src/HOL/Analysis/Measure_Space.thy
 author hoelzl Thu Sep 29 13:54:57 2016 +0200 (2016-09-29) changeset 63958 02de4a58e210 parent 63940 0d82c4c94014 child 63959 f77dca1abf1b permissions -rw-r--r--
HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
     1 (*  Title:      HOL/Analysis/Measure_Space.thy

     2     Author:     Lawrence C Paulson

     3     Author:     Johannes Hölzl, TU München

     4     Author:     Armin Heller, TU München

     5 *)

     6

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

     8

     9 theory Measure_Space

    10 imports

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

    12 begin

    13

    14 subsection "Relate extended reals and the indicator function"

    15

    16 lemma suminf_cmult_indicator:

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

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

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

    20 proof -

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

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

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

    24     by (auto simp: setsum.If_cases)

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

    26   proof (rule SUP_eqI)

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

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

    29   qed (insert assms, simp)

    30   ultimately show ?thesis using assms

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

    32 qed

    33

    34 lemma suminf_indicator:

    35   assumes "disjoint_family A"

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

    37 proof cases

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

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

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

    41   show ?thesis using * by simp

    42 qed simp

    43

    44 lemma setsum_indicator_disjoint_family:

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

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

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

    48 proof -

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

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

    51     by auto

    52   thus ?thesis

    53     unfolding indicator_def

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

    55 qed

    56

    57 text \<open>

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

    59   represent sigma algebras (with an arbitrary emeasure).

    60 \<close>

    61

    62 subsection "Extend binary sets"

    63

    64 lemma LIMSEQ_binaryset:

    65   assumes f: "f {} = 0"

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

    67 proof -

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

    69     proof

    70       fix n

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

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

    73     qed

    74   moreover

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

    76   ultimately

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

    78     by metis

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

    80     by simp

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

    82 qed

    83

    84 lemma binaryset_sums:

    85   assumes f: "f {} = 0"

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

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

    88

    89 lemma suminf_binaryset_eq:

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

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

    92   by (metis binaryset_sums sums_unique)

    93

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

    95

    96 text \<open>

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

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

    99 \<close>

   100

   101 definition subadditive where

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

   103

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

   105   by (auto simp add: subadditive_def)

   106

   107 definition countably_subadditive where

   108   "countably_subadditive M f \<longleftrightarrow>

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

   110

   111 lemma (in ring_of_sets) countably_subadditive_subadditive:

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

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

   114   shows  "subadditive M f"

   115 proof (auto simp add: subadditive_def)

   116   fix x y

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

   118   hence "disjoint_family (binaryset x y)"

   119     by (auto simp add: disjoint_family_on_def binaryset_def)

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

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

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

   123     using cs by (auto simp add: countably_subadditive_def)

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

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

   126     by (simp add: range_binaryset_eq UN_binaryset_eq)

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

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

   129 qed

   130

   131 definition additive where

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

   133

   134 definition increasing where

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

   136

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

   138

   139 lemma positiveD_empty:

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

   141   by (auto simp add: positive_def)

   142

   143 lemma additiveD:

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

   145   by (auto simp add: additive_def)

   146

   147 lemma increasingD:

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

   149   by (auto simp add: increasing_def)

   150

   151 lemma countably_additiveI[case_names countably]:

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

   153   \<Longrightarrow> countably_additive M f"

   154   by (simp add: countably_additive_def)

   155

   156 lemma (in ring_of_sets) disjointed_additive:

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

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

   159 proof (induct n)

   160   case (Suc n)

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

   162     by simp

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

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

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

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

   167   finally show ?case .

   168 qed simp

   169

   170 lemma (in ring_of_sets) additive_sum:

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

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

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

   174       and disj: "disjoint_family_on A S"

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

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

   177 proof induct

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

   179 next

   180   case (insert s S)

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

   182     by (auto simp add: disjoint_family_on_def neq_iff)

   183   moreover

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

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

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

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

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

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

   190     by (auto simp add: additive_def subset_insertI)

   191 qed

   192

   193 lemma (in ring_of_sets) additive_increasing:

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

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

   196   shows "increasing M f"

   197 proof (auto simp add: increasing_def)

   198   fix x y

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

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

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

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

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

   204   also have "... = f y"

   205     by (metis Un_Diff_cancel Un_absorb1 xy(3))

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

   207 qed

   208

   209 lemma (in ring_of_sets) subadditive:

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

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

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

   213 using S A

   214 proof (induct S)

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

   216 next

   217   case (insert x F)

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

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

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

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

   222     by simp

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

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

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

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

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

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

   229 qed

   230

   231 lemma (in ring_of_sets) countably_additive_additive:

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

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

   234   shows "additive M f"

   235 proof (auto simp add: additive_def)

   236   fix x y

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

   238   hence "disjoint_family (binaryset x y)"

   239     by (auto simp add: disjoint_family_on_def binaryset_def)

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

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

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

   243     using ca

   244     by (simp add: countably_additive_def)

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

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

   247     by (simp add: range_binaryset_eq UN_binaryset_eq)

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

   249     by (auto simp add: Un suminf_binaryset_eq positive_def)

   250 qed

   251

   252 lemma (in algebra) increasing_additive_bound:

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

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

   255       and inc: "increasing M f"

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

   257       and disj: "disjoint_family A"

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

   259 proof (safe intro!: suminf_le_const)

   260   fix N

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

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

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

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

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

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

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

   268

   269 lemma (in ring_of_sets) countably_additiveI_finite:

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

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

   272   shows "countably_additive M \<mu>"

   273 proof (rule countably_additiveI)

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

   275

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

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

   278

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

   280   proof (rule inj_onI, simp)

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

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

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

   284   qed

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

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

   287

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

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

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

   291   proof (rule finite_imageD)

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

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

   294       by (rule finite_subset) fact

   295   qed fact

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

   297     by (rule finite_subset)

   298

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

   300     using disj by (auto simp: disjoint_family_on_def)

   301

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

   303     by (rule suminf_finite) auto

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

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

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

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

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

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

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

   311 qed

   312

   313 lemma (in ring_of_sets) countably_additive_iff_continuous_from_below:

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

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

   316   shows "countably_additive M f \<longleftrightarrow>

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

   318   unfolding countably_additive_def

   319 proof safe

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

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

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

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

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

   325     by (auto simp: UN_disjointed_eq disjoint_family_disjointed)

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

   327     using f(1)[unfolded positive_def] dA

   328     by (auto intro!: summable_LIMSEQ)

   329   from LIMSEQ_Suc[OF this]

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

   331     unfolding lessThan_Suc_atMost .

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

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

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

   335 next

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

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

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

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

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

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

   342       using A * by auto

   343   qed (force intro!: incseq_SucI)

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

   345     using A

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

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

   348   ultimately

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

   350     unfolding sums_def by simp

   351   from sums_unique[OF this]

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

   353 qed

   354

   355 lemma (in ring_of_sets) continuous_from_above_iff_empty_continuous:

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

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

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

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

   360 proof safe

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

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

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

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

   365 next

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

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

   368

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

   370     using additive_increasing[OF f] unfolding increasing_def by simp

   371

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

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

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

   375     using A by (auto simp: decseq_def)

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

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

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

   379     using A by (auto intro!: f_mono)

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

   381     using A by (auto simp: top_unique)

   382   { fix i

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

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

   385       using A by (auto simp: top_unique) }

   386   note f_fin = this

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

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

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

   390       using A by auto

   391   qed

   392   from INF_Lim_ereal[OF decseq_f this]

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

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

   395     by auto

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

   397     using A(4) f_fin f_Int_fin

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

   399   moreover {

   400     fix n

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

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

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

   404       by auto

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

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

   407     by simp

   408   with LIMSEQ_INF[OF decseq_fA]

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

   410 qed

   411

   412 lemma (in ring_of_sets) empty_continuous_imp_continuous_from_below:

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

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

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

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

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

   418 proof -

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

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

   421   moreover

   422   { fix i

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

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

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

   426       by auto

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

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

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

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

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

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

   433     by (auto intro: ennreal_tendsto_const_minus)

   434 qed

   435

   436 lemma (in ring_of_sets) empty_continuous_imp_countably_additive:

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

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

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

   440   shows "countably_additive M f"

   441   using countably_additive_iff_continuous_from_below[OF f]

   442   using empty_continuous_imp_continuous_from_below[OF f fin] cont

   443   by blast

   444

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

   446

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

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

   449

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

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

   452

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

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

   455

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

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

   458

   459 lemma suminf_emeasure:

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

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

   462   by (simp add: countably_additive_def)

   463

   464 lemma sums_emeasure:

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

   466   unfolding sums_iff by (intro conjI suminf_emeasure) auto

   467

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

   469   by (metis sets.countably_additive_additive emeasure_positive emeasure_countably_additive)

   470

   471 lemma plus_emeasure:

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

   473   using additiveD[OF emeasure_additive] ..

   474

   475 lemma setsum_emeasure:

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

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

   478   by (metis sets.additive_sum emeasure_positive emeasure_additive)

   479

   480 lemma emeasure_mono:

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

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

   483

   484 lemma emeasure_space:

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

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

   487

   488 lemma emeasure_Diff:

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

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

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

   492 proof -

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

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

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

   496     by (subst plus_emeasure[symmetric]) auto

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

   498     using finite by simp

   499 qed

   500

   501 lemma emeasure_compl:

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

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

   504

   505 lemma Lim_emeasure_incseq:

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

   507   using emeasure_countably_additive

   508   by (auto simp add: sets.countably_additive_iff_continuous_from_below emeasure_positive

   509     emeasure_additive)

   510

   511 lemma incseq_emeasure:

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

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

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

   515

   516 lemma SUP_emeasure_incseq:

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

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

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

   520   by (simp add: LIMSEQ_unique)

   521

   522 lemma decseq_emeasure:

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

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

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

   526

   527 lemma INF_emeasure_decseq:

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

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

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

   531 proof -

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

   533     using A by (auto intro!: emeasure_mono)

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

   535

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

   537     by (simp add: ennreal_INF_const_minus)

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

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

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

   541   proof (rule SUP_emeasure_incseq)

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

   543       using A by auto

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

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

   546   qed

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

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

   549   finally show ?thesis

   550     by (rule ennreal_minus_cancel[rotated 3])

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

   552 qed

   553

   554 lemma INF_emeasure_decseq':

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

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

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

   558 proof -

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

   560     by (auto simp: less_top)

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

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

   563

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

   565   proof (rule INF_eq)

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

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

   568   qed auto

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

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

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

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

   573   finally show ?thesis .

   574 qed

   575

   576 lemma emeasure_INT_decseq_subset:

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

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

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

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

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

   582 proof cases

   583   assume "finite I"

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

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

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

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

   588   ultimately show ?thesis

   589     by simp

   590 next

   591   assume "infinite I"

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

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

   594     unfolding L_def

   595   proof (rule LeastI_ex)

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

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

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

   599   qed

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

   601     unfolding L_def by (intro Least_equality) auto

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

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

   604

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

   606   proof (intro INF_emeasure_decseq[symmetric])

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

   608       using L by (intro antimonoI F L_mono) auto

   609   qed (insert L fin, auto)

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

   611   proof (intro antisym INF_greatest)

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

   613       by (intro INF_lower2[of i]) auto

   614   qed (insert L, auto intro: INF_lower)

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

   616   proof (intro antisym INF_greatest)

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

   618       by (intro INF_lower2[of i]) auto

   619   qed (insert L, auto)

   620   finally show ?thesis .

   621 qed

   622

   623 lemma Lim_emeasure_decseq:

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

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

   626   using LIMSEQ_INF[OF decseq_emeasure, OF A]

   627   using INF_emeasure_decseq[OF A fin] by simp

   628

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

   630   assumes "P M"

   631   assumes cont: "sup_continuous F"

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

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

   634 proof -

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

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

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

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

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

   640   proof (rule incseq_SucI)

   641     fix i

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

   643     proof (induct i)

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

   645     next

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

   647     qed

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

   649       by auto

   650   qed

   651   ultimately show ?thesis

   652     by (subst SUP_emeasure_incseq) auto

   653 qed

   654

   655 lemma emeasure_lfp:

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

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

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

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

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

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

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

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

   664     unfolding SUP_apply[abs_def]

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

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

   667

   668 lemma emeasure_subadditive_finite:

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

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

   671

   672 lemma emeasure_subadditive:

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

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

   675

   676 lemma emeasure_subadditive_countably:

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

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

   679 proof -

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

   681     unfolding UN_disjointed_eq ..

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

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

   684     by (simp add:  disjoint_family_disjointed comp_def)

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

   686     using sets.range_disjointed_sets[OF assms] assms

   687     by (auto intro!: suminf_le emeasure_mono disjointed_subset)

   688   finally show ?thesis .

   689 qed

   690

   691 lemma emeasure_insert:

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

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

   694 proof -

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

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

   697 qed

   698

   699 lemma emeasure_insert_ne:

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

   701   by (rule emeasure_insert)

   702

   703 lemma emeasure_eq_setsum_singleton:

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

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

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

   707   by (auto simp: disjoint_family_on_def subset_eq)

   708

   709 lemma setsum_emeasure_cover:

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

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

   712   assumes disj: "disjoint_family_on B S"

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

   714 proof -

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

   716   proof (rule setsum_emeasure)

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

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

   719       unfolding disjoint_family_on_def by auto

   720   qed (insert assms, auto)

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

   722     using A by auto

   723   finally show ?thesis by simp

   724 qed

   725

   726 lemma emeasure_eq_0:

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

   728   by (metis emeasure_mono order_eq_iff zero_le)

   729

   730 lemma emeasure_UN_eq_0:

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

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

   733 proof -

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

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

   736   then show ?thesis

   737     by (auto intro: antisym zero_le)

   738 qed

   739

   740 lemma measure_eqI_finite:

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

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

   743   shows "M = N"

   744 proof (rule measure_eqI)

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

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

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

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

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

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

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

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

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

   754 qed simp

   755

   756 lemma measure_eqI_generator_eq:

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

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

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

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

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

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

   763   shows "M = N"

   764 proof -

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

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

   767   have "space M = \<Omega>"

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

   769     by blast

   770

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

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

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

   774     assume "D \<in> sets M"

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

   776       unfolding M

   777     proof (induct rule: sigma_sets_induct_disjoint)

   778       case (basic A)

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

   780       then show ?case using eq by auto

   781     next

   782       case empty then show ?case by simp

   783     next

   784       case (compl A)

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

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

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

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

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

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

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

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

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

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

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

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

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

   798       finally show ?case

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

   800     next

   801       case (union A)

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

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

   804       with A show ?case

   805         by auto

   806     qed }

   807   note * = this

   808   show "M = N"

   809   proof (rule measure_eqI)

   810     show "sets M = sets N"

   811       using M N by simp

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

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

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

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

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

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

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

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

   820       by (auto simp: subset_eq)

   821     have "disjoint_family ?D"

   822       by (auto simp: disjoint_family_disjointed)

   823     moreover

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

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

   826       fix i

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

   828         by (auto simp: disjointed_def)

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

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

   831     qed

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

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

   834   qed

   835 qed

   836

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

   838 proof (intro measure_eqI emeasure_measure_of_sigma)

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

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

   841     by (simp add: positive_def)

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

   843     by (simp add: emeasure_countably_additive)

   844 qed simp_all

   845

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

   847

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

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

   850

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

   852   by (simp add: null_sets_def)

   853

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

   855   unfolding null_sets_def by simp

   856

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

   858   unfolding null_sets_def by simp

   859

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

   861 proof (rule ring_of_setsI)

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

   863     using sets.sets_into_space by auto

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

   865     by auto

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

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

   868     by auto

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

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

   871     by (auto intro!: emeasure_subadditive emeasure_mono)

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

   873     using null_sets by auto

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

   875     by (auto intro!: antisym zero_le)

   876 qed

   877

   878 lemma UN_from_nat_into:

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

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

   881 proof -

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

   883     using I by simp

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

   885     by simp

   886   finally show ?thesis by simp

   887 qed

   888

   889 lemma null_sets_UN':

   890   assumes "countable I"

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

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

   893 proof cases

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

   895 next

   896   assume "I \<noteq> {}"

   897   show ?thesis

   898   proof (intro conjI CollectI null_setsI)

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

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

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

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

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

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

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

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

   907       by (intro antisym zero_le) simp

   908   qed

   909 qed

   910

   911 lemma null_sets_UN[intro]:

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

   913   by (rule null_sets_UN') auto

   914

   915 lemma null_set_Int1:

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

   917 proof (intro CollectI conjI null_setsI)

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

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

   920 qed (insert assms, auto)

   921

   922 lemma null_set_Int2:

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

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

   925

   926 lemma emeasure_Diff_null_set:

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

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

   929 proof -

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

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

   932   then show ?thesis

   933     unfolding * using assms

   934     by (subst emeasure_Diff) auto

   935 qed

   936

   937 lemma null_set_Diff:

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

   939 proof (intro CollectI conjI null_setsI)

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

   941 qed (insert assms, auto)

   942

   943 lemma emeasure_Un_null_set:

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

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

   946 proof -

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

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

   949   then show ?thesis

   950     unfolding * using assms

   951     by (subst plus_emeasure[symmetric]) auto

   952 qed

   953

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

   955

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

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

   958

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

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

   961

   962 syntax

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

   964

   965 translations

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

   967

   968 abbreviation

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

   970

   971 syntax

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

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

   974

   975 translations

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

   977

   978 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)"

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

   980

   981 lemma AE_I':

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

   983   unfolding eventually_ae_filter by auto

   984

   985 lemma AE_iff_null:

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

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

   988 proof

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

   990     unfolding eventually_ae_filter by auto

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

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

   993   then have "emeasure M ?P = 0"

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

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

   996 next

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

   998 qed

   999

  1000 lemma AE_iff_null_sets:

  1001   "N \<in> sets M \<Longrightarrow> N \<in> null_sets M \<longleftrightarrow> (AE x in M. x \<notin> N)"

  1002   using Int_absorb1[OF sets.sets_into_space, of N M]

  1003   by (subst AE_iff_null) (auto simp: Int_def[symmetric])

  1004

  1005 lemma AE_not_in:

  1006   "N \<in> null_sets M \<Longrightarrow> AE x in M. x \<notin> N"

  1007   by (metis AE_iff_null_sets null_setsD2)

  1008

  1009 lemma AE_iff_measurable:

  1010   "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"

  1011   using AE_iff_null[of _ P] by auto

  1012

  1013 lemma AE_E[consumes 1]:

  1014   assumes "AE x in M. P x"

  1015   obtains N where "{x \<in> space M. \<not> P x} \<subseteq> N" "emeasure M N = 0" "N \<in> sets M"

  1016   using assms unfolding eventually_ae_filter by auto

  1017

  1018 lemma AE_E2:

  1019   assumes "AE x in M. P x" "{x\<in>space M. P x} \<in> sets M"

  1020   shows "emeasure M {x\<in>space M. \<not> P x} = 0" (is "emeasure M ?P = 0")

  1021 proof -

  1022   have "{x\<in>space M. \<not> P x} = space M - {x\<in>space M. P x}" by auto

  1023   with AE_iff_null[of M P] assms show ?thesis by auto

  1024 qed

  1025

  1026 lemma AE_I:

  1027   assumes "{x \<in> space M. \<not> P x} \<subseteq> N" "emeasure M N = 0" "N \<in> sets M"

  1028   shows "AE x in M. P x"

  1029   using assms unfolding eventually_ae_filter by auto

  1030

  1031 lemma AE_mp[elim!]:

  1032   assumes AE_P: "AE x in M. P x" and AE_imp: "AE x in M. P x \<longrightarrow> Q x"

  1033   shows "AE x in M. Q x"

  1034 proof -

  1035   from AE_P obtain A where P: "{x\<in>space M. \<not> P x} \<subseteq> A"

  1036     and A: "A \<in> sets M" "emeasure M A = 0"

  1037     by (auto elim!: AE_E)

  1038

  1039   from AE_imp obtain B where imp: "{x\<in>space M. P x \<and> \<not> Q x} \<subseteq> B"

  1040     and B: "B \<in> sets M" "emeasure M B = 0"

  1041     by (auto elim!: AE_E)

  1042

  1043   show ?thesis

  1044   proof (intro AE_I)

  1045     have "emeasure M (A \<union> B) \<le> 0"

  1046       using emeasure_subadditive[of A M B] A B by auto

  1047     then show "A \<union> B \<in> sets M" "emeasure M (A \<union> B) = 0"

  1048       using A B by auto

  1049     show "{x\<in>space M. \<not> Q x} \<subseteq> A \<union> B"

  1050       using P imp by auto

  1051   qed

  1052 qed

  1053

  1054 (* depricated replace by laws about eventually *)

  1055 lemma

  1056   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"

  1057     and AE_disjI1: "AE x in M. P x \<Longrightarrow> AE x in M. P x \<or> Q x"

  1058     and AE_disjI2: "AE x in M. Q x \<Longrightarrow> AE x in M. P x \<or> Q x"

  1059     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"

  1060     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)"

  1061   by auto

  1062

  1063 lemma AE_impI:

  1064   "(P \<Longrightarrow> AE x in M. Q x) \<Longrightarrow> AE x in M. P \<longrightarrow> Q x"

  1065   by (cases P) auto

  1066

  1067 lemma AE_measure:

  1068   assumes AE: "AE x in M. P x" and sets: "{x\<in>space M. P x} \<in> sets M" (is "?P \<in> sets M")

  1069   shows "emeasure M {x\<in>space M. P x} = emeasure M (space M)"

  1070 proof -

  1071   from AE_E[OF AE] guess N . note N = this

  1072   with sets have "emeasure M (space M) \<le> emeasure M (?P \<union> N)"

  1073     by (intro emeasure_mono) auto

  1074   also have "\<dots> \<le> emeasure M ?P + emeasure M N"

  1075     using sets N by (intro emeasure_subadditive) auto

  1076   also have "\<dots> = emeasure M ?P" using N by simp

  1077   finally show "emeasure M ?P = emeasure M (space M)"

  1078     using emeasure_space[of M "?P"] by auto

  1079 qed

  1080

  1081 lemma AE_space: "AE x in M. x \<in> space M"

  1082   by (rule AE_I[where N="{}"]) auto

  1083

  1084 lemma AE_I2[simp, intro]:

  1085   "(\<And>x. x \<in> space M \<Longrightarrow> P x) \<Longrightarrow> AE x in M. P x"

  1086   using AE_space by force

  1087

  1088 lemma AE_Ball_mp:

  1089   "\<forall>x\<in>space M. P x \<Longrightarrow> AE x in M. P x \<longrightarrow> Q x \<Longrightarrow> AE x in M. Q x"

  1090   by auto

  1091

  1092 lemma AE_cong[cong]:

  1093   "(\<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)"

  1094   by auto

  1095

  1096 lemma AE_all_countable:

  1097   "(AE x in M. \<forall>i. P i x) \<longleftrightarrow> (\<forall>i::'i::countable. AE x in M. P i x)"

  1098 proof

  1099   assume "\<forall>i. AE x in M. P i x"

  1100   from this[unfolded eventually_ae_filter Bex_def, THEN choice]

  1101   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

  1102   have "{x\<in>space M. \<not> (\<forall>i. P i x)} \<subseteq> (\<Union>i. {x\<in>space M. \<not> P i x})" by auto

  1103   also have "\<dots> \<subseteq> (\<Union>i. N i)" using N by auto

  1104   finally have "{x\<in>space M. \<not> (\<forall>i. P i x)} \<subseteq> (\<Union>i. N i)" .

