src/HOL/Analysis/Sigma_Algebra.thy
 author immler Tue Jul 10 09:38:35 2018 +0200 (11 months ago) changeset 68607 67bb59e49834 parent 68403 223172b97d0b child 69164 74f1b0f10b2b permissions -rw-r--r--
make theorem, corollary, and proposition %important for HOL-Analysis manual
     1 (*  Title:      HOL/Analysis/Sigma_Algebra.thy

     2     Author:     Stefan Richter, Markus Wenzel, TU München

     3     Author:     Johannes Hölzl, TU München

     4     Plus material from the Hurd/Coble measure theory development,

     5     translated by Lawrence Paulson.

     6 *)

     7

     8 section \<open>Sigma Algebra\<close>

     9

    10 theory Sigma_Algebra

    11 imports

    12   Complex_Main

    13   "HOL-Library.Countable_Set"

    14   "HOL-Library.FuncSet"

    15   "HOL-Library.Indicator_Function"

    16   "HOL-Library.Extended_Nonnegative_Real"

    17   "HOL-Library.Disjoint_Sets"

    18 begin

    19

    20 text \<open>Sigma algebras are an elementary concept in measure

    21   theory. To measure --- that is to integrate --- functions, we first have

    22   to measure sets. Unfortunately, when dealing with a large universe,

    23   it is often not possible to consistently assign a measure to every

    24   subset. Therefore it is necessary to define the set of measurable

    25   subsets of the universe. A sigma algebra is such a set that has

    26   three very natural and desirable properties.\<close>

    27

    28 subsection \<open>Families of sets\<close>

    29

    30 locale%important subset_class =

    31   fixes \<Omega> :: "'a set" and M :: "'a set set"

    32   assumes space_closed: "M \<subseteq> Pow \<Omega>"

    33

    34 lemma (in subset_class) sets_into_space: "x \<in> M \<Longrightarrow> x \<subseteq> \<Omega>"

    35   by (metis PowD contra_subsetD space_closed)

    36

    37 subsubsection \<open>Semiring of sets\<close>

    38

    39 locale%important semiring_of_sets = subset_class +

    40   assumes empty_sets[iff]: "{} \<in> M"

    41   assumes Int[intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<inter> b \<in> M"

    42   assumes Diff_cover:

    43     "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> \<exists>C\<subseteq>M. finite C \<and> disjoint C \<and> a - b = \<Union>C"

    44

    45 lemma (in semiring_of_sets) finite_INT[intro]:

    46   assumes "finite I" "I \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M"

    47   shows "(\<Inter>i\<in>I. A i) \<in> M"

    48   using assms by (induct rule: finite_ne_induct) auto

    49

    50 lemma (in semiring_of_sets) Int_space_eq1 [simp]: "x \<in> M \<Longrightarrow> \<Omega> \<inter> x = x"

    51   by (metis Int_absorb1 sets_into_space)

    52

    53 lemma (in semiring_of_sets) Int_space_eq2 [simp]: "x \<in> M \<Longrightarrow> x \<inter> \<Omega> = x"

    54   by (metis Int_absorb2 sets_into_space)

    55

    56 lemma (in semiring_of_sets) sets_Collect_conj:

    57   assumes "{x\<in>\<Omega>. P x} \<in> M" "{x\<in>\<Omega>. Q x} \<in> M"

    58   shows "{x\<in>\<Omega>. Q x \<and> P x} \<in> M"

    59 proof -

    60   have "{x\<in>\<Omega>. Q x \<and> P x} = {x\<in>\<Omega>. Q x} \<inter> {x\<in>\<Omega>. P x}"

    61     by auto

    62   with assms show ?thesis by auto

    63 qed

    64

    65 lemma (in semiring_of_sets) sets_Collect_finite_All':

    66   assumes "\<And>i. i \<in> S \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M" "finite S" "S \<noteq> {}"

    67   shows "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} \<in> M"

    68 proof -

    69   have "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} = (\<Inter>i\<in>S. {x\<in>\<Omega>. P i x})"

    70     using \<open>S \<noteq> {}\<close> by auto

    71   with assms show ?thesis by auto

    72 qed

    73

    74 subsubsection \<open>Ring of sets\<close>

    75

    76 locale%important ring_of_sets = semiring_of_sets +

    77   assumes Un [intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<union> b \<in> M"

    78

    79 lemma (in ring_of_sets) finite_Union [intro]:

    80   "finite X \<Longrightarrow> X \<subseteq> M \<Longrightarrow> \<Union>X \<in> M"

    81   by (induct set: finite) (auto simp add: Un)

    82

    83 lemma (in ring_of_sets) finite_UN[intro]:

    84   assumes "finite I" and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M"

    85   shows "(\<Union>i\<in>I. A i) \<in> M"

    86   using assms by induct auto

    87

    88 lemma (in ring_of_sets) Diff [intro]:

    89   assumes "a \<in> M" "b \<in> M" shows "a - b \<in> M"

    90   using Diff_cover[OF assms] by auto

    91

    92 lemma ring_of_setsI:

    93   assumes space_closed: "M \<subseteq> Pow \<Omega>"

    94   assumes empty_sets[iff]: "{} \<in> M"

    95   assumes Un[intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<union> b \<in> M"

    96   assumes Diff[intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a - b \<in> M"

    97   shows "ring_of_sets \<Omega> M"

    98 proof

    99   fix a b assume ab: "a \<in> M" "b \<in> M"

   100   from ab show "\<exists>C\<subseteq>M. finite C \<and> disjoint C \<and> a - b = \<Union>C"

   101     by (intro exI[of _ "{a - b}"]) (auto simp: disjoint_def)

   102   have "a \<inter> b = a - (a - b)" by auto

   103   also have "\<dots> \<in> M" using ab by auto

   104   finally show "a \<inter> b \<in> M" .

   105 qed fact+

   106

   107 lemma ring_of_sets_iff: "ring_of_sets \<Omega> M \<longleftrightarrow> M \<subseteq> Pow \<Omega> \<and> {} \<in> M \<and> (\<forall>a\<in>M. \<forall>b\<in>M. a \<union> b \<in> M) \<and> (\<forall>a\<in>M. \<forall>b\<in>M. a - b \<in> M)"

   108 proof

   109   assume "ring_of_sets \<Omega> M"

   110   then interpret ring_of_sets \<Omega> M .

   111   show "M \<subseteq> Pow \<Omega> \<and> {} \<in> M \<and> (\<forall>a\<in>M. \<forall>b\<in>M. a \<union> b \<in> M) \<and> (\<forall>a\<in>M. \<forall>b\<in>M. a - b \<in> M)"

   112     using space_closed by auto

   113 qed (auto intro!: ring_of_setsI)

   114

   115 lemma (in ring_of_sets) insert_in_sets:

   116   assumes "{x} \<in> M" "A \<in> M" shows "insert x A \<in> M"

   117 proof -

   118   have "{x} \<union> A \<in> M" using assms by (rule Un)

   119   thus ?thesis by auto

   120 qed

   121

   122 lemma (in ring_of_sets) sets_Collect_disj:

   123   assumes "{x\<in>\<Omega>. P x} \<in> M" "{x\<in>\<Omega>. Q x} \<in> M"

   124   shows "{x\<in>\<Omega>. Q x \<or> P x} \<in> M"

   125 proof -

   126   have "{x\<in>\<Omega>. Q x \<or> P x} = {x\<in>\<Omega>. Q x} \<union> {x\<in>\<Omega>. P x}"

   127     by auto

   128   with assms show ?thesis by auto

   129 qed

   130

   131 lemma (in ring_of_sets) sets_Collect_finite_Ex:

   132   assumes "\<And>i. i \<in> S \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M" "finite S"

   133   shows "{x\<in>\<Omega>. \<exists>i\<in>S. P i x} \<in> M"

   134 proof -

   135   have "{x\<in>\<Omega>. \<exists>i\<in>S. P i x} = (\<Union>i\<in>S. {x\<in>\<Omega>. P i x})"

   136     by auto

   137   with assms show ?thesis by auto

   138 qed

   139

   140 subsubsection \<open>Algebra of sets\<close>

   141

   142 locale%important algebra = ring_of_sets +

   143   assumes top [iff]: "\<Omega> \<in> M"

   144

   145 lemma (in algebra) compl_sets [intro]:

   146   "a \<in> M \<Longrightarrow> \<Omega> - a \<in> M"

   147   by auto

   148

   149 proposition algebra_iff_Un:

   150   "algebra \<Omega> M \<longleftrightarrow>

   151     M \<subseteq> Pow \<Omega> \<and>

   152     {} \<in> M \<and>

   153     (\<forall>a \<in> M. \<Omega> - a \<in> M) \<and>

   154     (\<forall>a \<in> M. \<forall> b \<in> M. a \<union> b \<in> M)" (is "_ \<longleftrightarrow> ?Un")

   155 proof

   156   assume "algebra \<Omega> M"

   157   then interpret algebra \<Omega> M .

   158   show ?Un using sets_into_space by auto

   159 next

   160   assume ?Un

   161   then have "\<Omega> \<in> M" by auto

   162   interpret ring_of_sets \<Omega> M

   163   proof (rule ring_of_setsI)

   164     show \<Omega>: "M \<subseteq> Pow \<Omega>" "{} \<in> M"

   165       using \<open>?Un\<close> by auto

   166     fix a b assume a: "a \<in> M" and b: "b \<in> M"

   167     then show "a \<union> b \<in> M" using \<open>?Un\<close> by auto

   168     have "a - b = \<Omega> - ((\<Omega> - a) \<union> b)"

   169       using \<Omega> a b by auto

   170     then show "a - b \<in> M"

   171       using a b  \<open>?Un\<close> by auto

   172   qed

   173   show "algebra \<Omega> M" proof qed fact

   174 qed

   175

   176 proposition algebra_iff_Int:

   177      "algebra \<Omega> M \<longleftrightarrow>

   178        M \<subseteq> Pow \<Omega> & {} \<in> M &

   179        (\<forall>a \<in> M. \<Omega> - a \<in> M) &

   180        (\<forall>a \<in> M. \<forall> b \<in> M. a \<inter> b \<in> M)" (is "_ \<longleftrightarrow> ?Int")

   181 proof

   182   assume "algebra \<Omega> M"

   183   then interpret algebra \<Omega> M .

