightarrow> B \<in> null_sets M \<Longrightarrow> measure M (A - B) = measure M A"

  1612   by (simp add: measure_def emeasure_Diff_null_set)

  1613

  1614 lemma measure_eq_sum_singleton:

  1615   "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>

  1616     measure M S = (\<Sum>x\<in>S. measure M {x})"

  1617   using emeasure_eq_sum_singleton[of S M]

  1618   by (intro measure_eq_emeasure_eq_ennreal) (auto simp: sum_nonneg emeasure_eq_ennreal_measure)

  1619

  1620 lemma Lim_measure_incseq:

  1621   assumes A: "range A \<subseteq> sets M" "incseq A" and fin: "emeasure M (\<Union>i. A i) \<noteq> \<infinity>"

  1622   shows "(\<lambda>i. measure M (A i)) \<longlonglongrightarrow> measure M (\<Union>i. A i)"

  1623 proof (rule tendsto_ennrealD)

  1624   have "ennreal (measure M (\<Union>i. A i)) = emeasure M (\<Union>i. A i)"

  1625     using fin by (auto simp: emeasure_eq_ennreal_measure)

  1626   moreover have "ennreal (measure M (A i)) = emeasure M (A i)" for i

  1627     using assms emeasure_mono[of "A _" "\<Union>i. A i" M]

  1628     by (intro emeasure_eq_ennreal_measure[symmetric]) (auto simp: less_top UN_upper intro: le_less_trans)

  1629   ultimately show "(\<lambda>x. ennreal (Sigma_Algebra.measure M (A x))) \<longlonglongrightarrow> ennreal (Sigma_Algebra.measure M (\<Union>i. A i))"

  1630     using A by (auto intro!: Lim_emeasure_incseq)

  1631 qed auto

  1632

  1633 lemma Lim_measure_decseq:

  1634   assumes A: "range A \<subseteq> sets M" "decseq A" and fin: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"

  1635   shows "(\<lambda>n. measure M (A n)) \<longlonglongrightarrow> measure M (\<Inter>i. A i)"

  1636 proof (rule tendsto_ennrealD)

  1637   have "ennreal (measure M (\<Inter>i. A i)) = emeasure M (\<Inter>i. A i)"

  1638     using fin[of 0] A emeasure_mono[of "\<Inter>i. A i" "A 0" M]

  1639     by (auto intro!: emeasure_eq_ennreal_measure[symmetric] simp: INT_lower less_top intro: le_less_trans)

  1640   moreover have "ennreal (measure M (A i)) = emeasure M (A i)" for i

  1641     using A fin[of i] by (intro emeasure_eq_ennreal_measure[symmetric]) auto

  1642   ultimately show "(\<lambda>x. ennreal (Sigma_Algebra.measure M (A x))) \<longlonglongrightarrow> ennreal (Sigma_Algebra.measure M (\<Inter>i. A i))"

  1643     using fin A by (auto intro!: Lim_emeasure_decseq)

  1644 qed auto

  1645

  1646 subsection \<open>Set of measurable sets with finite measure\<close>

  1647

  1648 definition fmeasurable :: "'a measure \<Rightarrow> 'a set set"

  1649 where

  1650   "fmeasurable M = {A\<in>sets M. emeasure M A < \<infinity>}"

  1651

  1652 lemma fmeasurableD[dest, measurable_dest]: "A \<in> fmeasurable M \<Longrightarrow> A \<in> sets M"

  1653   by (auto simp: fmeasurable_def)

  1654

  1655 lemma fmeasurableD2: "A \<in> fmeasurable M \<Longrightarrow> emeasure M A \<noteq> top"

  1656   by (auto simp: fmeasurable_def)

  1657

  1658 lemma fmeasurableI: "A \<in> sets M \<Longrightarrow> emeasure M A < \<infinity> \<Longrightarrow> A \<in> fmeasurable M"

  1659   by (auto simp: fmeasurable_def)

  1660

  1661 lemma fmeasurableI_null_sets: "A \<in> null_sets M \<Longrightarrow> A \<in> fmeasurable M"

  1662   by (auto simp: fmeasurable_def)

  1663

  1664 lemma fmeasurableI2: "A \<in> fmeasurable M \<Longrightarrow> B \<subseteq> A \<Longrightarrow> B \<in> sets M \<Longrightarrow> B \<in> fmeasurable M"

  1665   using emeasure_mono[of B A M] by (auto simp: fmeasurable_def)

  1666

  1667 lemma measure_mono_fmeasurable:

  1668   "A \<subseteq> B \<Longrightarrow> A \<in> sets M \<Longrightarrow> B \<in> fmeasurable M \<Longrightarrow> measure M A \<le> measure M B"

  1669   by (auto simp: measure_def fmeasurable_def intro!: emeasure_mono enn2real_mono)

  1670

  1671 lemma emeasure_eq_measure2: "A \<in> fmeasurable M \<Longrightarrow> emeasure M A = measure M A"

  1672   by (simp add: emeasure_eq_ennreal_measure fmeasurable_def less_top)

  1673

  1674 interpretation fmeasurable: ring_of_sets "space M" "fmeasurable M"

  1675 proof (rule ring_of_setsI)

  1676   show "fmeasurable M \<subseteq> Pow (space M)" "{} \<in> fmeasurable M"

  1677     by (auto simp: fmeasurable_def dest: sets.sets_into_space)

  1678   fix a b assume *: "a \<in> fmeasurable M" "b \<in> fmeasurable M"

  1679   then have "emeasure M (a \<union> b) \<le> emeasure M a + emeasure M b"

  1680     by (intro emeasure_subadditive) auto

  1681   also have "\<dots> < top"

  1682     using * by (auto simp: fmeasurable_def)

  1683   finally show  "a \<union> b \<in> fmeasurable M"

  1684     using * by (auto intro: fmeasurableI)

  1685   show "a - b \<in> fmeasurable M"

  1686     using emeasure_mono[of "a - b" a M] * by (auto simp: fmeasurable_def Diff_subset)

  1687 qed

  1688

  1689 lemma fmeasurable_Diff: "A \<in> fmeasurable M \<Longrightarrow> B \<in> sets M \<Longrightarrow> A - B \<in> fmeasurable M"

  1690   using fmeasurableI2[of A M "A - B"] by auto

  1691

  1692 lemma fmeasurable_UN:

  1693   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"

  1694   shows "(\<Union>i\<in>I. F i) \<in> fmeasurable M"

  1695 proof (rule fmeasurableI2)

  1696   show "A \<in> fmeasurable M" "(\<Union>i\<in>I. F i) \<subseteq> A" using assms by auto

  1697   show "(\<Union>i\<in>I. F i) \<in> sets M"

  1698     using assms by (intro sets.countable_UN') auto

  1699 qed

  1700

  1701 lemma fmeasurable_INT:

  1702   assumes "countable I" "i \<in> I" "\<And>i. i \<in> I \<Longrightarrow> F i \<in> sets M" "F i \<in> fmeasurable M"

  1703   shows "(\<Inter>i\<in>I. F i) \<in> fmeasurable M"

  1704 proof (rule fmeasurableI2)

  1705   show "F i \<in> fmeasurable M" "(\<Inter>i\<in>I. F i) \<subseteq> F i"

  1706     using assms by auto

  1707   show "(\<Inter>i\<in>I. F i) \<in> sets M"

  1708     using assms by (intro sets.countable_INT') auto

  1709 qed

  1710

  1711 lemma measure_Un2:

  1712   "A \<in> fmeasurable M \<Longrightarrow> B \<in> fmeasurable M \<Longrightarrow> measure M (A \<union> B) = measure M A + measure M (B - A)"

  1713   using measure_Union[of M A "B - A"] by (auto simp: fmeasurableD2 fmeasurable.Diff)

  1714

  1715 lemma measure_Un3:

  1716   assumes "A \<in> fmeasurable M" "B \<in> fmeasurable M"

  1717   shows "measure M (A \<union> B) = measure M A + measure M B - measure M (A \<inter> B)"

  1718 proof -

  1719   have "measure M (A \<union> B) = measure M A + measure M (B - A)"

  1720     using assms by (rule measure_Un2)

  1721   also have "B - A = B - (A \<inter> B)"

  1722     by auto

  1723   also have "measure M (B - (A \<inter> B)) = measure M B - measure M (A \<inter> B)"

  1724     using assms by (intro measure_Diff) (auto simp: fmeasurable_def)

  1725   finally show ?thesis

  1726     by simp

  1727 qed

  1728

  1729 lemma measure_Un_AE:

  1730   "AE x in M. x \<notin> A \<or> x \<notin> B \<Longrightarrow> A \<in> fmeasurable M \<Longrightarrow> B \<in> fmeasurable M \<Longrightarrow>

  1731   measure M (A \<union> B) = measure M A + measure M B"

  1732   by (subst measure_Un2) (auto intro!: measure_eq_AE)

  1733

  1734 lemma measure_UNION_AE:

  1735   assumes I: "finite I"

  1736   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>

  1737     measure M (\<Union>i\<in>I. F i) = (\<Sum>i\<in>I. measure M (F i))"

  1738   unfolding AE_pairwise[OF countable_finite, OF I]

  1739   using I

  1740   apply (induction I rule: finite_induct)

  1741    apply simp

  1742   apply (simp add: pairwise_insert)

  1743   apply (subst measure_Un_AE)

  1744   apply auto

  1745   done

  1746

  1747 lemma measure_UNION':

  1748   "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>

  1749     measure M (\<Union>i\<in>I. F i) = (\<Sum>i\<in>I. measure M (F i))"

  1750   by (intro measure_UNION_AE) (auto simp: disjnt_def elim!: pairwise_mono intro!: always_eventually)

  1751

  1752 lemma measure_Union_AE:

  1753   "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>

  1754     measure M (\<Union>F) = (\<Sum>S\<in>F. measure M S)"

  1755   using measure_UNION_AE[of F "\<lambda>x. x" M] by simp

  1756

  1757 lemma measure_Union':

  1758   "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)"

  1759   using measure_UNION'[of F "\<lambda>x. x" M] by simp

  1760

  1761 lemma measure_Un_le:

  1762   assumes "A \<in> sets M" "B \<in> sets M" shows "measure M (A \<union> B) \<le> measure M A + measure M B"

  1763 proof cases

  1764   assume "A \<in> fmeasurable M \<and> B \<in> fmeasurable M"

  1765   with measure_subadditive[of A M B] assms show ?thesis

  1766     by (auto simp: fmeasurableD2)

  1767 next

  1768   assume "\<not> (A \<in> fmeasurable M \<and> B \<in> fmeasurable M)"

  1769   then have "A \<union> B \<notin> fmeasurable M"

  1770     using fmeasurableI2[of "A \<union> B" M A] fmeasurableI2[of "A \<union> B" M B] assms by auto

  1771   with assms show ?thesis

  1772     by (auto simp: fmeasurable_def measure_def less_top[symmetric])

  1773 qed

  1774

  1775 lemma measure_UNION_le:

  1776   "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))"

  1777 proof (induction I rule: finite_induct)

  1778   case (insert i I)

  1779   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)"

  1780     by (auto intro!: measure_Un_le)

  1781   also have "measure M (\<Union>i\<in>I. F i) \<le> (\<Sum>i\<in>I. measure M (F i))"

  1782     using insert by auto

  1783   finally show ?case

  1784     using insert by simp

  1785 qed simp

  1786

  1787 lemma measure_Union_le:

  1788   "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)"

  1789   using measure_UNION_le[of F "\<lambda>x. x" M] by simp

  1790

  1791 lemma

  1792   assumes "countable I" and I: "\<And>i. i \<in> I \<Longrightarrow> A i \<in> fmeasurable M"

  1793     and bound: "\<And>I'. I' \<subseteq> I \<Longrightarrow> finite I' \<Longrightarrow> measure M (\<Union>i\<in>I'. A i) \<le> B" and "0 \<le> B"

  1794   shows fmeasurable_UN_bound: "(\<Union>i\<in>I. A i) \<in> fmeasurable M" (is ?fm)

  1795     and measure_UN_bound: "measure M (\<Union>i\<in>I. A i) \<le> B" (is ?m)

  1796 proof -

  1797   have "?fm \<and> ?m"

  1798   proof cases

  1799     assume "I = {}" with \<open>0 \<le> B\<close> show ?thesis by simp

  1800   next

  1801     assume "I \<noteq> {}"

  1802     have "(\<Union>i\<in>I. A i) = (\<Union>i. (\<Union>n\<le>i. A (from_nat_into I n)))"

  1803       by (subst range_from_nat_into[symmetric, OF \<open>I \<noteq> {}\<close> \<open>countable I\<close>]) auto

  1804     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

  1805     also have "\<dots> = (SUP i. emeasure M (\<Union>n\<le>i. A (from_nat_into I n)))"

  1806       using I \<open>I \<noteq> {}\<close>[THEN from_nat_into] by (intro SUP_emeasure_incseq[symmetric]) (fastforce simp: incseq_Suc_iff)+

  1807     also have "\<dots> \<le> B"

  1808     proof (intro SUP_least)

  1809       fix i :: nat

  1810       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))"

  1811         using I \<open>I \<noteq> {}\<close>[THEN from_nat_into] by (intro emeasure_eq_measure2 fmeasurable.finite_UN) auto

  1812       also have "\<dots> = measure M (\<Union>n\<in>from_nat_into I  {..i}. A n)"

  1813         by simp

  1814       also have "\<dots> \<le> B"

  1815         by (intro ennreal_leI bound) (auto intro:  from_nat_into[OF \<open>I \<noteq> {}\<close>])

  1816       finally show "emeasure M (\<Union>n\<le>i. A (from_nat_into I n)) \<le> ennreal B" .

  1817     qed

  1818     finally have *: "emeasure M (\<Union>i\<in>I. A i) \<le> B" .