  1105   moreover from N have "(\<Union>i. N i) \<in> null_sets M"

  1106     by (intro null_sets_UN) auto

  1107   ultimately show "AE x in M. \<forall>i. P i x"

  1108     unfolding eventually_ae_filter by auto

  1109 qed auto

  1110

  1111 lemma AE_ball_countable:

  1112   assumes [intro]: "countable X"

  1113   shows "(AE x in M. \<forall>y\<in>X. P x y) \<longleftrightarrow> (\<forall>y\<in>X. AE x in M. P x y)"

  1114 proof

  1115   assume "\<forall>y\<in>X. AE x in M. P x y"

  1116   from this[unfolded eventually_ae_filter Bex_def, THEN bchoice]

  1117   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"

  1118     by auto

  1119   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})"

  1120     by auto

  1121   also have "\<dots> \<subseteq> (\<Union>y\<in>X. N y)"

  1122     using N by auto

  1123   finally have "{x\<in>space M. \<not> (\<forall>y\<in>X. P x y)} \<subseteq> (\<Union>y\<in>X. N y)" .

  1124   moreover from N have "(\<Union>y\<in>X. N y) \<in> null_sets M"

  1125     by (intro null_sets_UN') auto

  1126   ultimately show "AE x in M. \<forall>y\<in>X. P x y"

  1127     unfolding eventually_ae_filter by auto

  1128 qed auto

  1129

  1130 lemma AE_discrete_difference:

  1131   assumes X: "countable X"

  1132   assumes null: "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0"

  1133   assumes sets: "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"

  1134   shows "AE x in M. x \<notin> X"

  1135 proof -

  1136   have "(\<Union>x\<in>X. {x}) \<in> null_sets M"

  1137     using assms by (intro null_sets_UN') auto

  1138   from AE_not_in[OF this] show "AE x in M. x \<notin> X"

  1139     by auto

  1140 qed

  1141

  1142 lemma AE_finite_all:

  1143   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)"

  1144   using f by induct auto

  1145

  1146 lemma AE_finite_allI:

  1147   assumes "finite S"

  1148   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"

  1149   using AE_finite_all[OF \<open>finite S\<close>] by auto

  1150

  1151 lemma emeasure_mono_AE:

  1152   assumes imp: "AE x in M. x \<in> A \<longrightarrow> x \<in> B"

  1153     and B: "B \<in> sets M"

  1154   shows "emeasure M A \<le> emeasure M B"

  1155 proof cases

  1156   assume A: "A \<in> sets M"

  1157   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"

  1158     by (auto simp: eventually_ae_filter)

  1159   have "emeasure M A = emeasure M (A - N)"

  1160     using N A by (subst emeasure_Diff_null_set) auto

  1161   also have "emeasure M (A - N) \<le> emeasure M (B - N)"

  1162     using N A B sets.sets_into_space by (auto intro!: emeasure_mono)

  1163   also have "emeasure M (B - N) = emeasure M B"

  1164     using N B by (subst emeasure_Diff_null_set) auto

  1165   finally show ?thesis .

  1166 qed (simp add: emeasure_notin_sets)

  1167

  1168 lemma emeasure_eq_AE:

  1169   assumes iff: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B"

  1170   assumes A: "A \<in> sets M" and B: "B \<in> sets M"

  1171   shows "emeasure M A = emeasure M B"

  1172   using assms by (safe intro!: antisym emeasure_mono_AE) auto

  1173

  1174 lemma emeasure_Collect_eq_AE:

  1175   "AE x in M. P x \<longleftrightarrow> Q x \<Longrightarrow> Measurable.pred M Q \<Longrightarrow> Measurable.pred M P \<Longrightarrow>

  1176    emeasure M {x\<in>space M. P x} = emeasure M {x\<in>space M. Q x}"

  1177    by (intro emeasure_eq_AE) auto

  1178

  1179 lemma emeasure_eq_0_AE: "AE x in M. \<not> P x \<Longrightarrow> emeasure M {x\<in>space M. P x} = 0"

  1180   using AE_iff_measurable[OF _ refl, of M "\<lambda>x. \<not> P x"]

  1181   by (cases "{x\<in>space M. P x} \<in> sets M") (simp_all add: emeasure_notin_sets)

  1182

  1183 lemma emeasure_add_AE:

  1184   assumes [measurable]: "A \<in> sets M" "B \<in> sets M" "C \<in> sets M"

  1185   assumes 1: "AE x in M. x \<in> C \<longleftrightarrow> x \<in> A \<or> x \<in> B"

  1186   assumes 2: "AE x in M. \<not> (x \<in> A \<and> x \<in> B)"

  1187   shows "emeasure M C = emeasure M A + emeasure M B"

  1188 proof -

  1189   have "emeasure M C = emeasure M (A \<union> B)"

  1190     by (rule emeasure_eq_AE) (insert 1, auto)

  1191   also have "\<dots> = emeasure M A + emeasure M (B - A)"

  1192     by (subst plus_emeasure) auto

  1193   also have "emeasure M (B - A) = emeasure M B"

  1194     by (rule emeasure_eq_AE) (insert 2, auto)

  1195   finally show ?thesis .

  1196 qed

  1197

  1198 subsection \<open>\<open>\<sigma>\<close>-finite Measures\<close>

  1199

  1200 locale sigma_finite_measure =

  1201   fixes M :: "'a measure"

  1202   assumes sigma_finite_countable:

  1203     "\<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>)"

  1204

  1205 lemma (in sigma_finite_measure) sigma_finite:

  1206   obtains A :: "nat \<Rightarrow> 'a set"

  1207   where "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>"

  1208 proof -

  1209   obtain A :: "'a set set" where

  1210     [simp]: "countable A" and

  1211     A: "A \<subseteq> sets M" "(\<Union>A) = space M" "\<And>a. a \<in> A \<Longrightarrow> emeasure M a \<noteq> \<infinity>"

  1212     using sigma_finite_countable by metis

  1213   show thesis

  1214   proof cases

  1215     assume "A = {}" with \<open>(\<Union>A) = space M\<close> show thesis

  1216       by (intro that[of "\<lambda>_. {}"]) auto

  1217   next

  1218     assume "A \<noteq> {}"

  1219     show thesis

  1220     proof

  1221       show "range (from_nat_into A) \<subseteq> sets M"

  1222         using \<open>A \<noteq> {}\<close> A by auto

  1223       have "(\<Union>i. from_nat_into A i) = \<Union>A"

  1224         using range_from_nat_into[OF \<open>A \<noteq> {}\<close> \<open>countable A\<close>] by auto

  1225       with A show "(\<Union>i. from_nat_into A i) = space M"

  1226         by auto

  1227     qed (intro A from_nat_into \<open>A \<noteq> {}\<close>)

  1228   qed

  1229 qed

  1230

  1231 lemma (in sigma_finite_measure) sigma_finite_disjoint:

  1232   obtains A :: "nat \<Rightarrow> 'a set"

  1233   where "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>" "disjoint_family A"

  1234 proof -

  1235   obtain A :: "nat \<Rightarrow> 'a set" where

  1236     range: "range A \<subseteq> sets M" and

  1237     space: "(\<Union>i. A i) = space M" and

  1238     measure: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"

  1239     using sigma_finite by blast

  1240   show thesis

  1241   proof (rule that[of "disjointed A"])

  1242     show "range (disjointed A) \<subseteq> sets M"

  1243       by (rule sets.range_disjointed_sets[OF range])

  1244     show "(\<Union>i. disjointed A i) = space M"

  1245       and "disjoint_family (disjointed A)"

  1246       using disjoint_family_disjointed UN_disjointed_eq[of A] space range

  1247       by auto

  1248     show "emeasure M (disjointed A i) \<noteq> \<infinity>" for i

  1249     proof -

  1250       have "emeasure M (disjointed A i) \<le> emeasure M (A i)"

  1251         using range disjointed_subset[of A i] by (auto intro!: emeasure_mono)

  1252       then show ?thesis using measure[of i] by (auto simp: top_unique)

  1253     qed

  1254   qed

  1255 qed

  1256

  1257 lemma (in sigma_finite_measure) sigma_finite_incseq:

  1258   obtains A :: "nat \<Rightarrow> 'a set"

  1259   where "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>" "incseq A"

  1260 proof -

  1261   obtain F :: "nat \<Rightarrow> 'a set" where

  1262     F: "range F \<subseteq> sets M" "(\<Union>i. F i) = space M" "\<And>i. emeasure M (F i) \<noteq> \<infinity>"

  1263     using sigma_finite by blast

  1264   show thesis

  1265   proof (rule that[of "\<lambda>n. \<Union>i\<le>n. F i"])

  1266     show "range (\<lambda>n. \<Union>i\<le>n. F i) \<subseteq> sets M"

  1267       using F by (force simp: incseq_def)

  1268     show "(\<Union>n. \<Union>i\<le>n. F i) = space M"

  1269     proof -

  1270       from F have "\<And>x. x \<in> space M \<Longrightarrow> \<exists>i. x \<in> F i" by auto

  1271       with F show ?thesis by fastforce

  1272     qed

  1273     show "emeasure M (\<Union>i\<le>n. F i) \<noteq> \<infinity>" for n

  1274     proof -

  1275       have "emeasure M (\<Union>i\<le>n. F i) \<le> (\<Sum>i\<le>n. emeasure M (F i))"

  1276         using F by (auto intro!: emeasure_subadditive_finite)

  1277       also have "\<dots> < \<infinity>"

  1278         using F by (auto simp: setsum_Pinfty less_top)

  1279       finally show ?thesis by simp

  1280     qed

  1281     show "incseq (\<lambda>n. \<Union>i\<le>n. F i)"

  1282       by (force simp: incseq_def)

  1283   qed

  1284 qed

  1285

  1286 subsection \<open>Measure space induced by distribution of @{const measurable}-functions\<close>

  1287

  1288 definition distr :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b measure" where

  1289   "distr M N f = measure_of (space N) (sets N) (\<lambda>A. emeasure M (f - A \<inter> space M))"

  1290

  1291 lemma

  1292   shows sets_distr[simp, measurable_cong]: "sets (distr M N f) = sets N"

  1293     and space_distr[simp]: "space (distr M N f) = space N"

  1294   by (auto simp: distr_def)

  1295

  1296 lemma

  1297   shows measurable_distr_eq1[simp]: "measurable (distr Mf Nf f) Mf' = measurable Nf Mf'"

  1298     and measurable_distr_eq2[simp]: "measurable Mg' (distr Mg Ng g) = measurable Mg' Ng"

  1299   by (auto simp: measurable_def)

  1300

  1301 lemma distr_cong:

  1302   "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"

  1303   using sets_eq_imp_space_eq[of N L] by (simp add: distr_def Int_def cong: rev_conj_cong)

  1304

  1305 lemma emeasure_distr:

  1306   fixes f :: "'a \<Rightarrow> 'b"

  1307   assumes f: "f \<in> measurable M N" and A: "A \<in> sets N"

  1308   shows "emeasure (distr M N f) A = emeasure M (f - A \<inter> space M)" (is "_ = ?\<mu> A")

  1309   unfolding distr_def

  1310 proof (rule emeasure_measure_of_sigma)

  1311   show "positive (sets N) ?\<mu>"

  1312     by (auto simp: positive_def)

  1313

  1314   show "countably_additive (sets N) ?\<mu>"

  1315   proof (intro countably_additiveI)

  1316     fix A :: "nat \<Rightarrow> 'b set" assume "range A \<subseteq> sets N" "disjoint_family A"

  1317     then have A: "\<And>i. A i \<in> sets N" "(\<Union>i. A i) \<in> sets N" by auto

  1318     then have *: "range (\<lambda>i. f - (A i) \<inter> space M) \<subseteq> sets M"

  1319       using f by (auto simp: measurable_def)

  1320     moreover have "(\<Union>i. f -  A i \<inter> space M) \<in> sets M"

  1321       using * by blast

  1322     moreover have **: "disjoint_family (\<lambda>i. f - A i \<inter> space M)"

  1323       using \<open>disjoint_family A\<close> by (auto simp: disjoint_family_on_def)

  1324     ultimately show "(\<Sum>i. ?\<mu> (A i)) = ?\<mu> (\<Union>i. A i)"

  1325       using suminf_emeasure[OF _ **] A f

  1326       by (auto simp: comp_def vimage_UN)

  1327   qed

  1328   show "sigma_algebra (space N) (sets N)" ..

  1329 qed fact

  1330

  1331 lemma emeasure_Collect_distr:

  1332   assumes X[measurable]: "X \<in> measurable M N" "Measurable.pred N P"

  1333   shows "emeasure (distr M N X) {x\<in>space N. P x} = emeasure M {x\<in>space M. P (X x)}"

  1334   by (subst emeasure_distr)

  1335      (auto intro!: arg_cong2[where f=emeasure] X(1)[THEN measurable_space])

  1336

  1337 lemma emeasure_lfp2[consumes 1, case_names cont f measurable]:

  1338   assumes "P M"

  1339   assumes cont: "sup_continuous F"

  1340   assumes f: "\<And>M. P M \<Longrightarrow> f \<in> measurable M' M"

  1341   assumes *: "\<And>M A. P M \<Longrightarrow> (\<And>N. P N \<Longrightarrow> Measurable.pred N A) \<Longrightarrow> Measurable.pred M (F A)"

  1342   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)})"

  1343 proof (subst (1 2) emeasure_Collect_distr[symmetric, where X=f])

  1344   show "f \<in> measurable M' M"  "f \<in> measurable M' M"

  1345     using f[OF \<open>P M\<close>] by auto

  1346   { fix i show "Measurable.pred M ((F ^^ i) (\<lambda>x. False))"

  1347     using \<open>P M\<close> by (induction i arbitrary: M) (auto intro!: *) }

  1348   show "Measurable.pred M (lfp F)"

  1349     using \<open>P M\<close> cont * by (rule measurable_lfp_coinduct[of P])

  1350

  1351   have "emeasure (distr M' M f) {x \<in> space (distr M' M f). lfp F x} =

  1352     (SUP i. emeasure (distr M' M f) {x \<in> space (distr M' M f). (F ^^ i) (\<lambda>x. False) x})"

  1353     using \<open>P M\<close>

  1354   proof (coinduction arbitrary: M rule: emeasure_lfp')

  1355     case (measurable A N) then have "\<And>N. P N \<Longrightarrow> Measurable.pred (distr M' N f) A"

  1356       by metis

  1357     then have "\<And>N. P N \<Longrightarrow> Measurable.pred N A"

  1358       by simp

  1359     with \<open>P N\<close>[THEN *] show ?case

  1360       by auto

  1361   qed fact

  1362   then show "emeasure (distr M' M f) {x \<in> space M. lfp F x} =

  1363     (SUP i. emeasure (distr M' M f) {x \<in> space M. (F ^^ i) (\<lambda>x. False) x})"

  1364    by simp

  1365 qed

  1366

  1367 lemma distr_id[simp]: "distr N N (\<lambda>x. x) = N"

  1368   by (rule measure_eqI) (auto simp: emeasure_distr)

  1369

  1370 lemma measure_distr:

  1371   "f \<in> measurable M N \<Longrightarrow> S \<in> sets N \<Longrightarrow> measure (distr M N f) S = measure M (f - S \<inter> space M)"

  1372   by (simp add: emeasure_distr measure_def)

  1373

  1374 lemma distr_cong_AE:

  1375   assumes 1: "M = K" "sets N = sets L" and

  1376     2: "(AE x in M. f x = g x)" and "f \<in> measurable M N" and "g \<in> measurable K L"

  1377   shows "distr M N f = distr K L g"

  1378 proof (rule measure_eqI)

  1379   fix A assume "A \<in> sets (distr M N f)"

  1380   with assms show "emeasure (distr M N f) A = emeasure (distr K L g) A"

  1381     by (auto simp add: emeasure_distr intro!: emeasure_eq_AE measurable_sets)

  1382 qed (insert 1, simp)

  1383

  1384 lemma AE_distrD:

  1385   assumes f: "f \<in> measurable M M'"

  1386     and AE: "AE x in distr M M' f. P x"

  1387   shows "AE x in M. P (f x)"

  1388 proof -

  1389   from AE[THEN AE_E] guess N .