   184   show ?Int using sets_into_space by auto

   185 next

   186   assume ?Int

   187   show "algebra \<Omega> M"

   188   proof (unfold algebra_iff_Un, intro conjI ballI)

   189     show \<Omega>: "M \<subseteq> Pow \<Omega>" "{} \<in> M"

   190       using \<open>?Int\<close> by auto

   191     from \<open>?Int\<close> show "\<And>a. a \<in> M \<Longrightarrow> \<Omega> - a \<in> M" by auto

   192     fix a b assume M: "a \<in> M" "b \<in> M"

   193     hence "a \<union> b = \<Omega> - ((\<Omega> - a) \<inter> (\<Omega> - b))"

   194       using \<Omega> by blast

   195     also have "... \<in> M"

   196       using M \<open>?Int\<close> by auto

   197     finally show "a \<union> b \<in> M" .

   198   qed

   199 qed

   200

   201 lemma (in algebra) sets_Collect_neg:

   202   assumes "{x\<in>\<Omega>. P x} \<in> M"

   203   shows "{x\<in>\<Omega>. \<not> P x} \<in> M"

   204 proof -

   205   have "{x\<in>\<Omega>. \<not> P x} = \<Omega> - {x\<in>\<Omega>. P x}" by auto

   206   with assms show ?thesis by auto

   207 qed

   208

   209 lemma (in algebra) sets_Collect_imp:

   210   "{x\<in>\<Omega>. P x} \<in> M \<Longrightarrow> {x\<in>\<Omega>. Q x} \<in> M \<Longrightarrow> {x\<in>\<Omega>. Q x \<longrightarrow> P x} \<in> M"

   211   unfolding imp_conv_disj by (intro sets_Collect_disj sets_Collect_neg)

   212

   213 lemma (in algebra) sets_Collect_const:

   214   "{x\<in>\<Omega>. P} \<in> M"

   215   by (cases P) auto

   216

   217 lemma algebra_single_set:

   218   "X \<subseteq> S \<Longrightarrow> algebra S { {}, X, S - X, S }"

   219   by (auto simp: algebra_iff_Int)

   220

   221 subsubsection%unimportant \<open>Restricted algebras\<close>

   222

   223 abbreviation (in algebra)

   224   "restricted_space A \<equiv> ((\<inter>) A)  M"

   225

   226 lemma (in algebra) restricted_algebra:

   227   assumes "A \<in> M" shows "algebra A (restricted_space A)"

   228   using assms by (auto simp: algebra_iff_Int)

   229

   230 subsubsection \<open>Sigma Algebras\<close>

   231

   232 locale%important sigma_algebra = algebra +

   233   assumes countable_nat_UN [intro]: "\<And>A. range A \<subseteq> M \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"

   234

   235 lemma (in algebra) is_sigma_algebra:

   236   assumes "finite M"

   237   shows "sigma_algebra \<Omega> M"

   238 proof

   239   fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> M"

   240   then have "(\<Union>i. A i) = (\<Union>s\<in>M \<inter> range A. s)"

   241     by auto

   242   also have "(\<Union>s\<in>M \<inter> range A. s) \<in> M"

   243     using \<open>finite M\<close> by auto

   244   finally show "(\<Union>i. A i) \<in> M" .

   245 qed

   246

   247 lemma countable_UN_eq:

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

   249   shows "(range A \<subseteq> M \<longrightarrow> (\<Union>i. A i) \<in> M) \<longleftrightarrow>

   250     (range (A \<circ> from_nat) \<subseteq> M \<longrightarrow> (\<Union>i. (A \<circ> from_nat) i) \<in> M)"

   251 proof -

   252   let ?A' = "A \<circ> from_nat"

   253   have *: "(\<Union>i. ?A' i) = (\<Union>i. A i)" (is "?l = ?r")

   254   proof safe

   255     fix x i assume "x \<in> A i" thus "x \<in> ?l"

   256       by (auto intro!: exI[of _ "to_nat i"])

   257   next

   258     fix x i assume "x \<in> ?A' i" thus "x \<in> ?r"

   259       by (auto intro!: exI[of _ "from_nat i"])

   260   qed

   261   have **: "range ?A' = range A"

   262     using surj_from_nat

   263     by (auto simp: image_comp [symmetric] intro!: imageI)

   264   show ?thesis unfolding * ** ..

   265 qed

   266

   267 lemma (in sigma_algebra) countable_Union [intro]:

   268   assumes "countable X" "X \<subseteq> M" shows "\<Union>X \<in> M"

   269 proof cases

   270   assume "X \<noteq> {}"

   271   hence "\<Union>X = (\<Union>n. from_nat_into X n)"

   272     using assms by (auto intro: from_nat_into) (metis from_nat_into_surj)

   273   also have "\<dots> \<in> M" using assms

   274     by (auto intro!: countable_nat_UN) (metis \<open>X \<noteq> {}\<close> from_nat_into set_mp)

   275   finally show ?thesis .

   276 qed simp

   277

   278 lemma (in sigma_algebra) countable_UN[intro]:

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

   280   assumes "AX \<subseteq> M"

   281   shows  "(\<Union>x\<in>X. A x) \<in> M"

   282 proof -

   283   let ?A = "\<lambda>i. if i \<in> X then A i else {}"

   284   from assms have "range ?A \<subseteq> M" by auto

   285   with countable_nat_UN[of "?A \<circ> from_nat"] countable_UN_eq[of ?A M]

   286   have "(\<Union>x. ?A x) \<in> M" by auto

   287   moreover have "(\<Union>x. ?A x) = (\<Union>x\<in>X. A x)" by (auto split: if_split_asm)

   288   ultimately show ?thesis by simp

   289 qed

   290

   291 lemma (in sigma_algebra) countable_UN':

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

   293   assumes X: "countable X"

   294   assumes A: "AX \<subseteq> M"

   295   shows  "(\<Union>x\<in>X. A x) \<in> M"

   296 proof -

   297   have "(\<Union>x\<in>X. A x) = (\<Union>i\<in>to_nat_on X  X. A (from_nat_into X i))"

   298     using X by auto

   299   also have "\<dots> \<in> M"

   300     using A X

   301     by (intro countable_UN) auto

   302   finally show ?thesis .

   303 qed

   304

   305 lemma (in sigma_algebra) countable_UN'':

   306   "\<lbrakk> countable X; \<And>x y. x \<in> X \<Longrightarrow> A x \<in> M \<rbrakk> \<Longrightarrow> (\<Union>x\<in>X. A x) \<in> M"

   307 by(erule countable_UN')(auto)

   308

   309 lemma (in sigma_algebra) countable_INT [intro]:

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

   311   assumes A: "AX \<subseteq> M" "X \<noteq> {}"

   312   shows "(\<Inter>i\<in>X. A i) \<in> M"

   313 proof -

   314   from A have "\<forall>i\<in>X. A i \<in> M" by fast

   315   hence "\<Omega> - (\<Union>i\<in>X. \<Omega> - A i) \<in> M" by blast

   316   moreover

   317   have "(\<Inter>i\<in>X. A i) = \<Omega> - (\<Union>i\<in>X. \<Omega> - A i)" using space_closed A

   318     by blast

   319   ultimately show ?thesis by metis

   320 qed

   321

   322 lemma (in sigma_algebra) countable_INT':

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

   324   assumes X: "countable X" "X \<noteq> {}"

   325   assumes A: "AX \<subseteq> M"

   326   shows  "(\<Inter>x\<in>X. A x) \<in> M"

   327 proof -

   328   have "(\<Inter>x\<in>X. A x) = (\<Inter>i\<in>to_nat_on X  X. A (from_nat_into X i))"

   329     using X by auto

   330   also have "\<dots> \<in> M"

   331     using A X

   332     by (intro countable_INT) auto

   333   finally show ?thesis .

   334 qed

   335

   336 lemma (in sigma_algebra) countable_INT'':

   337   "UNIV \<in> M \<Longrightarrow> countable I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> F i \<in> M) \<Longrightarrow> (\<Inter>i\<in>I. F i) \<in> M"

   338   by (cases "I = {}") (auto intro: countable_INT')

   339

   340 lemma (in sigma_algebra) countable:

   341   assumes "\<And>a. a \<in> A \<Longrightarrow> {a} \<in> M" "countable A"

   342   shows "A \<in> M"

   343 proof -

   344   have "(\<Union>a\<in>A. {a}) \<in> M"

   345     using assms by (intro countable_UN') auto

   346   also have "(\<Union>a\<in>A. {a}) = A" by auto

   347   finally show ?thesis by auto

   348 qed

   349

   350 lemma ring_of_sets_Pow: "ring_of_sets sp (Pow sp)"

   351   by (auto simp: ring_of_sets_iff)

   352

   353 lemma algebra_Pow: "algebra sp (Pow sp)"

   354   by (auto simp: algebra_iff_Un)

   355

   356 lemma sigma_algebra_iff:

   357   "sigma_algebra \<Omega> M \<longleftrightarrow>

   358     algebra \<Omega> M \<and> (\<forall>A. range A \<subseteq> M \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"

   359   by (simp add: sigma_algebra_def sigma_algebra_axioms_def)

   360

   361 lemma sigma_algebra_Pow: "sigma_algebra sp (Pow sp)"

   362   by (auto simp: sigma_algebra_iff algebra_iff_Int)