  1819     then have ?fm

  1820       using I \<open>countable I\<close> by (intro fmeasurableI conjI) (auto simp: less_top[symmetric] top_unique)

  1821     with * \<open>0\<le>B\<close> show ?thesis

  1822       by (simp add: emeasure_eq_measure2)

  1823   qed

  1824   then show ?fm ?m by auto

  1825 qed

  1826

  1827 subsection \<open>Measure spaces with @{term "emeasure M (space M) < \<infinity>"}\<close>

  1828

  1829 locale finite_measure = sigma_finite_measure M for M +

  1830   assumes finite_emeasure_space: "emeasure M (space M) \<noteq> top"

  1831

  1832 lemma finite_measureI[Pure.intro!]:

  1833   "emeasure M (space M) \<noteq> \<infinity> \<Longrightarrow> finite_measure M"

  1834   proof qed (auto intro!: exI[of _ "{space M}"])

  1835

  1836 lemma (in finite_measure) emeasure_finite[simp, intro]: "emeasure M A \<noteq> top"

  1837   using finite_emeasure_space emeasure_space[of M A] by (auto simp: top_unique)

  1838

  1839 lemma (in finite_measure) fmeasurable_eq_sets: "fmeasurable M = sets M"

  1840   by (auto simp: fmeasurable_def less_top[symmetric])

  1841

  1842 lemma (in finite_measure) emeasure_eq_measure: "emeasure M A = ennreal (measure M A)"

  1843   by (intro emeasure_eq_ennreal_measure) simp

  1844

  1845 lemma (in finite_measure) emeasure_real: "\<exists>r. 0 \<le> r \<and> emeasure M A = ennreal r"

  1846   using emeasure_finite[of A] by (cases "emeasure M A" rule: ennreal_cases) auto

  1847

  1848 lemma (in finite_measure) bounded_measure: "measure M A \<le> measure M (space M)"

  1849   using emeasure_space[of M A] emeasure_real[of A] emeasure_real[of "space M"] by (auto simp: measure_def)

  1850

  1851 lemma (in finite_measure) finite_measure_Diff:

  1852   assumes sets: "A \<in> sets M" "B \<in> sets M" and "B \<subseteq> A"

  1853   shows "measure M (A - B) = measure M A - measure M B"

  1854   using measure_Diff[OF _ assms] by simp

  1855

  1856 lemma (in finite_measure) finite_measure_Union:

  1857   assumes sets: "A \<in> sets M" "B \<in> sets M" and "A \<inter> B = {}"

  1858   shows "measure M (A \<union> B) = measure M A + measure M B"

  1859   using measure_Union[OF _ _ assms] by simp

  1860

  1861 lemma (in finite_measure) finite_measure_finite_Union:

  1862   assumes measurable: "finite S" "AS \<subseteq> sets M" "disjoint_family_on A S"

  1863   shows "measure M (\<Union>i\<in>S. A i) = (\<Sum>i\<in>S. measure M (A i))"

  1864   using measure_finite_Union[OF assms] by simp

  1865

  1866 lemma (in finite_measure) finite_measure_UNION:

  1867   assumes A: "range A \<subseteq> sets M" "disjoint_family A"

  1868   shows "(\<lambda>i. measure M (A i)) sums (measure M (\<Union>i. A i))"

  1869   using measure_UNION[OF A] by simp

  1870

  1871 lemma (in finite_measure) finite_measure_mono:

  1872   assumes "A \<subseteq> B" "B \<in> sets M" shows "measure M A \<le> measure M B"

  1873   using emeasure_mono[OF assms] emeasure_real[of A] emeasure_real[of B] by (auto simp: measure_def)

  1874

  1875 lemma (in finite_measure) finite_measure_subadditive:

  1876   assumes m: "A \<in> sets M" "B \<in> sets M"

  1877   shows "measure M (A \<union> B) \<le> measure M A + measure M B"

  1878   using measure_subadditive[OF m] by simp

  1879

  1880 lemma (in finite_measure) finite_measure_subadditive_finite:

  1881   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))"

  1882   using measure_subadditive_finite[OF assms] by simp

  1883

  1884 lemma (in finite_measure) finite_measure_subadditive_countably:

  1885   "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))"

  1886   by (rule measure_subadditive_countably)

  1887      (simp_all add: ennreal_suminf_neq_top emeasure_eq_measure)

  1888

  1889 lemma (in finite_measure) finite_measure_eq_sum_singleton:

  1890   assumes "finite S" and *: "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"

  1891   shows "measure M S = (\<Sum>x\<in>S. measure M {x})"

  1892   using measure_eq_sum_singleton[OF assms] by simp

  1893

  1894 lemma (in finite_measure) finite_Lim_measure_incseq:

  1895   assumes A: "range A \<subseteq> sets M" "incseq A"

  1896   shows "(\<lambda>i. measure M (A i)) \<longlonglongrightarrow> measure M (\<Union>i. A i)"

  1897   using Lim_measure_incseq[OF A] by simp

  1898

  1899 lemma (in finite_measure) finite_Lim_measure_decseq:

  1900   assumes A: "range A \<subseteq> sets M" "decseq A"

  1901   shows "(\<lambda>n. measure M (A n)) \<longlonglongrightarrow> measure M (\<Inter>i. A i)"

  1902   using Lim_measure_decseq[OF A] by simp

  1903

  1904 lemma (in finite_measure) finite_measure_compl:

  1905   assumes S: "S \<in> sets M"

  1906   shows "measure M (space M - S) = measure M (space M) - measure M S"

  1907   using measure_Diff[OF _ sets.top S sets.sets_into_space] S by simp

  1908

  1909 lemma (in finite_measure) finite_measure_mono_AE:

  1910   assumes imp: "AE x in M. x \<in> A \<longrightarrow> x \<in> B" and B: "B \<in> sets M"

  1911   shows "measure M A \<le> measure M B"

  1912   using assms emeasure_mono_AE[OF imp B]

  1913   by (simp add: emeasure_eq_measure)

  1914

  1915 lemma (in finite_measure) finite_measure_eq_AE:

  1916   assumes iff: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B"

  1917   assumes A: "A \<in> sets M" and B: "B \<in> sets M"

  1918   shows "measure M A = measure M B"

  1919   using assms emeasure_eq_AE[OF assms] by (simp add: emeasure_eq_measure)

  1920

  1921 lemma (in finite_measure) measure_increasing: "increasing M (measure M)"

  1922   by (auto intro!: finite_measure_mono simp: increasing_def)

  1923

  1924 lemma (in finite_measure) measure_zero_union:

  1925   assumes "s \<in> sets M" "t \<in> sets M" "measure M t = 0"

  1926   shows "measure M (s \<union> t) = measure M s"

  1927 using assms

  1928 proof -

  1929   have "measure M (s \<union> t) \<le> measure M s"

  1930     using finite_measure_subadditive[of s t] assms by auto

  1931   moreover have "measure M (s \<union> t) \<ge> measure M s"

  1932     using assms by (blast intro: finite_measure_mono)

  1933   ultimately show ?thesis by simp

  1934 qed

  1935

  1936 lemma (in finite_measure) measure_eq_compl:

  1937   assumes "s \<in> sets M" "t \<in> sets M"

  1938   assumes "measure M (space M - s) = measure M (space M - t)"

  1939   shows "measure M s = measure M t"

  1940   using assms finite_measure_compl by auto

  1941

  1942 lemma (in finite_measure) measure_eq_bigunion_image:

  1943   assumes "range f \<subseteq> sets M" "range g \<subseteq> sets M"

  1944   assumes "disjoint_family f" "disjoint_family g"

  1945   assumes "\<And> n :: nat. measure M (f n) = measure M (g n)"

  1946   shows "measure M (\<Union>i. f i) = measure M (\<Union>i. g i)"

  1947 using assms

  1948 proof -

  1949   have a: "(\<lambda> i. measure M (f i)) sums (measure M (\<Union>i. f i))"

  1950     by (rule finite_measure_UNION[OF assms(1,3)])

  1951   have b: "(\<lambda> i. measure M (g i)) sums (measure M (\<Union>i. g i))"

  1952     by (rule finite_measure_UNION[OF assms(2,4)])

  1953   show ?thesis using sums_unique[OF b] sums_unique[OF a] assms by simp

  1954 qed

  1955

  1956 lemma (in finite_measure) measure_countably_zero:

  1957   assumes "range c \<subseteq> sets M"

  1958   assumes "\<And> i. measure M (c i) = 0"

  1959   shows "measure M (\<Union>i :: nat. c i) = 0"

  1960 proof (rule antisym)

  1961   show "measure M (\<Union>i :: nat. c i) \<le> 0"

  1962     using finite_measure_subadditive_countably[OF assms(1)] by (simp add: assms(2))

  1963 qed simp

  1964

  1965 lemma (in finite_measure) measure_space_inter:

  1966   assumes events:"s \<in> sets M" "t \<in> sets M"

  1967   assumes "measure M t = measure M (space M)"

  1968   shows "measure M (s \<inter> t) = measure M s"

  1969 proof -

  1970   have "measure M ((space M - s) \<union> (space M - t)) = measure M (space M - s)"

  1971     using events assms finite_measure_compl[of "t"] by (auto intro!: measure_zero_union)

  1972   also have "(space M - s) \<union> (space M - t) = space M - (s \<inter> t)"

  1973     by blast

  1974   finally show "measure M (s \<inter> t) = measure M s"

  1975     using events by (auto intro!: measure_eq_compl[of "s \<inter> t" s])

  1976 qed

  1977

  1978 lemma (in finite_measure) measure_equiprobable_finite_unions:

  1979   assumes s: "finite s" "\<And>x. x \<in> s \<Longrightarrow> {x} \<in> sets M"

  1980   assumes "\<And> x y. \<lbrakk>x \<in> s; y \<in> s\<rbrakk> \<Longrightarrow> measure M {x} = measure M {y}"

  1981   shows "measure M s = real (card s) * measure M {SOME x. x \<in> s}"

  1982 proof cases

  1983   assume "s \<noteq> {}"

  1984   then have "\<exists> x. x \<in> s" by blast

  1985   from someI_ex[OF this] assms

  1986   have prob_some: "\<And> x. x \<in> s \<Longrightarrow> measure M {x} = measure M {SOME y. y \<in> s}" by blast

  1987   have "measure M s = (\<Sum> x \<in> s. measure M {x})"

  1988     using finite_measure_eq_sum_singleton[OF s] by simp

  1989   also have "\<dots> = (\<Sum> x \<in> s. measure M {SOME y. y \<in> s})" using prob_some by auto

  1990   also have "\<dots> = real (card s) * measure M {(SOME x. x \<in> s)}"

  1991     using sum_constant assms by simp

  1992   finally show ?thesis by simp

  1993 qed simp

  1994

  1995 lemma (in finite_measure) measure_real_sum_image_fn:

  1996   assumes "e \<in> sets M"

  1997   assumes "\<And> x. x \<in> s \<Longrightarrow> e \<inter> f x \<in> sets M"

  1998   assumes "finite s"

  1999   assumes disjoint: "\<And> x y. \<lbrakk>x \<in> s ; y \<in> s ; x \<noteq> y\<rbrakk> \<Longrightarrow> f x \<inter> f y = {}"

  2000   assumes upper: "space M \<subseteq> (\<Union>i \<in> s. f i)"

  2001   shows "measure M e = (\<Sum> x \<in> s. measure M (e \<inter> f x))"

  2002 proof -

  2003   have "e \<subseteq> (\<Union>i\<in>s. f i)"

  2004     using \<open>e \<in> sets M\<close> sets.sets_into_space upper by blast

  2005   then have e: "e = (\<Union>i \<in> s. e \<inter> f i)"

  2006     by auto

  2007   hence "measure M e = measure M (\<Union>i \<in> s. e \<inter> f i)" by simp

  2008   also have "\<dots> = (\<Sum> x \<in> s. measure M (e \<inter> f x))"

  2009   proof (rule finite_measure_finite_Union)

  2010     show "finite s" by fact

  2011     show "(\<lambda>i. e \<inter> f i)s \<subseteq> sets M" using assms(2) by auto

  2012     show "disjoint_family_on (\<lambda>i. e \<inter> f i) s"

  2013       using disjoint by (auto simp: disjoint_family_on_def)

  2014   qed

  2015   finally show ?thesis .

  2016 qed

  2017

  2018 lemma (in finite_measure) measure_exclude:

  2019   assumes "A \<in> sets M" "B \<in> sets M"

  2020   assumes "measure M A = measure M (space M)" "A \<inter> B = {}"

  2021   shows "measure M B = 0"

  2022   using measure_space_inter[of B A] assms by (auto simp: ac_simps)

  2023 lemma (in finite_measure) finite_measure_distr:

  2024   assumes f: "f \<in> measurable M M'"

  2025   shows "finite_measure (distr M M' f)"

  2026 proof (rule finite_measureI)

  2027   have "f - space M' \<inter> space M = space M" using f by (auto dest: measurable_space)

  2028   with f show "emeasure (distr M M' f) (space (distr M M' f)) \<noteq> \<infinity>" by (auto simp: emeasure_distr)

  2029 qed

  2030

  2031 lemma emeasure_gfp[consumes 1, case_names cont measurable]:

  2032   assumes sets[simp]: "\<And>s. sets (M s) = sets N"

  2033   assumes "\<And>s. finite_measure (M s)"

  2034   assumes cont: "inf_continuous F" "inf_continuous f"

  2035   assumes meas: "\<And>P. Measurable.pred N P \<Longrightarrow> Measurable.pred N (F P)"

  2036   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"

  2037   assumes bound: "\<And>P. f P \<le> f (\<lambda>s. emeasure (M s) (space (M s)))"

  2038   shows "emeasure (M s) {x\<in>space N. gfp F x} = gfp f s"

  2039 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

  2040     P="Measurable.pred N", symmetric])

  2041   interpret finite_measure "M s" for s by fact

  2042   fix C assume "decseq C" "\<And>i. Measurable.pred N (C i)"

  2043   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}))"

  2044     unfolding INF_apply[abs_def]

  2045     by (subst INF_emeasure_decseq) (auto simp: antimono_def fun_eq_iff intro!: arg_cong2[where f=emeasure])

  2046 next

  2047   show "f x \<le> (\<lambda>s. emeasure (M s) {x \<in> space N. F top x})" for x

  2048     using bound[of x] sets_eq_imp_space_eq[OF sets] by (simp add: iter)

  2049 qed (auto simp add: iter le_fun_def INF_apply[abs_def] intro!: meas cont)

  2050

  2051 subsection \<open>Counting space\<close>

  2052

  2053 lemma strict_monoI_Suc:

  2054   assumes ord [simp]: "(\<And>n. f n < f (Suc n))" shows "strict_mono f"

  2055   unfolding strict_mono_def

  2056 proof safe

  2057   fix n m :: nat assume "n < m" then show "f n < f m"

  2058     by (induct m) (auto simp: less_Suc_eq intro: less_trans ord)

  2059 qed

  2060

  2061 lemma emeasure_count_space:

  2062   assumes "X \<subseteq> A" shows "emeasure (count_space A) X = (if finite X then of_nat (card X) else \<infinity>)"

  2063     (is "_ = ?M X")

  2064   unfolding count_space_def

  2065 proof (rule emeasure_measure_of_sigma)

  2066   show "X \<in> Pow A" using \<open>X \<subseteq> A\<close> by auto

  2067   show "sigma_algebra A (Pow A)" by (rule sigma_algebra_Pow)

  2068   show positive: "positive (Pow A) ?M"

  2069     by (auto simp: positive_def)

  2070   have additive: "additive (Pow A) ?M"

  2071     by (auto simp: card_Un_disjoint additive_def)

  2072

  2073   interpret ring_of_sets A "Pow A"

  2074     by (rule ring_of_setsI) auto

  2075   show "countably_additive (Pow A) ?M"

  2076     unfolding countably_additive_iff_continuous_from_below[OF positive additive]

  2077   proof safe

  2078     fix F :: "nat \<Rightarrow> 'a set" assume "incseq F"

  2079     show "(\<lambda>i. ?M (F i)) \<longlonglongrightarrow> ?M (\<Union>i. F i)"

  2080     proof cases

  2081       assume "\<exists>i. \<forall>j\<ge>i. F i = F j"

  2082       then guess i .. note i = this

  2083       { fix j from i \<open>incseq F\<close> have "F j \<subseteq> F i"

  2084           by (cases "i \<le> j") (auto simp: incseq_def) }

  2085       then have eq: "(\<Union>i. F i) = F i"

  2086         by auto

  2087       with i show ?thesis

  2088         by (auto intro!: Lim_transform_eventually[OF _ tendsto_const] eventually_sequentiallyI[where c=i])

  2089     next

  2090       assume "\<not> (\<exists>i. \<forall>j\<ge>i. F i = F j)"

  2091       then obtain f where f: "\<And>i. i \<le> f i" "\<And>i. F i \<noteq> F (f i)" by metis

  2092       then have "\<And>i. F i \<subseteq> F (f i)" using \<open>incseq F\<close> by (auto simp: incseq_def)

  2093       with f have *: "\<And>i. F i \<subset> F (f i)" by auto

  2094

  2095       have "incseq (\<lambda>i. ?M (F i))"

  2096         using \<open>incseq F\<close> unfolding incseq_def by (auto simp: card_mono dest: finite_subset)