  1390   with f show ?thesis

  1391     unfolding eventually_ae_filter

  1392     by (intro bexI[of _ "f - N \<inter> space M"])

  1393        (auto simp: emeasure_distr measurable_def)

  1394 qed

  1395

  1396 lemma AE_distr_iff:

  1397   assumes f[measurable]: "f \<in> measurable M N" and P[measurable]: "{x \<in> space N. P x} \<in> sets N"

  1398   shows "(AE x in distr M N f. P x) \<longleftrightarrow> (AE x in M. P (f x))"

  1399 proof (subst (1 2) AE_iff_measurable[OF _ refl])

  1400   have "f - {x\<in>space N. \<not> P x} \<inter> space M = {x \<in> space M. \<not> P (f x)}"

  1401     using f[THEN measurable_space] by auto

  1402   then show "(emeasure (distr M N f) {x \<in> space (distr M N f). \<not> P x} = 0) =

  1403     (emeasure M {x \<in> space M. \<not> P (f x)} = 0)"

  1404     by (simp add: emeasure_distr)

  1405 qed auto

  1406

  1407 lemma null_sets_distr_iff:

  1408   "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"

  1409   by (auto simp add: null_sets_def emeasure_distr)

  1410

  1411 lemma distr_distr:

  1412   "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)"

  1413   by (auto simp add: emeasure_distr measurable_space

  1414            intro!: arg_cong[where f="emeasure M"] measure_eqI)

  1415

  1416 subsection \<open>Real measure values\<close>

  1417

  1418 lemma ring_of_finite_sets: "ring_of_sets (space M) {A\<in>sets M. emeasure M A \<noteq> top}"

  1419 proof (rule ring_of_setsI)

  1420   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>

  1421     a \<union> b \<in> {A \<in> sets M. emeasure M A \<noteq> top}" for a b

  1422     using emeasure_subadditive[of a M b] by (auto simp: top_unique)

  1423

  1424   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>

  1425     a - b \<in> {A \<in> sets M. emeasure M A \<noteq> top}" for a b

  1426     using emeasure_mono[of "a - b" a M] by (auto simp: Diff_subset top_unique)

  1427 qed (auto dest: sets.sets_into_space)

  1428

  1429 lemma measure_nonneg[simp]: "0 \<le> measure M A"

  1430   unfolding measure_def by auto

  1431

  1432 lemma zero_less_measure_iff: "0 < measure M A \<longleftrightarrow> measure M A \<noteq> 0"

  1433   using measure_nonneg[of M A] by (auto simp add: le_less)

  1434

  1435 lemma measure_le_0_iff: "measure M X \<le> 0 \<longleftrightarrow> measure M X = 0"

  1436   using measure_nonneg[of M X] by linarith

  1437

  1438 lemma measure_empty[simp]: "measure M {} = 0"

  1439   unfolding measure_def by (simp add: zero_ennreal.rep_eq)

  1440

  1441 lemma emeasure_eq_ennreal_measure:

  1442   "emeasure M A \<noteq> top \<Longrightarrow> emeasure M A = ennreal (measure M A)"

  1443   by (cases "emeasure M A" rule: ennreal_cases) (auto simp: measure_def)

  1444

  1445 lemma measure_zero_top: "emeasure M A = top \<Longrightarrow> measure M A = 0"

  1446   by (simp add: measure_def enn2ereal_top)

  1447

  1448 lemma measure_eq_emeasure_eq_ennreal: "0 \<le> x \<Longrightarrow> emeasure M A = ennreal x \<Longrightarrow> measure M A = x"

  1449   using emeasure_eq_ennreal_measure[of M A]

  1450   by (cases "A \<in> M") (auto simp: measure_notin_sets emeasure_notin_sets)

  1451

  1452 lemma enn2real_plus:"a < top \<Longrightarrow> b < top \<Longrightarrow> enn2real (a + b) = enn2real a + enn2real b"

  1453   by (simp add: enn2real_def plus_ennreal.rep_eq real_of_ereal_add less_top

  1454            del: real_of_ereal_enn2ereal)

  1455

  1456 lemma measure_Union:

  1457   "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>

  1458     measure M (A \<union> B) = measure M A + measure M B"

  1459   by (simp add: measure_def plus_emeasure[symmetric] enn2real_plus less_top)

  1460

  1461 lemma disjoint_family_on_insert:

  1462   "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"

  1463   by (fastforce simp: disjoint_family_on_def)

  1464

  1465 lemma measure_finite_Union:

  1466   "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>

  1467     measure M (\<Union>i\<in>S. A i) = (\<Sum>i\<in>S. measure M (A i))"

  1468   by (induction S rule: finite_induct)

  1469      (auto simp: disjoint_family_on_insert measure_Union setsum_emeasure[symmetric] sets.countable_UN'[OF countable_finite])

  1470

  1471 lemma measure_Diff:

  1472   assumes finite: "emeasure M A \<noteq> \<infinity>"

  1473   and measurable: "A \<in> sets M" "B \<in> sets M" "B \<subseteq> A"

  1474   shows "measure M (A - B) = measure M A - measure M B"

  1475 proof -

  1476   have "emeasure M (A - B) \<le> emeasure M A" "emeasure M B \<le> emeasure M A"

  1477     using measurable by (auto intro!: emeasure_mono)

  1478   hence "measure M ((A - B) \<union> B) = measure M (A - B) + measure M B"

  1479     using measurable finite by (rule_tac measure_Union) (auto simp: top_unique)

  1480   thus ?thesis using \<open>B \<subseteq> A\<close> by (auto simp: Un_absorb2)

  1481 qed

  1482

  1483 lemma measure_UNION:

  1484   assumes measurable: "range A \<subseteq> sets M" "disjoint_family A"

  1485   assumes finite: "emeasure M (\<Union>i. A i) \<noteq> \<infinity>"

  1486   shows "(\<lambda>i. measure M (A i)) sums (measure M (\<Union>i. A i))"

  1487 proof -

  1488   have "(\<lambda>i. emeasure M (A i)) sums (emeasure M (\<Union>i. A i))"

  1489     unfolding suminf_emeasure[OF measurable, symmetric] by (simp add: summable_sums)

  1490   moreover

  1491   { fix i

  1492     have "emeasure M (A i) \<le> emeasure M (\<Union>i. A i)"

  1493       using measurable by (auto intro!: emeasure_mono)

  1494     then have "emeasure M (A i) = ennreal ((measure M (A i)))"

  1495       using finite by (intro emeasure_eq_ennreal_measure) (auto simp: top_unique) }

  1496   ultimately show ?thesis using finite

  1497     by (subst (asm) (2) emeasure_eq_ennreal_measure) simp_all

  1498 qed

  1499

  1500 lemma measure_subadditive:

  1501   assumes measurable: "A \<in> sets M" "B \<in> sets M"

  1502   and fin: "emeasure M A \<noteq> \<infinity>" "emeasure M B \<noteq> \<infinity>"

  1503   shows "measure M (A \<union> B) \<le> measure M A + measure M B"

  1504 proof -

  1505   have "emeasure M (A \<union> B) \<noteq> \<infinity>"

  1506     using emeasure_subadditive[OF measurable] fin by (auto simp: top_unique)

  1507   then show "(measure M (A \<union> B)) \<le> (measure M A) + (measure M B)"

  1508     using emeasure_subadditive[OF measurable] fin

  1509     apply simp

  1510     apply (subst (asm) (2 3 4) emeasure_eq_ennreal_measure)

  1511     apply (auto simp add: ennreal_plus[symmetric] simp del: ennreal_plus)

  1512     done

  1513 qed

  1514

  1515 lemma measure_subadditive_finite:

  1516   assumes A: "finite I" "AI \<subseteq> sets M" and fin: "\<And>i. i \<in> I \<Longrightarrow> emeasure M (A i) \<noteq> \<infinity>"

  1517   shows "measure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. measure M (A i))"

  1518 proof -

  1519   { have "emeasure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. emeasure M (A i))"

  1520       using emeasure_subadditive_finite[OF A] .

  1521     also have "\<dots> < \<infinity>"

  1522       using fin by (simp add: less_top A)

  1523     finally have "emeasure M (\<Union>i\<in>I. A i) \<noteq> top" by simp }

  1524   note * = this

  1525   show ?thesis

  1526     using emeasure_subadditive_finite[OF A] fin

  1527     unfolding emeasure_eq_ennreal_measure[OF *]

  1528     by (simp_all add: setsum_nonneg emeasure_eq_ennreal_measure)

  1529 qed

  1530

  1531 lemma measure_subadditive_countably:

  1532   assumes A: "range A \<subseteq> sets M" and fin: "(\<Sum>i. emeasure M (A i)) \<noteq> \<infinity>"

  1533   shows "measure M (\<Union>i. A i) \<le> (\<Sum>i. measure M (A i))"

  1534 proof -

  1535   from fin have **: "\<And>i. emeasure M (A i) \<noteq> top"

  1536     using ennreal_suminf_lessD[of "\<lambda>i. emeasure M (A i)"] by (simp add: less_top)

  1537   { have "emeasure M (\<Union>i. A i) \<le> (\<Sum>i. emeasure M (A i))"

  1538       using emeasure_subadditive_countably[OF A] .

  1539     also have "\<dots> < \<infinity>"

  1540       using fin by (simp add: less_top)

  1541     finally have "emeasure M (\<Union>i. A i) \<noteq> top" by simp }

  1542   then have "ennreal (measure M (\<Union>i. A i)) = emeasure M (\<Union>i. A i)"

  1543     by (rule emeasure_eq_ennreal_measure[symmetric])

  1544   also have "\<dots> \<le> (\<Sum>i. emeasure M (A i))"

  1545     using emeasure_subadditive_countably[OF A] .

  1546   also have "\<dots> = ennreal (\<Sum>i. measure M (A i))"

  1547     using fin unfolding emeasure_eq_ennreal_measure[OF **]

  1548     by (subst suminf_ennreal) (auto simp: **)

  1549   finally show ?thesis

  1550     apply (rule ennreal_le_iff[THEN iffD1, rotated])

  1551     apply (intro suminf_nonneg allI measure_nonneg summable_suminf_not_top)

  1552     using fin

  1553     apply (simp add: emeasure_eq_ennreal_measure[OF **])

  1554     done

  1555 qed

  1556

  1557 lemma measure_eq_setsum_singleton:

  1558   "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>

  1559     measure M S = (\<Sum>x\<in>S. measure M {x})"

  1560   using emeasure_eq_setsum_singleton[of S M]

  1561   by (intro measure_eq_emeasure_eq_ennreal) (auto simp: setsum_nonneg emeasure_eq_ennreal_measure)

  1562

  1563 lemma Lim_measure_incseq:

  1564   assumes A: "range A \<subseteq> sets M" "incseq A" and fin: "emeasure M (\<Union>i. A i) \<noteq> \<infinity>"

  1565   shows "(\<lambda>i. measure M (A i)) \<longlonglongrightarrow> measure M (\<Union>i. A i)"

  1566 proof (rule tendsto_ennrealD)

  1567   have "ennreal (measure M (\<Union>i. A i)) = emeasure M (\<Union>i. A i)"

  1568     using fin by (auto simp: emeasure_eq_ennreal_measure)

  1569   moreover have "ennreal (measure M (A i)) = emeasure M (A i)" for i

  1570     using assms emeasure_mono[of "A _" "\<Union>i. A i" M]

  1571     by (intro emeasure_eq_ennreal_measure[symmetric]) (auto simp: less_top UN_upper intro: le_less_trans)

  1572   ultimately show "(\<lambda>x. ennreal (Sigma_Algebra.measure M (A x))) \<longlonglongrightarrow> ennreal (Sigma_Algebra.measure M (\<Union>i. A i))"

  1573     using A by (auto intro!: Lim_emeasure_incseq)

  1574 qed auto

  1575

  1576 lemma Lim_measure_decseq:

  1577   assumes A: "range A \<subseteq> sets M" "decseq A" and fin: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"

  1578   shows "(\<lambda>n. measure M (A n)) \<longlonglongrightarrow> measure M (\<Inter>i. A i)"

  1579 proof (rule tendsto_ennrealD)

  1580   have "ennreal (measure M (\<Inter>i. A i)) = emeasure M (\<Inter>i. A i)"

  1581     using fin[of 0] A emeasure_mono[of "\<Inter>i. A i" "A 0" M]

  1582     by (auto intro!: emeasure_eq_ennreal_measure[symmetric] simp: INT_lower less_top intro: le_less_trans)

  1583   moreover have "ennreal (measure M (A i)) = emeasure M (A i)" for i

  1584     using A fin[of i] by (intro emeasure_eq_ennreal_measure[symmetric]) auto

  1585   ultimately show "(\<lambda>x. ennreal (Sigma_Algebra.measure M (A x))) \<longlonglongrightarrow> ennreal (Sigma_Algebra.measure M (\<Inter>i. A i))"

  1586     using fin A by (auto intro!: Lim_emeasure_decseq)

  1587 qed auto

  1588

  1589 subsection \<open>Set of measurable sets with finite measure\<close>

  1590

  1591 definition fmeasurable :: "'a measure \<Rightarrow> 'a set set"

  1592 where

  1593   "fmeasurable M = {A\<in>sets M. emeasure M A < \<infinity>}"

  1594

  1595 lemma fmeasurableD[dest, measurable_dest]: "A \<in> fmeasurable M \<Longrightarrow> A \<in> sets M"

  1596   by (auto simp: fmeasurable_def)

  1597

  1598 lemma fmeasurableI: "A \<in> sets M \<Longrightarrow> emeasure M A < \<infinity> \<Longrightarrow> A \<in> fmeasurable M"

  1599   by (auto simp: fmeasurable_def)

  1600

  1601 lemma fmeasurableI_null_sets: "A \<in> null_sets M \<Longrightarrow> A \<in> fmeasurable M"

  1602   by (auto simp: fmeasurable_def)

  1603

  1604 lemma fmeasurableI2: "A \<in> fmeasurable M \<Longrightarrow> B \<subseteq> A \<Longrightarrow> B \<in> sets M \<Longrightarrow> B \<in> fmeasurable M"

  1605   using emeasure_mono[of B A M] by (auto simp: fmeasurable_def)

  1606

  1607 lemma measure_mono_fmeasurable:

  1608   "A \<subseteq> B \<Longrightarrow> A \<in> sets M \<Longrightarrow> B \<in> fmeasurable M \<Longrightarrow> measure M A \<le> measure M B"

  1609   by (auto simp: measure_def fmeasurable_def intro!: emeasure_mono enn2real_mono)

  1610

  1611 lemma emeasure_eq_measure2: "A \<in> fmeasurable M \<Longrightarrow> emeasure M A = measure M A"

  1612   by (simp add: emeasure_eq_ennreal_measure fmeasurable_def less_top)

  1613

  1614 interpretation fmeasurable: ring_of_sets "space M" "fmeasurable M"

  1615 proof (rule ring_of_setsI)

  1616   show "fmeasurable M \<subseteq> Pow (space M)" "{} \<in> fmeasurable M"

  1617     by (auto simp: fmeasurable_def dest: sets.sets_into_space)

  1618   fix a b assume *: "a \<in> fmeasurable M" "b \<in> fmeasurable M"

  1619   then have "emeasure M (a \<union> b) \<le> emeasure M a + emeasure M b"

  1620     by (intro emeasure_subadditive) auto

  1621   also have "\<dots> < top"

  1622     using * by (auto simp: fmeasurable_def)

  1623   finally show  "a \<union> b \<in> fmeasurable M"

  1624     using * by (auto intro: fmeasurableI)

  1625   show "a - b \<in> fmeasurable M"

  1626     using emeasure_mono[of "a - b" a M] * by (auto simp: fmeasurable_def Diff_subset)

  1627 qed

  1628

  1629 lemma fmeasurable_Diff: "A \<in> fmeasurable M \<Longrightarrow> B \<in> sets M \<Longrightarrow> A - B \<in> fmeasurable M"

  1630   using fmeasurableI2[of A M "A - B"] by auto

  1631

  1632 lemma fmeasurable_UN:

  1633   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"

  1634   shows "(\<Union>i\<in>I. F i) \<in> fmeasurable M"

  1635 proof (rule fmeasurableI2)

  1636   show "A \<in> fmeasurable M" "(\<Union>i\<in>I. F i) \<subseteq> A" using assms by auto

  1637   show "(\<Union>i\<in>I. F i) \<in> sets M"

  1638     using assms by (intro sets.countable_UN') auto

  1639 qed

  1640

  1641 lemma fmeasurable_INT:

  1642   assumes "countable I" "i \<in> I" "\<And>i. i \<in> I \<Longrightarrow> F i \<in> sets M" "F i \<in> fmeasurable M"

  1643   shows "(\<Inter>i\<in>I. F i) \<in> fmeasurable M"

  1644 proof (rule fmeasurableI2)

  1645   show "F i \<in> fmeasurable M" "(\<Inter>i\<in>I. F i) \<subseteq> F i"

  1646     using assms by auto

  1647   show "(\<Inter>i\<in>I. F i) \<in> sets M"

  1648     using assms by (intro sets.countable_INT') auto

  1649 qed

  1650

  1651 subsection \<open>Measure spaces with @{term "emeasure M (space M) < \<infinity>"}\<close>

  1652

  1653 locale finite_measure = sigma_finite_measure M for M +

  1654   assumes finite_emeasure_space: "emeasure M (space M) \<noteq> top"

  1655

  1656 lemma finite_measureI[Pure.intro!]:

  1657   "emeasure M (space M) \<noteq> \<infinity> \<Longrightarrow> finite_measure M"

  1658   proof qed (auto intro!: exI[of _ "{space M}"])

  1659

  1660 lemma (in finite_measure) emeasure_finite[simp, intro]: "emeasure M A \<noteq> top"

  1661   using finite_emeasure_space emeasure_space[of M A] by (auto simp: top_unique)

  1662

  1663 lemma (in finite_measure) fmeasurable_eq_sets: "fmeasurable M = sets M"

  1664   by (auto simp: fmeasurable_def less_top[symmetric])

  1665

  1666 lemma (in finite_measure) emeasure_eq_measure: "emeasure M A = ennreal (measure M A)"

  1667   by (intro emeasure_eq_ennreal_measure) simp

  1668

  1669 lemma (in finite_measure) emeasure_real: "\<exists>r. 0 \<le> r \<and> emeasure M A = ennreal r"

  1670   using emeasure_finite[of A] by (cases "emeasure M A" rule: ennreal_cases) auto

  1671

  1672 lemma (in finite_measure) bounded_measure: "measure M A \<le> measure M (space M)"

  1673   using emeasure_space[of M A] emeasure_real[of A] emeasure_real[of "space M"] by (auto simp: measure_def)

  1674

  1675 lemma (in finite_measure) finite_measure_Diff:

  1676   assumes sets: "A \<in> sets M" "B \<in> sets M" and "B \<subseteq> A"

  1677   shows "measure M (A - B) = measure M A - measure M B"

  1678   using measure_Diff[OF _ assms] by simp

  1679

  1680 lemma (in finite_measure) finite_measure_Union:

  1681   assumes sets: "A \<in> sets M" "B \<in> sets M" and "A \<inter> B = {}"

  1682   shows "measure M (A \<union> B) = measure M A + measure M B"

  1683   using measure_Union[OF _ _ assms] by simp

  1684

  1685 lemma (in finite_measure) finite_measure_finite_Union:

  1686   assumes measurable: "finite S" "AS \<subseteq> sets M" "disjoint_family_on A S"

  1687   shows "measure M (\<Union>i\<in>S. A i) = (\<Sum>i\<in>S. measure M (A i))"

  1688   using measure_finite_Union[OF assms] by simp

  1689

  1690 lemma (in finite_measure) finite_measure_UNION:

  1691   assumes A: "range A \<subseteq> sets M" "disjoint_family A"

  1692   shows "(\<lambda>i. measure M (A i)) sums (measure M (\<Union>i. A i))"

  1693   using measure_UNION[OF A] by simp

  1694

  1695 lemma (in finite_measure) finite_measure_mono:

  1696   assumes "A \<subseteq> B" "B \<in> sets M" shows "measure M A \<le> measure M B"

  1697   using emeasure_mono[OF assms] emeasure_real[of A] emeasure_real[of B] by (auto simp: measure_def)

  1698

  1699 lemma (in finite_measure) finite_measure_subadditive:

  1700   assumes m: "A \<in> sets M" "B \<in> sets M"

  1701   shows "measure M (A \<union> B) \<le> measure M A + measure M B"

  1702   using measure_subadditive[OF m] by simp

  1703

  1704 lemma (in finite_measure) finite_measure_subadditive_finite:

  1705   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))"

  1706   using measure_subadditive_finite[OF assms] by simp

  1707

  1708 lemma (in finite_measure) finite_measure_subadditive_countably:

  1709   "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))"

  1710   by (rule measure_subadditive_countably)

  1711      (simp_all add: ennreal_suminf_neq_top emeasure_eq_measure)

  1712

  1713 lemma (in finite_measure) finite_measure_eq_setsum_singleton:

  1714   assumes "finite S" and *: "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"

  1715   shows "measure M S = (\<Sum>x\<in>S. measure M {x})"

  1716   using measure_eq_setsum_singleton[OF assms] by simp

  1717

  1718 lemma (in finite_measure) finite_Lim_measure_incseq:

  1719   assumes A: "range A \<subseteq> sets M" "incseq A"

  1720   shows "(\<lambda>i. measure M (A i)) \<longlonglongrightarrow> measure M (\<Union>i. A i)"

  1721   using Lim_measure_incseq[OF A] by simp

  1722

  1723 lemma (in finite_measure) finite_Lim_measure_decseq:

  1724   assumes A: "range A \<subseteq> sets M" "decseq A"

  1725   shows "(\<lambda>n. measure M (A n)) \<longlonglongrightarrow> measure M (\<Inter>i. A i)"

  1726   using Lim_measure_decseq[OF A] by simp

  1727

  1728 lemma (in finite_measure) finite_measure_compl:

  1729   assumes S: "S \<in> sets M"

  1730   shows "measure M (space M - S) = measure M (space M) - measure M S"

  1731   using measure_Diff[OF _ sets.top S sets.sets_into_space] S by simp

  1732

  1733 lemma (in finite_measure) finite_measure_mono_AE:

  1734   assumes imp: "AE x in M. x \<in> A \<longrightarrow> x \<in> B" and B: "B \<in> sets M"

  1735   shows "measure M A \<le> measure M B"

  1736   using assms emeasure_mono_AE[OF imp B]

  1737   by (simp add: emeasure_eq_measure)

  1738

  1739 lemma (in finite_measure) finite_measure_eq_AE:

  1740   assumes iff: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B"

  1741   assumes A: "A \<in> sets M" and B: "B \<in> sets M"

  1742   shows "measure M A = measure M B"

  1743   using assms emeasure_eq_AE[OF assms] by (simp add: emeasure_eq_measure)

  1744

  1745 lemma (in finite_measure) measure_increasing: "increasing M (measure M)"

  1746   by (auto intro!: finite_measure_mono simp: increasing_def)

  1747

  1748 lemma (in finite_measure) measure_zero_union:

  1749   assumes "s \<in> sets M" "t \<in> sets M" "measure M t = 0"

  1750   shows "measure M (s \<union> t) = measure M s"

  1751 using assms

  1752 proof -

  1753   have "measure M (s \<union> t) \<le> measure M s"

  1754     using finite_measure_subadditive[of s t] assms by auto

  1755   moreover have "measure M (s \<union> t) \<ge> measure M s"

  1756     using assms by (blast intro: finite_measure_mono)

  1757   ultimately show ?thesis by simp

  1758 qed

  1759

  1760 lemma (in finite_measure) measure_eq_compl:

  1761   assumes "s \<in> sets M" "t \<in> sets M"

  1762   assumes "measure M (space M - s) = measure M (space M - t)"

  1763   shows "measure M s = measure M t"

  1764   using assms finite_measure_compl by auto

  1765

  1766 lemma (in finite_measure) measure_eq_bigunion_image:

  1767   assumes "range f \<subseteq> sets M" "range g \<subseteq> sets M"

  1768   assumes "disjoint_family f" "disjoint_family g"

  1769   assumes "\<And> n :: nat. measure M (f n) = measure M (g n)"

  1770   shows "measure M (\<Union>i. f i) = measure M (\<Union>i. g i)"

  1771 using assms

  1772 proof -

  1773   have a: "(\<lambda> i. measure M (f i)) sums (measure M (\<Union>i. f i))"

  1774     by (rule finite_measure_UNION[OF assms(1,3)])

  1775   have b: "(\<lambda> i. measure M (g i)) sums (measure M (\<Union>i. g i))"

  1776     by (rule finite_measure_UNION[OF assms(2,4)])

  1777   show ?thesis using sums_unique[OF b] sums_unique[OF a] assms by simp

  1778 qed

  1779

  1780 lemma (in finite_measure) measure_countably_zero:

  1781   assumes "range c \<subseteq> sets M"

  1782   assumes "\<And> i. measure M (c i) = 0"

  1783   shows "measure M (\<Union>i :: nat. c i) = 0"

  1784 proof (rule antisym)

  1785   show "measure M (\<Union>i :: nat. c i) \<le> 0"

  1786     using finite_measure_subadditive_countably[OF assms(1)] by (simp add: assms(2))

  1787 qed simp

  1788

  1789 lemma (in finite_measure) measure_space_inter:

  1790   assumes events:"s \<in> sets M" "t \<in> sets M"

  1791   assumes "measure M t = measure M (space M)"

  1792   shows "measure M (s \<inter> t) = measure M s"

  1793 proof -

  1794   have "measure M ((space M - s) \<union> (space M - t)) = measure M (space M - s)"

  1795     using events assms finite_measure_compl[of "t"] by (auto intro!: measure_zero_union)

  1796   also have "(space M - s) \<union> (space M - t) = space M - (s \<inter> t)"

  1797     by blast

  1798   finally show "measure M (s \<inter> t) = measure M s"

  1799     using events by (auto intro!: measure_eq_compl[of "s \<inter> t" s])

  1800 qed

  1801

  1802 lemma (in finite_measure) measure_equiprobable_finite_unions:

  1803   assumes s: "finite s" "\<And>x. x \<in> s \<Longrightarrow> {x} \<in> sets M"

  1804   assumes "\<And> x y. \<lbrakk>x \<in> s; y \<in> s\<rbrakk> \<Longrightarrow> measure M {x} = measure M {y}"

  1805   shows "measure M s = real (card s) * measure M {SOME x. x \<in> s}"

  1806 proof cases

  1807   assume "s \<noteq> {}"

  1808   then have "\<exists> x. x \<in> s" by blast

  1809   from someI_ex[OF this] assms

  1810   have prob_some: "\<And> x. x \<in> s \<Longrightarrow> measure M {x} = measure M {SOME y. y \<in> s}" by blast

  1811   have "measure M s = (\<Sum> x \<in> s. measure M {x})"

  1812     using finite_measure_eq_setsum_singleton[OF s] by simp

  1813   also have "\<dots> = (\<Sum> x \<in> s. measure M {SOME y. y \<in> s})" using prob_some by auto

  1814   also have "\<dots> = real (card s) * measure M {(SOME x. x \<in> s)}"

  1815     using setsum_constant assms by simp

  1816   finally show ?thesis by simp

  1817 qed simp

  1818

  1819 lemma (in finite_measure) measure_real_sum_image_fn:

  1820   assumes "e \<in> sets M"

  1821   assumes "\<And> x. x \<in> s \<Longrightarrow> e \<inter> f x \<in> sets M"

  1822   assumes "finite s"

  1823   assumes disjoint: "\<And> x y. \<lbrakk>x \<in> s ; y \<in> s ; x \<noteq> y\<rbrakk> \<Longrightarrow> f x \<inter> f y = {}"

  1824   assumes upper: "space M \<subseteq> (\<Union>i \<in> s. f i)"

  1825   shows "measure M e = (\<Sum> x \<in> s. measure M (e \<inter> f x))"

  1826 proof -

  1827   have "e \<subseteq> (\<Union>i\<in>s. f i)"

  1828     using \<open>e \<in> sets M\<close> sets.sets_into_space upper by blast

  1829   then have e: "e = (\<Union>i \<in> s. e \<inter> f i)"

  1830     by auto

  1831   hence "measure M e = measure M (\<Union>i \<in> s. e \<inter> f i)" by simp

  1832   also have "\<dots> = (\<Sum> x \<in> s. measure M (e \<inter> f x))"

  1833   proof (rule finite_measure_finite_Union)

  1834     show "finite s" by fact

  1835     show "(\<lambda>i. e \<inter> f i)s \<subseteq> sets M" using assms(2) by auto

  1836     show "disjoint_family_on (\<lambda>i. e \<inter> f i) s"

  1837       using disjoint by (auto simp: disjoint_family_on_def)

  1838   qed

  1839   finally show ?thesis .

  1840 qed

  1841

  1842 lemma (in finite_measure) measure_exclude:

  1843   assumes "A \<in> sets M" "B \<in> sets M"

  1844   assumes "measure M A = measure M (space M)" "A \<inter> B = {}"

  1845   shows "measure M B = 0"

  1846   using measure_space_inter[of B A] assms by (auto simp: ac_simps)

  1847 lemma (in finite_measure) finite_measure_distr:

  1848   assumes f: "f \<in> measurable M M'"

  1849   shows "finite_measure (distr M M' f)"

  1850 proof (rule finite_measureI)

  1851   have "f - space M' \<inter> space M = space M" using f by (auto dest: measurable_space)

  1852   with f show "emeasure (distr M M' f) (space (distr M M' f)) \<noteq> \<infinity>" by (auto simp: emeasure_distr)

  1853 qed

  1854

  1855 lemma emeasure_gfp[consumes 1, case_names cont measurable]:

  1856   assumes sets[simp]: "\<And>s. sets (M s) = sets N"

  1857   assumes "\<And>s. finite_measure (M s)"

  1858   assumes cont: "inf_continuous F" "inf_continuous f"

  1859   assumes meas: "\<And>P. Measurable.pred N P \<Longrightarrow> Measurable.pred N (F P)"

  1860   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"

  1861   assumes bound: "\<And>P. f P \<le> f (\<lambda>s. emeasure (M s) (space (M s)))"

  1862   shows "emeasure (M s) {x\<in>space N. gfp F x} = gfp f s"

  1863 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

  1864     P="Measurable.pred N", symmetric])

  1865   interpret finite_measure "M s" for s by fact

  1866   fix C assume "decseq C" "\<And>i. Measurable.pred N (C i)"

  1867   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}))"

  1868     unfolding INF_apply[abs_def]

  1869     by (subst INF_emeasure_decseq) (auto simp: antimono_def fun_eq_iff intro!: arg_cong2[where f=emeasure])

  1870 next

  1871   show "f x \<le> (\<lambda>s. emeasure (M s) {x \<in> space N. F top x})" for x

  1872     using bound[of x] sets_eq_imp_space_eq[OF sets] by (simp add: iter)

  1873 qed (auto simp add: iter le_fun_def INF_apply[abs_def] intro!: meas cont)

  1874

  1875 subsection \<open>Counting space\<close>

  1876

  1877 lemma strict_monoI_Suc:

  1878   assumes ord [simp]: "(\<And>n. f n < f (Suc n))" shows "strict_mono f"

  1879   unfolding strict_mono_def

  1880 proof safe

  1881   fix n m :: nat assume "n < m" then show "f n < f m"

  1882     by (induct m) (auto simp: less_Suc_eq intro: less_trans ord)

  1883 qed

  1884

  1885 lemma emeasure_count_space:

  1886   assumes "X \<subseteq> A" shows "emeasure (count_space A) X = (if finite X then of_nat (card X) else \<infinity>)"

  1887     (is "_ = ?M X")

  1888   unfolding count_space_def

  1889 proof (rule emeasure_measure_of_sigma)

  1890   show "X \<in> Pow A" using \<open>X \<subseteq> A\<close> by auto

  1891   show "sigma_algebra A (Pow A)" by (rule sigma_algebra_Pow)

  1892   show positive: "positive (Pow A) ?M"

  1893     by (auto simp: positive_def)

  1894   have additive: "additive (Pow A) ?M"

  1895     by (auto simp: card_Un_disjoint additive_def)

  1896

  1897   interpret ring_of_sets A "Pow A"

  1898     by (rule ring_of_setsI) auto

  1899   show "countably_additive (Pow A) ?M"

  1900     unfolding countably_additive_iff_continuous_from_below[OF positive additive]

  1901   proof safe

  1902     fix F :: "nat \<Rightarrow> 'a set" assume "incseq F"

  1903     show "(\<lambda>i. ?M (F i)) \<longlonglongrightarrow> ?M (\<Union>i. F i)"

  1904     proof cases

  1905       assume "\<exists>i. \<forall>j\<ge>i. F i = F j"

  1906       then guess i .. note i = this

  1907       { fix j from i \<open>incseq F\<close> have "F j \<subseteq> F i"

  1908           by (cases "i \<le> j") (auto simp: incseq_def) }

  1909       then have eq: "(\<Union>i. F i) = F i"

  1910         by auto

  1911       with i show ?thesis

  1912         by (auto intro!: Lim_transform_eventually[OF _ tendsto_const] eventually_sequentiallyI[where c=i])

  1913     next

  1914       assume "\<not> (\<exists>i. \<forall>j\<ge>i. F i = F j)"

  1915       then obtain f where f: "\<And>i. i \<le> f i" "\<And>i. F i \<noteq> F (f i)" by metis

  1916       then have "\<And>i. F i \<subseteq> F (f i)" using \<open>incseq F\<close> by (auto simp: incseq_def)

  1917       with f have *: "\<And>i. F i \<subset> F (f i)" by auto

  1918

  1919       have "incseq (\<lambda>i. ?M (F i))"

  1920         using \<open>incseq F\<close> unfolding incseq_def by (auto simp: card_mono dest: finite_subset)

  1921       then have "(\<lambda>i. ?M (F i)) \<longlonglongrightarrow> (SUP n. ?M (F n))"

  1922         by (rule LIMSEQ_SUP)

  1923

  1924       moreover have "(SUP n. ?M (F n)) = top"

  1925       proof (rule ennreal_SUP_eq_top)

  1926         fix n :: nat show "\<exists>k::nat\<in>UNIV. of_nat n \<le> ?M (F k)"

  1927         proof (induct n)

  1928           case (Suc n)

  1929           then guess k .. note k = this

  1930           moreover have "finite (F k) \<Longrightarrow> finite (F (f k)) \<Longrightarrow> card (F k) < card (F (f k))"

  1931             using \<open>F k \<subset> F (f k)\<close> by (simp add: psubset_card_mono)

  1932           moreover have "finite (F (f k)) \<Longrightarrow> finite (F k)"

  1933             using \<open>k \<le> f k\<close> \<open>incseq F\<close> by (auto simp: incseq_def dest: finite_subset)

  1934           ultimately show ?case

  1935             by (auto intro!: exI[of _ "f k"] simp del: of_nat_Suc)

  1936         qed auto

  1937       qed

  1938

  1939       moreover

  1940       have "inj (\<lambda>n. F ((f ^^ n) 0))"

  1941         by (intro strict_mono_imp_inj_on strict_monoI_Suc) (simp add: *)

  1942       then have 1: "infinite (range (\<lambda>i. F ((f ^^ i) 0)))"

  1943         by (rule range_inj_infinite)

  1944       have "infinite (Pow (\<Union>i. F i))"

  1945         by (rule infinite_super[OF _ 1]) auto

  1946       then have "infinite (\<Union>i. F i)"

  1947         by auto

  1948

  1949       ultimately show ?thesis by auto

  1950     qed

  1951   qed

  1952 qed

  1953

  1954 lemma distr_bij_count_space:

  1955   assumes f: "bij_betw f A B"

  1956   shows "distr (count_space A) (count_space B) f = count_space B"

  1957 proof (rule measure_eqI)

  1958   have f': "f \<in> measurable (count_space A) (count_space B)"

  1959     using f unfolding Pi_def bij_betw_def by auto

  1960   fix X assume "X \<in> sets (distr (count_space A) (count_space B) f)"

  1961   then have X: "X \<in> sets (count_space B)" by auto

  1962   moreover from X have "f - X \<inter> A = the_inv_into A f  X"

  1963     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])

  1964   moreover have "inj_on (the_inv_into A f) B"

  1965     using X f by (auto simp: bij_betw_def inj_on_the_inv_into)

  1966   with X have "inj_on (the_inv_into A f) X"

  1967     by (auto intro: subset_inj_on)

  1968   ultimately show "emeasure (distr (count_space A) (count_space B) f) X = emeasure (count_space B) X"

  1969     using f unfolding emeasure_distr[OF f' X]

  1970     by (subst (1 2) emeasure_count_space) (auto simp: card_image dest: finite_imageD)

  1971 qed simp

  1972

  1973 lemma emeasure_count_space_finite[simp]:

  1974   "X \<subseteq> A \<Longrightarrow> finite X \<Longrightarrow> emeasure (count_space A) X = of_nat (card X)"

  1975   using emeasure_count_space[of X A] by simp

  1976

  1977 lemma emeasure_count_space_infinite[simp]:

  1978   "X \<subseteq> A \<Longrightarrow> infinite X \<Longrightarrow> emeasure (count_space A) X = \<infinity>"

  1979   using emeasure_count_space[of X A] by simp

  1980

  1981 lemma measure_count_space: "measure (count_space A) X = (if X \<subseteq> A then of_nat (card X) else 0)"

  1982   by (cases "finite X") (auto simp: measure_notin_sets ennreal_of_nat_eq_real_of_nat

  1983                                     measure_zero_top measure_eq_emeasure_eq_ennreal)

  1984

  1985 lemma emeasure_count_space_eq_0:

  1986   "emeasure (count_space A) X = 0 \<longleftrightarrow> (X \<subseteq> A \<longrightarrow> X = {})"

  1987 proof cases

  1988   assume X: "X \<subseteq> A"

  1989   then show ?thesis

  1990   proof (intro iffI impI)

  1991     assume "emeasure (count_space A) X = 0"

  1992     with X show "X = {}"

  1993       by (subst (asm) emeasure_count_space) (auto split: if_split_asm)

  1994   qed simp

  1995 qed (simp add: emeasure_notin_sets)

  1996

  1997 lemma space_empty: "space M = {} \<Longrightarrow> M = count_space {}"

  1998   by (rule measure_eqI) (simp_all add: space_empty_iff)

  1999

  2000 lemma null_sets_count_space: "null_sets (count_space A) = { {} }"

  2001   unfolding null_sets_def by (auto simp add: emeasure_count_space_eq_0)

  2002

  2003 lemma AE_count_space: "(AE x in count_space A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x)"

  2004   unfolding eventually_ae_filter by (auto simp add: null_sets_count_space)

  2005

  2006 lemma sigma_finite_measure_count_space_countable:

  2007   assumes A: "countable A"

  2008   shows "sigma_finite_measure (count_space A)"

  2009   proof qed (insert A, auto intro!: exI[of _ "(\<lambda>a. {a})  A"])

  2010

  2011 lemma sigma_finite_measure_count_space:

  2012   fixes A :: "'a::countable set" shows "sigma_finite_measure (count_space A)"

  2013   by (rule sigma_finite_measure_count_space_countable) auto

  2014

  2015 lemma finite_measure_count_space:

  2016   assumes [simp]: "finite A"

  2017   shows "finite_measure (count_space A)"

  2018   by rule simp

  2019

  2020 lemma sigma_finite_measure_count_space_finite:

  2021   assumes A: "finite A" shows "sigma_finite_measure (count_space A)"

  2022 proof -

  2023   interpret finite_measure "count_space A" using A by (rule finite_measure_count_space)

  2024   show "sigma_finite_measure (count_space A)" ..