   363

   364 lemma (in sigma_algebra) sets_Collect_countable_All:

   365   assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"

   366   shows "{x\<in>\<Omega>. \<forall>i::'i::countable. P i x} \<in> M"

   367 proof -

   368   have "{x\<in>\<Omega>. \<forall>i::'i::countable. P i x} = (\<Inter>i. {x\<in>\<Omega>. P i x})" by auto

   369   with assms show ?thesis by auto

   370 qed

   371

   372 lemma (in sigma_algebra) sets_Collect_countable_Ex:

   373   assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"

   374   shows "{x\<in>\<Omega>. \<exists>i::'i::countable. P i x} \<in> M"

   375 proof -

   376   have "{x\<in>\<Omega>. \<exists>i::'i::countable. P i x} = (\<Union>i. {x\<in>\<Omega>. P i x})" by auto

   377   with assms show ?thesis by auto

   378 qed

   379

   380 lemma (in sigma_algebra) sets_Collect_countable_Ex':

   381   assumes "\<And>i. i \<in> I \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M"

   382   assumes "countable I"

   383   shows "{x\<in>\<Omega>. \<exists>i\<in>I. P i x} \<in> M"

   384 proof -

   385   have "{x\<in>\<Omega>. \<exists>i\<in>I. P i x} = (\<Union>i\<in>I. {x\<in>\<Omega>. P i x})" by auto

   386   with assms show ?thesis

   387     by (auto intro!: countable_UN')

   388 qed

   389

   390 lemma (in sigma_algebra) sets_Collect_countable_All':

   391   assumes "\<And>i. i \<in> I \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M"

   392   assumes "countable I"

   393   shows "{x\<in>\<Omega>. \<forall>i\<in>I. P i x} \<in> M"

   394 proof -

   395   have "{x\<in>\<Omega>. \<forall>i\<in>I. P i x} = (\<Inter>i\<in>I. {x\<in>\<Omega>. P i x}) \<inter> \<Omega>" by auto

   396   with assms show ?thesis

   397     by (cases "I = {}") (auto intro!: countable_INT')

   398 qed

   399

   400 lemma (in sigma_algebra) sets_Collect_countable_Ex1':

   401   assumes "\<And>i. i \<in> I \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M"

   402   assumes "countable I"

   403   shows "{x\<in>\<Omega>. \<exists>!i\<in>I. P i x} \<in> M"

   404 proof -

   405   have "{x\<in>\<Omega>. \<exists>!i\<in>I. P i x} = {x\<in>\<Omega>. \<exists>i\<in>I. P i x \<and> (\<forall>j\<in>I. P j x \<longrightarrow> i = j)}"

   406     by auto

   407   with assms show ?thesis

   408     by (auto intro!: sets_Collect_countable_All' sets_Collect_countable_Ex' sets_Collect_conj sets_Collect_imp sets_Collect_const)

   409 qed

   410

   411 lemmas (in sigma_algebra) sets_Collect =

   412   sets_Collect_imp sets_Collect_disj sets_Collect_conj sets_Collect_neg sets_Collect_const

   413   sets_Collect_countable_All sets_Collect_countable_Ex sets_Collect_countable_All

   414

   415 lemma (in sigma_algebra) sets_Collect_countable_Ball:

   416   assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"

   417   shows "{x\<in>\<Omega>. \<forall>i::'i::countable\<in>X. P i x} \<in> M"

   418   unfolding Ball_def by (intro sets_Collect assms)

   419

   420 lemma (in sigma_algebra) sets_Collect_countable_Bex:

   421   assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"

   422   shows "{x\<in>\<Omega>. \<exists>i::'i::countable\<in>X. P i x} \<in> M"

   423   unfolding Bex_def by (intro sets_Collect assms)

   424

   425 lemma sigma_algebra_single_set:

   426   assumes "X \<subseteq> S"

   427   shows "sigma_algebra S { {}, X, S - X, S }"

   428   using algebra.is_sigma_algebra[OF algebra_single_set[OF \<open>X \<subseteq> S\<close>]] by simp

   429

   430 subsubsection%unimportant \<open>Binary Unions\<close>

   431

   432 definition binary :: "'a \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"

   433   where "binary a b =  (\<lambda>x. b)(0 := a)"

   434

   435 lemma range_binary_eq: "range(binary a b) = {a,b}"

   436   by (auto simp add: binary_def)

   437

   438 lemma Un_range_binary: "a \<union> b = (\<Union>i::nat. binary a b i)"

   439   by (simp add: range_binary_eq cong del: strong_SUP_cong)

   440

   441 lemma Int_range_binary: "a \<inter> b = (\<Inter>i::nat. binary a b i)"

   442   by (simp add: range_binary_eq cong del: strong_INF_cong)

   443

   444 lemma sigma_algebra_iff2:

   445      "sigma_algebra \<Omega> M \<longleftrightarrow>

   446        M \<subseteq> Pow \<Omega> \<and>

   447        {} \<in> M \<and> (\<forall>s \<in> M. \<Omega> - s \<in> M) \<and>

   448        (\<forall>A. range A \<subseteq> M \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"

   449   by (auto simp add: range_binary_eq sigma_algebra_def sigma_algebra_axioms_def

   450          algebra_iff_Un Un_range_binary)

   451

   452 subsubsection \<open>Initial Sigma Algebra\<close>

   453

   454 text%important \<open>Sigma algebras can naturally be created as the closure of any set of

   455   M with regard to the properties just postulated.\<close>

   456

   457 inductive_set%important sigma_sets :: "'a set \<Rightarrow> 'a set set \<Rightarrow> 'a set set"

   458   for sp :: "'a set" and A :: "'a set set"

   459   where

   460     Basic[intro, simp]: "a \<in> A \<Longrightarrow> a \<in> sigma_sets sp A"

   461   | Empty: "{} \<in> sigma_sets sp A"

   462   | Compl: "a \<in> sigma_sets sp A \<Longrightarrow> sp - a \<in> sigma_sets sp A"

   463   | Union: "(\<And>i::nat. a i \<in> sigma_sets sp A) \<Longrightarrow> (\<Union>i. a i) \<in> sigma_sets sp A"

   464

   465 lemma (in sigma_algebra) sigma_sets_subset:

   466   assumes a: "a \<subseteq> M"

   467   shows "sigma_sets \<Omega> a \<subseteq> M"

   468 proof

   469   fix x

   470   assume "x \<in> sigma_sets \<Omega> a"

   471   from this show "x \<in> M"

   472     by (induct rule: sigma_sets.induct, auto) (metis a subsetD)

   473 qed

   474

   475 lemma sigma_sets_into_sp: "A \<subseteq> Pow sp \<Longrightarrow> x \<in> sigma_sets sp A \<Longrightarrow> x \<subseteq> sp"

   476   by (erule sigma_sets.induct, auto)

   477

   478 lemma sigma_algebra_sigma_sets:

   479      "a \<subseteq> Pow \<Omega> \<Longrightarrow> sigma_algebra \<Omega> (sigma_sets \<Omega> a)"

   480   by (auto simp add: sigma_algebra_iff2 dest: sigma_sets_into_sp

   481            intro!: sigma_sets.Union sigma_sets.Empty sigma_sets.Compl)

   482

   483 lemma sigma_sets_least_sigma_algebra:

   484   assumes "A \<subseteq> Pow S"

   485   shows "sigma_sets S A = \<Inter>{B. A \<subseteq> B \<and> sigma_algebra S B}"

   486 proof safe

   487   fix B X assume "A \<subseteq> B" and sa: "sigma_algebra S B"

   488     and X: "X \<in> sigma_sets S A"

   489   from sigma_algebra.sigma_sets_subset[OF sa, simplified, OF \<open>A \<subseteq> B\<close>] X

   490   show "X \<in> B" by auto

   491 next

   492   fix X assume "X \<in> \<Inter>{B. A \<subseteq> B \<and> sigma_algebra S B}"

   493   then have [intro!]: "\<And>B. A \<subseteq> B \<Longrightarrow> sigma_algebra S B \<Longrightarrow> X \<in> B"

   494      by simp

   495   have "A \<subseteq> sigma_sets S A" using assms by auto

   496   moreover have "sigma_algebra S (sigma_sets S A)"

   497     using assms by (intro sigma_algebra_sigma_sets[of A]) auto

   498   ultimately show "X \<in> sigma_sets S A" by auto

   499 qed

   500

   501 lemma sigma_sets_top: "sp \<in> sigma_sets sp A"

   502   by (metis Diff_empty sigma_sets.Compl sigma_sets.Empty)

   503

   504 lemma sigma_sets_Un:

   505   "a \<in> sigma_sets sp A \<Longrightarrow> b \<in> sigma_sets sp A \<Longrightarrow> a \<union> b \<in> sigma_sets sp A"

   506 apply (simp add: Un_range_binary range_binary_eq)

   507 apply (rule Union, simp add: binary_def)

   508 done

   509

   510 lemma sigma_sets_Inter:

   511   assumes Asb: "A \<subseteq> Pow sp"

   512   shows "(\<And>i::nat. a i \<in> sigma_sets sp A) \<Longrightarrow> (\<Inter>i. a i) \<in> sigma_sets sp A"

   513 proof -

   514   assume ai: "\<And>i::nat. a i \<in> sigma_sets sp A"

   515   hence "\<And>i::nat. sp-(a i) \<in> sigma_sets sp A"

   516     by (rule sigma_sets.Compl)

   517   hence "(\<Union>i. sp-(a i)) \<in> sigma_sets sp A"

   518     by (rule sigma_sets.Union)

   519   hence "sp-(\<Union>i. sp-(a i)) \<in> sigma_sets sp A"

   520     by (rule sigma_sets.Compl)

   521   also have "sp-(\<Union>i. sp-(a i)) = sp Int (\<Inter>i. a i)"

   522     by auto

   523   also have "... = (\<Inter>i. a i)" using ai

   524     by (blast dest: sigma_sets_into_sp [OF Asb])

   525   finally show ?thesis .