  2097       then have "(\<lambda>i. ?M (F i)) \<longlonglongrightarrow> (SUP n. ?M (F n))"

  2098         by (rule LIMSEQ_SUP)

  2099

  2100       moreover have "(SUP n. ?M (F n)) = top"

  2101       proof (rule ennreal_SUP_eq_top)

  2102         fix n :: nat show "\<exists>k::nat\<in>UNIV. of_nat n \<le> ?M (F k)"

  2103         proof (induct n)

  2104           case (Suc n)

  2105           then guess k .. note k = this

  2106           moreover have "finite (F k) \<Longrightarrow> finite (F (f k)) \<Longrightarrow> card (F k) < card (F (f k))"

  2107             using \<open>F k \<subset> F (f k)\<close> by (simp add: psubset_card_mono)

  2108           moreover have "finite (F (f k)) \<Longrightarrow> finite (F k)"

  2109             using \<open>k \<le> f k\<close> \<open>incseq F\<close> by (auto simp: incseq_def dest: finite_subset)

  2110           ultimately show ?case

  2111             by (auto intro!: exI[of _ "f k"] simp del: of_nat_Suc)

  2112         qed auto

  2113       qed

  2114

  2115       moreover

  2116       have "inj (\<lambda>n. F ((f ^^ n) 0))"

  2117         by (intro strict_mono_imp_inj_on strict_monoI_Suc) (simp add: *)

  2118       then have 1: "infinite (range (\<lambda>i. F ((f ^^ i) 0)))"

  2119         by (rule range_inj_infinite)

  2120       have "infinite (Pow (\<Union>i. F i))"

  2121         by (rule infinite_super[OF _ 1]) auto

  2122       then have "infinite (\<Union>i. F i)"

  2123         by auto

  2124

  2125       ultimately show ?thesis by auto

  2126     qed

  2127   qed

  2128 qed

  2129

  2130 lemma distr_bij_count_space:

  2131   assumes f: "bij_betw f A B"

  2132   shows "distr (count_space A) (count_space B) f = count_space B"

  2133 proof (rule measure_eqI)

  2134   have f': "f \<in> measurable (count_space A) (count_space B)"

  2135     using f unfolding Pi_def bij_betw_def by auto

  2136   fix X assume "X \<in> sets (distr (count_space A) (count_space B) f)"

  2137   then have X: "X \<in> sets (count_space B)" by auto

  2138   moreover from X have "f - X \<inter> A = the_inv_into A f  X"

  2139     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])

  2140   moreover have "inj_on (the_inv_into A f) B"

  2141     using X f by (auto simp: bij_betw_def inj_on_the_inv_into)

  2142   with X have "inj_on (the_inv_into A f) X"

  2143     by (auto intro: subset_inj_on)

  2144   ultimately show "emeasure (distr (count_space A) (count_space B) f) X = emeasure (count_space B) X"

  2145     using f unfolding emeasure_distr[OF f' X]

  2146     by (subst (1 2) emeasure_count_space) (auto simp: card_image dest: finite_imageD)

  2147 qed simp

  2148

  2149 lemma emeasure_count_space_finite[simp]:

  2150   "X \<subseteq> A \<Longrightarrow> finite X \<Longrightarrow> emeasure (count_space A) X = of_nat (card X)"

  2151   using emeasure_count_space[of X A] by simp

  2152

  2153 lemma emeasure_count_space_infinite[simp]:

  2154   "X \<subseteq> A \<Longrightarrow> infinite X \<Longrightarrow> emeasure (count_space A) X = \<infinity>"

  2155   using emeasure_count_space[of X A] by simp

  2156

  2157 lemma measure_count_space: "measure (count_space A) X = (if X \<subseteq> A then of_nat (card X) else 0)"

  2158   by (cases "finite X") (auto simp: measure_notin_sets ennreal_of_nat_eq_real_of_nat

  2159                                     measure_zero_top measure_eq_emeasure_eq_ennreal)

  2160

  2161 lemma emeasure_count_space_eq_0:

  2162   "emeasure (count_space A) X = 0 \<longleftrightarrow> (X \<subseteq> A \<longrightarrow> X = {})"

  2163 proof cases

  2164   assume X: "X \<subseteq> A"

  2165   then show ?thesis

  2166   proof (intro iffI impI)

  2167     assume "emeasure (count_space A) X = 0"

  2168     with X show "X = {}"

  2169       by (subst (asm) emeasure_count_space) (auto split: if_split_asm)

  2170   qed simp

  2171 qed (simp add: emeasure_notin_sets)

  2172

  2173 lemma null_sets_count_space: "null_sets (count_space A) = { {} }"

  2174   unfolding null_sets_def by (auto simp add: emeasure_count_space_eq_0)

  2175

  2176 lemma AE_count_space: "(AE x in count_space A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x)"

  2177   unfolding eventually_ae_filter by (auto simp add: null_sets_count_space)

  2178

  2179 lemma sigma_finite_measure_count_space_countable:

  2180   assumes A: "countable A"

  2181   shows "sigma_finite_measure (count_space A)"

  2182   proof qed (insert A, auto intro!: exI[of _ "(\<lambda>a. {a})  A"])

  2183

  2184 lemma sigma_finite_measure_count_space:

  2185   fixes A :: "'a::countable set" shows "sigma_finite_measure (count_space A)"

  2186   by (rule sigma_finite_measure_count_space_countable) auto

  2187

  2188 lemma finite_measure_count_space:

  2189   assumes [simp]: "finite A"

  2190   shows "finite_measure (count_space A)"

  2191   by rule simp

  2192

  2193 lemma sigma_finite_measure_count_space_finite:

  2194   assumes A: "finite A" shows "sigma_finite_measure (count_space A)"

  2195 proof -

  2196   interpret finite_measure "count_space A" using A by (rule finite_measure_count_space)

  2197   show "sigma_finite_measure (count_space A)" ..

  2198 qed

  2199

  2200 subsection \<open>Measure restricted to space\<close>

  2201

  2202 lemma emeasure_restrict_space:

  2203   assumes "\<Omega> \<inter> space M \<in> sets M" "A \<subseteq> \<Omega>"

  2204   shows "emeasure (restrict_space M \<Omega>) A = emeasure M A"

  2205 proof (cases "A \<in> sets M")

  2206   case True

  2207   show ?thesis

  2208   proof (rule emeasure_measure_of[OF restrict_space_def])

  2209     show "op \<inter> \<Omega>  sets M \<subseteq> Pow (\<Omega> \<inter> space M)" "A \<in> sets (restrict_space M \<Omega>)"

  2210       using \<open>A \<subseteq> \<Omega>\<close> \<open>A \<in> sets M\<close> sets.space_closed by (auto simp: sets_restrict_space)

  2211     show "positive (sets (restrict_space M \<Omega>)) (emeasure M)"

  2212       by (auto simp: positive_def)

  2213     show "countably_additive (sets (restrict_space M \<Omega>)) (emeasure M)"

  2214     proof (rule countably_additiveI)

  2215       fix A :: "nat \<Rightarrow> _" assume "range A \<subseteq> sets (restrict_space M \<Omega>)" "disjoint_family A"

  2216       with assms have "\<And>i. A i \<in> sets M" "\<And>i. A i \<subseteq> space M" "disjoint_family A"

  2217         by (fastforce simp: sets_restrict_space_iff[OF assms(1)] image_subset_iff

  2218                       dest: sets.sets_into_space)+

  2219       then show "(\<Sum>i. emeasure M (A i)) = emeasure M (\<Union>i. A i)"

  2220         by (subst suminf_emeasure) (auto simp: disjoint_family_subset)

  2221     qed

  2222   qed

  2223 next

  2224   case False

  2225   with assms have "A \<notin> sets (restrict_space M \<Omega>)"

  2226     by (simp add: sets_restrict_space_iff)

  2227   with False show ?thesis

  2228     by (simp add: emeasure_notin_sets)

  2229 qed

  2230

  2231 lemma measure_restrict_space:

  2232   assumes "\<Omega> \<inter> space M \<in> sets M" "A \<subseteq> \<Omega>"

  2233   shows "measure (restrict_space M \<Omega>) A = measure M A"

  2234   using emeasure_restrict_space[OF assms] by (simp add: measure_def)

  2235

  2236 lemma AE_restrict_space_iff:

  2237   assumes "\<Omega> \<inter> space M \<in> sets M"

  2238   shows "(AE x in restrict_space M \<Omega>. P x) \<longleftrightarrow> (AE x in M. x \<in> \<Omega> \<longrightarrow> P x)"

  2239 proof -

  2240   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)"

  2241     by auto

  2242   { fix X assume X: "X \<in> sets M" "emeasure M X = 0"

  2243     then have "emeasure M (\<Omega> \<inter> space M \<inter> X) \<le> emeasure M X"

  2244       by (intro emeasure_mono) auto

  2245     then have "emeasure M (\<Omega> \<inter> space M \<inter> X) = 0"

  2246       using X by (auto intro!: antisym) }

  2247   with assms show ?thesis

  2248     unfolding eventually_ae_filter

  2249     by (auto simp add: space_restrict_space null_sets_def sets_restrict_space_iff

  2250                        emeasure_restrict_space cong: conj_cong

  2251              intro!: ex_cong[where f="\<lambda>X. (\<Omega> \<inter> space M) \<inter> X"])

  2252 qed

  2253

  2254 lemma restrict_restrict_space:

  2255   assumes "A \<inter> space M \<in> sets M" "B \<inter> space M \<in> sets M"

  2256   shows "restrict_space (restrict_space M A) B = restrict_space M (A \<inter> B)" (is "?l = ?r")

  2257 proof (rule measure_eqI[symmetric])

  2258   show "sets ?r = sets ?l"

  2259     unfolding sets_restrict_space image_comp by (intro image_cong) auto

  2260 next

  2261   fix X assume "X \<in> sets (restrict_space M (A \<inter> B))"

  2262   then obtain Y where "Y \<in> sets M" "X = Y \<inter> A \<inter> B"

  2263     by (auto simp: sets_restrict_space)

  2264   with assms sets.Int[OF assms] show "emeasure ?r X = emeasure ?l X"

  2265     by (subst (1 2) emeasure_restrict_space)

  2266        (auto simp: space_restrict_space sets_restrict_space_iff emeasure_restrict_space ac_simps)

  2267 qed

  2268

  2269 lemma restrict_count_space: "restrict_space (count_space B) A = count_space (A \<inter> B)"

  2270 proof (rule measure_eqI)

  2271   show "sets (restrict_space (count_space B) A) = sets (count_space (A \<inter> B))"

  2272     by (subst sets_restrict_space) auto

  2273   moreover fix X assume "X \<in> sets (restrict_space (count_space B) A)"

  2274   ultimately have "X \<subseteq> A \<inter> B" by auto

  2275   then show "emeasure (restrict_space (count_space B) A) X = emeasure (count_space (A \<inter> B)) X"

  2276     by (cases "finite X") (auto simp add: emeasure_restrict_space)

  2277 qed

  2278

  2279 lemma sigma_finite_measure_restrict_space:

  2280   assumes "sigma_finite_measure M"

  2281   and A: "A \<in> sets M"

  2282   shows "sigma_finite_measure (restrict_space M A)"

  2283 proof -

  2284   interpret sigma_finite_measure M by fact

  2285   from sigma_finite_countable obtain C

  2286     where C: "countable C" "C \<subseteq> sets M" "(\<Union>C) = space M" "\<forall>a\<in>C. emeasure M a \<noteq> \<infinity>"

  2287     by blast

  2288   let ?C = "op \<inter> A  C"

  2289   from C have "countable ?C" "?C \<subseteq> sets (restrict_space M A)" "(\<Union>?C) = space (restrict_space M A)"

  2290     by(auto simp add: sets_restrict_space space_restrict_space)

  2291   moreover {

  2292     fix a

  2293     assume "a \<in> ?C"

  2294     then obtain a' where "a = A \<inter> a'" "a' \<in> C" ..

  2295     then have "emeasure (restrict_space M A) a \<le> emeasure M a'"

  2296       using A C by(auto simp add: emeasure_restrict_space intro: emeasure_mono)

  2297     also have "\<dots> < \<infinity>" using C(4)[rule_format, of a'] \<open>a' \<in> C\<close> by (simp add: less_top)

  2298     finally have "emeasure (restrict_space M A) a \<noteq> \<infinity>" by simp }

  2299   ultimately show ?thesis

  2300     by unfold_locales (rule exI conjI|assumption|blast)+

  2301 qed

  2302

  2303 lemma finite_measure_restrict_space:

  2304   assumes "finite_measure M"

  2305   and A: "A \<in> sets M"

  2306   shows "finite_measure (restrict_space M A)"

  2307 proof -

  2308   interpret finite_measure M by fact

  2309   show ?thesis

  2310     by(rule finite_measureI)(simp add: emeasure_restrict_space A space_restrict_space)

  2311 qed

  2312

  2313 lemma restrict_distr:

  2314   assumes [measurable]: "f \<in> measurable M N"

  2315   assumes [simp]: "\<Omega> \<inter> space N \<in> sets N" and restrict: "f \<in> space M \<rightarrow> \<Omega>"

  2316   shows "restrict_space (distr M N f) \<Omega> = distr M (restrict_space N \<Omega>) f"

  2317   (is "?l = ?r")

  2318 proof (rule measure_eqI)

  2319   fix A assume "A \<in> sets (restrict_space (distr M N f) \<Omega>)"

  2320   with restrict show "emeasure ?l A = emeasure ?r A"

  2321     by (subst emeasure_distr)

  2322        (auto simp: sets_restrict_space_iff emeasure_restrict_space emeasure_distr

  2323              intro!: measurable_restrict_space2)

  2324 qed (simp add: sets_restrict_space)

  2325

  2326 lemma measure_eqI_restrict_generator:

  2327   assumes E: "Int_stable E" "E \<subseteq> Pow \<Omega>" "\<And>X. X \<in> E \<Longrightarrow> emeasure M X = emeasure N X"

  2328   assumes sets_eq: "sets M = sets N" and \<Omega>: "\<Omega> \<in> sets M"

  2329   assumes "sets (restrict_space M \<Omega>) = sigma_sets \<Omega> E"

  2330   assumes "sets (restrict_space N \<Omega>) = sigma_sets \<Omega> E"

  2331   assumes ae: "AE x in M. x \<in> \<Omega>" "AE x in N. x \<in> \<Omega>"

  2332   assumes A: "countable A" "A \<noteq> {}" "A \<subseteq> E" "\<Union>A = \<Omega>" "\<And>a. a \<in> A \<Longrightarrow> emeasure M a \<noteq> \<infinity>"

  2333   shows "M = N"

  2334 proof (rule measure_eqI)

  2335   fix X assume X: "X \<in> sets M"

  2336   then have "emeasure M X = emeasure (restrict_space M \<Omega>) (X \<inter> \<Omega>)"

  2337     using ae \<Omega> by (auto simp add: emeasure_restrict_space intro!: emeasure_eq_AE)

  2338   also have "restrict_space M \<Omega> = restrict_space N \<Omega>"

  2339   proof (rule measure_eqI_generator_eq)

  2340     fix X assume "X \<in> E"

  2341     then show "emeasure (restrict_space M \<Omega>) X = emeasure (restrict_space N \<Omega>) X"

  2342       using E \<Omega> by (subst (1 2) emeasure_restrict_space) (auto simp: sets_eq sets_eq[THEN sets_eq_imp_space_eq])

  2343   next

  2344     show "range (from_nat_into A) \<subseteq> E" "(\<Union>i. from_nat_into A i) = \<Omega>"

  2345       using A by (auto cong del: strong_SUP_cong)

  2346   next

  2347     fix i

  2348     have "emeasure (restrict_space M \<Omega>) (from_nat_into A i) = emeasure M (from_nat_into A i)"

  2349       using A \<Omega> by (subst emeasure_restrict_space)

  2350                    (auto simp: sets_eq sets_eq[THEN sets_eq_imp_space_eq] intro: from_nat_into)

  2351     with A show "emeasure (restrict_space M \<Omega>) (from_nat_into A i) \<noteq> \<infinity>"

  2352       by (auto intro: from_nat_into)

  2353   qed fact+

  2354   also have "emeasure (restrict_space N \<Omega>) (X \<inter> \<Omega>) = emeasure N X"

  2355     using X ae \<Omega> by (auto simp add: emeasure_restrict_space sets_eq intro!: emeasure_eq_AE)

  2356   finally show "emeasure M X = emeasure N X" .