  2025 qed

  2026

  2027 subsection \<open>Measure restricted to space\<close>

  2028

  2029 lemma emeasure_restrict_space:

  2030   assumes "\<Omega> \<inter> space M \<in> sets M" "A \<subseteq> \<Omega>"

  2031   shows "emeasure (restrict_space M \<Omega>) A = emeasure M A"

  2032 proof (cases "A \<in> sets M")

  2033   case True

  2034   show ?thesis

  2035   proof (rule emeasure_measure_of[OF restrict_space_def])

  2036     show "op \<inter> \<Omega>  sets M \<subseteq> Pow (\<Omega> \<inter> space M)" "A \<in> sets (restrict_space M \<Omega>)"

  2037       using \<open>A \<subseteq> \<Omega>\<close> \<open>A \<in> sets M\<close> sets.space_closed by (auto simp: sets_restrict_space)

  2038     show "positive (sets (restrict_space M \<Omega>)) (emeasure M)"

  2039       by (auto simp: positive_def)

  2040     show "countably_additive (sets (restrict_space M \<Omega>)) (emeasure M)"

  2041     proof (rule countably_additiveI)

  2042       fix A :: "nat \<Rightarrow> _" assume "range A \<subseteq> sets (restrict_space M \<Omega>)" "disjoint_family A"

  2043       with assms have "\<And>i. A i \<in> sets M" "\<And>i. A i \<subseteq> space M" "disjoint_family A"

  2044         by (fastforce simp: sets_restrict_space_iff[OF assms(1)] image_subset_iff

  2045                       dest: sets.sets_into_space)+

  2046       then show "(\<Sum>i. emeasure M (A i)) = emeasure M (\<Union>i. A i)"

  2047         by (subst suminf_emeasure) (auto simp: disjoint_family_subset)

  2048     qed

  2049   qed

  2050 next

  2051   case False

  2052   with assms have "A \<notin> sets (restrict_space M \<Omega>)"

  2053     by (simp add: sets_restrict_space_iff)

  2054   with False show ?thesis

  2055     by (simp add: emeasure_notin_sets)

  2056 qed

  2057

  2058 lemma measure_restrict_space:

  2059   assumes "\<Omega> \<inter> space M \<in> sets M" "A \<subseteq> \<Omega>"

  2060   shows "measure (restrict_space M \<Omega>) A = measure M A"

  2061   using emeasure_restrict_space[OF assms] by (simp add: measure_def)

  2062

  2063 lemma AE_restrict_space_iff:

  2064   assumes "\<Omega> \<inter> space M \<in> sets M"

  2065   shows "(AE x in restrict_space M \<Omega>. P x) \<longleftrightarrow> (AE x in M. x \<in> \<Omega> \<longrightarrow> P x)"

  2066 proof -

  2067   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)"

  2068     by auto

  2069   { fix X assume X: "X \<in> sets M" "emeasure M X = 0"

  2070     then have "emeasure M (\<Omega> \<inter> space M \<inter> X) \<le> emeasure M X"

  2071       by (intro emeasure_mono) auto

  2072     then have "emeasure M (\<Omega> \<inter> space M \<inter> X) = 0"

  2073       using X by (auto intro!: antisym) }

  2074   with assms show ?thesis

  2075     unfolding eventually_ae_filter

  2076     by (auto simp add: space_restrict_space null_sets_def sets_restrict_space_iff

  2077                        emeasure_restrict_space cong: conj_cong

  2078              intro!: ex_cong[where f="\<lambda>X. (\<Omega> \<inter> space M) \<inter> X"])

  2079 qed

  2080

  2081 lemma restrict_restrict_space:

  2082   assumes "A \<inter> space M \<in> sets M" "B \<inter> space M \<in> sets M"

  2083   shows "restrict_space (restrict_space M A) B = restrict_space M (A \<inter> B)" (is "?l = ?r")

  2084 proof (rule measure_eqI[symmetric])

  2085   show "sets ?r = sets ?l"

  2086     unfolding sets_restrict_space image_comp by (intro image_cong) auto

  2087 next

  2088   fix X assume "X \<in> sets (restrict_space M (A \<inter> B))"

  2089   then obtain Y where "Y \<in> sets M" "X = Y \<inter> A \<inter> B"

  2090     by (auto simp: sets_restrict_space)

  2091   with assms sets.Int[OF assms] show "emeasure ?r X = emeasure ?l X"

  2092     by (subst (1 2) emeasure_restrict_space)

  2093        (auto simp: space_restrict_space sets_restrict_space_iff emeasure_restrict_space ac_simps)

  2094 qed

  2095

  2096 lemma restrict_count_space: "restrict_space (count_space B) A = count_space (A \<inter> B)"

  2097 proof (rule measure_eqI)

  2098   show "sets (restrict_space (count_space B) A) = sets (count_space (A \<inter> B))"

  2099     by (subst sets_restrict_space) auto

  2100   moreover fix X assume "X \<in> sets (restrict_space (count_space B) A)"

  2101   ultimately have "X \<subseteq> A \<inter> B" by auto

  2102   then show "emeasure (restrict_space (count_space B) A) X = emeasure (count_space (A \<inter> B)) X"

  2103     by (cases "finite X") (auto simp add: emeasure_restrict_space)

  2104 qed

  2105

  2106 lemma sigma_finite_measure_restrict_space:

  2107   assumes "sigma_finite_measure M"

  2108   and A: "A \<in> sets M"

  2109   shows "sigma_finite_measure (restrict_space M A)"

  2110 proof -

  2111   interpret sigma_finite_measure M by fact

  2112   from sigma_finite_countable obtain C

  2113     where C: "countable C" "C \<subseteq> sets M" "(\<Union>C) = space M" "\<forall>a\<in>C. emeasure M a \<noteq> \<infinity>"

  2114     by blast

  2115   let ?C = "op \<inter> A  C"

  2116   from C have "countable ?C" "?C \<subseteq> sets (restrict_space M A)" "(\<Union>?C) = space (restrict_space M A)"

  2117     by(auto simp add: sets_restrict_space space_restrict_space)

  2118   moreover {

  2119     fix a

  2120     assume "a \<in> ?C"

  2121     then obtain a' where "a = A \<inter> a'" "a' \<in> C" ..

  2122     then have "emeasure (restrict_space M A) a \<le> emeasure M a'"

  2123       using A C by(auto simp add: emeasure_restrict_space intro: emeasure_mono)

  2124     also have "\<dots> < \<infinity>" using C(4)[rule_format, of a'] \<open>a' \<in> C\<close> by (simp add: less_top)

  2125     finally have "emeasure (restrict_space M A) a \<noteq> \<infinity>" by simp }

  2126   ultimately show ?thesis

  2127     by unfold_locales (rule exI conjI|assumption|blast)+

  2128 qed

  2129

  2130 lemma finite_measure_restrict_space:

  2131   assumes "finite_measure M"

  2132   and A: "A \<in> sets M"

  2133   shows "finite_measure (restrict_space M A)"

  2134 proof -

  2135   interpret finite_measure M by fact

  2136   show ?thesis

  2137     by(rule finite_measureI)(simp add: emeasure_restrict_space A space_restrict_space)

  2138 qed

  2139

  2140 lemma restrict_distr:

  2141   assumes [measurable]: "f \<in> measurable M N"

  2142   assumes [simp]: "\<Omega> \<inter> space N \<in> sets N" and restrict: "f \<in> space M \<rightarrow> \<Omega>"

  2143   shows "restrict_space (distr M N f) \<Omega> = distr M (restrict_space N \<Omega>) f"

  2144   (is "?l = ?r")

  2145 proof (rule measure_eqI)

  2146   fix A assume "A \<in> sets (restrict_space (distr M N f) \<Omega>)"

  2147   with restrict show "emeasure ?l A = emeasure ?r A"

  2148     by (subst emeasure_distr)

  2149        (auto simp: sets_restrict_space_iff emeasure_restrict_space emeasure_distr

  2150              intro!: measurable_restrict_space2)

  2151 qed (simp add: sets_restrict_space)

  2152

  2153 lemma measure_eqI_restrict_generator:

  2154   assumes E: "Int_stable E" "E \<subseteq> Pow \<Omega>" "\<And>X. X \<in> E \<Longrightarrow> emeasure M X = emeasure N X"

  2155   assumes sets_eq: "sets M = sets N" and \<Omega>: "\<Omega> \<in> sets M"

  2156   assumes "sets (restrict_space M \<Omega>) = sigma_sets \<Omega> E"

  2157   assumes "sets (restrict_space N \<Omega>) = sigma_sets \<Omega> E"

  2158   assumes ae: "AE x in M. x \<in> \<Omega>" "AE x in N. x \<in> \<Omega>"

  2159   assumes A: "countable A" "A \<noteq> {}" "A \<subseteq> E" "\<Union>A = \<Omega>" "\<And>a. a \<in> A \<Longrightarrow> emeasure M a \<noteq> \<infinity>"

  2160   shows "M = N"

  2161 proof (rule measure_eqI)

  2162   fix X assume X: "X \<in> sets M"

  2163   then have "emeasure M X = emeasure (restrict_space M \<Omega>) (X \<inter> \<Omega>)"

  2164     using ae \<Omega> by (auto simp add: emeasure_restrict_space intro!: emeasure_eq_AE)

  2165   also have "restrict_space M \<Omega> = restrict_space N \<Omega>"

  2166   proof (rule measure_eqI_generator_eq)

  2167     fix X assume "X \<in> E"

  2168     then show "emeasure (restrict_space M \<Omega>) X = emeasure (restrict_space N \<Omega>) X"

  2169       using E \<Omega> by (subst (1 2) emeasure_restrict_space) (auto simp: sets_eq sets_eq[THEN sets_eq_imp_space_eq])

  2170   next

  2171     show "range (from_nat_into A) \<subseteq> E" "(\<Union>i. from_nat_into A i) = \<Omega>"

  2172       using A by (auto cong del: strong_SUP_cong)

  2173   next

  2174     fix i

  2175     have "emeasure (restrict_space M \<Omega>) (from_nat_into A i) = emeasure M (from_nat_into A i)"

  2176       using A \<Omega> by (subst emeasure_restrict_space)

  2177                    (auto simp: sets_eq sets_eq[THEN sets_eq_imp_space_eq] intro: from_nat_into)

  2178     with A show "emeasure (restrict_space M \<Omega>) (from_nat_into A i) \<noteq> \<infinity>"

  2179       by (auto intro: from_nat_into)

  2180   qed fact+

  2181   also have "emeasure (restrict_space N \<Omega>) (X \<inter> \<Omega>) = emeasure N X"

  2182     using X ae \<Omega> by (auto simp add: emeasure_restrict_space sets_eq intro!: emeasure_eq_AE)

  2183   finally show "emeasure M X = emeasure N X" .

  2184 qed fact

  2185

  2186 subsection \<open>Null measure\<close>

  2187

  2188 definition "null_measure M = sigma (space M) (sets M)"

  2189

  2190 lemma space_null_measure[simp]: "space (null_measure M) = space M"

  2191   by (simp add: null_measure_def)

  2192

  2193 lemma sets_null_measure[simp, measurable_cong]: "sets (null_measure M) = sets M"

  2194   by (simp add: null_measure_def)

  2195

  2196 lemma emeasure_null_measure[simp]: "emeasure (null_measure M) X = 0"

  2197   by (cases "X \<in> sets M", rule emeasure_measure_of)

  2198      (auto simp: positive_def countably_additive_def emeasure_notin_sets null_measure_def

  2199            dest: sets.sets_into_space)

  2200

  2201 lemma measure_null_measure[simp]: "measure (null_measure M) X = 0"

  2202   by (intro measure_eq_emeasure_eq_ennreal) auto

  2203

  2204 lemma null_measure_idem [simp]: "null_measure (null_measure M) = null_measure M"

  2205   by(rule measure_eqI) simp_all

  2206

  2207 subsection \<open>Scaling a measure\<close>

  2208

  2209 definition scale_measure :: "ennreal \<Rightarrow> 'a measure \<Rightarrow> 'a measure"

  2210 where

  2211   "scale_measure r M = measure_of (space M) (sets M) (\<lambda>A. r * emeasure M A)"

  2212

  2213 lemma space_scale_measure: "space (scale_measure r M) = space M"

  2214   by (simp add: scale_measure_def)

  2215

  2216 lemma sets_scale_measure [simp, measurable_cong]: "sets (scale_measure r M) = sets M"

  2217   by (simp add: scale_measure_def)

  2218

  2219 lemma emeasure_scale_measure [simp]:

  2220   "emeasure (scale_measure r M) A = r * emeasure M A"

  2221   (is "_ = ?\<mu> A")

  2222 proof(cases "A \<in> sets M")

  2223   case True

  2224   show ?thesis unfolding scale_measure_def

  2225   proof(rule emeasure_measure_of_sigma)

  2226     show "sigma_algebra (space M) (sets M)" ..

  2227     show "positive (sets M) ?\<mu>" by (simp add: positive_def)

  2228     show "countably_additive (sets M) ?\<mu>"

  2229     proof (rule countably_additiveI)

  2230       fix A :: "nat \<Rightarrow> _"  assume *: "range A \<subseteq> sets M" "disjoint_family A"

  2231       have "(\<Sum>i. ?\<mu> (A i)) = r * (\<Sum>i. emeasure M (A i))"

  2232         by simp

  2233       also have "\<dots> = ?\<mu> (\<Union>i. A i)" using * by(simp add: suminf_emeasure)

  2234       finally show "(\<Sum>i. ?\<mu> (A i)) = ?\<mu> (\<Union>i. A i)" .

  2235     qed

  2236   qed(fact True)

  2237 qed(simp add: emeasure_notin_sets)

  2238

  2239 lemma scale_measure_1 [simp]: "scale_measure 1 M = M"

  2240   by(rule measure_eqI) simp_all

  2241

  2242 lemma scale_measure_0[simp]: "scale_measure 0 M = null_measure M"

  2243   by(rule measure_eqI) simp_all

  2244

  2245 lemma measure_scale_measure [simp]: "0 \<le> r \<Longrightarrow> measure (scale_measure r M) A = r * measure M A"

  2246   using emeasure_scale_measure[of r M A]

  2247     emeasure_eq_ennreal_measure[of M A]

  2248     measure_eq_emeasure_eq_ennreal[of _ "scale_measure r M" A]

  2249   by (cases "emeasure (scale_measure r M) A = top")

  2250      (auto simp del: emeasure_scale_measure

  2251            simp: ennreal_top_eq_mult_iff ennreal_mult_eq_top_iff measure_zero_top ennreal_mult[symmetric])

  2252

  2253 lemma scale_scale_measure [simp]:

  2254   "scale_measure r (scale_measure r' M) = scale_measure (r * r') M"

  2255   by (rule measure_eqI) (simp_all add: max_def mult.assoc)

  2256

  2257 lemma scale_null_measure [simp]: "scale_measure r (null_measure M) = null_measure M"

  2258   by (rule measure_eqI) simp_all

  2259

  2260

  2261 subsection \<open>Complete lattice structure on measures\<close>

  2262

  2263 lemma (in finite_measure) finite_measure_Diff':

  2264   "A \<in> sets M \<Longrightarrow> B \<in> sets M \<Longrightarrow> measure M (A - B) = measure M A - measure M (A \<inter> B)"

  2265   using finite_measure_Diff[of A "A \<inter> B"] by (auto simp: Diff_Int)

  2266

  2267 lemma (in finite_measure) finite_measure_Union':

  2268   "A \<in> sets M \<Longrightarrow> B \<in> sets M \<Longrightarrow> measure M (A \<union> B) = measure M A + measure M (B - A)"

  2269   using finite_measure_Union[of A "B - A"] by auto

  2270

  2271 lemma finite_unsigned_Hahn_decomposition:

  2272   assumes "finite_measure M" "finite_measure N" and [simp]: "sets N = sets M"

  2273   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)"

  2274 proof -

  2275   interpret M: finite_measure M by fact

  2276   interpret N: finite_measure N by fact

  2277

  2278   define d where "d X = measure M X - measure N X" for X

  2279

  2280   have [intro]: "bdd_above (dsets M)"

  2281     using sets.sets_into_space[of _ M]

  2282     by (intro bdd_aboveI[where M="measure M (space M)"])

  2283        (auto simp: d_def field_simps subset_eq intro!: add_increasing M.finite_measure_mono)

  2284

  2285   define \<gamma> where "\<gamma> = (SUP X:sets M. d X)"

  2286   have le_\<gamma>[intro]: "X \<in> sets M \<Longrightarrow> d X \<le> \<gamma>" for X

  2287     by (auto simp: \<gamma>_def intro!: cSUP_upper)

  2288

  2289   have "\<exists>f. \<forall>n. f n \<in> sets M \<and> d (f n) > \<gamma> - 1 / 2^n"

  2290   proof (intro choice_iff[THEN iffD1] allI)

  2291     fix n

  2292     have "\<exists>X\<in>sets M. \<gamma> - 1 / 2^n < d X"

  2293       unfolding \<gamma>_def by (intro less_cSUP_iff[THEN iffD1]) auto

  2294     then show "\<exists>y. y \<in> sets M \<and> \<gamma> - 1 / 2 ^ n < d y"

  2295       by auto

  2296   qed

  2297   then obtain E where [measurable]: "E n \<in> sets M" and E: "d (E n) > \<gamma> - 1 / 2^n" for n

  2298     by auto

  2299

  2300   define F where "F m n = (if m \<le> n then \<Inter>i\<in>{m..n}. E i else space M)" for m n

  2301

  2302   have [measurable]: "m \<le> n \<Longrightarrow> F m n \<in> sets M" for m n

  2303     by (auto simp: F_def)

  2304

  2305   have 1: "\<gamma> - 2 / 2 ^ m + 1 / 2 ^ n \<le> d (F m n)" if "m \<le> n" for m n

  2306     using that

  2307   proof (induct rule: dec_induct)

  2308     case base with E[of m] show ?case

  2309       by (simp add: F_def field_simps)

  2310   next

  2311     case (step i)

  2312     have F_Suc: "F m (Suc i) = F m i \<inter> E (Suc i)"

  2313       using \<open>m \<le> i\<close> by (auto simp: F_def le_Suc_eq)

  2314

  2315     have "\<gamma> + (\<gamma> - 2 / 2^m + 1 / 2 ^ Suc i) \<le> (\<gamma> - 1 / 2^Suc i) + (\<gamma> - 2 / 2^m + 1 / 2^i)"

  2316       by (simp add: field_simps)

  2317     also have "\<dots> \<le> d (E (Suc i)) + d (F m i)"

  2318       using E[of "Suc i"] by (intro add_mono step) auto

  2319     also have "\<dots> = d (E (Suc i)) + d (F m i - E (Suc i)) + d (F m (Suc i))"

  2320       using \<open>m \<le> i\<close> by (simp add: d_def field_simps F_Suc M.finite_measure_Diff' N.finite_measure_Diff')

  2321     also have "\<dots> = d (E (Suc i) \<union> F m i) + d (F m (Suc i))"

  2322       using \<open>m \<le> i\<close> by (simp add: d_def field_simps M.finite_measure_Union' N.finite_measure_Union')

  2323     also have "\<dots> \<le> \<gamma> + d (F m (Suc i))"

  2324       using \<open>m \<le> i\<close> by auto

  2325     finally show ?case

  2326       by auto

  2327   qed

  2328

  2329   define F' where "F' m = (\<Inter>i\<in>{m..}. E i)" for m

  2330   have F'_eq: "F' m = (\<Inter>i. F m (i + m))" for m

  2331     by (fastforce simp: le_iff_add[of m] F'_def F_def)

  2332

  2333   have [measurable]: "F' m \<in> sets M" for m

  2334     by (auto simp: F'_def)

  2335

  2336   have \<gamma>_le: "\<gamma> - 0 \<le> d (\<Union>m. F' m)"

  2337   proof (rule LIMSEQ_le)

  2338     show "(\<lambda>n. \<gamma> - 2 / 2 ^ n) \<longlonglongrightarrow> \<gamma> - 0"

  2339       by (intro tendsto_intros LIMSEQ_divide_realpow_zero) auto

  2340     have "incseq F'"

  2341       by (auto simp: incseq_def F'_def)

  2342     then show "(\<lambda>m. d (F' m)) \<longlonglongrightarrow> d (\<Union>m. F' m)"

  2343       unfolding d_def

  2344       by (intro tendsto_diff M.finite_Lim_measure_incseq N.finite_Lim_measure_incseq) auto

  2345

  2346     have "\<gamma> - 2 / 2 ^ m + 0 \<le> d (F' m)" for m

  2347     proof (rule LIMSEQ_le)

  2348       have *: "decseq (\<lambda>n. F m (n + m))"

  2349         by (auto simp: decseq_def F_def)

  2350       show "(\<lambda>n. d (F m n)) \<longlonglongrightarrow> d (F' m)"

  2351         unfolding d_def F'_eq

  2352         by (rule LIMSEQ_offset[where k=m])

  2353            (auto intro!: tendsto_diff M.finite_Lim_measure_decseq N.finite_Lim_measure_decseq *)

  2354       show "(\<lambda>n. \<gamma> - 2 / 2 ^ m + 1 / 2 ^ n) \<longlonglongrightarrow> \<gamma> - 2 / 2 ^ m + 0"

  2355         by (intro tendsto_add LIMSEQ_divide_realpow_zero tendsto_const) auto

  2356       show "\<exists>N. \<forall>n\<ge>N. \<gamma> - 2 / 2 ^ m + 1 / 2 ^ n \<le> d (F m n)"

  2357         using 1[of m] by (intro exI[of _ m]) auto

  2358     qed

  2359     then show "\<exists>N. \<forall>n\<ge>N. \<gamma> - 2 / 2 ^ n \<le> d (F' n)"

  2360       by auto

  2361   qed

  2362

  2363   show ?thesis

  2364   proof (safe intro!: bexI[of _ "\<Union>m. F' m"])

  2365     fix X assume [measurable]: "X \<in> sets M" and X: "X \<subseteq> (\<Union>m. F' m)"

  2366     have "d (\<Union>m. F' m) - d X = d ((\<Union>m. F' m) - X)"

  2367       using X by (auto simp: d_def M.finite_measure_Diff N.finite_measure_Diff)

  2368     also have "\<dots> \<le> \<gamma>"

  2369       by auto

  2370     finally have "0 \<le> d X"

  2371       using \<gamma>_le by auto

  2372     then show "emeasure N X \<le> emeasure M X"

  2373       by (auto simp: d_def M.emeasure_eq_measure N.emeasure_eq_measure)

  2374   next

  2375     fix X assume [measurable]: "X \<in> sets M" and X: "X \<inter> (\<Union>m. F' m) = {}"

  2376     then have "d (\<Union>m. F' m) + d X = d (X \<union> (\<Union>m. F' m))"

  2377       by (auto simp: d_def M.finite_measure_Union N.finite_measure_Union)

  2378     also have "\<dots> \<le> \<gamma>"

  2379       by auto

  2380     finally have "d X \<le> 0"

  2381       using \<gamma>_le by auto

  2382     then show "emeasure M X \<le> emeasure N X"

  2383       by (auto simp: d_def M.emeasure_eq_measure N.emeasure_eq_measure)

  2384   qed auto

  2385 qed

  2386

  2387 lemma unsigned_Hahn_decomposition:

  2388   assumes [simp]: "sets N = sets M" and [measurable]: "A \<in> sets M"

  2389     and [simp]: "emeasure M A \<noteq> top" "emeasure N A \<noteq> top"

  2390   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)"

  2391 proof -

  2392   have "\<exists>Y\<in>sets (restrict_space M A).