   526 qed

   527

   528 lemma sigma_sets_INTER:

   529   assumes Asb: "A \<subseteq> Pow sp"

   530       and ai: "\<And>i::nat. i \<in> S \<Longrightarrow> a i \<in> sigma_sets sp A" and non: "S \<noteq> {}"

   531   shows "(\<Inter>i\<in>S. a i) \<in> sigma_sets sp A"

   532 proof -

   533   from ai have "\<And>i. (if i\<in>S then a i else sp) \<in> sigma_sets sp A"

   534     by (simp add: sigma_sets.intros(2-) sigma_sets_top)

   535   hence "(\<Inter>i. (if i\<in>S then a i else sp)) \<in> sigma_sets sp A"

   536     by (rule sigma_sets_Inter [OF Asb])

   537   also have "(\<Inter>i. (if i\<in>S then a i else sp)) = (\<Inter>i\<in>S. a i)"

   538     by auto (metis ai non sigma_sets_into_sp subset_empty subset_iff Asb)+

   539   finally show ?thesis .

   540 qed

   541

   542 lemma sigma_sets_UNION:

   543   "countable B \<Longrightarrow> (\<And>b. b \<in> B \<Longrightarrow> b \<in> sigma_sets X A) \<Longrightarrow> (\<Union>B) \<in> sigma_sets X A"

   544   apply (cases "B = {}")

   545   apply (simp add: sigma_sets.Empty)

   546   using from_nat_into [of B] range_from_nat_into [of B] sigma_sets.Union [of "from_nat_into B" X A]

   547   apply simp

   548   apply auto

   549   apply (metis Sup_bot_conv(1) Union_empty \<open>\<lbrakk>B \<noteq> {}; countable B\<rbrakk> \<Longrightarrow> range (from_nat_into B) = B\<close>)

   550   done

   551

   552 lemma (in sigma_algebra) sigma_sets_eq:

   553      "sigma_sets \<Omega> M = M"

   554 proof

   555   show "M \<subseteq> sigma_sets \<Omega> M"

   556     by (metis Set.subsetI sigma_sets.Basic)

   557   next

   558   show "sigma_sets \<Omega> M \<subseteq> M"

   559     by (metis sigma_sets_subset subset_refl)

   560 qed

   561

   562 lemma sigma_sets_eqI:

   563   assumes A: "\<And>a. a \<in> A \<Longrightarrow> a \<in> sigma_sets M B"

   564   assumes B: "\<And>b. b \<in> B \<Longrightarrow> b \<in> sigma_sets M A"

   565   shows "sigma_sets M A = sigma_sets M B"

   566 proof (intro set_eqI iffI)

   567   fix a assume "a \<in> sigma_sets M A"

   568   from this A show "a \<in> sigma_sets M B"

   569     by induct (auto intro!: sigma_sets.intros(2-) del: sigma_sets.Basic)

   570 next

   571   fix b assume "b \<in> sigma_sets M B"

   572   from this B show "b \<in> sigma_sets M A"

   573     by induct (auto intro!: sigma_sets.intros(2-) del: sigma_sets.Basic)

   574 qed

   575

   576 lemma sigma_sets_subseteq: assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"

   577 proof

   578   fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"

   579     by induct (insert \<open>A \<subseteq> B\<close>, auto intro: sigma_sets.intros(2-))

   580 qed

   581

   582 lemma sigma_sets_mono: assumes "A \<subseteq> sigma_sets X B" shows "sigma_sets X A \<subseteq> sigma_sets X B"

   583 proof

   584   fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"

   585     by induct (insert \<open>A \<subseteq> sigma_sets X B\<close>, auto intro: sigma_sets.intros(2-))

   586 qed

   587

   588 lemma sigma_sets_mono': assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"

   589 proof

   590   fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"

   591     by induct (insert \<open>A \<subseteq> B\<close>, auto intro: sigma_sets.intros(2-))

   592 qed

   593

   594 lemma sigma_sets_superset_generator: "A \<subseteq> sigma_sets X A"

   595   by (auto intro: sigma_sets.Basic)

   596

   597 lemma (in sigma_algebra) restriction_in_sets:

   598   fixes A :: "nat \<Rightarrow> 'a set"

   599   assumes "S \<in> M"

   600   and *: "range A \<subseteq> (\<lambda>A. S \<inter> A)  M" (is "_ \<subseteq> ?r")

   601   shows "range A \<subseteq> M" "(\<Union>i. A i) \<in> (\<lambda>A. S \<inter> A)  M"

   602 proof -

   603   { fix i have "A i \<in> ?r" using * by auto

   604     hence "\<exists>B. A i = B \<inter> S \<and> B \<in> M" by auto

   605     hence "A i \<subseteq> S" "A i \<in> M" using \<open>S \<in> M\<close> by auto }

   606   thus "range A \<subseteq> M" "(\<Union>i. A i) \<in> (\<lambda>A. S \<inter> A)  M"

   607     by (auto intro!: image_eqI[of _ _ "(\<Union>i. A i)"])

   608 qed

   609

   610 lemma (in sigma_algebra) restricted_sigma_algebra:

   611   assumes "S \<in> M"

   612   shows "sigma_algebra S (restricted_space S)"

   613   unfolding sigma_algebra_def sigma_algebra_axioms_def

   614 proof safe

   615   show "algebra S (restricted_space S)" using restricted_algebra[OF assms] .

   616 next

   617   fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> restricted_space S"

   618   from restriction_in_sets[OF assms this[simplified]]

   619   show "(\<Union>i. A i) \<in> restricted_space S" by simp

   620 qed

   621

   622 lemma sigma_sets_Int:

   623   assumes "A \<in> sigma_sets sp st" "A \<subseteq> sp"

   624   shows "(\<inter>) A  sigma_sets sp st = sigma_sets A ((\<inter>) A  st)"

   625 proof (intro equalityI subsetI)

   626   fix x assume "x \<in> (\<inter>) A  sigma_sets sp st"

   627   then obtain y where "y \<in> sigma_sets sp st" "x = y \<inter> A" by auto

   628   then have "x \<in> sigma_sets (A \<inter> sp) ((\<inter>) A  st)"

   629   proof (induct arbitrary: x)

   630     case (Compl a)

   631     then show ?case

   632       by (force intro!: sigma_sets.Compl simp: Diff_Int_distrib ac_simps)

   633   next

   634     case (Union a)

   635     then show ?case

   636       by (auto intro!: sigma_sets.Union

   637                simp add: UN_extend_simps simp del: UN_simps)

   638   qed (auto intro!: sigma_sets.intros(2-))

   639   then show "x \<in> sigma_sets A ((\<inter>) A  st)"

   640     using \<open>A \<subseteq> sp\<close> by (simp add: Int_absorb2)

   641 next

   642   fix x assume "x \<in> sigma_sets A ((\<inter>) A  st)"

   643   then show "x \<in> (\<inter>) A  sigma_sets sp st"

   644   proof induct

   645     case (Compl a)

   646     then obtain x where "a = A \<inter> x" "x \<in> sigma_sets sp st" by auto

   647     then show ?case using \<open>A \<subseteq> sp\<close>

   648       by (force simp add: image_iff intro!: bexI[of _ "sp - x"] sigma_sets.Compl)

   649   next

   650     case (Union a)

   651     then have "\<forall>i. \<exists>x. x \<in> sigma_sets sp st \<and> a i = A \<inter> x"

   652       by (auto simp: image_iff Bex_def)

   653     from choice[OF this] guess f ..

   654     then show ?case

   655       by (auto intro!: bexI[of _ "(\<Union>x. f x)"] sigma_sets.Union

   656                simp add: image_iff)

   657   qed (auto intro!: sigma_sets.intros(2-))

   658 qed

   659

   660 lemma sigma_sets_empty_eq: "sigma_sets A {} = {{}, A}"

   661 proof (intro set_eqI iffI)

   662   fix a assume "a \<in> sigma_sets A {}" then show "a \<in> {{}, A}"

   663     by induct blast+

   664 qed (auto intro: sigma_sets.Empty sigma_sets_top)

   665

   666 lemma sigma_sets_single[simp]: "sigma_sets A {A} = {{}, A}"

   667 proof (intro set_eqI iffI)

   668   fix x assume "x \<in> sigma_sets A {A}"

   669   then show "x \<in> {{}, A}"

   670     by induct blast+

   671 next

   672   fix x assume "x \<in> {{}, A}"

   673   then show "x \<in> sigma_sets A {A}"

   674     by (auto intro: sigma_sets.Empty sigma_sets_top)

   675 qed

   676

   677 lemma sigma_sets_sigma_sets_eq:

   678   "M \<subseteq> Pow S \<Longrightarrow> sigma_sets S (sigma_sets S M) = sigma_sets S M"

   679   by (rule sigma_algebra.sigma_sets_eq[OF sigma_algebra_sigma_sets, of M S]) auto

   680

   681 lemma sigma_sets_singleton:

   682   assumes "X \<subseteq> S"

   683   shows "sigma_sets S { X } = { {}, X, S - X, S }"

   684 proof -

   685   interpret sigma_algebra S "{ {}, X, S - X, S }"

   686     by (rule sigma_algebra_single_set) fact

   687   have "sigma_sets S { X } \<subseteq> sigma_sets S { {}, X, S - X, S }"

   688     by (rule sigma_sets_subseteq) simp

   689   moreover have "\<dots> = { {}, X, S - X, S }"

   690     using sigma_sets_eq by simp

   691   moreover

   692   { fix A assume "A \<in> { {}, X, S - X, S }"

   693     then have "A \<in> sigma_sets S { X }"