  2357 qed fact

  2358

  2359 subsection \<open>Null measure\<close>

  2360

  2361 definition "null_measure M = sigma (space M) (sets M)"

  2362

  2363 lemma space_null_measure[simp]: "space (null_measure M) = space M"

  2364   by (simp add: null_measure_def)

  2365

  2366 lemma sets_null_measure[simp, measurable_cong]: "sets (null_measure M) = sets M"

  2367   by (simp add: null_measure_def)

  2368

  2369 lemma emeasure_null_measure[simp]: "emeasure (null_measure M) X = 0"

  2370   by (cases "X \<in> sets M", rule emeasure_measure_of)

  2371      (auto simp: positive_def countably_additive_def emeasure_notin_sets null_measure_def

  2372            dest: sets.sets_into_space)

  2373

  2374 lemma measure_null_measure[simp]: "measure (null_measure M) X = 0"

  2375   by (intro measure_eq_emeasure_eq_ennreal) auto

  2376

  2377 lemma null_measure_idem [simp]: "null_measure (null_measure M) = null_measure M"

  2378   by(rule measure_eqI) simp_all

  2379

  2380 subsection \<open>Scaling a measure\<close>

  2381

  2382 definition scale_measure :: "ennreal \<Rightarrow> 'a measure \<Rightarrow> 'a measure"

  2383 where

  2384   "scale_measure r M = measure_of (space M) (sets M) (\<lambda>A. r * emeasure M A)"

  2385

  2386 lemma space_scale_measure: "space (scale_measure r M) = space M"

  2387   by (simp add: scale_measure_def)

  2388

  2389 lemma sets_scale_measure [simp, measurable_cong]: "sets (scale_measure r M) = sets M"

  2390   by (simp add: scale_measure_def)

  2391

  2392 lemma emeasure_scale_measure [simp]:

  2393   "emeasure (scale_measure r M) A = r * emeasure M A"

  2394   (is "_ = ?\<mu> A")

  2395 proof(cases "A \<in> sets M")

  2396   case True

  2397   show ?thesis unfolding scale_measure_def

  2398   proof(rule emeasure_measure_of_sigma)

  2399     show "sigma_algebra (space M) (sets M)" ..

  2400     show "positive (sets M) ?\<mu>" by (simp add: positive_def)

  2401     show "countably_additive (sets M) ?\<mu>"

  2402     proof (rule countably_additiveI)

  2403       fix A :: "nat \<Rightarrow> _"  assume *: "range A \<subseteq> sets M" "disjoint_family A"

  2404       have "(\<Sum>i. ?\<mu> (A i)) = r * (\<Sum>i. emeasure M (A i))"

  2405         by simp

  2406       also have "\<dots> = ?\<mu> (\<Union>i. A i)" using * by(simp add: suminf_emeasure)

  2407       finally show "(\<Sum>i. ?\<mu> (A i)) = ?\<mu> (\<Union>i. A i)" .

  2408     qed

  2409   qed(fact True)

  2410 qed(simp add: emeasure_notin_sets)

  2411

  2412 lemma scale_measure_1 [simp]: "scale_measure 1 M = M"

  2413   by(rule measure_eqI) simp_all

  2414

  2415 lemma scale_measure_0[simp]: "scale_measure 0 M = null_measure M"

  2416   by(rule measure_eqI) simp_all

  2417

  2418 lemma measure_scale_measure [simp]: "0 \<le> r \<Longrightarrow> measure (scale_measure r M) A = r * measure M A"

  2419   using emeasure_scale_measure[of r M A]

  2420     emeasure_eq_ennreal_measure[of M A]

  2421     measure_eq_emeasure_eq_ennreal[of _ "scale_measure r M" A]

  2422   by (cases "emeasure (scale_measure r M) A = top")

  2423      (auto simp del: emeasure_scale_measure

  2424            simp: ennreal_top_eq_mult_iff ennreal_mult_eq_top_iff measure_zero_top ennreal_mult[symmetric])

  2425

  2426 lemma scale_scale_measure [simp]:

  2427   "scale_measure r (scale_measure r' M) = scale_measure (r * r') M"

  2428   by (rule measure_eqI) (simp_all add: max_def mult.assoc)

  2429

  2430 lemma scale_null_measure [simp]: "scale_measure r (null_measure M) = null_measure M"

  2431   by (rule measure_eqI) simp_all

  2432

  2433

  2434 subsection \<open>Complete lattice structure on measures\<close>

  2435

  2436 lemma (in finite_measure) finite_measure_Diff':

  2437   "A \<in> sets M \<Longrightarrow> B \<in> sets M \<Longrightarrow> measure M (A - B) = measure M A - measure M (A \<inter> B)"

  2438   using finite_measure_Diff[of A "A \<inter> B"] by (auto simp: Diff_Int)

  2439

  2440 lemma (in finite_measure) finite_measure_Union':

  2441   "A \<in> sets M \<Longrightarrow> B \<in> sets M \<Longrightarrow> measure M (A \<union> B) = measure M A + measure M (B - A)"

  2442   using finite_measure_Union[of A "B - A"] by auto

  2443

  2444 lemma finite_unsigned_Hahn_decomposition:

  2445   assumes "finite_measure M" "finite_measure N" and [simp]: "sets N = sets M"

  2446   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)"

  2447 proof -

  2448   interpret M: finite_measure M by fact

  2449   interpret N: finite_measure N by fact

  2450

  2451   define d where "d X = measure M X - measure N X" for X

  2452

  2453   have [intro]: "bdd_above (dsets M)"

  2454     using sets.sets_into_space[of _ M]

  2455     by (intro bdd_aboveI[where M="measure M (space M)"])

  2456        (auto simp: d_def field_simps subset_eq intro!: add_increasing M.finite_measure_mono)

  2457

  2458   define \<gamma> where "\<gamma> = (SUP X:sets M. d X)"

  2459   have le_\<gamma>[intro]: "X \<in> sets M \<Longrightarrow> d X \<le> \<gamma>" for X

  2460     by (auto simp: \<gamma>_def intro!: cSUP_upper)

  2461

  2462   have "\<exists>f. \<forall>n. f n \<in> sets M \<and> d (f n) > \<gamma> - 1 / 2^n"

  2463   proof (intro choice_iff[THEN iffD1] allI)

  2464     fix n

  2465     have "\<exists>X\<in>sets M. \<gamma> - 1 / 2^n < d X"

  2466       unfolding \<gamma>_def by (intro less_cSUP_iff[THEN iffD1]) auto

  2467     then show "\<exists>y. y \<in> sets M \<and> \<gamma> - 1 / 2 ^ n < d y"

  2468       by auto

  2469   qed

  2470   then obtain E where [measurable]: "E n \<in> sets M" and E: "d (E n) > \<gamma> - 1 / 2^n" for n

  2471     by auto

  2472

  2473   define F where "F m n = (if m \<le> n then \<Inter>i\<in>{m..n}. E i else space M)" for m n

  2474

  2475   have [measurable]: "m \<le> n \<Longrightarrow> F m n \<in> sets M" for m n

  2476     by (auto simp: F_def)

  2477

  2478   have 1: "\<gamma> - 2 / 2 ^ m + 1 / 2 ^ n \<le> d (F m n)" if "m \<le> n" for m n

  2479     using that

  2480   proof (induct rule: dec_induct)

  2481     case base with E[of m] show ?case

  2482       by (simp add: F_def field_simps)

  2483   next

  2484     case (step i)

  2485     have F_Suc: "F m (Suc i) = F m i \<inter> E (Suc i)"

  2486       using \<open>m \<le> i\<close> by (auto simp: F_def le_Suc_eq)

  2487

  2488     have "\<gamma> + (\<gamma> - 2 / 2^m + 1 / 2 ^ Suc i) \<le> (\<gamma> - 1 / 2^Suc i) + (\<gamma> - 2 / 2^m + 1 / 2^i)"

  2489       by (simp add: field_simps)

  2490     also have "\<dots> \<le> d (E (Suc i)) + d (F m i)"

  2491       using E[of "Suc i"] by (intro add_mono step) auto

  2492     also have "\<dots> = d (E (Suc i)) + d (F m i - E (Suc i)) + d (F m (Suc i))"

  2493       using \<open>m \<le> i\<close> by (simp add: d_def field_simps F_Suc M.finite_measure_Diff' N.finite_measure_Diff')

  2494     also have "\<dots> = d (E (Suc i) \<union> F m i) + d (F m (Suc i))"

  2495       using \<open>m \<le> i\<close> by (simp add: d_def field_simps M.finite_measure_Union' N.finite_measure_Union')

  2496     also have "\<dots> \<le> \<gamma> + d (F m (Suc i))"

  2497       using \<open>m \<le> i\<close> by auto

  2498     finally show ?case

  2499       by auto

  2500   qed

  2501

  2502   define F' where "F' m = (\<Inter>i\<in>{m..}. E i)" for m

  2503   have F'_eq: "F' m = (\<Inter>i. F m (i + m))" for m

  2504     by (fastforce simp: le_iff_add[of m] F'_def F_def)

  2505

  2506   have [measurable]: "F' m \<in> sets M" for m

  2507     by (auto simp: F'_def)

  2508

  2509   have \<gamma>_le: "\<gamma> - 0 \<le> d (\<Union>m. F' m)"

  2510   proof (rule LIMSEQ_le)

  2511     show "(\<lambda>n. \<gamma> - 2 / 2 ^ n) \<longlonglongrightarrow> \<gamma> - 0"

  2512       by (intro tendsto_intros LIMSEQ_divide_realpow_zero) auto

  2513     have "incseq F'"

  2514       by (auto simp: incseq_def F'_def)

  2515     then show "(\<lambda>m. d (F' m)) \<longlonglongrightarrow> d (\<Union>m. F' m)"

  2516       unfolding d_def

  2517       by (intro tendsto_diff M.finite_Lim_measure_incseq N.finite_Lim_measure_incseq) auto

  2518

  2519     have "\<gamma> - 2 / 2 ^ m + 0 \<le> d (F' m)" for m

  2520     proof (rule LIMSEQ_le)

  2521       have *: "decseq (\<lambda>n. F m (n + m))"

  2522         by (auto simp: decseq_def F_def)

  2523       show "(\<lambda>n. d (F m n)) \<longlonglongrightarrow> d (F' m)"

  2524         unfolding d_def F'_eq

  2525         by (rule LIMSEQ_offset[where k=m])

  2526            (auto intro!: tendsto_diff M.finite_Lim_measure_decseq N.finite_Lim_measure_decseq *)

  2527       show "(\<lambda>n. \<gamma> - 2 / 2 ^ m + 1 / 2 ^ n) \<longlonglongrightarrow> \<gamma> - 2 / 2 ^ m + 0"

  2528         by (intro tendsto_add LIMSEQ_divide_realpow_zero tendsto_const) auto

  2529       show "\<exists>N. \<forall>n\<ge>N. \<gamma> - 2 / 2 ^ m + 1 / 2 ^ n \<le> d (F m n)"

  2530         using 1[of m] by (intro exI[of _ m]) auto

  2531     qed

  2532     then show "\<exists>N. \<forall>n\<ge>N. \<gamma> - 2 / 2 ^ n \<le> d (F' n)"

  2533       by auto

  2534   qed

  2535

  2536   show ?thesis

  2537   proof (safe intro!: bexI[of _ "\<Union>m. F' m"])

  2538     fix X assume [measurable]: "X \<in> sets M" and X: "X \<subseteq> (\<Union>m. F' m)"

  2539     have "d (\<Union>m. F' m) - d X = d ((\<Union>m. F' m) - X)"

  2540       using X by (auto simp: d_def M.finite_measure_Diff N.finite_measure_Diff)

  2541     also have "\<dots> \<le> \<gamma>"

  2542       by auto

  2543     finally have "0 \<le> d X"

  2544       using \<gamma>_le by auto

  2545     then show "emeasure N X \<le> emeasure M X"

  2546       by (auto simp: d_def M.emeasure_eq_measure N.emeasure_eq_measure)

  2547   next

  2548     fix X assume [measurable]: "X \<in> sets M" and X: "X \<inter> (\<Union>m. F' m) = {}"

  2549     then have "d (\<Union>m. F' m) + d X = d (X \<union> (\<Union>m. F' m))"

  2550       by (auto simp: d_def M.finite_measure_Union N.finite_measure_Union)

  2551     also have "\<dots> \<le> \<gamma>"

  2552       by auto

  2553     finally have "d X \<le> 0"

  2554       using \<gamma>_le by auto

  2555     then show "emeasure M X \<le> emeasure N X"

  2556       by (auto simp: d_def M.emeasure_eq_measure N.emeasure_eq_measure)

  2557   qed auto

  2558 qed

  2559

  2560 lemma unsigned_Hahn_decomposition:

  2561   assumes [simp]: "sets N = sets M" and [measurable]: "A \<in> sets M"

  2562     and [simp]: "emeasure M A \<noteq> top" "emeasure N A \<noteq> top"

  2563   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)"

  2564 proof -

  2565   have "\<exists>Y\<in>sets (restrict_space M A).

  2566     (\<forall>X\<in>sets (restrict_space M A). X \<subseteq> Y \<longrightarrow> (restrict_space N A) X \<le> (restrict_space M A) X) \<and>

  2567     (\<forall>X\<in>sets (restrict_space M A). X \<inter> Y = {} \<longrightarrow> (restrict_space M A) X \<le> (restrict_space N A) X)"

  2568   proof (rule finite_unsigned_Hahn_decomposition)

  2569     show "finite_measure (restrict_space M A)" "finite_measure (restrict_space N A)"

  2570       by (auto simp: space_restrict_space emeasure_restrict_space less_top intro!: finite_measureI)

  2571   qed (simp add: sets_restrict_space)

  2572   then guess Y ..

  2573   then show ?thesis

  2574     apply (intro bexI[of _ Y] conjI ballI conjI)

  2575     apply (simp_all add: sets_restrict_space emeasure_restrict_space)

  2576     apply safe

  2577     subgoal for X Z

  2578       by (erule ballE[of _ _ X]) (auto simp add: Int_absorb1)

  2579     subgoal for X Z

  2580       by (erule ballE[of _ _ X])  (auto simp add: Int_absorb1 ac_simps)

  2581     apply auto

  2582     done

  2583 qed

  2584

  2585 text \<open>

  2586   Define a lexicographical order on @{type measure}, in the order space, sets and measure. The parts

  2587   of the lexicographical order are point-wise ordered.