  2393     (\<forall>X\<in>sets (restrict_space M A). X \<subseteq> Y \<longrightarrow> (restrict_space N A) X \<le> (restrict_space M A) X) \<and>

  2394     (\<forall>X\<in>sets (restrict_space M A). X \<inter> Y = {} \<longrightarrow> (restrict_space M A) X \<le> (restrict_space N A) X)"

  2395   proof (rule finite_unsigned_Hahn_decomposition)

  2396     show "finite_measure (restrict_space M A)" "finite_measure (restrict_space N A)"

  2397       by (auto simp: space_restrict_space emeasure_restrict_space less_top intro!: finite_measureI)

  2398   qed (simp add: sets_restrict_space)

  2399   then guess Y ..

  2400   then show ?thesis

  2401     apply (intro bexI[of _ Y] conjI ballI conjI)

  2402     apply (simp_all add: sets_restrict_space emeasure_restrict_space)

  2403     apply safe

  2404     subgoal for X Z

  2405       by (erule ballE[of _ _ X]) (auto simp add: Int_absorb1)

  2406     subgoal for X Z

  2407       by (erule ballE[of _ _ X])  (auto simp add: Int_absorb1 ac_simps)

  2408     apply auto

  2409     done

  2410 qed

  2411

  2412 text \<open>

  2413   Define a lexicographical order on @{type measure}, in the order space, sets and measure. The parts

  2414   of the lexicographical order are point-wise ordered.

  2415 \<close>

  2416

  2417 instantiation measure :: (type) order_bot

  2418 begin

  2419

  2420 inductive less_eq_measure :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> bool" where

  2421   "space M \<subset> space N \<Longrightarrow> less_eq_measure M N"

  2422 | "space M = space N \<Longrightarrow> sets M \<subset> sets N \<Longrightarrow> less_eq_measure M N"

  2423 | "space M = space N \<Longrightarrow> sets M = sets N \<Longrightarrow> emeasure M \<le> emeasure N \<Longrightarrow> less_eq_measure M N"

  2424

  2425 lemma le_measure_iff:

  2426   "M \<le> N \<longleftrightarrow> (if space M = space N then

  2427     if sets M = sets N then emeasure M \<le> emeasure N else sets M \<subseteq> sets N else space M \<subseteq> space N)"

  2428   by (auto elim: less_eq_measure.cases intro: less_eq_measure.intros)

  2429

  2430 definition less_measure :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> bool" where

  2431   "less_measure M N \<longleftrightarrow> (M \<le> N \<and> \<not> N \<le> M)"

  2432

  2433 definition bot_measure :: "'a measure" where

  2434   "bot_measure = sigma {} {}"

  2435

  2436 lemma

  2437   shows space_bot[simp]: "space bot = {}"

  2438     and sets_bot[simp]: "sets bot = {{}}"

  2439     and emeasure_bot[simp]: "emeasure bot X = 0"

  2440   by (auto simp: bot_measure_def sigma_sets_empty_eq emeasure_sigma)

  2441

  2442 instance

  2443 proof standard

  2444   show "bot \<le> a" for a :: "'a measure"

  2445     by (simp add: le_measure_iff bot_measure_def sigma_sets_empty_eq emeasure_sigma le_fun_def)

  2446 qed (auto simp: le_measure_iff less_measure_def split: if_split_asm intro: measure_eqI)

  2447

  2448 end

  2449

  2450 lemma le_measure: "sets M = sets N \<Longrightarrow> M \<le> N \<longleftrightarrow> (\<forall>A\<in>sets M. emeasure M A \<le> emeasure N A)"

  2451   apply (auto simp: le_measure_iff le_fun_def dest: sets_eq_imp_space_eq)

  2452   subgoal for X

  2453     by (cases "X \<in> sets M") (auto simp: emeasure_notin_sets)

  2454   done

  2455

  2456 definition sup_measure' :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> 'a measure"

  2457 where

  2458   "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))"

  2459

  2460 lemma assumes [simp]: "sets B = sets A"

  2461   shows space_sup_measure'[simp]: "space (sup_measure' A B) = space A"

  2462     and sets_sup_measure'[simp]: "sets (sup_measure' A B) = sets A"

  2463   using sets_eq_imp_space_eq[OF assms] by (simp_all add: sup_measure'_def)

  2464

  2465 lemma emeasure_sup_measure':

  2466   assumes sets_eq[simp]: "sets B = sets A" and [simp, intro]: "X \<in> sets A"

  2467   shows "emeasure (sup_measure' A B) X = (SUP Y:sets A. emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y))"

  2468     (is "_ = ?S X")

  2469 proof -

  2470   note sets_eq_imp_space_eq[OF sets_eq, simp]

  2471   show ?thesis

  2472     using sup_measure'_def

  2473   proof (rule emeasure_measure_of)

  2474     let ?d = "\<lambda>X Y. emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y)"

  2475     show "countably_additive (sets (sup_measure' A B)) (\<lambda>X. SUP Y : sets A. emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y))"

  2476     proof (rule countably_additiveI, goal_cases)

  2477       case (1 X)

  2478       then have [measurable]: "\<And>i. X i \<in> sets A" and "disjoint_family X"

  2479         by auto

  2480       have "(\<Sum>i. ?S (X i)) = (SUP Y:sets A. \<Sum>i. ?d (X i) Y)"

  2481       proof (rule ennreal_suminf_SUP_eq_directed)

  2482         fix J :: "nat set" and a b assume "finite J" and [measurable]: "a \<in> sets A" "b \<in> sets A"

  2483         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

  2484         proof cases

  2485           assume "emeasure A (X i) = top \<or> emeasure B (X i) = top"

  2486           then show ?thesis

  2487           proof

  2488             assume "emeasure A (X i) = top" then show ?thesis

  2489               by (intro bexI[of _ "X i"]) auto

  2490           next

  2491             assume "emeasure B (X i) = top" then show ?thesis

  2492               by (intro bexI[of _ "{}"]) auto

  2493           qed

  2494         next

  2495           assume finite: "\<not> (emeasure A (X i) = top \<or> emeasure B (X i) = top)"

  2496           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)"

  2497             using unsigned_Hahn_decomposition[of B A "X i"] by simp

  2498           then obtain Y where [measurable]: "Y \<in> sets A" and [simp]: "Y \<subseteq> X i"

  2499             and B_le_A: "\<And>C. C \<in> sets A \<Longrightarrow> C \<subseteq> Y \<Longrightarrow> B C \<le> A C"

  2500             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"

  2501             by auto

  2502

  2503           show ?thesis

  2504           proof (intro bexI[of _ Y] ballI conjI)

  2505             fix a assume [measurable]: "a \<in> sets A"

  2506             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"

  2507               for a Y by auto

  2508             then have "?d (X i) a =

  2509               (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))"

  2510               by (subst (1 2) plus_emeasure) (auto simp: Diff_eq[symmetric])

  2511             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))"

  2512               by (intro add_mono order_refl B_le_A A_le_B) (auto simp: Diff_eq[symmetric])

  2513             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))"

  2514               by (simp add: ac_simps)

  2515             also have "\<dots> \<le> A (X i \<inter> Y) + B (X i \<inter> - Y)"

  2516               by (subst (1 2) plus_emeasure) (auto simp: Diff_eq[symmetric] *)

  2517             finally show "?d (X i) a \<le> ?d (X i) Y" .

  2518           qed auto

  2519         qed

  2520         then obtain C where [measurable]: "C i \<in> sets A" and "C i \<subseteq> X i"

  2521           and C: "\<And>a. a \<in> sets A \<Longrightarrow> ?d (X i) a \<le> ?d (X i) (C i)" for i

  2522           by metis

  2523         have *: "X i \<inter> (\<Union>i. C i) = X i \<inter> C i" for i

  2524         proof safe

  2525           fix x j assume "x \<in> X i" "x \<in> C j"

  2526           moreover have "i = j \<or> X i \<inter> X j = {}"

  2527             using \<open>disjoint_family X\<close> by (auto simp: disjoint_family_on_def)

  2528           ultimately show "x \<in> C i"

  2529             using \<open>C i \<subseteq> X i\<close> \<open>C j \<subseteq> X j\<close> by auto

  2530         qed auto

  2531         have **: "X i \<inter> - (\<Union>i. C i) = X i \<inter> - C i" for i

  2532         proof safe

  2533           fix x j assume "x \<in> X i" "x \<notin> C i" "x \<in> C j"

  2534           moreover have "i = j \<or> X i \<inter> X j = {}"

  2535             using \<open>disjoint_family X\<close> by (auto simp: disjoint_family_on_def)

  2536           ultimately show False

  2537             using \<open>C i \<subseteq> X i\<close> \<open>C j \<subseteq> X j\<close> by auto

  2538         qed auto

  2539         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"

  2540           apply (intro bexI[of _ "\<Union>i. C i"])

  2541           unfolding * **

  2542           apply (intro C ballI conjI)

  2543           apply auto

  2544           done

  2545       qed

  2546       also have "\<dots> = ?S (\<Union>i. X i)"

  2547         unfolding UN_extend_simps(4)

  2548         by (auto simp add: suminf_add[symmetric] Diff_eq[symmetric] simp del: UN_simps

  2549                  intro!: SUP_cong arg_cong2[where f="op +"] suminf_emeasure

  2550                          disjoint_family_on_bisimulation[OF \<open>disjoint_family X\<close>])

  2551       finally show "(\<Sum>i. ?S (X i)) = ?S (\<Union>i. X i)" .

  2552     qed

  2553   qed (auto dest: sets.sets_into_space simp: positive_def intro!: SUP_const)

  2554 qed

  2555

  2556 lemma le_emeasure_sup_measure'1:

  2557   assumes "sets B = sets A" "X \<in> sets A" shows "emeasure A X \<le> emeasure (sup_measure' A B) X"

  2558   by (subst emeasure_sup_measure'[OF assms]) (auto intro!: SUP_upper2[of "X"] assms)

  2559

  2560 lemma le_emeasure_sup_measure'2:

  2561   assumes "sets B = sets A" "X \<in> sets A" shows "emeasure B X \<le> emeasure (sup_measure' A B) X"

  2562   by (subst emeasure_sup_measure'[OF assms]) (auto intro!: SUP_upper2[of "{}"] assms)

  2563

  2564 lemma emeasure_sup_measure'_le2:

  2565   assumes [simp]: "sets B = sets C" "sets A = sets C" and [measurable]: "X \<in> sets C"

  2566   assumes A: "\<And>Y. Y \<subseteq> X \<Longrightarrow> Y \<in> sets A \<Longrightarrow> emeasure A Y \<le> emeasure C Y"

  2567   assumes B: "\<And>Y. Y \<subseteq> X \<Longrightarrow> Y \<in> sets A \<Longrightarrow> emeasure B Y \<le> emeasure C Y"

  2568   shows "emeasure (sup_measure' A B) X \<le> emeasure C X"

  2569 proof (subst emeasure_sup_measure')

  2570   show "(SUP Y:sets A. emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y)) \<le> emeasure C X"

  2571     unfolding \<open>sets A = sets C\<close>

  2572   proof (intro SUP_least)

  2573     fix Y assume [measurable]: "Y \<in> sets C"

  2574     have [simp]: "X \<inter> Y \<union> (X - Y) = X"

  2575       by auto

  2576     have "emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y) \<le> emeasure C (X \<inter> Y) + emeasure C (X \<inter> - Y)"

  2577       by (intro add_mono A B) (auto simp: Diff_eq[symmetric])

  2578     also have "\<dots> = emeasure C X"

  2579       by (subst plus_emeasure) (auto simp: Diff_eq[symmetric])

  2580     finally show "emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y) \<le> emeasure C X" .

  2581   qed

  2582 qed simp_all

  2583

  2584 definition sup_lexord :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b::order) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a"

  2585 where

  2586   "sup_lexord A B k s c =

  2587     (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)"

  2588

  2589 lemma sup_lexord:

  2590   "(k A < k B \<Longrightarrow> P B) \<Longrightarrow> (k B < k A \<Longrightarrow> P A) \<Longrightarrow> (k A = k B \<Longrightarrow> P c) \<Longrightarrow>

  2591     (\<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)"

  2592   by (auto simp: sup_lexord_def)

  2593

  2594 lemmas le_sup_lexord = sup_lexord[where P="\<lambda>a. c \<le> a" for c]

  2595

  2596 lemma sup_lexord1: "k A = k B \<Longrightarrow> sup_lexord A B k s c = c"

  2597   by (simp add: sup_lexord_def)

  2598

  2599 lemma sup_lexord_commute: "sup_lexord A B k s c = sup_lexord B A k s c"

  2600   by (auto simp: sup_lexord_def)

  2601

  2602 lemma sigma_sets_le_sets_iff: "(sigma_sets (space x) \<A> \<subseteq> sets x) = (\<A> \<subseteq> sets x)"

  2603   using sets.sigma_sets_subset[of \<A> x] by auto

  2604

  2605 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))"

  2606   by (cases "\<Omega> = space x")

  2607      (simp_all add: eq_commute[of _ "sets x"] le_measure_iff emeasure_sigma le_fun_def

  2608                     sigma_sets_superset_generator sigma_sets_le_sets_iff)

  2609

  2610 instantiation measure :: (type) semilattice_sup

  2611 begin

  2612

  2613 definition sup_measure :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> 'a measure"

  2614 where

  2615   "sup_measure A B =

  2616     sup_lexord A B space (sigma (space A \<union> space B) {})

  2617       (sup_lexord A B sets (sigma (space A) (sets A \<union> sets B)) (sup_measure' A B))"

  2618

  2619 instance

  2620 proof

  2621   fix x y z :: "'a measure"

  2622   show "x \<le> sup x y"

  2623     unfolding sup_measure_def

  2624   proof (intro le_sup_lexord)

  2625     assume "space x = space y"

  2626     then have *: "sets x \<union> sets y \<subseteq> Pow (space x)"

  2627       using sets.space_closed by auto

  2628     assume "\<not> sets y \<subseteq> sets x" "\<not> sets x \<subseteq> sets y"

  2629     then have "sets x \<subset> sets x \<union> sets y"

  2630       by auto

  2631     also have "\<dots> \<le> sigma (space x) (sets x \<union> sets y)"

  2632       by (subst sets_measure_of[OF *]) (rule sigma_sets_superset_generator)

  2633     finally show "x \<le> sigma (space x) (sets x \<union> sets y)"

  2634       by (simp add: space_measure_of[OF *] less_eq_measure.intros(2))

  2635   next

  2636     assume "\<not> space y \<subseteq> space x" "\<not> space x \<subseteq> space y"

  2637     then show "x \<le> sigma (space x \<union> space y) {}"

  2638       by (intro less_eq_measure.intros) auto

  2639   next

  2640     assume "sets x = sets y" then show "x \<le> sup_measure' x y"

  2641       by (simp add: le_measure le_emeasure_sup_measure'1)

  2642   qed (auto intro: less_eq_measure.intros)

  2643   show "y \<le> sup x y"

  2644     unfolding sup_measure_def

  2645   proof (intro le_sup_lexord)

  2646     assume **: "space x = space y"

  2647     then have *: "sets x \<union> sets y \<subseteq> Pow (space y)"

  2648       using sets.space_closed by auto

  2649     assume "\<not> sets y \<subseteq> sets x" "\<not> sets x \<subseteq> sets y"

  2650     then have "sets y \<subset> sets x \<union> sets y"

  2651       by auto

  2652     also have "\<dots> \<le> sigma (space y) (sets x \<union> sets y)"

  2653       by (subst sets_measure_of[OF *]) (rule sigma_sets_superset_generator)

  2654     finally show "y \<le> sigma (space x) (sets x \<union> sets y)"

  2655       by (simp add: ** space_measure_of[OF *] less_eq_measure.intros(2))

  2656   next

  2657     assume "\<not> space y \<subseteq> space x" "\<not> space x \<subseteq> space y"

  2658     then show "y \<le> sigma (space x \<union> space y) {}"

  2659       by (intro less_eq_measure.intros) auto

  2660   next

  2661     assume "sets x = sets y" then show "y \<le> sup_measure' x y"

  2662       by (simp add: le_measure le_emeasure_sup_measure'2)

  2663   qed (auto intro: less_eq_measure.intros)

  2664   show "x \<le> y \<Longrightarrow> z \<le> y \<Longrightarrow> sup x z \<le> y"

  2665     unfolding sup_measure_def

  2666   proof (intro sup_lexord[where P="\<lambda>x. x \<le> y"])

  2667     assume "x \<le> y" "z \<le> y" and [simp]: "space x = space z" "sets x = sets z"

  2668     from \<open>x \<le> y\<close> show "sup_measure' x z \<le> y"

  2669     proof cases

  2670       case 1 then show ?thesis

  2671         by (intro less_eq_measure.intros(1)) simp

  2672     next

  2673       case 2 then show ?thesis

  2674         by (intro less_eq_measure.intros(2)) simp_all

  2675     next

  2676       case 3 with \<open>z \<le> y\<close> \<open>x \<le> y\<close> show ?thesis

  2677         by (auto simp add: le_measure intro!: emeasure_sup_measure'_le2)

  2678     qed

  2679   next

  2680     assume **: "x \<le> y" "z \<le> y" "space x = space z" "\<not> sets z \<subseteq> sets x" "\<not> sets x \<subseteq> sets z"

  2681     then have *: "sets x \<union> sets z \<subseteq> Pow (space x)"

  2682       using sets.space_closed by auto

  2683     show "sigma (space x) (sets x \<union> sets z) \<le> y"

  2684       unfolding sigma_le_iff[OF *] using ** by (auto simp: le_measure_iff split: if_split_asm)

  2685   next

  2686     assume "x \<le> y" "z \<le> y" "\<not> space z \<subseteq> space x" "\<not> space x \<subseteq> space z"

  2687     then have "space x \<subseteq> space y" "space z \<subseteq> space y"

  2688       by (auto simp: le_measure_iff split: if_split_asm)

  2689     then show "sigma (space x \<union> space z) {} \<le> y"

  2690       by (simp add: sigma_le_iff)

  2691   qed

  2692 qed

  2693

  2694 end

  2695

  2696 lemma space_empty_eq_bot: "space a = {} \<longleftrightarrow> a = bot"

  2697   using space_empty[of a] by (auto intro!: measure_eqI)

  2698

  2699 lemma sets_eq_iff_bounded: "A \<le> B \<Longrightarrow> B \<le> C \<Longrightarrow> sets A = sets C \<Longrightarrow> sets B = sets A"

  2700   by (auto dest: sets_eq_imp_space_eq simp add: le_measure_iff split: if_split_asm)

  2701

  2702 lemma sets_sup: "sets A = sets M \<Longrightarrow> sets B = sets M \<Longrightarrow> sets (sup A B) = sets M"

  2703   by (auto simp add: sup_measure_def sup_lexord_def dest: sets_eq_imp_space_eq)

  2704

  2705 lemma le_measureD1: "A \<le> B \<Longrightarrow> space A \<le> space B"

  2706   by (auto simp: le_measure_iff split: if_split_asm)

  2707

  2708 lemma le_measureD2: "A \<le> B \<Longrightarrow> space A = space B \<Longrightarrow> sets A \<le> sets B"

  2709   by (auto simp: le_measure_iff split: if_split_asm)

  2710

  2711 lemma le_measureD3: "A \<le> B \<Longrightarrow> sets A = sets B \<Longrightarrow> emeasure A X \<le> emeasure B X"

  2712   by (auto simp: le_measure_iff le_fun_def dest: sets_eq_imp_space_eq split: if_split_asm)

  2713

  2714 lemma UN_space_closed: "UNION S sets \<subseteq> Pow (UNION S space)"

  2715   using sets.space_closed by auto

  2716

  2717 definition Sup_lexord :: "('a \<Rightarrow> 'b::complete_lattice) \<Rightarrow> ('a set \<Rightarrow> 'a) \<Rightarrow> ('a set \<Rightarrow> 'a) \<Rightarrow> 'a set \<Rightarrow> 'a"

  2718 where

  2719   "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)"

  2720

  2721 lemma Sup_lexord:

  2722   "(\<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>

  2723     P (Sup_lexord k c s A)"

  2724   by (auto simp: Sup_lexord_def Let_def)