   694       by (auto intro: sigma_sets.intros(2-) sigma_sets_top) }

   695   ultimately have "sigma_sets S { X } = sigma_sets S { {}, X, S - X, S }"

   696     by (intro antisym) auto

   697   with sigma_sets_eq show ?thesis by simp

   698 qed

   699

   700 lemma restricted_sigma:

   701   assumes S: "S \<in> sigma_sets \<Omega> M" and M: "M \<subseteq> Pow \<Omega>"

   702   shows "algebra.restricted_space (sigma_sets \<Omega> M) S =

   703     sigma_sets S (algebra.restricted_space M S)"

   704 proof -

   705   from S sigma_sets_into_sp[OF M]

   706   have "S \<in> sigma_sets \<Omega> M" "S \<subseteq> \<Omega>" by auto

   707   from sigma_sets_Int[OF this]

   708   show ?thesis by simp

   709 qed

   710

   711 lemma sigma_sets_vimage_commute:

   712   assumes X: "X \<in> \<Omega> \<rightarrow> \<Omega>'"

   713   shows "{X - A \<inter> \<Omega> |A. A \<in> sigma_sets \<Omega>' M'}

   714        = sigma_sets \<Omega> {X - A \<inter> \<Omega> |A. A \<in> M'}" (is "?L = ?R")

   715 proof

   716   show "?L \<subseteq> ?R"

   717   proof clarify

   718     fix A assume "A \<in> sigma_sets \<Omega>' M'"

   719     then show "X - A \<inter> \<Omega> \<in> ?R"

   720     proof induct

   721       case Empty then show ?case

   722         by (auto intro!: sigma_sets.Empty)

   723     next

   724       case (Compl B)

   725       have [simp]: "X - (\<Omega>' - B) \<inter> \<Omega> = \<Omega> - (X - B \<inter> \<Omega>)"

   726         by (auto simp add: funcset_mem [OF X])

   727       with Compl show ?case

   728         by (auto intro!: sigma_sets.Compl)

   729     next

   730       case (Union F)

   731       then show ?case

   732         by (auto simp add: vimage_UN UN_extend_simps(4) simp del: UN_simps

   733                  intro!: sigma_sets.Union)

   734     qed auto

   735   qed

   736   show "?R \<subseteq> ?L"

   737   proof clarify

   738     fix A assume "A \<in> ?R"

   739     then show "\<exists>B. A = X - B \<inter> \<Omega> \<and> B \<in> sigma_sets \<Omega>' M'"

   740     proof induct

   741       case (Basic B) then show ?case by auto

   742     next

   743       case Empty then show ?case

   744         by (auto intro!: sigma_sets.Empty exI[of _ "{}"])

   745     next

   746       case (Compl B)

   747       then obtain A where A: "B = X - A \<inter> \<Omega>" "A \<in> sigma_sets \<Omega>' M'" by auto

   748       then have [simp]: "\<Omega> - B = X - (\<Omega>' - A) \<inter> \<Omega>"

   749         by (auto simp add: funcset_mem [OF X])

   750       with A(2) show ?case

   751         by (auto intro: sigma_sets.Compl)

   752     next

   753       case (Union F)

   754       then have "\<forall>i. \<exists>B. F i = X - B \<inter> \<Omega> \<and> B \<in> sigma_sets \<Omega>' M'" by auto

   755       from choice[OF this] guess A .. note A = this

   756       with A show ?case

   757         by (auto simp: vimage_UN[symmetric] intro: sigma_sets.Union)

   758     qed

   759   qed

   760 qed

   761

   762 lemma (in ring_of_sets) UNION_in_sets:

   763   fixes A:: "nat \<Rightarrow> 'a set"

   764   assumes A: "range A \<subseteq> M"

   765   shows  "(\<Union>i\<in>{0..<n}. A i) \<in> M"

   766 proof (induct n)

   767   case 0 show ?case by simp

   768 next

   769   case (Suc n)

   770   thus ?case

   771     by (simp add: atLeastLessThanSuc) (metis A Un UNIV_I image_subset_iff)

   772 qed

   773

   774 lemma (in ring_of_sets) range_disjointed_sets:

   775   assumes A: "range A \<subseteq> M"

   776   shows  "range (disjointed A) \<subseteq> M"

   777 proof (auto simp add: disjointed_def)

   778   fix n

   779   show "A n - (\<Union>i\<in>{0..<n}. A i) \<in> M" using UNION_in_sets

   780     by (metis A Diff UNIV_I image_subset_iff)

   781 qed

   782

   783 lemma (in algebra) range_disjointed_sets':

   784   "range A \<subseteq> M \<Longrightarrow> range (disjointed A) \<subseteq> M"

   785   using range_disjointed_sets .

   786

   787 lemma sigma_algebra_disjoint_iff:

   788   "sigma_algebra \<Omega> M \<longleftrightarrow> algebra \<Omega> M \<and>

   789     (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"

   790 proof (auto simp add: sigma_algebra_iff)

   791   fix A :: "nat \<Rightarrow> 'a set"

   792   assume M: "algebra \<Omega> M"

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

   794      and UnA: "\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M"

   795   hence "range (disjointed A) \<subseteq> M \<longrightarrow>

   796          disjoint_family (disjointed A) \<longrightarrow>

   797          (\<Union>i. disjointed A i) \<in> M" by blast

   798   hence "(\<Union>i. disjointed A i) \<in> M"

   799     by (simp add: algebra.range_disjointed_sets'[of \<Omega>] M A disjoint_family_disjointed)

   800   thus "(\<Union>i::nat. A i) \<in> M" by (simp add: UN_disjointed_eq)

   801 qed

   802

   803 subsubsection%unimportant \<open>Ring generated by a semiring\<close>

   804

   805 definition (in semiring_of_sets)

   806   "generated_ring = { \<Union>C | C. C \<subseteq> M \<and> finite C \<and> disjoint C }"

   807

   808 lemma (in semiring_of_sets) generated_ringE[elim?]:

   809   assumes "a \<in> generated_ring"

   810   obtains C where "finite C" "disjoint C" "C \<subseteq> M" "a = \<Union>C"

   811   using assms unfolding generated_ring_def by auto

   812

   813 lemma (in semiring_of_sets) generated_ringI[intro?]:

   814   assumes "finite C" "disjoint C" "C \<subseteq> M" "a = \<Union>C"

   815   shows "a \<in> generated_ring"

   816   using assms unfolding generated_ring_def by auto

   817

   818 lemma (in semiring_of_sets) generated_ringI_Basic:

   819   "A \<in> M \<Longrightarrow> A \<in> generated_ring"

   820   by (rule generated_ringI[of "{A}"]) (auto simp: disjoint_def)

   821

   822 lemma (in semiring_of_sets) generated_ring_disjoint_Un[intro]:

   823   assumes a: "a \<in> generated_ring" and b: "b \<in> generated_ring"

   824   and "a \<inter> b = {}"

   825   shows "a \<union> b \<in> generated_ring"

   826 proof -

   827   from a guess Ca .. note Ca = this

   828   from b guess Cb .. note Cb = this

   829   show ?thesis

   830   proof

   831     show "disjoint (Ca \<union> Cb)"

   832       using \<open>a \<inter> b = {}\<close> Ca Cb by (auto intro!: disjoint_union)

   833   qed (insert Ca Cb, auto)

   834 qed

   835

   836 lemma (in semiring_of_sets) generated_ring_empty: "{} \<in> generated_ring"

   837   by (auto simp: generated_ring_def disjoint_def)

   838

   839 lemma (in semiring_of_sets) generated_ring_disjoint_Union:

   840   assumes "finite A" shows "A \<subseteq> generated_ring \<Longrightarrow> disjoint A \<Longrightarrow> \<Union>A \<in> generated_ring"

   841   using assms by (induct A) (auto simp: disjoint_def intro!: generated_ring_disjoint_Un generated_ring_empty)

   842

   843 lemma (in semiring_of_sets) generated_ring_disjoint_UNION:

   844   "finite I \<Longrightarrow> disjoint (A  I) \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> A i \<in> generated_ring) \<Longrightarrow> UNION I A \<in> generated_ring"

   845   by (intro generated_ring_disjoint_Union) auto

   846

   847 lemma (in semiring_of_sets) generated_ring_Int:

   848   assumes a: "a \<in> generated_ring" and b: "b \<in> generated_ring"

   849   shows "a \<inter> b \<in> generated_ring"

   850 proof -

   851   from a guess Ca .. note Ca = this

   852   from b guess Cb .. note Cb = this

   853   define C where "C = (\<lambda>(a,b). a \<inter> b) (Ca\<times>Cb)"

   854   show ?thesis

   855   proof

   856     show "disjoint C"

   857     proof (simp add: disjoint_def C_def, intro ballI impI)

   858       fix a1 b1 a2 b2 assume sets: "a1 \<in> Ca" "b1 \<in> Cb" "a2 \<in> Ca" "b2 \<in> Cb"

   859       assume "a1 \<inter> b1 \<noteq> a2 \<inter> b2"

   860       then have "a1 \<noteq> a2 \<or> b1 \<noteq> b2" by auto

   861       then show "(a1 \<inter> b1) \<inter> (a2 \<inter> b2) = {}"

   862       proof

   863         assume "a1 \<noteq> a2"

   864         with sets Ca have "a1 \<inter> a2 = {}"

   865           by (auto simp: disjoint_def)

   866         then show ?thesis by auto

   867       next

   868         assume "b1 \<noteq> b2"

   869         with sets Cb have "b1 \<inter> b2 = {}"

   870           by (auto simp: disjoint_def)

   871         then show ?thesis by auto

   872       qed

   873     qed

   874   qed (insert Ca Cb, auto simp: C_def)

   875 qed

   876

   877 lemma (in semiring_of_sets) generated_ring_Inter:

   878   assumes "finite A" "A \<noteq> {}" shows "A \<subseteq> generated_ring \<Longrightarrow> \<Inter>A \<in> generated_ring"

   879   using assms by (induct A rule: finite_ne_induct) (auto intro: generated_ring_Int)

   880

   881 lemma (in semiring_of_sets) generated_ring_INTER:

   882   "finite I \<Longrightarrow> I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> A i \<in> generated_ring) \<Longrightarrow> INTER I A \<in> generated_ring"

   883   by (intro generated_ring_Inter) auto

   884

   885 lemma (in semiring_of_sets) generating_ring:

   886   "ring_of_sets \<Omega> generated_ring"

   887 proof (rule ring_of_setsI)

   888   let ?R = generated_ring

   889   show "?R \<subseteq> Pow \<Omega>"

   890     using sets_into_space by (auto simp: generated_ring_def generated_ring_empty)

   891   show "{} \<in> ?R" by (rule generated_ring_empty)

   892

   893   { fix a assume a: "a \<in> ?R" then guess Ca .. note Ca = this

   894     fix b assume b: "b \<in> ?R" then guess Cb .. note Cb = this

   895

   896     show "a - b \<in> ?R"

   897     proof cases

   898       assume "Cb = {}" with Cb \<open>a \<in> ?R\<close> show ?thesis

   899         by simp

   900     next

   901       assume "Cb \<noteq> {}"

   902       with Ca Cb have "a - b = (\<Union>a'\<in>Ca. \<Inter>b'\<in>Cb. a' - b')" by auto

   903       also have "\<dots> \<in> ?R"

   904       proof (intro generated_ring_INTER generated_ring_disjoint_UNION)

   905         fix a b assume "a \<in> Ca" "b \<in> Cb"

   906         with Ca Cb Diff_cover[of a b] show "a - b \<in> ?R"

   907           by (auto simp add: generated_ring_def)

   908             (metis DiffI Diff_eq_empty_iff empty_iff)

   909       next

   910         show "disjoint ((\<lambda>a'. \<Inter>b'\<in>Cb. a' - b')Ca)"

   911           using Ca by (auto simp add: disjoint_def \<open>Cb \<noteq> {}\<close>)

   912       next

   913         show "finite Ca" "finite Cb" "Cb \<noteq> {}" by fact+

   914       qed

   915       finally show "a - b \<in> ?R" .

   916     qed }

   917   note Diff = this

   918

   919   fix a b assume sets: "a \<in> ?R" "b \<in> ?R"

   920   have "a \<union> b = (a - b) \<union> (a \<inter> b) \<union> (b - a)" by auto

   921   also have "\<dots> \<in> ?R"

   922     by (intro sets generated_ring_disjoint_Un generated_ring_Int Diff) auto

   923   finally show "a \<union> b \<in> ?R" .

   924 qed

   925

   926 lemma (in semiring_of_sets) sigma_sets_generated_ring_eq: "sigma_sets \<Omega> generated_ring = sigma_sets \<Omega> M"

   927 proof

   928   interpret M: sigma_algebra \<Omega> "sigma_sets \<Omega> M"

   929     using space_closed by (rule sigma_algebra_sigma_sets)

   930   show "sigma_sets \<Omega> generated_ring \<subseteq> sigma_sets \<Omega> M"

   931     by (blast intro!: sigma_sets_mono elim: generated_ringE)

   932 qed (auto intro!: generated_ringI_Basic sigma_sets_mono)

   933

   934 subsubsection%unimportant \<open>A Two-Element Series\<close>

   935

   936 definition binaryset :: "'a set \<Rightarrow> 'a set \<Rightarrow> nat \<Rightarrow> 'a set"

   937   where "binaryset A B = (\<lambda>x. {})(0 := A, Suc 0 := B)"

   938

   939 lemma range_binaryset_eq: "range(binaryset A B) = {A,B,{}}"

   940   apply (simp add: binaryset_def)

   941   apply (rule set_eqI)

   942   apply (auto simp add: image_iff)

   943   done

   944

   945 lemma UN_binaryset_eq: "(\<Union>i. binaryset A B i) = A \<union> B"

   946   by (simp add: range_binaryset_eq cong del: strong_SUP_cong)

   947

   948 subsubsection \<open>Closed CDI\<close>

   949

   950 definition%important closed_cdi where

   951   "closed_cdi \<Omega> M \<longleftrightarrow>

   952    M \<subseteq> Pow \<Omega> &

   953    (\<forall>s \<in> M. \<Omega> - s \<in> M) &

   954    (\<forall>A. (range A \<subseteq> M) & (A 0 = {}) & (\<forall>n. A n \<subseteq> A (Suc n)) \<longrightarrow>

   955         (\<Union>i. A i) \<in> M) &

   956    (\<forall>A. (range A \<subseteq> M) & disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"

   957

   958 inductive_set

   959   smallest_ccdi_sets :: "'a set \<Rightarrow> 'a set set \<Rightarrow> 'a set set"

   960   for \<Omega> M

   961   where

   962     Basic [intro]:

   963       "a \<in> M \<Longrightarrow> a \<in> smallest_ccdi_sets \<Omega> M"

   964   | Compl [intro]:

   965       "a \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> \<Omega> - a \<in> smallest_ccdi_sets \<Omega> M"

   966   | Inc:

   967       "range A \<in> Pow(smallest_ccdi_sets \<Omega> M) \<Longrightarrow> A 0 = {} \<Longrightarrow> (\<And>n. A n \<subseteq> A (Suc n))

   968        \<Longrightarrow> (\<Union>i. A i) \<in> smallest_ccdi_sets \<Omega> M"

   969   | Disj:

   970       "range A \<in> Pow(smallest_ccdi_sets \<Omega> M) \<Longrightarrow> disjoint_family A

   971        \<Longrightarrow> (\<Union>i::nat. A i) \<in> smallest_ccdi_sets \<Omega> M"

   972

   973 lemma (in subset_class) smallest_closed_cdi1: "M \<subseteq> smallest_ccdi_sets \<Omega> M"

   974   by auto

   975

   976 lemma (in subset_class) smallest_ccdi_sets: "smallest_ccdi_sets \<Omega> M \<subseteq> Pow \<Omega>"

   977   apply (rule subsetI)

   978   apply (erule smallest_ccdi_sets.induct)

   979   apply (auto intro: range_subsetD dest: sets_into_space)

   980   done

   981

   982 lemma (in subset_class) smallest_closed_cdi2: "closed_cdi \<Omega> (smallest_ccdi_sets \<Omega> M)"

   983   apply (auto simp add: closed_cdi_def smallest_ccdi_sets)

   984   apply (blast intro: smallest_ccdi_sets.Inc smallest_ccdi_sets.Disj) +

   985   done

   986

   987 lemma closed_cdi_subset: "closed_cdi \<Omega> M \<Longrightarrow> M \<subseteq> Pow \<Omega>"

   988   by (simp add: closed_cdi_def)

   989

   990 lemma closed_cdi_Compl: "closed_cdi \<Omega> M \<Longrightarrow> s \<in> M \<Longrightarrow> \<Omega> - s \<in> M"

   991   by (simp add: closed_cdi_def)

   992

   993 lemma closed_cdi_Inc:

   994   "closed_cdi \<Omega> M \<Longrightarrow> range A \<subseteq> M \<Longrightarrow> A 0 = {} \<Longrightarrow> (!!n. A n \<subseteq> A (Suc n)) \<Longrightarrow> (\<Union>i. A i) \<in> M"

   995   by (simp add: closed_cdi_def)

   996

   997 lemma closed_cdi_Disj:

   998   "closed_cdi \<Omega> M \<Longrightarrow> range A \<subseteq> M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"

   999   by (simp add: closed_cdi_def)

  1000

  1001 lemma closed_cdi_Un:

  1002   assumes cdi: "closed_cdi \<Omega> M" and empty: "{} \<in> M"

  1003       and A: "A \<in> M" and B: "B \<in> M"

  1004       and disj: "A \<inter> B = {}"

  1005     shows "A \<union> B \<in> M"

  1006 proof -

  1007   have ra: "range (binaryset A B) \<subseteq> M"

  1008    by (simp add: range_binaryset_eq empty A B)

  1009  have di:  "disjoint_family (binaryset A B)" using disj

  1010    by (simp add: disjoint_family_on_def binaryset_def Int_commute)

  1011  from closed_cdi_Disj [OF cdi ra di]

  1012  show ?thesis

  1013    by (simp add: UN_binaryset_eq)

  1014 qed

  1015

  1016 lemma (in algebra) smallest_ccdi_sets_Un:

  1017   assumes A: "A \<in> smallest_ccdi_sets \<Omega> M" and B: "B \<in> smallest_ccdi_sets \<Omega> M"

  1018       and disj: "A \<inter> B = {}"

  1019     shows "A \<union> B \<in> smallest_ccdi_sets \<Omega> M"

  1020 proof -

  1021   have ra: "range (binaryset A B) \<in> Pow (smallest_ccdi_sets \<Omega> M)"

  1022     by (simp add: range_binaryset_eq  A B smallest_ccdi_sets.Basic)

  1023   have di:  "disjoint_family (binaryset A B)" using disj

  1024     by (simp add: disjoint_family_on_def binaryset_def Int_commute)

  1025   from Disj [OF ra di]

  1026   show ?thesis

  1027     by (simp add: UN_binaryset_eq)

  1028 qed

  1029

  1030 lemma (in algebra) smallest_ccdi_sets_Int1:

  1031   assumes a: "a \<in> M"

  1032   shows "b \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets \<Omega> M"

  1033 proof (induct rule: smallest_ccdi_sets.induct)

  1034   case (Basic x)

  1035   thus ?case

  1036     by (metis a Int smallest_ccdi_sets.Basic)

  1037 next

  1038   case (Compl x)

  1039   have "a \<inter> (\<Omega> - x) = \<Omega> - ((\<Omega> - a) \<union> (a \<inter> x))"

  1040     by blast

  1041   also have "... \<in> smallest_ccdi_sets \<Omega> M"

  1042     by (metis smallest_ccdi_sets.Compl a Compl(2) Diff_Int2 Diff_Int_distrib2

  1043            Diff_disjoint Int_Diff Int_empty_right smallest_ccdi_sets_Un

  1044            smallest_ccdi_sets.Basic smallest_ccdi_sets.Compl)

  1045   finally show ?case .