  2588 \<close>

  2589

  2590 instantiation measure :: (type) order_bot

  2591 begin

  2592

  2593 inductive less_eq_measure :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> bool" where

  2594   "space M \<subset> space N \<Longrightarrow> less_eq_measure M N"

  2595 | "space M = space N \<Longrightarrow> sets M \<subset> sets N \<Longrightarrow> less_eq_measure M N"

  2596 | "space M = space N \<Longrightarrow> sets M = sets N \<Longrightarrow> emeasure M \<le> emeasure N \<Longrightarrow> less_eq_measure M N"

  2597

  2598 lemma le_measure_iff:

  2599   "M \<le> N \<longleftrightarrow> (if space M = space N then

  2600     if sets M = sets N then emeasure M \<le> emeasure N else sets M \<subseteq> sets N else space M \<subseteq> space N)"

  2601   by (auto elim: less_eq_measure.cases intro: less_eq_measure.intros)

  2602

  2603 definition less_measure :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> bool" where

  2604   "less_measure M N \<longleftrightarrow> (M \<le> N \<and> \<not> N \<le> M)"

  2605

  2606 definition bot_measure :: "'a measure" where

  2607   "bot_measure = sigma {} {}"

  2608

  2609 lemma

  2610   shows space_bot[simp]: "space bot = {}"

  2611     and sets_bot[simp]: "sets bot = {{}}"

  2612     and emeasure_bot[simp]: "emeasure bot X = 0"

  2613   by (auto simp: bot_measure_def sigma_sets_empty_eq emeasure_sigma)

  2614

  2615 instance

  2616 proof standard

  2617   show "bot \<le> a" for a :: "'a measure"

  2618     by (simp add: le_measure_iff bot_measure_def sigma_sets_empty_eq emeasure_sigma le_fun_def)

  2619 qed (auto simp: le_measure_iff less_measure_def split: if_split_asm intro: measure_eqI)

  2620

  2621 end

  2622

  2623 lemma le_measure: "sets M = sets N \<Longrightarrow> M \<le> N \<longleftrightarrow> (\<forall>A\<in>sets M. emeasure M A \<le> emeasure N A)"

  2624   apply (auto simp: le_measure_iff le_fun_def dest: sets_eq_imp_space_eq)

  2625   subgoal for X

  2626     by (cases "X \<in> sets M") (auto simp: emeasure_notin_sets)

  2627   done

  2628

  2629 definition sup_measure' :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> 'a measure"

  2630 where

  2631   "sup_measure' A B = measure_of (space A) (sets A) (\<lambda>X. SUP Y:sets A. emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y))"

  2632

  2633 lemma assumes [simp]: "sets B = sets A"

  2634   shows space_sup_measure'[simp]: "space (sup_measure' A B) = space A"

  2635     and sets_sup_measure'[simp]: "sets (sup_measure' A B) = sets A"

  2636   using sets_eq_imp_space_eq[OF assms] by (simp_all add: sup_measure'_def)

  2637

  2638 lemma emeasure_sup_measure':

  2639   assumes sets_eq[simp]: "sets B = sets A" and [simp, intro]: "X \<in> sets A"

  2640   shows "emeasure (sup_measure' A B) X = (SUP Y:sets A. emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y))"

  2641     (is "_ = ?S X")

  2642 proof -

  2643   note sets_eq_imp_space_eq[OF sets_eq, simp]

  2644   show ?thesis

  2645     using sup_measure'_def

  2646   proof (rule emeasure_measure_of)

  2647     let ?d = "\<lambda>X Y. emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y)"

  2648     show "countably_additive (sets (sup_measure' A B)) (\<lambda>X. SUP Y : sets A. emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y))"

  2649     proof (rule countably_additiveI, goal_cases)

  2650       case (1 X)

  2651       then have [measurable]: "\<And>i. X i \<in> sets A" and "disjoint_family X"

  2652         by auto

  2653       have "(\<Sum>i. ?S (X i)) = (SUP Y:sets A. \<Sum>i. ?d (X i) Y)"

  2654       proof (rule ennreal_suminf_SUP_eq_directed)

  2655         fix J :: "nat set" and a b assume "finite J" and [measurable]: "a \<in> sets A" "b \<in> sets A"

  2656         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

  2657         proof cases

  2658           assume "emeasure A (X i) = top \<or> emeasure B (X i) = top"

  2659           then show ?thesis

  2660           proof

  2661             assume "emeasure A (X i) = top" then show ?thesis

  2662               by (intro bexI[of _ "X i"]) auto

  2663           next

  2664             assume "emeasure B (X i) = top" then show ?thesis

  2665               by (intro bexI[of _ "{}"]) auto

  2666           qed

  2667         next

  2668           assume finite: "\<not> (emeasure A (X i) = top \<or> emeasure B (X i) = top)"

  2669           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)"

  2670             using unsigned_Hahn_decomposition[of B A "X i"] by simp

  2671           then obtain Y where [measurable]: "Y \<in> sets A" and [simp]: "Y \<subseteq> X i"

  2672             and B_le_A: "\<And>C. C \<in> sets A \<Longrightarrow> C \<subseteq> Y \<Longrightarrow> B C \<le> A C"

  2673             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"

  2674             by auto

  2675

  2676           show ?thesis

  2677           proof (intro bexI[of _ Y] ballI conjI)

  2678             fix a assume [measurable]: "a \<in> sets A"

  2679             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"

  2680               for a Y by auto

  2681             then have "?d (X i) a =

  2682               (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))"

  2683               by (subst (1 2) plus_emeasure) (auto simp: Diff_eq[symmetric])

  2684             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))"

  2685               by (intro add_mono order_refl B_le_A A_le_B) (auto simp: Diff_eq[symmetric])

  2686             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))"

  2687               by (simp add: ac_simps)

  2688             also have "\<dots> \<le> A (X i \<inter> Y) + B (X i \<inter> - Y)"

  2689               by (subst (1 2) plus_emeasure) (auto simp: Diff_eq[symmetric] *)

  2690             finally show "?d (X i) a \<le> ?d (X i) Y" .

  2691           qed auto

  2692         qed

  2693         then obtain C where [measurable]: "C i \<in> sets A" and "C i \<subseteq> X i"

  2694           and C: "\<And>a. a \<in> sets A \<Longrightarrow> ?d (X i) a \<le> ?d (X i) (C i)" for i

  2695           by metis

  2696         have *: "X i \<inter> (\<Union>i. C i) = X i \<inter> C i" for i

  2697         proof safe

  2698           fix x j assume "x \<in> X i" "x \<in> C j"

  2699           moreover have "i = j \<or> X i \<inter> X j = {}"

  2700             using \<open>disjoint_family X\<close> by (auto simp: disjoint_family_on_def)

  2701           ultimately show "x \<in> C i"

  2702             using \<open>C i \<subseteq> X i\<close> \<open>C j \<subseteq> X j\<close> by auto

  2703         qed auto

  2704         have **: "X i \<inter> - (\<Union>i. C i) = X i \<inter> - C i" for i

  2705         proof safe

  2706           fix x j assume "x \<in> X i" "x \<notin> C i" "x \<in> C j"

  2707           moreover have "i = j \<or> X i \<inter> X j = {}"

  2708             using \<open>disjoint_family X\<close> by (auto simp: disjoint_family_on_def)

  2709           ultimately show False

  2710             using \<open>C i \<subseteq> X i\<close> \<open>C j \<subseteq> X j\<close> by auto

  2711         qed auto

  2712         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"

  2713           apply (intro bexI[of _ "\<Union>i. C i"])

  2714           unfolding * **

  2715           apply (intro C ballI conjI)

  2716           apply auto

  2717           done

  2718       qed

  2719       also have "\<dots> = ?S (\<Union>i. X i)"

  2720         unfolding UN_extend_simps(4)

  2721         by (auto simp add: suminf_add[symmetric] Diff_eq[symmetric] simp del: UN_simps

  2722                  intro!: SUP_cong arg_cong2[where f="op +"] suminf_emeasure

  2723                          disjoint_family_on_bisimulation[OF \<open>disjoint_family X\<close>])

  2724       finally show "(\<Sum>i. ?S (X i)) = ?S (\<Union>i. X i)" .

  2725     qed

  2726   qed (auto dest: sets.sets_into_space simp: positive_def intro!: SUP_const)

  2727 qed

  2728

  2729 lemma le_emeasure_sup_measure'1:

  2730   assumes "sets B = sets A" "X \<in> sets A" shows "emeasure A X \<le> emeasure (sup_measure' A B) X"

  2731   by (subst emeasure_sup_measure'[OF assms]) (auto intro!: SUP_upper2[of "X"] assms)

  2732

  2733 lemma le_emeasure_sup_measure'2:

  2734   assumes "sets B = sets A" "X \<in> sets A" shows "emeasure B X \<le> emeasure (sup_measure' A B) X"

  2735   by (subst emeasure_sup_measure'[OF assms]) (auto intro!: SUP_upper2[of "{}"] assms)

  2736

  2737 lemma emeasure_sup_measure'_le2:

  2738   assumes [simp]: "sets B = sets C" "sets A = sets C" and [measurable]: "X \<in> sets C"

  2739   assumes A: "\<And>Y. Y \<subseteq> X \<Longrightarrow> Y \<in> sets A \<Longrightarrow> emeasure A Y \<le> emeasure C Y"

  2740   assumes B: "\<And>Y. Y \<subseteq> X \<Longrightarrow> Y \<in> sets A \<Longrightarrow> emeasure B Y \<le> emeasure C Y"

  2741   shows "emeasure (sup_measure' A B) X \<le> emeasure C X"

  2742 proof (subst emeasure_sup_measure')

  2743   show "(SUP Y:sets A. emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y)) \<le> emeasure C X"

  2744     unfolding \<open>sets A = sets C\<close>

  2745   proof (intro SUP_least)

  2746     fix Y assume [measurable]: "Y \<in> sets C"

  2747     have [simp]: "X \<inter> Y \<union> (X - Y) = X"

  2748       by auto

  2749     have "emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y) \<le> emeasure C (X \<inter> Y) + emeasure C (X \<inter> - Y)"

  2750       by (intro add_mono A B) (auto simp: Diff_eq[symmetric])

  2751     also have "\<dots> = emeasure C X"

  2752       by (subst plus_emeasure) (auto simp: Diff_eq[symmetric])

  2753     finally show "emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y) \<le> emeasure C X" .

  2754   qed

  2755 qed simp_all

  2756

  2757 definition sup_lexord :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b::order) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a"

  2758 where

  2759   "sup_lexord A B k s c =

  2760     (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)"

  2761

  2762 lemma sup_lexord:

  2763   "(k A < k B \<Longrightarrow> P B) \<Longrightarrow> (k B < k A \<Longrightarrow> P A) \<Longrightarrow> (k A = k B \<Longrightarrow> P c) \<Longrightarrow>

  2764     (\<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)"

  2765   by (auto simp: sup_lexord_def)

  2766

  2767 lemmas le_sup_lexord = sup_lexord[where P="\<lambda>a. c \<le> a" for c]

  2768

  2769 lemma sup_lexord1: "k A = k B \<Longrightarrow> sup_lexord A B k s c = c"

  2770   by (simp add: sup_lexord_def)

  2771

  2772 lemma sup_lexord_commute: "sup_lexord A B k s c = sup_lexord B A k s c"

  2773   by (auto simp: sup_lexord_def)

  2774

  2775 lemma sigma_sets_le_sets_iff: "(sigma_sets (space x) \<A> \<subseteq> sets x) = (\<A> \<subseteq> sets x)"

  2776   using sets.sigma_sets_subset[of \<A> x] by auto

  2777

  2778 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))"

  2779   by (cases "\<Omega> = space x")

  2780      (simp_all add: eq_commute[of _ "sets x"] le_measure_iff emeasure_sigma le_fun_def

  2781                     sigma_sets_superset_generator sigma_sets_le_sets_iff)

  2782

  2783 instantiation measure :: (type) semilattice_sup

  2784 begin

  2785

  2786 definition sup_measure :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> 'a measure"

  2787 where

  2788   "sup_measure A B =

  2789     sup_lexord A B space (sigma (space A \<union> space B) {})

  2790       (sup_lexord A B sets (sigma (space A) (sets A \<union> sets B)) (sup_measure' A B))"

  2791

  2792 instance

  2793 proof

  2794   fix x y z :: "'a measure"

  2795   show "x \<le> sup x y"

  2796     unfolding sup_measure_def

  2797   proof (intro le_sup_lexord)

  2798     assume "space x = space y"

  2799     then have *: "sets x \<union> sets y \<subseteq> Pow (space x)"

  2800       using sets.space_closed by auto

  2801     assume "\<not> sets y \<subseteq> sets x" "\<not> sets x \<subseteq> sets y"

  2802     then have "sets x \<subset> sets x \<union> sets y"

  2803       by auto

  2804     also have "\<dots> \<le> sigma (space x) (sets x \<union> sets y)"

  2805       by (subst sets_measure_of[OF *]) (rule sigma_sets_superset_generator)

  2806     finally show "x \<le> sigma (space x) (sets x \<union> sets y)"

  2807       by (simp add: space_measure_of[OF *] less_eq_measure.intros(2))

  2808   next

  2809     assume "\<not> space y \<subseteq> space x" "\<not> space x \<subseteq> space y"

  2810     then show "x \<le> sigma (space x \<union> space y) {}"

  2811       by (intro less_eq_measure.intros) auto

  2812   next

  2813     assume "sets x = sets y" then show "x \<le> sup_measure' x y"

  2814       by (simp add: le_measure le_emeasure_sup_measure'1)

  2815   qed (auto intro: less_eq_measure.intros)

  2816   show "y \<le> sup x y"

  2817     unfolding sup_measure_def

  2818   proof (intro le_sup_lexord)

  2819     assume **: "space x = space y"

  2820     then have *: "sets x \<union> sets y \<subseteq> Pow (space y)"

  2821       using sets.space_closed by auto

  2822     assume "\<not> sets y \<subseteq> sets x" "\<not> sets x \<subseteq> sets y"

  2823     then have "sets y \<subset> sets x \<union> sets y"

  2824       by auto

  2825     also have "\<dots> \<le> sigma (space y) (sets x \<union> sets y)"

  2826       by (subst sets_measure_of[OF *]) (rule sigma_sets_superset_generator)

  2827     finally show "y \<le> sigma (space x) (sets x \<union> sets y)"

  2828       by (simp add: ** space_measure_of[OF *] less_eq_measure.intros(2))

  2829   next

  2830     assume "\<not> space y \<subseteq> space x" "\<not> space x \<subseteq> space y"

  2831     then show "y \<le> sigma (space x \<union> space y) {}"

  2832       by (intro less_eq_measure.intros) auto

  2833   next

  2834     assume "sets x = sets y" then show "y \<le> sup_measure' x y"

  2835       by (simp add: le_measure le_emeasure_sup_measure'2)

  2836   qed (auto intro: less_eq_measure.intros)

  2837   show "x \<le> y \<Longrightarrow> z \<le> y \<Longrightarrow> sup x z \<le> y"

  2838     unfolding sup_measure_def

  2839   proof (intro sup_lexord[where P="\<lambda>x. x \<le> y"])

  2840     assume "x \<le> y" "z \<le> y" and [simp]: "space x = space z" "sets x = sets z"

  2841     from \<open>x \<le> y\<close> show "sup_measure' x z \<le> y"

  2842     proof cases

  2843       case 1 then show ?thesis

  2844         by (intro less_eq_measure.intros(1)) simp

  2845     next

  2846       case 2 then show ?thesis

  2847         by (intro less_eq_measure.intros(2)) simp_all

  2848     next

  2849       case 3 with \<open>z \<le> y\<close> \<open>x \<le> y\<close> show ?thesis