  2725

  2726 lemma Sup_lexord1:

  2727   assumes A: "A \<noteq> {}" "(\<And>a. a \<in> A \<Longrightarrow> k a = (\<Union>a\<in>A. k a))" "P (c A)"

  2728   shows "P (Sup_lexord k c s A)"

  2729   unfolding Sup_lexord_def Let_def

  2730 proof (clarsimp, safe)

  2731   show "\<forall>a\<in>A. k a \<noteq> (\<Union>x\<in>A. k x) \<Longrightarrow> P (s A)"

  2732     by (metis assms(1,2) ex_in_conv)

  2733 next

  2734   fix a assume "a \<in> A" "k a = (\<Union>x\<in>A. k x)"

  2735   then have "{a \<in> A. k a = (\<Union>x\<in>A. k x)} = {a \<in> A. k a = k a}"

  2736     by (metis A(2)[symmetric])

  2737   then show "P (c {a \<in> A. k a = (\<Union>x\<in>A. k x)})"

  2738     by (simp add: A(3))

  2739 qed

  2740

  2741 instantiation measure :: (type) complete_lattice

  2742 begin

  2743

  2744 interpretation sup_measure: comm_monoid_set sup "bot :: 'a measure"

  2745   by standard (auto intro!: antisym)

  2746

  2747 lemma sup_measure_F_mono':

  2748   "finite J \<Longrightarrow> finite I \<Longrightarrow> sup_measure.F id I \<le> sup_measure.F id (I \<union> J)"

  2749 proof (induction J rule: finite_induct)

  2750   case empty then show ?case

  2751     by simp

  2752 next

  2753   case (insert i J)

  2754   show ?case

  2755   proof cases

  2756     assume "i \<in> I" with insert show ?thesis

  2757       by (auto simp: insert_absorb)

  2758   next

  2759     assume "i \<notin> I"

  2760     have "sup_measure.F id I \<le> sup_measure.F id (I \<union> J)"

  2761       by (intro insert)

  2762     also have "\<dots> \<le> sup_measure.F id (insert i (I \<union> J))"

  2763       using insert \<open>i \<notin> I\<close> by (subst sup_measure.insert) auto

  2764     finally show ?thesis

  2765       by auto

  2766   qed

  2767 qed

  2768

  2769 lemma sup_measure_F_mono: "finite I \<Longrightarrow> J \<subseteq> I \<Longrightarrow> sup_measure.F id J \<le> sup_measure.F id I"

  2770   using sup_measure_F_mono'[of I J] by (auto simp: finite_subset Un_absorb1)

  2771

  2772 lemma sets_sup_measure_F:

  2773   "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"

  2774   by (induction I rule: finite_ne_induct) (simp_all add: sets_sup)

  2775

  2776 definition Sup_measure' :: "'a measure set \<Rightarrow> 'a measure"

  2777 where

  2778   "Sup_measure' M = measure_of (\<Union>a\<in>M. space a) (\<Union>a\<in>M. sets a)

  2779     (\<lambda>X. (SUP P:{P. finite P \<and> P \<subseteq> M }. sup_measure.F id P X))"

  2780

  2781 lemma space_Sup_measure'2: "space (Sup_measure' M) = (\<Union>m\<in>M. space m)"

  2782   unfolding Sup_measure'_def by (intro space_measure_of[OF UN_space_closed])

  2783

  2784 lemma sets_Sup_measure'2: "sets (Sup_measure' M) = sigma_sets (\<Union>m\<in>M. space m) (\<Union>m\<in>M. sets m)"

  2785   unfolding Sup_measure'_def by (intro sets_measure_of[OF UN_space_closed])

  2786

  2787 lemma sets_Sup_measure':

  2788   assumes sets_eq[simp]: "\<And>m. m \<in> M \<Longrightarrow> sets m = sets A" and "M \<noteq> {}"

  2789   shows "sets (Sup_measure' M) = sets A"

  2790   using sets_eq[THEN sets_eq_imp_space_eq, simp] \<open>M \<noteq> {}\<close> by (simp add: Sup_measure'_def)

  2791

  2792 lemma space_Sup_measure':

  2793   assumes sets_eq[simp]: "\<And>m. m \<in> M \<Longrightarrow> sets m = sets A" and "M \<noteq> {}"

  2794   shows "space (Sup_measure' M) = space A"

  2795   using sets_eq[THEN sets_eq_imp_space_eq, simp] \<open>M \<noteq> {}\<close>

  2796   by (simp add: Sup_measure'_def )

  2797

  2798 lemma emeasure_Sup_measure':

  2799   assumes sets_eq[simp]: "\<And>m. m \<in> M \<Longrightarrow> sets m = sets A" and "X \<in> sets A" "M \<noteq> {}"

  2800   shows "emeasure (Sup_measure' M) X = (SUP P:{P. finite P \<and> P \<subseteq> M}. sup_measure.F id P X)"

  2801     (is "_ = ?S X")

  2802   using Sup_measure'_def

  2803 proof (rule emeasure_measure_of)

  2804   note sets_eq[THEN sets_eq_imp_space_eq, simp]

  2805   have *: "sets (Sup_measure' M) = sets A" "space (Sup_measure' M) = space A"

  2806     using \<open>M \<noteq> {}\<close> by (simp_all add: Sup_measure'_def)

  2807   let ?\<mu> = "sup_measure.F id"

  2808   show "countably_additive (sets (Sup_measure' M)) ?S"

  2809   proof (rule countably_additiveI, goal_cases)

  2810     case (1 F)

  2811     then have **: "range F \<subseteq> sets A"

  2812       by (auto simp: *)

  2813     show "(\<Sum>i. ?S (F i)) = ?S (\<Union>i. F i)"

  2814     proof (subst ennreal_suminf_SUP_eq_directed)

  2815       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}"

  2816       have "(i \<noteq> {} \<longrightarrow> sets (?\<mu> i) = sets A) \<and> (j \<noteq> {} \<longrightarrow> sets (?\<mu> j) = sets A) \<and>

  2817         (i \<noteq> {} \<or> j \<noteq> {} \<longrightarrow> sets (?\<mu> (i \<union> j)) = sets A)"

  2818         using ij by (intro impI sets_sup_measure_F conjI) auto

  2819       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

  2820         using ij

  2821         by (cases "i = {}"; cases "j = {}")

  2822            (auto intro!: le_measureD3 sup_measure_F_mono simp: sets_sup_measure_F

  2823                  simp del: id_apply)

  2824       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)"

  2825         by (safe intro!: bexI[of _ "i \<union> j"]) auto

  2826     next

  2827       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))"

  2828       proof (intro SUP_cong refl)

  2829         fix i assume i: "i \<in> {P. finite P \<and> P \<subseteq> M}"

  2830         show "(\<Sum>n. ?\<mu> i (F n)) = ?\<mu> i (UNION UNIV F)"

  2831         proof cases

  2832           assume "i \<noteq> {}" with i ** show ?thesis

  2833             apply (intro suminf_emeasure \<open>disjoint_family F\<close>)

  2834             apply (subst sets_sup_measure_F[OF _ _ sets_eq])

  2835             apply auto

  2836             done

  2837         qed simp

  2838       qed

  2839     qed

  2840   qed

  2841   show "positive (sets (Sup_measure' M)) ?S"

  2842     by (auto simp: positive_def bot_ennreal[symmetric])

  2843   show "X \<in> sets (Sup_measure' M)"

  2844     using assms * by auto

  2845 qed (rule UN_space_closed)

  2846

  2847 definition Sup_measure :: "'a measure set \<Rightarrow> 'a measure"

  2848 where

  2849   "Sup_measure = Sup_lexord space (Sup_lexord sets Sup_measure'

  2850     (\<lambda>U. sigma (\<Union>u\<in>U. space u) (\<Union>u\<in>U. sets u))) (\<lambda>U. sigma (\<Union>u\<in>U. space u) {})"

  2851

  2852 definition Inf_measure :: "'a measure set \<Rightarrow> 'a measure"

  2853 where

  2854   "Inf_measure A = Sup {x. \<forall>a\<in>A. x \<le> a}"

  2855

  2856 definition inf_measure :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> 'a measure"

  2857 where

  2858   "inf_measure a b = Inf {a, b}"

  2859

  2860 definition top_measure :: "'a measure"

  2861 where

  2862   "top_measure = Inf {}"

  2863

  2864 instance

  2865 proof

  2866   note UN_space_closed [simp]

  2867   show upper: "x \<le> Sup A" if x: "x \<in> A" for x :: "'a measure" and A

  2868     unfolding Sup_measure_def

  2869   proof (intro Sup_lexord[where P="\<lambda>y. x \<le> y"])

  2870     assume "\<And>a. a \<in> A \<Longrightarrow> space a \<noteq> (\<Union>a\<in>A. space a)"

  2871     from this[OF \<open>x \<in> A\<close>] \<open>x \<in> A\<close> show "x \<le> sigma (\<Union>a\<in>A. space a) {}"

  2872       by (intro less_eq_measure.intros) auto

  2873   next

  2874     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}"

  2875       and neq: "\<And>aa. aa \<in> S \<Longrightarrow> sets aa \<noteq> (\<Union>a\<in>S. sets a)"

  2876     have sp_a: "space a = (UNION S space)"

  2877       using \<open>a\<in>A\<close> by (auto simp: S)

  2878     show "x \<le> sigma (UNION S space) (UNION S sets)"

  2879     proof cases

  2880       assume [simp]: "space x = space a"

  2881       have "sets x \<subset> (\<Union>a\<in>S. sets a)"

  2882         using \<open>x\<in>A\<close> neq[of x] by (auto simp: S)

  2883       also have "\<dots> \<subseteq> sigma_sets (\<Union>x\<in>S. space x) (\<Union>x\<in>S. sets x)"

  2884         by (rule sigma_sets_superset_generator)

  2885       finally show ?thesis

  2886         by (intro less_eq_measure.intros(2)) (simp_all add: sp_a)

  2887     next

  2888       assume "space x \<noteq> space a"

  2889       moreover have "space x \<le> space a"

  2890         unfolding a using \<open>x\<in>A\<close> by auto

  2891       ultimately show ?thesis

  2892         by (intro less_eq_measure.intros) (simp add: less_le sp_a)

  2893     qed

  2894   next

  2895     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}"

  2896       and "b \<in> S" and b: "sets b = (\<Union>a\<in>S. sets a)" and S': "S' = {a' \<in> S. sets a' = sets b}"

  2897     then have "S' \<noteq> {}" "space b = space a"

  2898       by auto

  2899     have sets_eq: "\<And>x. x \<in> S' \<Longrightarrow> sets x = sets b"

  2900       by (auto simp: S')

  2901     note sets_eq[THEN sets_eq_imp_space_eq, simp]

  2902     have *: "sets (Sup_measure' S') = sets b" "space (Sup_measure' S') = space b"

  2903       using \<open>S' \<noteq> {}\<close> by (simp_all add: Sup_measure'_def sets_eq)

  2904     show "x \<le> Sup_measure' S'"

  2905     proof cases

  2906       assume "x \<in> S"

  2907       with \<open>b \<in> S\<close> have "space x = space b"

  2908         by (simp add: S)

  2909       show ?thesis

  2910       proof cases

  2911         assume "x \<in> S'"

  2912         show "x \<le> Sup_measure' S'"

  2913         proof (intro le_measure[THEN iffD2] ballI)

  2914           show "sets x = sets (Sup_measure' S')"

  2915             using \<open>x\<in>S'\<close> * by (simp add: S')

  2916           fix X assume "X \<in> sets x"

  2917           show "emeasure x X \<le> emeasure (Sup_measure' S') X"

  2918           proof (subst emeasure_Sup_measure'[OF _ \<open>X \<in> sets x\<close>])

  2919             show "emeasure x X \<le> (SUP P : {P. finite P \<and> P \<subseteq> S'}. emeasure (sup_measure.F id P) X)"

  2920               using \<open>x\<in>S'\<close> by (intro SUP_upper2[where i="{x}"]) auto

  2921           qed (insert \<open>x\<in>S'\<close> S', auto)

  2922         qed

  2923       next

  2924         assume "x \<notin> S'"

  2925         then have "sets x \<noteq> sets b"

  2926           using \<open>x\<in>S\<close> by (auto simp: S')

  2927         moreover have "sets x \<le> sets b"

  2928           using \<open>x\<in>S\<close> unfolding b by auto

  2929         ultimately show ?thesis

  2930           using * \<open>x \<in> S\<close>

  2931           by (intro less_eq_measure.intros(2))

  2932              (simp_all add: * \<open>space x = space b\<close> less_le)

  2933       qed

  2934     next

  2935       assume "x \<notin> S"

  2936       with \<open>x\<in>A\<close> \<open>x \<notin> S\<close> \<open>space b = space a\<close> show ?thesis

  2937         by (intro less_eq_measure.intros)

  2938            (simp_all add: * less_le a SUP_upper S)

  2939     qed

  2940   qed

  2941   show least: "Sup A \<le> x" if x: "\<And>z. z \<in> A \<Longrightarrow> z \<le> x" for x :: "'a measure" and A

  2942     unfolding Sup_measure_def

  2943   proof (intro Sup_lexord[where P="\<lambda>y. y \<le> x"])

  2944     assume "\<And>a. a \<in> A \<Longrightarrow> space a \<noteq> (\<Union>a\<in>A. space a)"

  2945     show "sigma (UNION A space) {} \<le> x"

  2946       using x[THEN le_measureD1] by (subst sigma_le_iff) auto

  2947   next

  2948     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}"

  2949       "\<And>a. a \<in> S \<Longrightarrow> sets a \<noteq> (\<Union>a\<in>S. sets a)"

  2950     have "UNION S space \<subseteq> space x"

  2951       using S le_measureD1[OF x] by auto

  2952     moreover

  2953     have "UNION S space = space a"

  2954       using \<open>a\<in>A\<close> S by auto

  2955     then have "space x = UNION S space \<Longrightarrow> UNION S sets \<subseteq> sets x"

  2956       using \<open>a \<in> A\<close> le_measureD2[OF x] by (auto simp: S)

  2957     ultimately show "sigma (UNION S space) (UNION S sets) \<le> x"

  2958       by (subst sigma_le_iff) simp_all

  2959   next

  2960     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}"

  2961       and "b \<in> S" and b: "sets b = (\<Union>a\<in>S. sets a)" and S': "S' = {a' \<in> S. sets a' = sets b}"

  2962     then have "S' \<noteq> {}" "space b = space a"

  2963       by auto

  2964     have sets_eq: "\<And>x. x \<in> S' \<Longrightarrow> sets x = sets b"

  2965       by (auto simp: S')

  2966     note sets_eq[THEN sets_eq_imp_space_eq, simp]

  2967     have *: "sets (Sup_measure' S') = sets b" "space (Sup_measure' S') = space b"

  2968       using \<open>S' \<noteq> {}\<close> by (simp_all add: Sup_measure'_def sets_eq)

  2969     show "Sup_measure' S' \<le> x"

  2970     proof cases

  2971       assume "space x = space a"

  2972       show ?thesis

  2973       proof cases

  2974         assume **: "sets x = sets b"

  2975         show ?thesis

  2976         proof (intro le_measure[THEN iffD2] ballI)

  2977           show ***: "sets (Sup_measure' S') = sets x"

  2978             by (simp add: * **)

  2979           fix X assume "X \<in> sets (Sup_measure' S')"

  2980           show "emeasure (Sup_measure' S') X \<le> emeasure x X"

  2981             unfolding ***

  2982           proof (subst emeasure_Sup_measure'[OF _ \<open>X \<in> sets (Sup_measure' S')\<close>])

  2983             show "(SUP P : {P. finite P \<and> P \<subseteq> S'}. emeasure (sup_measure.F id P) X) \<le> emeasure x X"

  2984             proof (safe intro!: SUP_least)

  2985               fix P assume P: "finite P" "P \<subseteq> S'"

  2986               show "emeasure (sup_measure.F id P) X \<le> emeasure x X"

  2987               proof cases

  2988                 assume "P = {}" then show ?thesis

  2989                   by auto

  2990               next

  2991                 assume "P \<noteq> {}"

  2992                 from P have "finite P" "P \<subseteq> A"

  2993                   unfolding S' S by (simp_all add: subset_eq)

  2994                 then have "sup_measure.F id P \<le> x"

  2995                   by (induction P) (auto simp: x)

  2996                 moreover have "sets (sup_measure.F id P) = sets x"

  2997                   using \<open>finite P\<close> \<open>P \<noteq> {}\<close> \<open>P \<subseteq> S'\<close> \<open>sets x = sets b\<close>

  2998                   by (intro sets_sup_measure_F) (auto simp: S')

  2999                 ultimately show "emeasure (sup_measure.F id P) X \<le> emeasure x X"

  3000                   by (rule le_measureD3)

  3001               qed

  3002             qed

  3003             show "m \<in> S' \<Longrightarrow> sets m = sets (Sup_measure' S')" for m

  3004               unfolding * by (simp add: S')

  3005           qed fact

  3006         qed

  3007       next

  3008         assume "sets x \<noteq> sets b"

  3009         moreover have "sets b \<le> sets x"

  3010           unfolding b S using x[THEN le_measureD2] \<open>space x = space a\<close> by auto

  3011         ultimately show "Sup_measure' S' \<le> x"

  3012           using \<open>space x = space a\<close> \<open>b \<in> S\<close>

  3013           by (intro less_eq_measure.intros(2)) (simp_all add: * S)

  3014       qed

  3015     next

  3016       assume "space x \<noteq> space a"

  3017       then have "space a < space x"

  3018         using le_measureD1[OF x[OF \<open>a\<in>A\<close>]] by auto

  3019       then show "Sup_measure' S' \<le> x"

  3020         by (intro less_eq_measure.intros) (simp add: * \<open>space b = space a\<close>)

  3021     qed

  3022   qed

  3023   show "Sup {} = (bot::'a measure)" "Inf {} = (top::'a measure)"

  3024     by (auto intro!: antisym least simp: top_measure_def)

  3025   show lower: "x \<in> A \<Longrightarrow> Inf A \<le> x" for x :: "'a measure" and A

  3026     unfolding Inf_measure_def by (intro least) auto

  3027   show greatest: "(\<And>z. z \<in> A \<Longrightarrow> x \<le> z) \<Longrightarrow> x \<le> Inf A" for x :: "'a measure" and A

  3028     unfolding Inf_measure_def by (intro upper) auto

  3029   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"

  3030     by (auto simp: inf_measure_def intro!: lower greatest)

  3031 qed

  3032

  3033 end

  3034

  3035 lemma sets_SUP:

  3036   assumes "\<And>x. x \<in> I \<Longrightarrow> sets (M x) = sets N"

  3037   shows "I \<noteq> {} \<Longrightarrow> sets (SUP i:I. M i) = sets N"

  3038   unfolding Sup_measure_def

  3039   using assms assms[THEN sets_eq_imp_space_eq]

  3040     sets_Sup_measure'[where A=N and M="MI"]

  3041   by (intro Sup_lexord1[where P="\<lambda>x. sets x = sets N"]) auto

  3042

  3043 lemma emeasure_SUP:

  3044   assumes sets: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets N" "X \<in> sets N" "I \<noteq> {}"

  3045   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)"

  3046 proof -

  3047   interpret sup_measure: comm_monoid_set sup "bot :: 'b measure"

  3048     by standard (auto intro!: antisym)

  3049   have eq: "finite J \<Longrightarrow> sup_measure.F id J = (SUP i:J. i)" for J :: "'b measure set"

  3050     by (induction J rule: finite_induct) auto

  3051   have 1: "J \<noteq> {} \<Longrightarrow> J \<subseteq> I \<Longrightarrow> sets (SUP x:J. M x) = sets N" for J

  3052     by (intro sets_SUP sets) (auto )

  3053   from \<open>I \<noteq> {}\<close> obtain i where "i\<in>I" by auto

  3054   have "Sup_measure' (MI) X = (SUP P:{P. finite P \<and> P \<subseteq> MI}. sup_measure.F id P X)"

  3055     using sets by (intro emeasure_Sup_measure') auto

  3056   also have "Sup_measure' (MI) = (SUP i:I. M i)"

  3057     unfolding Sup_measure_def using \<open>I \<noteq> {}\<close> sets sets(1)[THEN sets_eq_imp_space_eq]

  3058     by (intro Sup_lexord1[where P="\<lambda>x. _ = x"]) auto

  3059   also have "(SUP P:{P. finite P \<and> P \<subseteq> MI}. sup_measure.F id P X) =

  3060     (SUP J:{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. (SUP i:J. M i) X)"

  3061   proof (intro SUP_eq)

  3062     fix J assume "J \<in> {P. finite P \<and> P \<subseteq> MI}"

  3063     then obtain J' where J': "J' \<subseteq> I" "finite J'" and J: "J = MJ'" and "finite J"

  3064       using finite_subset_image[of J M I] by auto

  3065     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"

  3066     proof cases

  3067       assume "J' = {}" with \<open>i \<in> I\<close> show ?thesis

  3068         by (auto simp add: J)

  3069     next

  3070       assume "J' \<noteq> {}" with J J' show ?thesis

  3071         by (intro bexI[of _ "J'"]) (auto simp add: eq simp del: id_apply)

  3072     qed

  3073   next

  3074     fix J assume J: "J \<in> {P. P \<noteq> {} \<and> finite P \<and> P \<subseteq> I}"

  3075     show "\<exists>J'\<in>{J. finite J \<and> J \<subseteq> MI}. (SUP i:J. M i) X \<le> sup_measure.F id J' X"

  3076       using J by (intro bexI[of _ "MJ"]) (auto simp add: eq simp del: id_apply)

  3077   qed

  3078   finally show ?thesis .