  1046 next

  1047   case (Inc A)

  1048   have 1: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) = a \<inter> (\<Union>i. A i)"

  1049     by blast

  1050   have "range (\<lambda>i. a \<inter> A i) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Inc

  1051     by blast

  1052   moreover have "(\<lambda>i. a \<inter> A i) 0 = {}"

  1053     by (simp add: Inc)

  1054   moreover have "!!n. (\<lambda>i. a \<inter> A i) n \<subseteq> (\<lambda>i. a \<inter> A i) (Suc n)" using Inc

  1055     by blast

  1056   ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets \<Omega> M"

  1057     by (rule smallest_ccdi_sets.Inc)

  1058   show ?case

  1059     by (metis 1 2)

  1060 next

  1061   case (Disj A)

  1062   have 1: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) = a \<inter> (\<Union>i. A i)"

  1063     by blast

  1064   have "range (\<lambda>i. a \<inter> A i) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Disj

  1065     by blast

  1066   moreover have "disjoint_family (\<lambda>i. a \<inter> A i)" using Disj

  1067     by (auto simp add: disjoint_family_on_def)

  1068   ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets \<Omega> M"

  1069     by (rule smallest_ccdi_sets.Disj)

  1070   show ?case

  1071     by (metis 1 2)

  1072 qed

  1073

  1074

  1075 lemma (in algebra) smallest_ccdi_sets_Int:

  1076   assumes b: "b \<in> smallest_ccdi_sets \<Omega> M"

  1077   shows "a \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets \<Omega> M"

  1078 proof (induct rule: smallest_ccdi_sets.induct)

  1079   case (Basic x)

  1080   thus ?case

  1081     by (metis b smallest_ccdi_sets_Int1)

  1082 next

  1083   case (Compl x)

  1084   have "(\<Omega> - x) \<inter> b = \<Omega> - (x \<inter> b \<union> (\<Omega> - b))"

  1085     by blast

  1086   also have "... \<in> smallest_ccdi_sets \<Omega> M"

  1087     by (metis Compl(2) Diff_disjoint Int_Diff Int_commute Int_empty_right b

  1088            smallest_ccdi_sets.Compl smallest_ccdi_sets_Un)

  1089   finally show ?case .

  1090 next

  1091   case (Inc A)

  1092   have 1: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) = (\<Union>i. A i) \<inter> b"

  1093     by blast

  1094   have "range (\<lambda>i. A i \<inter> b) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Inc

  1095     by blast

  1096   moreover have "(\<lambda>i. A i \<inter> b) 0 = {}"

  1097     by (simp add: Inc)

  1098   moreover have "!!n. (\<lambda>i. A i \<inter> b) n \<subseteq> (\<lambda>i. A i \<inter> b) (Suc n)" using Inc

  1099     by blast

  1100   ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets \<Omega> M"

  1101     by (rule smallest_ccdi_sets.Inc)

  1102   show ?case

  1103     by (metis 1 2)

  1104 next

  1105   case (Disj A)

  1106   have 1: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) = (\<Union>i. A i) \<inter> b"

  1107     by blast

  1108   have "range (\<lambda>i. A i \<inter> b) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Disj

  1109     by blast

  1110   moreover have "disjoint_family (\<lambda>i. A i \<inter> b)" using Disj

  1111     by (auto simp add: disjoint_family_on_def)

  1112   ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets \<Omega> M"

  1113     by (rule smallest_ccdi_sets.Disj)

  1114   show ?case

  1115     by (metis 1 2)

  1116 qed

  1117

  1118 lemma (in algebra) sigma_property_disjoint_lemma:

  1119   assumes sbC: "M \<subseteq> C"

  1120       and ccdi: "closed_cdi \<Omega> C"

  1121   shows "sigma_sets \<Omega> M \<subseteq> C"

  1122 proof -

  1123   have "smallest_ccdi_sets \<Omega> M \<in> {B . M \<subseteq> B \<and> sigma_algebra \<Omega> B}"

  1124     apply (auto simp add: sigma_algebra_disjoint_iff algebra_iff_Int

  1125             smallest_ccdi_sets_Int)

  1126     apply (metis Union_Pow_eq Union_upper subsetD smallest_ccdi_sets)

  1127     apply (blast intro: smallest_ccdi_sets.Disj)

  1128     done

  1129   hence "sigma_sets (\<Omega>) (M) \<subseteq> smallest_ccdi_sets \<Omega> M"

  1130     by clarsimp

  1131        (drule sigma_algebra.sigma_sets_subset [where a="M"], auto)

  1132   also have "...  \<subseteq> C"

  1133     proof

  1134       fix x

  1135       assume x: "x \<in> smallest_ccdi_sets \<Omega> M"

  1136       thus "x \<in> C"

  1137         proof (induct rule: smallest_ccdi_sets.induct)

  1138           case (Basic x)

  1139           thus ?case

  1140             by (metis Basic subsetD sbC)

  1141         next

  1142           case (Compl x)

  1143           thus ?case

  1144             by (blast intro: closed_cdi_Compl [OF ccdi, simplified])

  1145         next

  1146           case (Inc A)

  1147           thus ?case

  1148                by (auto intro: closed_cdi_Inc [OF ccdi, simplified])

  1149         next

  1150           case (Disj A)

  1151           thus ?case

  1152                by (auto intro: closed_cdi_Disj [OF ccdi, simplified])

  1153         qed

  1154     qed

  1155   finally show ?thesis .

  1156 qed

  1157

  1158 lemma (in algebra) sigma_property_disjoint:

  1159   assumes sbC: "M \<subseteq> C"

  1160       and compl: "!!s. s \<in> C \<inter> sigma_sets (\<Omega>) (M) \<Longrightarrow> \<Omega> - s \<in> C"

  1161       and inc: "!!A. range A \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)

  1162                      \<Longrightarrow> A 0 = {} \<Longrightarrow> (!!n. A n \<subseteq> A (Suc n))

  1163                      \<Longrightarrow> (\<Union>i. A i) \<in> C"

  1164       and disj: "!!A. range A \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)

  1165                       \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> C"

  1166   shows "sigma_sets (\<Omega>) (M) \<subseteq> C"

  1167 proof -

  1168   have "sigma_sets (\<Omega>) (M) \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)"

  1169     proof (rule sigma_property_disjoint_lemma)

  1170       show "M \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)"

  1171         by (metis Int_greatest Set.subsetI sbC sigma_sets.Basic)

  1172     next

  1173       show "closed_cdi \<Omega> (C \<inter> sigma_sets (\<Omega>) (M))"

  1174         by (simp add: closed_cdi_def compl inc disj)

  1175            (metis PowI Set.subsetI le_infI2 sigma_sets_into_sp space_closed

  1176              IntE sigma_sets.Compl range_subsetD sigma_sets.Union)

  1177     qed

  1178   thus ?thesis

  1179     by blast

  1180 qed

  1181

  1182 subsubsection \<open>Dynkin systems\<close>

  1183

  1184 locale%important dynkin_system = subset_class +

  1185   assumes space: "\<Omega> \<in> M"

  1186     and   compl[intro!]: "\<And>A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"

  1187     and   UN[intro!]: "\<And>A. disjoint_family A \<Longrightarrow> range A \<subseteq> M

  1188                            \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"

  1189

  1190 lemma (in dynkin_system) empty[intro, simp]: "{} \<in> M"

  1191   using space compl[of "\<Omega>"] by simp

  1192

  1193 lemma (in dynkin_system) diff:

  1194   assumes sets: "D \<in> M" "E \<in> M" and "D \<subseteq> E"

  1195   shows "E - D \<in> M"

  1196 proof -

  1197   let ?f = "\<lambda>x. if x = 0 then D else if x = Suc 0 then \<Omega> - E else {}"

  1198   have "range ?f = {D, \<Omega> - E, {}}"

  1199     by (auto simp: image_iff)

  1200   moreover have "D \<union> (\<Omega> - E) = (\<Union>i. ?f i)"

  1201     by (auto simp: image_iff split: if_split_asm)

  1202   moreover

  1203   have "disjoint_family ?f" unfolding disjoint_family_on_def

  1204     using \<open>D \<in> M\<close>[THEN sets_into_space] \<open>D \<subseteq> E\<close> by auto

  1205   ultimately have "\<Omega> - (D \<union> (\<Omega> - E)) \<in> M"

  1206     using sets by auto

  1207   also have "\<Omega> - (D \<union> (\<Omega> - E)) = E - D"

  1208     using assms sets_into_space by auto

  1209   finally show ?thesis .