  2850         by (auto simp add: le_measure intro!: emeasure_sup_measure'_le2)

  2851     qed

  2852   next

  2853     assume **: "x \<le> y" "z \<le> y" "space x = space z" "\<not> sets z \<subseteq> sets x" "\<not> sets x \<subseteq> sets z"

  2854     then have *: "sets x \<union> sets z \<subseteq> Pow (space x)"

  2855       using sets.space_closed by auto

  2856     show "sigma (space x) (sets x \<union> sets z) \<le> y"

  2857       unfolding sigma_le_iff[OF *] using ** by (auto simp: le_measure_iff split: if_split_asm)

  2858   next

  2859     assume "x \<le> y" "z \<le> y" "\<not> space z \<subseteq> space x" "\<not> space x \<subseteq> space z"

  2860     then have "space x \<subseteq> space y" "space z \<subseteq> space y"

  2861       by (auto simp: le_measure_iff split: if_split_asm)

  2862     then show "sigma (space x \<union> space z) {} \<le> y"

  2863       by (simp add: sigma_le_iff)

  2864   qed

  2865 qed

  2866

  2867 end

  2868

  2869 lemma space_empty_eq_bot: "space a = {} \<longleftrightarrow> a = bot"

  2870   using space_empty[of a] by (auto intro!: measure_eqI)

  2871

  2872 lemma sets_eq_iff_bounded: "A \<le> B \<Longrightarrow> B \<le> C \<Longrightarrow> sets A = sets C \<Longrightarrow> sets B = sets A"

  2873   by (auto dest: sets_eq_imp_space_eq simp add: le_measure_iff split: if_split_asm)

  2874

  2875 lemma sets_sup: "sets A = sets M \<Longrightarrow> sets B = sets M \<Longrightarrow> sets (sup A B) = sets M"

  2876   by (auto simp add: sup_measure_def sup_lexord_def dest: sets_eq_imp_space_eq)

  2877

  2878 lemma le_measureD1: "A \<le> B \<Longrightarrow> space A \<le> space B"

  2879   by (auto simp: le_measure_iff split: if_split_asm)

  2880

  2881 lemma le_measureD2: "A \<le> B \<Longrightarrow> space A = space B \<Longrightarrow> sets A \<le> sets B"

  2882   by (auto simp: le_measure_iff split: if_split_asm)

  2883

  2884 lemma le_measureD3: "A \<le> B \<Longrightarrow> sets A = sets B \<Longrightarrow> emeasure A X \<le> emeasure B X"

  2885   by (auto simp: le_measure_iff le_fun_def dest: sets_eq_imp_space_eq split: if_split_asm)

  2886

  2887 lemma UN_space_closed: "UNION S sets \<subseteq> Pow (UNION S space)"

  2888   using sets.space_closed by auto

  2889

  2890 definition Sup_lexord :: "('a \<Rightarrow> 'b::complete_lattice) \<Rightarrow> ('a set \<Rightarrow> 'a) \<Rightarrow> ('a set \<Rightarrow> 'a) \<Rightarrow> 'a set \<Rightarrow> 'a"

  2891 where

  2892   "Sup_lexord k c s A = (let U = (SUP a:A. k a) in if \<exists>a\<in>A. k a = U then c {a\<in>A. k a = U} else s A)"

  2893

  2894 lemma Sup_lexord:

  2895   "(\<And>a S. a \<in> A \<Longrightarrow> k a = (SUP a:A. k a) \<Longrightarrow> S = {a'\<in>A. k a' = k a} \<Longrightarrow> P (c S)) \<Longrightarrow> ((\<And>a. a \<in> A \<Longrightarrow> k a \<noteq> (SUP a:A. k a)) \<Longrightarrow> P (s A)) \<Longrightarrow>

  2896     P (Sup_lexord k c s A)"

  2897   by (auto simp: Sup_lexord_def Let_def)

  2898

  2899 lemma Sup_lexord1:

  2900   assumes A: "A \<noteq> {}" "(\<And>a. a \<in> A \<Longrightarrow> k a = (\<Union>a\<in>A. k a))" "P (c A)"

  2901   shows "P (Sup_lexord k c s A)"

  2902   unfolding Sup_lexord_def Let_def

  2903 proof (clarsimp, safe)

  2904   show "\<forall>a\<in>A. k a \<noteq> (\<Union>x\<in>A. k x) \<Longrightarrow> P (s A)"

  2905     by (metis assms(1,2) ex_in_conv)

  2906 next

  2907   fix a assume "a \<in> A" "k a = (\<Union>x\<in>A. k x)"

  2908   then have "{a \<in> A. k a = (\<Union>x\<in>A. k x)} = {a \<in> A. k a = k a}"

  2909     by (metis A(2)[symmetric])

  2910   then show "P (c {a \<in> A. k a = (\<Union>x\<in>A. k x)})"

  2911     by (simp add: A(3))

  2912 qed

  2913

  2914 instantiation measure :: (type) complete_lattice

  2915 begin

  2916

  2917 interpretation sup_measure: comm_monoid_set sup "bot :: 'a measure"

  2918   by standard (auto intro!: antisym)

  2919

  2920 lemma sup_measure_F_mono':

  2921   "finite J \<Longrightarrow> finite I \<Longrightarrow> sup_measure.F id I \<le> sup_measure.F id (I \<union> J)"

  2922 proof (induction J rule: finite_induct)

  2923   case empty then show ?case

  2924     by simp

  2925 next

  2926   case (insert i J)

  2927   show ?case

  2928   proof cases

  2929     assume "i \<in> I" with insert show ?thesis

  2930       by (auto simp: insert_absorb)

  2931   next

  2932     assume "i \<notin> I"

  2933     have "sup_measure.F id I \<le> sup_measure.F id (I \<union> J)"

  2934       by (intro insert)

  2935     also have "\<dots> \<le> sup_measure.F id (insert i (I \<union> J))"

  2936       using insert \<open>i \<notin> I\<close> by (subst sup_measure.insert) auto

  2937     finally show ?thesis

  2938       by auto

  2939   qed

  2940 qed

  2941

  2942 lemma sup_measure_F_mono: "finite I \<Longrightarrow> J \<subseteq> I \<Longrightarrow> sup_measure.F id J \<le> sup_measure.F id I"

  2943   using sup_measure_F_mono'[of I J] by (auto simp: finite_subset Un_absorb1)

  2944

  2945 lemma sets_sup_measure_F:

  2946   "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"

  2947   by (induction I rule: finite_ne_induct) (simp_all add: sets_sup)

  2948

  2949 definition Sup_measure' :: "'a measure set \<Rightarrow> 'a measure"

  2950 where

  2951   "Sup_measure' M = measure_of (\<Union>a\<in>M. space a) (\<Union>a\<in>M. sets a)

  2952     (\<lambda>X. (SUP P:{P. finite P \<and> P \<subseteq> M }. sup_measure.F id P X))"

  2953

  2954 lemma space_Sup_measure'2: "space (Sup_measure' M) = (\<Union>m\<in>M. space m)"

  2955   unfolding Sup_measure'_def by (intro space_measure_of[OF UN_space_closed])

  2956

  2957 lemma sets_Sup_measure'2: "sets (Sup_measure' M) = sigma_sets (\<Union>m\<in>M. space m) (\<Union>m\<in>M. sets m)"

  2958   unfolding Sup_measure'_def by (intro sets_measure_of[OF UN_space_closed])

  2959

  2960 lemma sets_Sup_measure':

  2961   assumes sets_eq[simp]: "\<And>m. m \<in> M \<Longrightarrow> sets m = sets A" and "M \<noteq> {}"

  2962   shows "sets (Sup_measure' M) = sets A"

  2963   using sets_eq[THEN sets_eq_imp_space_eq, simp] \<open>M \<noteq> {}\<close> by (simp add: Sup_measure'_def)

  2964

  2965 lemma space_Sup_measure':

  2966   assumes sets_eq[simp]: "\<And>m. m \<in> M \<Longrightarrow> sets m = sets A" and "M \<noteq> {}"

  2967   shows "space (Sup_measure' M) = space A"

  2968   using sets_eq[THEN sets_eq_imp_space_eq, simp] \<open>M \<noteq> {}\<close>

  2969   by (simp add: Sup_measure'_def )

  2970

  2971 lemma emeasure_Sup_measure':

  2972   assumes sets_eq[simp]: "\<And>m. m \<in> M \<Longrightarrow> sets m = sets A" and "X \<in> sets A" "M \<noteq> {}"

  2973   shows "emeasure (Sup_measure' M) X = (SUP P:{P. finite P \<and> P \<subseteq> M}. sup_measure.F id P X)"

  2974     (is "_ = ?S X")

  2975   using Sup_measure'_def

  2976 proof (rule emeasure_measure_of)

  2977   note sets_eq[THEN sets_eq_imp_space_eq, simp]

  2978   have *: "sets (Sup_measure' M) = sets A" "space (Sup_measure' M) = space A"

  2979     using \<open>M \<noteq> {}\<close> by (simp_all add: Sup_measure'_def)

  2980   let ?\<mu> = "sup_measure.F id"

  2981   show "countably_additive (sets (Sup_measure' M)) ?S"

  2982   proof (rule countably_additiveI, goal_cases)

  2983     case (1 F)

  2984     then have **: "range F \<subseteq> sets A"

  2985       by (auto simp: *)

  2986     show "(\<Sum>i. ?S (F i)) = ?S (\<Union>i. F i)"

  2987     proof (subst ennreal_suminf_SUP_eq_directed)

  2988       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}"

  2989       have "(i \<noteq> {} \<longrightarrow> sets (?\<mu> i) = sets A) \<and> (j \<noteq> {} \<longrightarrow> sets (?\<mu> j) = sets A) \<and>

  2990         (i \<noteq> {} \<or> j \<noteq> {} \<longrightarrow> sets (?\<mu> (i \<union> j)) = sets A)"

  2991         using ij by (intro impI sets_sup_measure_F conjI) auto

  2992       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

  2993         using ij

  2994         by (cases "i = {}"; cases "j = {}")

  2995            (auto intro!: le_measureD3 sup_measure_F_mono simp: sets_sup_measure_F

  2996                  simp del: id_apply)

  2997       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)"

  2998         by (safe intro!: bexI[of _ "i \<union> j"]) auto

  2999     next

  3000       show "(SUP P : {P. finite P \<and> P \<subseteq> M}. \<Sum>n. ?\<mu> P (F n)) = (SUP P : {P. finite P \<and> P \<subseteq> M}. ?\<mu> P (UNION UNIV F))"

  3001       proof (intro SUP_cong refl)

  3002         fix i assume i: "i \<in> {P. finite P \<and> P \<subseteq> M}"

  3003         show "(\<Sum>n. ?\<mu> i (F n)) = ?\<mu> i (UNION UNIV F)"

  3004         proof cases

  3005           assume "i \<noteq> {}" with i ** show ?thesis

  3006             apply (intro suminf_emeasure \<open>disjoint_family F\<close>)

  3007             apply (subst sets_sup_measure_F[OF _ _ sets_eq])

  3008             apply auto

  3009             done

  3010         qed simp

  3011       qed

  3012     qed

  3013   qed

  3014   show "positive (sets (Sup_measure' M)) ?S"

  3015     by (auto simp: positive_def bot_ennreal[symmetric])

  3016   show "X \<in> sets (Sup_measure' M)"

  3017     using assms * by auto

  3018 qed (rule UN_space_closed)

  3019

  3020 definition Sup_measure :: "'a measure set \<Rightarrow> 'a measure"

  3021 where

  3022   "Sup_measure = Sup_lexord space (Sup_lexord sets Sup_measure'

  3023     (\<lambda>U. sigma (\<Union>u\<in>U. space u) (\<Union>u\<in>U. sets u))) (\<lambda>U. sigma (\<Union>u\<in>U. space u) {})"

  3024

  3025 definition Inf_measure :: "'a measure set \<Rightarrow> 'a measure"

  3026 where

  3027   "Inf_measure A = Sup {x. \<forall>a\<in>A. x \<le> a}"

  3028

  3029 definition inf_measure :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> 'a measure"

  3030 where

  3031   "inf_measure a b = Inf {a, b}"

  3032

  3033 definition top_measure :: "'a measure"

  3034 where

  3035   "top_measure = Inf {}"

  3036

  3037 instance

  3038 proof

  3039   note UN_space_closed [simp]

  3040   show upper: "x \<le> Sup A" if x: "x \<in> A" for x :: "'a measure" and A

  3041     unfolding Sup_measure_def

  3042   proof (intro Sup_lexord[where P="\<lambda>y. x \<le> y"])

  3043     assume "\<And>a. a \<in> A \<Longrightarrow> space a \<noteq> (\<Union>a\<in>A. space a)"

  3044     from this[OF \<open>x \<in> A\<close>] \<open>x \<in> A\<close> show "x \<le> sigma (\<Union>a\<in>A. space a) {}"

  3045       by (intro less_eq_measure.intros) auto

  3046   next

  3047     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}"

  3048       and neq: "\<And>aa. aa \<in> S \<Longrightarrow> sets aa \<noteq> (\<Union>a\<in>S. sets a)"

  3049     have sp_a: "space a = (UNION S space)"

  3050       using \<open>a\<in>A\<close> by (auto simp: S)

  3051     show "x \<le> sigma (UNION S space) (UNION S sets)"

  3052     proof cases

  3053       assume [simp]: "space x = space a"

  3054       have "sets x \<subset> (\<Union>a\<in>S. sets a)"

  3055         using \<open>x\<in>A\<close> neq[of x] by (auto simp: S)

  3056       also have "\<dots> \<subseteq> sigma_sets (\<Union>x\<in>S. space x) (\<Union>x\<in>S. sets x)"

  3057         by (rule sigma_sets_superset_generator)

  3058       finally show ?thesis

  3059         by (intro less_eq_measure.intros(2)) (simp_all add: sp_a)

  3060     next

  3061       assume "space x \<noteq> space a"

  3062       moreover have "space x \<le> space a"

  3063         unfolding a using \<open>x\<in>A\<close> by auto

  3064       ultimately show ?thesis

  3065         by (intro less_eq_measure.intros) (simp add: less_le sp_a)

  3066     qed

  3067   next

  3068     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}"

  3069       and "b \<in> S" and b: "sets b = (\<Union>a\<in>S. sets a)" and S': "S' = {a' \<in> S. sets a' = sets b}"

  3070     then have "S' \<noteq> {}" "space b = space a"

  3071       by auto

  3072     have sets_eq: "\<And>x. x \<in> S' \<Longrightarrow> sets x = sets b"

  3073       by (auto simp: S')

  3074     note sets_eq[THEN sets_eq_imp_space_eq, simp]

  3075     have *: "sets (Sup_measure' S') = sets b" "space (Sup_measure' S') = space b"

  3076       using \<open>S' \<noteq> {}\<close> by (simp_all add: Sup_measure'_def sets_eq)

  3077     show "x \<le> Sup_measure' S'"

  3078     proof cases

  3079       assume "x \<in> S"

  3080       with \<open>b \<in> S\<close> have "space x = space b"

  3081         by (simp add: S)

  3082       show ?thesis

  3083       proof cases

  3084         assume "x \<in> S'"

  3085         show "x \<le> Sup_measure' S'"

  3086         proof (intro le_measure[THEN iffD2] ballI)

  3087           show "sets x = sets (Sup_measure' S')"

  3088             using \<open>x\<in>S'\<close> * by (simp add: S')

  3089           fix X assume "X \<in> sets x"

  3090           show "emeasure x X \<le> emeasure (Sup_measure' S') X"