  3079 qed

  3080

  3081 lemma emeasure_SUP_chain:

  3082   assumes sets: "\<And>i. i \<in> A \<Longrightarrow> sets (M i) = sets N" "X \<in> sets N"

  3083   assumes ch: "Complete_Partial_Order.chain op \<le> (M  A)" and "A \<noteq> {}"

  3084   shows "emeasure (SUP i:A. M i) X = (SUP i:A. emeasure (M i) X)"

  3085 proof (subst emeasure_SUP[OF sets \<open>A \<noteq> {}\<close>])

  3086   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)"

  3087   proof (rule SUP_eq)

  3088     fix J assume "J \<in> {J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> A}"

  3089     then have J: "Complete_Partial_Order.chain op \<le> (M  J)" "finite J" "J \<noteq> {}" and "J \<subseteq> A"

  3090       using ch[THEN chain_subset, of "MJ"] by auto

  3091     with in_chain_finite[OF J(1)] obtain j where "j \<in> J" "(SUP j:J. M j) = M j"

  3092       by auto

  3093     with \<open>J \<subseteq> A\<close> show "\<exists>j\<in>A. emeasure (SUPREMUM J M) X \<le> emeasure (M j) X"

  3094       by auto

  3095   next

  3096     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"

  3097       by (intro bexI[of _ "{j}"]) auto

  3098   qed

  3099 qed

  3100

  3101 subsubsection \<open>Supremum of a set of $\sigma$-algebras\<close>

  3102

  3103 lemma space_Sup_eq_UN: "space (Sup M) = (\<Union>x\<in>M. space x)"

  3104   unfolding Sup_measure_def

  3105   apply (intro Sup_lexord[where P="\<lambda>x. space x = _"])

  3106   apply (subst space_Sup_measure'2)

  3107   apply auto []

  3108   apply (subst space_measure_of[OF UN_space_closed])

  3109   apply auto

  3110   done

  3111

  3112 lemma sets_Sup_eq:

  3113   assumes *: "\<And>m. m \<in> M \<Longrightarrow> space m = X" and "M \<noteq> {}"

  3114   shows "sets (Sup M) = sigma_sets X (\<Union>x\<in>M. sets x)"

  3115   unfolding Sup_measure_def

  3116   apply (rule Sup_lexord1)

  3117   apply fact

  3118   apply (simp add: assms)

  3119   apply (rule Sup_lexord)

  3120   subgoal premises that for a S

  3121     unfolding that(3) that(2)[symmetric]

  3122     using that(1)

  3123     apply (subst sets_Sup_measure'2)

  3124     apply (intro arg_cong2[where f=sigma_sets])

  3125     apply (auto simp: *)

  3126     done

  3127   apply (subst sets_measure_of[OF UN_space_closed])

  3128   apply (simp add:  assms)

  3129   done

  3130

  3131 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)"

  3132   by (subst sets_Sup_eq[where X=X]) auto

  3133

  3134 lemma Sup_lexord_rel:

  3135   assumes "\<And>i. i \<in> I \<Longrightarrow> k (A i) = k (B i)"

  3136     "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))}))"

  3137     "R (s (AI)) (s (BI))"

  3138   shows "R (Sup_lexord k c s (AI)) (Sup_lexord k c s (BI))"

  3139 proof -

  3140   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))}"

  3141     using assms(1) by auto

  3142   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))}"

  3143     by auto

  3144   ultimately show ?thesis

  3145     using assms by (auto simp: Sup_lexord_def Let_def)

  3146 qed

  3147

  3148 lemma sets_SUP_cong:

  3149   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)"

  3150   unfolding Sup_measure_def

  3151   using eq eq[THEN sets_eq_imp_space_eq]

  3152   apply (intro Sup_lexord_rel[where R="\<lambda>x y. sets x = sets y"])

  3153   apply simp

  3154   apply simp

  3155   apply (simp add: sets_Sup_measure'2)

  3156   apply (intro arg_cong2[where f="\<lambda>x y. sets (sigma x y)"])

  3157   apply auto

  3158   done

  3159

  3160 lemma sets_Sup_in_sets:

  3161   assumes "M \<noteq> {}"

  3162   assumes "\<And>m. m \<in> M \<Longrightarrow> space m = space N"

  3163   assumes "\<And>m. m \<in> M \<Longrightarrow> sets m \<subseteq> sets N"

  3164   shows "sets (Sup M) \<subseteq> sets N"

  3165 proof -

  3166   have *: "UNION M space = space N"

  3167     using assms by auto

  3168   show ?thesis

  3169     unfolding * using assms by (subst sets_Sup_eq[of M "space N"]) (auto intro!: sets.sigma_sets_subset)

  3170 qed

  3171

  3172 lemma measurable_Sup1:

  3173   assumes m: "m \<in> M" and f: "f \<in> measurable m N"

  3174     and const_space: "\<And>m n. m \<in> M \<Longrightarrow> n \<in> M \<Longrightarrow> space m = space n"

  3175   shows "f \<in> measurable (Sup M) N"

  3176 proof -

  3177   have "space (Sup M) = space m"

  3178     using m by (auto simp add: space_Sup_eq_UN dest: const_space)

  3179   then show ?thesis

  3180     using m f unfolding measurable_def by (auto intro: in_sets_Sup[OF const_space])

  3181 qed

  3182

  3183 lemma measurable_Sup2:

  3184   assumes M: "M \<noteq> {}"

  3185   assumes f: "\<And>m. m \<in> M \<Longrightarrow> f \<in> measurable N m"

  3186     and const_space: "\<And>m n. m \<in> M \<Longrightarrow> n \<in> M \<Longrightarrow> space m = space n"

  3187   shows "f \<in> measurable N (Sup M)"

  3188 proof -

  3189   from M obtain m where "m \<in> M" by auto

  3190   have space_eq: "\<And>n. n \<in> M \<Longrightarrow> space n = space m"

  3191     by (intro const_space \<open>m \<in> M\<close>)

  3192   have "f \<in> measurable N (sigma (\<Union>m\<in>M. space m) (\<Union>m\<in>M. sets m))"

  3193   proof (rule measurable_measure_of)

  3194     show "f \<in> space N \<rightarrow> UNION M space"

  3195       using measurable_space[OF f] M by auto

  3196   qed (auto intro: measurable_sets f dest: sets.sets_into_space)

  3197   also have "measurable N (sigma (\<Union>m\<in>M. space m) (\<Union>m\<in>M. sets m)) = measurable N (Sup M)"

  3198     apply (intro measurable_cong_sets refl)

  3199     apply (subst sets_Sup_eq[OF space_eq M])

  3200     apply simp

  3201     apply (subst sets_measure_of[OF UN_space_closed])

  3202     apply (simp add: space_eq M)

  3203     done

  3204   finally show ?thesis .

  3205 qed

  3206

  3207 lemma sets_Sup_sigma:

  3208   assumes [simp]: "M \<noteq> {}" and M: "\<And>m. m \<in> M \<Longrightarrow> m \<subseteq> Pow \<Omega>"

  3209   shows "sets (SUP m:M. sigma \<Omega> m) = sets (sigma \<Omega> (\<Union>M))"

  3210 proof -

  3211   { fix a m assume "a \<in> sigma_sets \<Omega> m" "m \<in> M"

  3212     then have "a \<in> sigma_sets \<Omega> (\<Union>M)"

  3213      by induction (auto intro: sigma_sets.intros(2-)) }

  3214   then show "sets (SUP m:M. sigma \<Omega> m) = sets (sigma \<Omega> (\<Union>M))"

  3215     apply (subst sets_Sup_eq[where X="\<Omega>"])

  3216     apply (auto simp add: M) []

  3217     apply auto []

  3218     apply (simp add: space_measure_of_conv M Union_least)

  3219     apply (rule sigma_sets_eqI)

  3220     apply auto

  3221     done

  3222 qed

  3223

  3224 lemma Sup_sigma:

  3225   assumes [simp]: "M \<noteq> {}" and M: "\<And>m. m \<in> M \<Longrightarrow> m \<subseteq> Pow \<Omega>"

  3226   shows "(SUP m:M. sigma \<Omega> m) = (sigma \<Omega> (\<Union>M))"

  3227 proof (intro antisym SUP_least)

  3228   have *: "\<Union>M \<subseteq> Pow \<Omega>"

  3229     using M by auto

  3230   show "sigma \<Omega> (\<Union>M) \<le> (SUP m:M. sigma \<Omega> m)"

  3231   proof (intro less_eq_measure.intros(3))

  3232     show "space (sigma \<Omega> (\<Union>M)) = space (SUP m:M. sigma \<Omega> m)"

  3233       "sets (sigma \<Omega> (\<Union>M)) = sets (SUP m:M. sigma \<Omega> m)"

  3234       using sets_Sup_sigma[OF assms] sets_Sup_sigma[OF assms, THEN sets_eq_imp_space_eq]

  3235       by auto

  3236   qed (simp add: emeasure_sigma le_fun_def)

  3237   fix m assume "m \<in> M" then show "sigma \<Omega> m \<le> sigma \<Omega> (\<Union>M)"

  3238     by (subst sigma_le_iff) (auto simp add: M *)

  3239 qed

  3240

  3241 lemma SUP_sigma_sigma:

  3242   "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)"

  3243   using Sup_sigma[of "fM" \<Omega>] by auto

  3244

  3245 lemma sets_vimage_Sup_eq:

  3246   assumes *: "M \<noteq> {}" "f \<in> X \<rightarrow> Y" "\<And>m. m \<in> M \<Longrightarrow> space m = Y"

  3247   shows "sets (vimage_algebra X f (Sup M)) = sets (SUP m : M. vimage_algebra X f m)"

  3248   (is "?IS = ?SI")

  3249 proof

  3250   show "?IS \<subseteq> ?SI"

  3251     apply (intro sets_image_in_sets measurable_Sup2)

  3252     apply (simp add: space_Sup_eq_UN *)

  3253     apply (simp add: *)

  3254     apply (intro measurable_Sup1)

  3255     apply (rule imageI)

  3256     apply assumption

  3257     apply (rule measurable_vimage_algebra1)

  3258     apply (auto simp: *)

  3259     done

  3260   show "?SI \<subseteq> ?IS"

  3261     apply (intro sets_Sup_in_sets)

  3262     apply (auto simp: *) []

  3263     apply (auto simp: *) []

  3264     apply (elim imageE)

  3265     apply simp

  3266     apply (rule sets_image_in_sets)

  3267     apply simp

  3268     apply (simp add: measurable_def)

  3269     apply (simp add: * space_Sup_eq_UN sets_vimage_algebra2)

  3270     apply (auto intro: in_sets_Sup[OF *(3)])

  3271     done

  3272 qed

  3273

  3274 lemma restrict_space_eq_vimage_algebra':

  3275   "sets (restrict_space M \<Omega>) = sets (vimage_algebra (\<Omega> \<inter> space M) (\<lambda>x. x) M)"

  3276 proof -

  3277   have *: "{A \<inter> (\<Omega> \<inter> space M) |A. A \<in> sets M} = {A \<inter> \<Omega> |A. A \<in> sets M}"

  3278     using sets.sets_into_space[of _ M] by blast

  3279

  3280   show ?thesis

  3281     unfolding restrict_space_def

  3282     by (subst sets_measure_of)

  3283        (auto simp add: image_subset_iff sets_vimage_algebra * dest: sets.sets_into_space intro!: arg_cong2[where f=sigma_sets])

  3284 qed

  3285

  3286 lemma sigma_le_sets:

  3287   assumes [simp]: "A \<subseteq> Pow X" shows "sets (sigma X A) \<subseteq> sets N \<longleftrightarrow> X \<in> sets N \<and> A \<subseteq> sets N"

  3288 proof

  3289   have "X \<in> sigma_sets X A" "A \<subseteq> sigma_sets X A"

  3290     by (auto intro: sigma_sets_top)

  3291   moreover assume "sets (sigma X A) \<subseteq> sets N"

  3292   ultimately show "X \<in> sets N \<and> A \<subseteq> sets N"

  3293     by auto

  3294 next

  3295   assume *: "X \<in> sets N \<and> A \<subseteq> sets N"

  3296   { fix Y assume "Y \<in> sigma_sets X A" from this * have "Y \<in> sets N"

  3297       by induction auto }

  3298   then show "sets (sigma X A) \<subseteq> sets N"

  3299     by auto

  3300 qed

  3301

  3302 lemma measurable_iff_sets:

  3303   "f \<in> measurable M N \<longleftrightarrow> (f \<in> space M \<rightarrow> space N \<and> sets (vimage_algebra (space M) f N) \<subseteq> sets M)"

  3304 proof -

  3305   have *: "{f - A \<inter> space M |A. A \<in> sets N} \<subseteq> Pow (space M)"

  3306     by auto

  3307   show ?thesis

  3308     unfolding measurable_def

  3309     by (auto simp add: vimage_algebra_def sigma_le_sets[OF *])

  3310 qed

  3311

  3312 lemma sets_vimage_algebra_space: "X \<in> sets (vimage_algebra X f M)"

  3313   using sets.top[of "vimage_algebra X f M"] by simp

  3314

  3315 lemma measurable_mono:

  3316   assumes N: "sets N' \<le> sets N" "space N = space N'"

  3317   assumes M: "sets M \<le> sets M'" "space M = space M'"

  3318   shows "measurable M N \<subseteq> measurable M' N'"

  3319   unfolding measurable_def

  3320 proof safe

  3321   fix f A assume "f \<in> space M \<rightarrow> space N" "A \<in> sets N'"

  3322   moreover assume "\<forall>y\<in>sets N. f - y \<inter> space M \<in> sets M" note this[THEN bspec, of A]

  3323   ultimately show "f - A \<inter> space M' \<in> sets M'"

  3324     using assms by auto

  3325 qed (insert N M, auto)

  3326

  3327 lemma measurable_Sup_measurable:

  3328   assumes f: "f \<in> space N \<rightarrow> A"

  3329   shows "f \<in> measurable N (Sup {M. space M = A \<and> f \<in> measurable N M})"

  3330 proof (rule measurable_Sup2)

  3331   show "{M. space M = A \<and> f \<in> measurable N M} \<noteq> {}"

  3332     using f unfolding ex_in_conv[symmetric]

  3333     by (intro exI[of _ "sigma A {}"]) (auto intro!: measurable_measure_of)

  3334 qed auto

  3335

  3336 lemma (in sigma_algebra) sigma_sets_subset':

  3337   assumes a: "a \<subseteq> M" "\<Omega>' \<in> M"

  3338   shows "sigma_sets \<Omega>' a \<subseteq> M"

  3339 proof

  3340   show "x \<in> M" if x: "x \<in> sigma_sets \<Omega>' a" for x

  3341     using x by (induct rule: sigma_sets.induct) (insert a, auto)

  3342 qed

  3343

  3344 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)"

  3345   by (intro in_sets_Sup[where X=Y]) auto

  3346

  3347 lemma measurable_SUP1:

  3348   "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>

  3349     f \<in> measurable (SUP i:I. M i) N"

  3350   by (auto intro: measurable_Sup1)

  3351

  3352 lemma sets_image_in_sets':

  3353   assumes X: "X \<in> sets N"

  3354   assumes f: "\<And>A. A \<in> sets M \<Longrightarrow> f - A \<inter> X \<in> sets N"

  3355   shows "sets (vimage_algebra X f M) \<subseteq> sets N"

  3356   unfolding sets_vimage_algebra

  3357   by (rule sets.sigma_sets_subset') (auto intro!: measurable_sets X f)

  3358

  3359 lemma mono_vimage_algebra:

  3360   "sets M \<le> sets N \<Longrightarrow> sets (vimage_algebra X f M) \<subseteq> sets (vimage_algebra X f N)"

  3361   using sets.top[of "sigma X {f - A \<inter> X |A. A \<in> sets N}"]

  3362   unfolding vimage_algebra_def

  3363   apply (subst (asm) space_measure_of)

  3364   apply auto []

  3365   apply (subst sigma_le_sets)

  3366   apply auto

  3367   done

  3368

  3369 lemma mono_restrict_space: "sets M \<le> sets N \<Longrightarrow> sets (restrict_space M X) \<subseteq> sets (restrict_space N X)"

  3370   unfolding sets_restrict_space by (rule image_mono)

  3371

  3372 lemma sets_eq_bot: "sets M = {{}} \<longleftrightarrow> M = bot"

  3373   apply safe

  3374   apply (intro measure_eqI)

  3375   apply auto

  3376   done

  3377

  3378 lemma sets_eq_bot2: "{{}} = sets M \<longleftrightarrow> M = bot"

  3379   using sets_eq_bot[of M] by blast

  3380

  3381

  3382 lemma (in finite_measure) countable_support:

  3383   "countable {x. measure M {x} \<noteq> 0}"

  3384 proof cases

  3385   assume "measure M (space M) = 0"

  3386   with bounded_measure measure_le_0_iff have "{x. measure M {x} \<noteq> 0} = {}"

  3387     by auto

  3388   then show ?thesis

  3389     by simp

  3390 next

  3391   let ?M = "measure M (space M)" and ?m = "\<lambda>x. measure M {x}"

  3392   assume "?M \<noteq> 0"

  3393   then have *: "{x. ?m x \<noteq> 0} = (\<Union>n. {x. ?M / Suc n < ?m x})"

  3394     using reals_Archimedean[of "?m x / ?M" for x]

  3395     by (auto simp: field_simps not_le[symmetric] divide_le_0_iff measure_le_0_iff)

  3396   have **: "\<And>n. finite {x. ?M / Suc n < ?m x}"

  3397   proof (rule ccontr)

  3398     fix n assume "infinite {x. ?M / Suc n < ?m x}" (is "infinite ?X")

  3399     then obtain X where "finite X" "card X = Suc (Suc n)" "X \<subseteq> ?X"

  3400       by (metis infinite_arbitrarily_large)

  3401     from this(3) have *: "\<And>x. x \<in> X \<Longrightarrow> ?M / Suc n \<le> ?m x"

  3402       by auto

  3403     { fix x assume "x \<in> X"

  3404       from \<open>?M \<noteq> 0\<close> *[OF this] have "?m x \<noteq> 0" by (auto simp: field_simps measure_le_0_iff)

  3405       then have "{x} \<in> sets M" by (auto dest: measure_notin_sets) }

  3406     note singleton_sets = this

  3407     have "?M < (\<Sum>x\<in>X. ?M / Suc n)"

  3408       using \<open>?M \<noteq> 0\<close>

  3409       by (simp add: \<open>card X = Suc (Suc n)\<close> field_simps less_le)

  3410     also have "\<dots> \<le> (\<Sum>x\<in>X. ?m x)"

  3411       by (rule setsum_mono) fact

  3412     also have "\<dots> = measure M (\<Union>x\<in>X. {x})"

  3413       using singleton_sets \<open>finite X\<close>

  3414       by (intro finite_measure_finite_Union[symmetric]) (auto simp: disjoint_family_on_def)

  3415     finally have "?M < measure M (\<Union>x\<in>X. {x})" .

  3416     moreover have "measure M (\<Union>x\<in>X. {x}) \<le> ?M"

  3417       using singleton_sets[THEN sets.sets_into_space] by (intro finite_measure_mono) auto

  3418     ultimately show False by simp

  3419   qed

  3420   show ?thesis

  3421     unfolding * by (intro countable_UN countableI_type countable_finite[OF **])

  3422 qed

  3423

  3424 end
`