  3091           proof (subst emeasure_Sup_measure'[OF _ \<open>X \<in> sets x\<close>])

  3092             show "emeasure x X \<le> (SUP P : {P. finite P \<and> P \<subseteq> S'}. emeasure (sup_measure.F id P) X)"

  3093               using \<open>x\<in>S'\<close> by (intro SUP_upper2[where i="{x}"]) auto

  3094           qed (insert \<open>x\<in>S'\<close> S', auto)

  3095         qed

  3096       next

  3097         assume "x \<notin> S'"

  3098         then have "sets x \<noteq> sets b"

  3099           using \<open>x\<in>S\<close> by (auto simp: S')

  3100         moreover have "sets x \<le> sets b"

  3101           using \<open>x\<in>S\<close> unfolding b by auto

  3102         ultimately show ?thesis

  3103           using * \<open>x \<in> S\<close>

  3104           by (intro less_eq_measure.intros(2))

  3105              (simp_all add: * \<open>space x = space b\<close> less_le)

  3106       qed

  3107     next

  3108       assume "x \<notin> S"

  3109       with \<open>x\<in>A\<close> \<open>x \<notin> S\<close> \<open>space b = space a\<close> show ?thesis

  3110         by (intro less_eq_measure.intros)

  3111            (simp_all add: * less_le a SUP_upper S)

  3112     qed

  3113   qed

  3114   show least: "Sup A \<le> x" if x: "\<And>z. z \<in> A \<Longrightarrow> z \<le> x" for x :: "'a measure" and A

  3115     unfolding Sup_measure_def

  3116   proof (intro Sup_lexord[where P="\<lambda>y. y \<le> x"])

  3117     assume "\<And>a. a \<in> A \<Longrightarrow> space a \<noteq> (\<Union>a\<in>A. space a)"

  3118     show "sigma (UNION A space) {} \<le> x"

  3119       using x[THEN le_measureD1] by (subst sigma_le_iff) auto

  3120   next

  3121     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}"

  3122       "\<And>a. a \<in> S \<Longrightarrow> sets a \<noteq> (\<Union>a\<in>S. sets a)"

  3123     have "UNION S space \<subseteq> space x"

  3124       using S le_measureD1[OF x] by auto

  3125     moreover

  3126     have "UNION S space = space a"

  3127       using \<open>a\<in>A\<close> S by auto

  3128     then have "space x = UNION S space \<Longrightarrow> UNION S sets \<subseteq> sets x"

  3129       using \<open>a \<in> A\<close> le_measureD2[OF x] by (auto simp: S)

  3130     ultimately show "sigma (UNION S space) (UNION S sets) \<le> x"

  3131       by (subst sigma_le_iff) simp_all

  3132   next

  3133     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}"

  3134       and "b \<in> S" and b: "sets b = (\<Union>a\<in>S. sets a)" and S': "S' = {a' \<in> S. sets a' = sets b}"

  3135     then have "S' \<noteq> {}" "space b = space a"

  3136       by auto

  3137     have sets_eq: "\<And>x. x \<in> S' \<Longrightarrow> sets x = sets b"

  3138       by (auto simp: S')

  3139     note sets_eq[THEN sets_eq_imp_space_eq, simp]

  3140     have *: "sets (Sup_measure' S') = sets b" "space (Sup_measure' S') = space b"

  3141       using \<open>S' \<noteq> {}\<close> by (simp_all add: Sup_measure'_def sets_eq)

  3142     show "Sup_measure' S' \<le> x"

  3143     proof cases

  3144       assume "space x = space a"

  3145       show ?thesis

  3146       proof cases

  3147         assume **: "sets x = sets b"

  3148         show ?thesis

  3149         proof (intro le_measure[THEN iffD2] ballI)

  3150           show ***: "sets (Sup_measure' S') = sets x"

  3151             by (simp add: * **)

  3152           fix X assume "X \<in> sets (Sup_measure' S')"

  3153           show "emeasure (Sup_measure' S') X \<le> emeasure x X"

  3154             unfolding ***

  3155           proof (subst emeasure_Sup_measure'[OF _ \<open>X \<in> sets (Sup_measure' S')\<close>])

  3156             show "(SUP P : {P. finite P \<and> P \<subseteq> S'}. emeasure (sup_measure.F id P) X) \<le> emeasure x X"

  3157             proof (safe intro!: SUP_least)

  3158               fix P assume P: "finite P" "P \<subseteq> S'"

  3159               show "emeasure (sup_measure.F id P) X \<le> emeasure x X"

  3160               proof cases

  3161                 assume "P = {}" then show ?thesis

  3162                   by auto

  3163               next

  3164                 assume "P \<noteq> {}"

  3165                 from P have "finite P" "P \<subseteq> A"

  3166                   unfolding S' S by (simp_all add: subset_eq)

  3167                 then have "sup_measure.F id P \<le> x"

  3168                   by (induction P) (auto simp: x)

  3169                 moreover have "sets (sup_measure.F id P) = sets x"

  3170                   using \<open>finite P\<close> \<open>P \<noteq> {}\<close> \<open>P \<subseteq> S'\<close> \<open>sets x = sets b\<close>

  3171                   by (intro sets_sup_measure_F) (auto simp: S')

  3172                 ultimately show "emeasure (sup_measure.F id P) X \<le> emeasure x X"

  3173                   by (rule le_measureD3)

  3174               qed

  3175             qed

  3176             show "m \<in> S' \<Longrightarrow> sets m = sets (Sup_measure' S')" for m

  3177               unfolding * by (simp add: S')

  3178           qed fact

  3179         qed

  3180       next

  3181         assume "sets x \<noteq> sets b"

  3182         moreover have "sets b \<le> sets x"

  3183           unfolding b S using x[THEN le_measureD2] \<open>space x = space a\<close> by auto

  3184         ultimately show "Sup_measure' S' \<le> x"

  3185           using \<open>space x = space a\<close> \<open>b \<in> S\<close>

  3186           by (intro less_eq_measure.intros(2)) (simp_all add: * S)

  3187       qed

  3188     next

  3189       assume "space x \<noteq> space a"

  3190       then have "space a < space x"

  3191         using le_measureD1[OF x[OF \<open>a\<in>A\<close>]] by auto

  3192       then show "Sup_measure' S' \<le> x"

  3193         by (intro less_eq_measure.intros) (simp add: * \<open>space b = space a\<close>)

  3194     qed

  3195   qed

  3196   show "Sup {} = (bot::'a measure)" "Inf {} = (top::'a measure)"

  3197     by (auto intro!: antisym least simp: top_measure_def)

  3198   show lower: "x \<in> A \<Longrightarrow> Inf A \<le> x" for x :: "'a measure" and A

  3199     unfolding Inf_measure_def by (intro least) auto

  3200   show greatest: "(\<And>z. z \<in> A \<Longrightarrow> x \<le> z) \<Longrightarrow> x \<le> Inf A" for x :: "'a measure" and A

  3201     unfolding Inf_measure_def by (intro upper) auto

  3202   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"

  3203     by (auto simp: inf_measure_def intro!: lower greatest)

  3204 qed

  3205

  3206 end

  3207

  3208 lemma sets_SUP:

  3209   assumes "\<And>x. x \<in> I \<Longrightarrow> sets (M x) = sets N"

  3210   shows "I \<noteq> {} \<Longrightarrow> sets (SUP i:I. M i) = sets N"

  3211   unfolding Sup_measure_def

  3212   using assms assms[THEN sets_eq_imp_space_eq]

  3213     sets_Sup_measure'[where A=N and M="MI"]

  3214   by (intro Sup_lexord1[where P="\<lambda>x. sets x = sets N"]) auto

  3215

  3216 lemma emeasure_SUP:

  3217   assumes sets: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets N" "X \<in> sets N" "I \<noteq> {}"

  3218   shows "emeasure (SUP i:I. M i) X = (SUP J:{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emeasure (SUP i:J. M i) X)"

  3219 proof -

  3220   interpret sup_measure: comm_monoid_set sup "bot :: 'b measure"

  3221     by standard (auto intro!: antisym)

  3222   have eq: "finite J \<Longrightarrow> sup_measure.F id J = (SUP i:J. i)" for J :: "'b measure set"

  3223     by (induction J rule: finite_induct) auto

  3224   have 1: "J \<noteq> {} \<Longrightarrow> J \<subseteq> I \<Longrightarrow> sets (SUP x:J. M x) = sets N" for J

  3225     by (intro sets_SUP sets) (auto )

  3226   from \<open>I \<noteq> {}\<close> obtain i where "i\<in>I" by auto

  3227   have "Sup_measure' (MI) X = (SUP P:{P. finite P \<and> P \<subseteq> MI}. sup_measure.F id P X)"

  3228     using sets by (intro emeasure_Sup_measure') auto

  3229   also have "Sup_measure' (MI) = (SUP i:I. M i)"

  3230     unfolding Sup_measure_def using \<open>I \<noteq> {}\<close> sets sets(1)[THEN sets_eq_imp_space_eq]

  3231     by (intro Sup_lexord1[where P="\<lambda>x. _ = x"]) auto

  3232   also have "(SUP P:{P. finite P \<and> P \<subseteq> MI}. sup_measure.F id P X) =

  3233     (SUP J:{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. (SUP i:J. M i) X)"

  3234   proof (intro SUP_eq)

  3235     fix J assume "J \<in> {P. finite P \<and> P \<subseteq> MI}"

  3236     then obtain J' where J': "J' \<subseteq> I" "finite J'" and J: "J = MJ'" and "finite J"

  3237       using finite_subset_image[of J M I] by auto

  3238     show "\<exists>j\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. sup_measure.F id J X \<le> (SUP i:j. M i) X"

  3239     proof cases

  3240       assume "J' = {}" with \<open>i \<in> I\<close> show ?thesis

  3241         by (auto simp add: J)

  3242     next

  3243       assume "J' \<noteq> {}" with J J' show ?thesis

  3244         by (intro bexI[of _ "J'"]) (auto simp add: eq simp del: id_apply)

  3245     qed

  3246   next

  3247     fix J assume J: "J \<in> {P. P \<noteq> {} \<and> finite P \<and> P \<subseteq> I}"

  3248     show "\<exists>J'\<in>{J. finite J \<and> J \<subseteq> MI}. (SUP i:J. M i) X \<le> sup_measure.F id J' X"

  3249       using J by (intro bexI[of _ "MJ"]) (auto simp add: eq simp del: id_apply)

  3250   qed

  3251   finally show ?thesis .

  3252 qed

  3253

  3254 lemma emeasure_SUP_chain:

  3255   assumes sets: "\<And>i. i \<in> A \<Longrightarrow> sets (M i) = sets N" "X \<in> sets N"

  3256   assumes ch: "Complete_Partial_Order.chain op \<le> (M  A)" and "A \<noteq> {}"

  3257   shows "emeasure (SUP i:A. M i) X = (SUP i:A. emeasure (M i) X)"

  3258 proof (subst emeasure_SUP[OF sets \<open>A \<noteq> {}\<close>])

  3259   show "(SUP J:{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> A}. emeasure (SUPREMUM J M) X) = (SUP i:A. emeasure (M i) X)"

  3260   proof (rule SUP_eq)

  3261     fix J assume "J \<in> {J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> A}"

  3262     then have J: "Complete_Partial_Order.chain op \<le> (M  J)" "finite J" "J \<noteq> {}" and "J \<subseteq> A"

  3263       using ch[THEN chain_subset, of "MJ"] by auto

  3264     with in_chain_finite[OF J(1)] obtain j where "j \<in> J" "(SUP j:J. M j) = M j"

  3265       by auto

  3266     with \<open>J \<subseteq> A\<close> show "\<exists>j\<in>A. emeasure (SUPREMUM J M) X \<le> emeasure (M j) X"

  3267       by auto

  3268   next

  3269     fix j assume "j\<in>A" then show "\<exists>i\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> A}. emeasure (M j) X \<le> emeasure (SUPREMUM i M) X"

  3270       by (intro bexI[of _ "{j}"]) auto

  3271   qed

  3272 qed

  3273

  3274 subsubsection \<open>Supremum of a set of $\sigma$-algebras\<close>

  3275

  3276 lemma space_Sup_eq_UN: "space (Sup M) = (\<Union>x\<in>M. space x)"

  3277   unfolding Sup_measure_def

  3278   apply (intro Sup_lexord[where P="\<lambda>x. space x = _"])

  3279   apply (subst space_Sup_measure'2)

  3280   apply auto []

  3281   apply (subst space_measure_of[OF UN_space_closed])

  3282   apply auto

  3283   done

  3284

  3285 lemma sets_Sup_eq:

  3286   assumes *: "\<And>m. m \<in> M \<Longrightarrow> space m = X" and "M \<noteq> {}"

  3287   shows "sets (Sup M) = sigma_sets X (\<Union>x\<in>M. sets x)"

  3288   unfolding Sup_measure_def

  3289   apply (rule Sup_lexord1)

  3290   apply fact

  3291   apply (simp add: assms)

  3292   apply (rule Sup_lexord)

  3293   subgoal premises that for a S

  3294     unfolding that(3) that(2)[symmetric]

  3295     using that(1)

  3296     apply (subst sets_Sup_measure'2)

  3297     apply (intro arg_cong2[where f=sigma_sets])

  3298     apply (auto simp: *)

  3299     done

  3300   apply (subst sets_measure_of[OF UN_space_closed])

  3301   apply (simp add:  assms)

  3302   done

  3303

  3304 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)"

  3305   by (subst sets_Sup_eq[where X=X]) auto

  3306

  3307 lemma Sup_lexord_rel:

  3308   assumes "\<And>i. i \<in> I \<Longrightarrow> k (A i) = k (B i)"

  3309     "R (c (A  {a \<in> I. k (B a) = (SUP x:I. k (B x))})) (c (B  {a \<in> I. k (B a) = (SUP x:I. k (B x))}))"

  3310     "R (s (AI)) (s (BI))"

  3311   shows "R (Sup_lexord k c s (AI)) (Sup_lexord k c s (BI))"

  3312 proof -

  3313   have "A  {a \<in> I. k (B a) = (SUP x:I. k (B x))} =  {a \<in> A  I. k a = (SUP x:I. k (B x))}"

  3314     using assms(1) by auto

  3315   moreover have "B  {a \<in> I. k (B a) = (SUP x:I. k (B x))} =  {a \<in> B  I. k a = (SUP x:I. k (B x))}"

  3316     by auto

  3317   ultimately show ?thesis

  3318     using assms by (auto simp: Sup_lexord_def Let_def)

  3319 qed

  3320

  3321 lemma sets_SUP_cong:

  3322   assumes eq: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets (N i)" shows "sets (SUP i:I. M i) = sets (SUP i:I. N i)"

  3323   unfolding Sup_measure_def

  3324   using eq eq[THEN sets_eq_imp_space_eq]

  3325   apply (intro Sup_lexord_rel[where R="\<lambda>x y. sets x = sets y"])

  3326   apply simp

  3327   apply simp

  3328   apply (simp add: sets_Sup_measure'2)

  3329   apply (intro arg_cong2[where f="\<lambda>x y. sets (sigma x y)"])

  3330   apply auto

  3331   done

  3332

  3333 lemma sets_Sup_in_sets:

  3334   assumes "M \<noteq> {}"

  3335   assumes "\<And>m. m \<in> M \<Longrightarrow> space m = space N"

  3336   assumes "\<And>m. m \<in> M \<Longrightarrow> sets m \<subseteq> sets N"

  3337   shows "sets (Sup M) \<subseteq> sets N"

  3338 proof -

  3339   have *: "UNION M space = space N"

  3340     using assms by auto

  3341   show ?thesis

  3342     unfolding * using assms by (subst sets_Sup_eq[of M "space N"]) (auto intro!: sets.sigma_sets_subset)

  3343 qed

  3344

  3345 lemma measurable_Sup1:

  3346   assumes m: "m \<in> M" and f: "f \<in> measurable m N"

  3347     and const_space: "\<And>m n. m \<in> M \<Longrightarrow> n \<in> M \<Longrightarrow> space m = space n"

  3348   shows "f \<in> measurable (Sup M) N"

  3349 proof -

  3350   have "space (Sup M) = space m"

  3351     using m by (auto simp add: space_Sup_eq_UN dest: const_space)

  3352   then show ?thesis

  3353     using m f unfolding measurable_def by (auto intro: in_sets_Sup[OF const_space])

  3354 qed

  3355

  3356 lemma measurable_Sup2:

  3357   assumes M: "M \<noteq> {}"

  3358   assumes f: "\<And>m. m \<in> M \<Longrightarrow> f \<in> measurable N m"

  3359     and const_space: "\<And>m n. m \<in> M \<Longrightarrow> n \<in> M \<Longrightarrow> space m = space n"

  3360   shows "f \<in> measurable N (Sup M)"

  3361 proof -

  3362   from M obtain m where "m \<in> M" by auto

  3363   have space_eq: "\<And>n. n \<in> M \<Longrightarrow> space n = space m"

  3364     by (intro const_space \<open>m \<in> M\<close>)

  3365   have "f \<in> measurable N (sigma (\<Union>m\<in>M. space m) (\<Union>m\<in>M. sets m))"

  3366   proof (rule measurable_measure_of)

  3367     show "f \<in> space N \<rightarrow> UNION M space"

  3368       using measurable_space[OF f] M by auto

  3369   qed (auto intro: measurable_sets f dest: sets.sets_into_space)

  3370   also have "measurable N (sigma (\<Union>m\<in>M. space m) (\<Union>m\<in>M. sets m)) = measurable N (Sup M)"

  3371     apply (intro measurable_cong_sets refl)

  3372     apply (subst sets_Sup_eq[OF space_eq M])

  3373     apply simp

  3374     apply (subst sets_measure_of[OF UN_space_closed])

  3375     apply (simp add: space_eq M)

  3376     done

  3377   finally show ?thesis .

  3378 qed

  3379

  3380 lemma sets_Sup_sigma:

  3381   assumes [simp]: "M \<noteq> {}" and M: "\<And>m. m \<in> M \<Longrightarrow> m \<subseteq> Pow \<Omega>"

  3382   shows "sets (SUP m:M. sigma \<Omega> m) = sets (sigma \<Omega> (\<Union>M))"

  3383 proof -

  3384   { fix a m assume "a \<in> sigma_sets \<Omega> m" "m \<in> M"

  3385     then have "a \<in> sigma_sets \<Omega> (\<Union>M)"

  3386      by induction (auto intro: sigma_sets.intros(2-)) }

  3387   then show "sets (SUP m:M. sigma \<Omega> m) = sets (sigma \<Omega> (\<Union>M))"

  3388     apply (subst sets_Sup_eq[where X="\<Omega>"])

  3389     apply (auto simp add: M) []

  3390     apply auto []

  3391     apply (simp add: space_measure_of_conv M Union_least)

  3392     apply (rule sigma_sets_eqI)

  3393     apply auto

  3394     done

  3395 qed

  3396

  3397 lemma Sup_sigma:

  3398   assumes [simp]: "M \<noteq> {}" and M: "\<And>m. m \<in> M \<Longrightarrow> m \<subseteq> Pow \<Omega>"

  3399   shows "(SUP m:M. sigma \<Omega> m) = (sigma \<Omega> (\<Union>M))"

  3400 proof (intro antisym SUP_least)

  3401   have *: "\<Union>M \<subseteq> Pow \<Omega>"

  3402     using M by auto

  3403   show "sigma \<Omega> (\<Union>M) \<le> (SUP m:M. sigma \<Omega> m)"

  3404   proof (intro less_eq_measure.intros(3))

  3405     show "space (sigma \<Omega> (\<Union>M)) = space (SUP m:M. sigma \<Omega> m)"

  3406       "sets (sigma \<Omega> (\<Union>M)) = sets (SUP m:M. sigma \<Omega> m)"

  3407       using sets_Sup_sigma[OF assms] sets_Sup_sigma[OF assms, THEN sets_eq_imp_space_eq]

  3408       by auto

  3409   qed (simp add: emeasure_sigma le_fun_def)

  3410   fix m assume "m \<in> M" then show "sigma \<Omega> m \<le> sigma \<Omega> (\<Union>M)"

  3411     by (subst sigma_le_iff) (auto simp add: M *)

  3412 qed

  3413

  3414 lemma SUP_sigma_sigma:

  3415   "M \<noteq> {} \<Longrightarrow> (\<And>m. m \<in> M \<Longrightarrow> f m \<subseteq> Pow \<Omega>) \<Longrightarrow> (SUP m:M. sigma \<Omega> (f m)) = sigma \<Omega> (\<Union>m\<in>M. f m)"

  3416   using Sup_sigma[of "fM" \<Omega>] by auto

  3417

  3418 lemma sets_vimage_Sup_eq:

  3419   assumes *: "M \<noteq> {}" "f \<in> X \<rightarrow> Y" "\<And>m. m \<in> M \<Longrightarrow> space m = Y"

  3420   shows "sets (vimage_algebra X f (Sup M)) = sets (SUP m : M. vimage_algebra X f m)"

  3421   (is "?IS = ?SI")

  3422 proof

  3423   show "?IS \<subseteq> ?SI"

  3424     apply (intro sets_image_in_sets measurable_Sup2)

  3425     apply (simp add: space_Sup_eq_UN *)

  3426     apply (simp add: *)

  3427     apply (intro measurable_Sup1)

  3428     apply (rule imageI)

  3429     apply assumption

  3430     apply (rule measurable_vimage_algebra1)

  3431     apply (auto simp: *)

  3432     done

  3433   show "?SI \<subseteq> ?IS"

  3434     apply (intro sets_Sup_in_sets)

  3435     apply (auto simp: *) []

  3436     apply (auto simp: *) []

  3437     apply (elim imageE)

  3438     apply simp

  3439     apply (rule sets_image_in_sets)

  3440     apply simp

  3441     apply (simp add: measurable_def)

  3442     apply (simp add: * space_Sup_eq_UN sets_vimage_algebra2)

  3443     apply (auto intro: in_sets_Sup[OF *(3)])

  3444     done

  3445 qed

  3446

  3447 lemma restrict_space_eq_vimage_algebra':

  3448   "sets (restrict_space M \<Omega>) = sets (vimage_algebra (\<Omega> \<inter> space M) (\<lambda>x. x) M)"

  3449 proof -

  3450   have *: "{A \<inter> (\<Omega> \<inter> space M) |A. A \<in> sets M} = {A \<inter> \<Omega> |A. A \<in> sets M}"

  3451     using sets.sets_into_space[of _ M] by blast

  3452

  3453   show ?thesis

  3454     unfolding restrict_space_def

  3455     by (subst sets_measure_of)

  3456        (auto simp add: image_subset_iff sets_vimage_algebra * dest: sets.sets_into_space intro!: arg_cong2[where f=sigma_sets])

  3457 qed

  3458

  3459 lemma sigma_le_sets:

  3460   assumes [simp]: "A \<subseteq> Pow X" shows "sets (sigma X A) \<subseteq> sets N \<longleftrightarrow> X \<in> sets N \<and> A \<subseteq> sets N"

  3461 proof

  3462   have "X \<in> sigma_sets X A" "A \<subseteq> sigma_sets X A"

  3463     by (auto intro: sigma_sets_top)

  3464   moreover assume "sets (sigma X A) \<subseteq> sets N"

  3465   ultimately show "X \<in> sets N \<and> A \<subseteq> sets N"

  3466     by auto

  3467 next

  3468   assume *: "X \<in> sets N \<and> A \<subseteq> sets N"

  3469   { fix Y assume "Y \<in> sigma_sets X A" from this * have "Y \<in> sets N"

  3470       by induction auto }

  3471   then show "sets (sigma X A) \<subseteq> sets N"

  3472     by auto

  3473 qed

  3474

  3475 lemma measurable_iff_sets:

  3476   "f \<in> measurable M N \<longleftrightarrow> (f \<in> space M \<rightarrow> space N \<and> sets (vimage_algebra (space M) f N) \<subseteq> sets M)"

  3477 proof -

  3478   have *: "{f - A \<inter> space M |A. A \<in> sets N} \<subseteq> Pow (space M)"

  3479     by auto

  3480   show ?thesis

  3481     unfolding measurable_def

  3482     by (auto simp add: vimage_algebra_def sigma_le_sets[OF *])

  3483 qed

  3484

  3485 lemma sets_vimage_algebra_space: "X \<in> sets (vimage_algebra X f M)"

  3486   using sets.top[of "vimage_algebra X f M"] by simp

  3487

  3488 lemma measurable_mono:

  3489   assumes N: "sets N' \<le> sets N" "space N = space N'"

  3490   assumes M: "sets M \<le> sets M'" "space M = space M'"

  3491   shows "measurable M N \<subseteq> measurable M' N'"

  3492   unfolding measurable_def

  3493 proof safe

  3494   fix f A assume "f \<in> space M \<rightarrow> space N" "A \<in> sets N'"

  3495   moreover assume "\<forall>y\<in>sets N. f - y \<inter> space M \<in> sets M" note this[THEN bspec, of A]

  3496   ultimately show "f - A \<inter> space M' \<in> sets M'"

  3497     using assms by auto

  3498 qed (insert N M, auto)

  3499

  3500 lemma measurable_Sup_measurable:

  3501   assumes f: "f \<in> space N \<rightarrow> A"

  3502   shows "f \<in> measurable N (Sup {M. space M = A \<and> f \<in> measurable N M})"

  3503 proof (rule measurable_Sup2)

  3504   show "{M. space M = A \<and> f \<in> measurable N M} \<noteq> {}"

  3505     using f unfolding ex_in_conv[symmetric]

  3506     by (intro exI[of _ "sigma A {}"]) (auto intro!: measurable_measure_of)

  3507 qed auto

  3508

  3509 lemma (in sigma_algebra) sigma_sets_subset':

  3510   assumes a: "a \<subseteq> M" "\<Omega>' \<in> M"

  3511   shows "sigma_sets \<Omega>' a \<subseteq> M"

  3512 proof

  3513   show "x \<in> M" if x: "x \<in> sigma_sets \<Omega>' a" for x

  3514     using x by (induct rule: sigma_sets.induct) (insert a, auto)

  3515 qed

  3516

  3517 lemma in_sets_SUP: "i \<in> I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> space (M i) = Y) \<Longrightarrow> X \<in> sets (M i) \<Longrightarrow> X \<in> sets (SUP i:I. M i)"

  3518   by (intro in_sets_Sup[where X=Y]) auto

  3519

  3520 lemma measurable_SUP1:

  3521   "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>

  3522     f \<in> measurable (SUP i:I. M i) N"

  3523   by (auto intro: measurable_Sup1)

  3524

  3525 lemma sets_image_in_sets':

  3526   assumes X: "X \<in> sets N"

  3527   assumes f: "\<And>A. A \<in> sets M \<Longrightarrow> f - A \<inter> X \<in> sets N"

  3528   shows "sets (vimage_algebra X f M) \<subseteq> sets N"

  3529   unfolding sets_vimage_algebra

  3530   by (rule sets.sigma_sets_subset') (auto intro!: measurable_sets X f)

  3531

  3532 lemma mono_vimage_algebra:

  3533   "sets M \<le> sets N \<Longrightarrow> sets (vimage_algebra X f M) \<subseteq> sets (vimage_algebra X f N)"

  3534   using sets.top[of "sigma X {f - A \<inter> X |A. A \<in> sets N}"]

  3535   unfolding vimage_algebra_def

  3536   apply (subst (asm) space_measure_of)

  3537   apply auto []

  3538   apply (subst sigma_le_sets)

  3539   apply auto

  3540   done

  3541

  3542 lemma mono_restrict_space: "sets M \<le> sets N \<Longrightarrow> sets (restrict_space M X) \<subseteq> sets (restrict_space N X)"

  3543   unfolding sets_restrict_space by (rule image_mono)

  3544

  3545 lemma sets_eq_bot: "sets M = {{}} \<longleftrightarrow> M = bot"

  3546   apply safe

  3547   apply (intro measure_eqI)

  3548   apply auto

  3549   done

  3550

  3551 lemma sets_eq_bot2: "{{}} = sets M \<longleftrightarrow> M = bot"

  3552   using sets_eq_bot[of M] by blast

  3553

  3554

  3555 lemma (in finite_measure) countable_support:

  3556   "countable {x. measure M {x} \<noteq> 0}"

  3557 proof cases

  3558   assume "measure M (space M) = 0"

  3559   with bounded_measure measure_le_0_iff have "{x. measure M {x} \<noteq> 0} = {}"

  3560     by auto

  3561   then show ?thesis

  3562     by simp

  3563 next

  3564   let ?M = "measure M (space M)" and ?m = "\<lambda>x. measure M {x}"

  3565   assume "?M \<noteq> 0"

  3566   then have *: "{x. ?m x \<noteq> 0} = (\<Union>n. {x. ?M / Suc n < ?m x})"

  3567     using reals_Archimedean[of "?m x / ?M" for x]

  3568     by (auto simp: field_simps not_le[symmetric] divide_le_0_iff measure_le_0_iff)

  3569   have **: "\<And>n. finite {x. ?M / Suc n < ?m x}"

  3570   proof (rule ccontr)

  3571     fix n assume "infinite {x. ?M / Suc n < ?m x}" (is "infinite ?X")

  3572     then obtain X where "finite X" "card X = Suc (Suc n)" "X \<subseteq> ?X"

  3573       by (metis infinite_arbitrarily_large)

  3574     from this(3) have *: "\<And>x. x \<in> X \<Longrightarrow> ?M / Suc n \<le> ?m x"

  3575       by auto

  3576     { fix x assume "x \<in> X"

  3577       from \<open>?M \<noteq> 0\<close> *[OF this] have "?m x \<noteq> 0" by (auto simp: field_simps measure_le_0_iff)

  3578       then have "{x} \<in> sets M" by (auto dest: measure_notin_sets) }

  3579     note singleton_sets = this

  3580     have "?M < (\<Sum>x\<in>X. ?M / Suc n)"

  3581       using \<open>?M \<noteq> 0\<close>

  3582       by (simp add: \<open>card X = Suc (Suc n)\<close> field_simps less_le)

  3583     also have "\<dots> \<le> (\<Sum>x\<in>X. ?m x)"

  3584       by (rule sum_mono) fact

  3585     also have "\<dots> = measure M (\<Union>x\<in>X. {x})"

  3586       using singleton_sets \<open>finite X\<close>

  3587       by (intro finite_measure_finite_Union[symmetric]) (auto simp: disjoint_family_on_def)

  3588     finally have "?M < measure M (\<Union>x\<in>X. {x})" .

  3589     moreover have "measure M (\<Union>x\<in>X. {x}) \<le> ?M"

  3590       using singleton_sets[THEN sets.sets_into_space] by (intro finite_measure_mono) auto

  3591     ultimately show False by simp

  3592   qed

  3593   show ?thesis

  3594     unfolding * by (intro countable_UN countableI_type countable_finite[OF **])

  3595 qed

  3596

  3597 end