  1210 qed

  1211

  1212 lemma dynkin_systemI:

  1213   assumes "\<And> A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>" "\<Omega> \<in> M"

  1214   assumes "\<And> A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"

  1215   assumes "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> M

  1216           \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"

  1217   shows "dynkin_system \<Omega> M"

  1218   using assms by (auto simp: dynkin_system_def dynkin_system_axioms_def subset_class_def)

  1219

  1220 lemma dynkin_systemI':

  1221   assumes 1: "\<And> A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>"

  1222   assumes empty: "{} \<in> M"

  1223   assumes Diff: "\<And> A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"

  1224   assumes 2: "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> M

  1225           \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"

  1226   shows "dynkin_system \<Omega> M"

  1227 proof -

  1228   from Diff[OF empty] have "\<Omega> \<in> M" by auto

  1229   from 1 this Diff 2 show ?thesis

  1230     by (intro dynkin_systemI) auto

  1231 qed

  1232

  1233 lemma dynkin_system_trivial:

  1234   shows "dynkin_system A (Pow A)"

  1235   by (rule dynkin_systemI) auto

  1236

  1237 lemma sigma_algebra_imp_dynkin_system:

  1238   assumes "sigma_algebra \<Omega> M" shows "dynkin_system \<Omega> M"

  1239 proof -

  1240   interpret sigma_algebra \<Omega> M by fact

  1241   show ?thesis using sets_into_space by (fastforce intro!: dynkin_systemI)

  1242 qed

  1243

  1244 subsubsection "Intersection sets systems"

  1245

  1246 definition%important "Int_stable M \<longleftrightarrow> (\<forall> a \<in> M. \<forall> b \<in> M. a \<inter> b \<in> M)"

  1247

  1248 lemma (in algebra) Int_stable: "Int_stable M"

  1249   unfolding Int_stable_def by auto

  1250

  1251 lemma Int_stableI_image:

  1252   "(\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> \<exists>k\<in>I. A i \<inter> A j = A k) \<Longrightarrow> Int_stable (A  I)"

  1253   by (auto simp: Int_stable_def image_def)

  1254

  1255 lemma Int_stableI:

  1256   "(\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<inter> b \<in> A) \<Longrightarrow> Int_stable A"

  1257   unfolding Int_stable_def by auto

  1258

  1259 lemma Int_stableD:

  1260   "Int_stable M \<Longrightarrow> a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<inter> b \<in> M"

  1261   unfolding Int_stable_def by auto

  1262

  1263 lemma (in dynkin_system) sigma_algebra_eq_Int_stable:

  1264   "sigma_algebra \<Omega> M \<longleftrightarrow> Int_stable M"

  1265 proof

  1266   assume "sigma_algebra \<Omega> M" then show "Int_stable M"

  1267     unfolding sigma_algebra_def using algebra.Int_stable by auto

  1268 next

  1269   assume "Int_stable M"

  1270   show "sigma_algebra \<Omega> M"

  1271     unfolding sigma_algebra_disjoint_iff algebra_iff_Un

  1272   proof (intro conjI ballI allI impI)

  1273     show "M \<subseteq> Pow (\<Omega>)" using sets_into_space by auto

  1274   next

  1275     fix A B assume "A \<in> M" "B \<in> M"

  1276     then have "A \<union> B = \<Omega> - ((\<Omega> - A) \<inter> (\<Omega> - B))"

  1277               "\<Omega> - A \<in> M" "\<Omega> - B \<in> M"

  1278       using sets_into_space by auto

  1279     then show "A \<union> B \<in> M"

  1280       using \<open>Int_stable M\<close> unfolding Int_stable_def by auto

  1281   qed auto

  1282 qed

  1283

  1284 subsubsection "Smallest Dynkin systems"

  1285

  1286 definition%important dynkin where

  1287   "dynkin \<Omega> M =  (\<Inter>{D. dynkin_system \<Omega> D \<and> M \<subseteq> D})"

  1288

  1289 lemma dynkin_system_dynkin:

  1290   assumes "M \<subseteq> Pow (\<Omega>)"

  1291   shows "dynkin_system \<Omega> (dynkin \<Omega> M)"

  1292 proof (rule dynkin_systemI)

  1293   fix A assume "A \<in> dynkin \<Omega> M"

  1294   moreover

  1295   { fix D assume "A \<in> D" and d: "dynkin_system \<Omega> D"

  1296     then have "A \<subseteq> \<Omega>" by (auto simp: dynkin_system_def subset_class_def) }

  1297   moreover have "{D. dynkin_system \<Omega> D \<and> M \<subseteq> D} \<noteq> {}"

  1298     using assms dynkin_system_trivial by fastforce

  1299   ultimately show "A \<subseteq> \<Omega>"

  1300     unfolding dynkin_def using assms

  1301     by auto

  1302 next

  1303   show "\<Omega> \<in> dynkin \<Omega> M"

  1304     unfolding dynkin_def using dynkin_system.space by fastforce

  1305 next

  1306   fix A assume "A \<in> dynkin \<Omega> M"

  1307   then show "\<Omega> - A \<in> dynkin \<Omega> M"

  1308     unfolding dynkin_def using dynkin_system.compl by force

  1309 next

  1310   fix A :: "nat \<Rightarrow> 'a set"

  1311   assume A: "disjoint_family A" "range A \<subseteq> dynkin \<Omega> M"

  1312   show "(\<Union>i. A i) \<in> dynkin \<Omega> M" unfolding dynkin_def

  1313   proof (simp, safe)

  1314     fix D assume "dynkin_system \<Omega> D" "M \<subseteq> D"

  1315     with A have "(\<Union>i. A i) \<in> D"

  1316       by (intro dynkin_system.UN) (auto simp: dynkin_def)

  1317     then show "(\<Union>i. A i) \<in> D" by auto

  1318   qed

  1319 qed

  1320

  1321 lemma dynkin_Basic[intro]: "A \<in> M \<Longrightarrow> A \<in> dynkin \<Omega> M"

  1322   unfolding dynkin_def by auto

  1323

  1324 lemma (in dynkin_system) restricted_dynkin_system:

  1325   assumes "D \<in> M"

  1326   shows "dynkin_system \<Omega> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}"

  1327 proof (rule dynkin_systemI, simp_all)

  1328   have "\<Omega> \<inter> D = D"

  1329     using \<open>D \<in> M\<close> sets_into_space by auto

  1330   then show "\<Omega> \<inter> D \<in> M"

  1331     using \<open>D \<in> M\<close> by auto

  1332 next

  1333   fix A assume "A \<subseteq> \<Omega> \<and> A \<inter> D \<in> M"

  1334   moreover have "(\<Omega> - A) \<inter> D = (\<Omega> - (A \<inter> D)) - (\<Omega> - D)"

  1335     by auto

  1336   ultimately show "\<Omega> - A \<subseteq> \<Omega> \<and> (\<Omega> - A) \<inter> D \<in> M"

  1337     using  \<open>D \<in> M\<close> by (auto intro: diff)

  1338 next

  1339   fix A :: "nat \<Rightarrow> 'a set"

  1340   assume "disjoint_family A" "range A \<subseteq> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}"

  1341   then have "\<And>i. A i \<subseteq> \<Omega>" "disjoint_family (\<lambda>i. A i \<inter> D)"

  1342     "range (\<lambda>i. A i \<inter> D) \<subseteq> M" "(\<Union>x. A x) \<inter> D = (\<Union>x. A x \<inter> D)"

  1343     by ((fastforce simp: disjoint_family_on_def)+)

  1344   then show "(\<Union>x. A x) \<subseteq> \<Omega> \<and> (\<Union>x. A x) \<inter> D \<in> M"

  1345     by (auto simp del: UN_simps)

  1346 qed

  1347

  1348 lemma (in dynkin_system) dynkin_subset:

  1349   assumes "N \<subseteq> M"

  1350   shows "dynkin \<Omega> N \<subseteq> M"

  1351 proof -

  1352   have "dynkin_system \<Omega> M" ..

  1353   then have "dynkin_system \<Omega> M"

  1354     using assms unfolding dynkin_system_def dynkin_system_axioms_def subset_class_def by simp

  1355   with \<open>N \<subseteq> M\<close> show ?thesis by (auto simp add: dynkin_def)

  1356 qed

  1357

  1358 lemma sigma_eq_dynkin:

  1359   assumes sets: "M \<subseteq> Pow \<Omega>"

  1360   assumes "Int_stable M"

  1361   shows "sigma_sets \<Omega> M = dynkin \<Omega> M"

  1362 proof -

  1363   have "dynkin \<Omega> M \<subseteq> sigma_sets (\<Omega>) (M)"

  1364     using sigma_algebra_imp_dynkin_system

  1365     unfolding dynkin_def sigma_sets_least_sigma_algebra[OF sets] by auto

  1366   moreover

  1367   interpret dynkin_system \<Omega> "dynkin \<Omega> M"

  1368     using dynkin_system_dynkin[OF sets] .

  1369   have "sigma_algebra \<Omega> (dynkin \<Omega> M)"

  1370     unfolding sigma_algebra_eq_Int_stable Int_stable_def

  1371   proof (intro ballI)

  1372     fix A B assume "A \<in> dynkin \<Omega> M" "B \<in> dynkin \<Omega> M"

  1373     let ?D = "\<lambda>E. {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> E \<in> dynkin \<Omega> M}"

  1374     have "M \<subseteq> ?D B"

  1375     proof

  1376       fix E assume "E \<in> M"

  1377       then have "M \<subseteq> ?D E" "E \<in> dynkin \<Omega> M"

  1378         using sets_into_space \<open>Int_stable M\<close> by (auto simp: Int_stable_def)

  1379       then have "dynkin \<Omega> M \<subseteq> ?D E"

  1380         using restricted_dynkin_system \<open>E \<in> dynkin \<Omega> M\<close>

  1381         by (intro dynkin_system.dynkin_subset) simp_all

  1382       then have "B \<in> ?D E"

  1383         using \<open>B \<in> dynkin \<Omega> M\<close> by auto

  1384       then have "E \<inter> B \<in> dynkin \<Omega> M"

  1385         by (subst Int_commute) simp

  1386       then show "E \<in> ?D B"

  1387         using sets \<open>E \<in> M\<close> by auto

  1388     qed

  1389     then have "dynkin \<Omega> M \<subseteq> ?D B"

  1390       using restricted_dynkin_system \<open>B \<in> dynkin \<Omega> M\<close>

  1391       by (intro dynkin_system.dynkin_subset) simp_all

  1392     then show "A \<inter> B \<in> dynkin \<Omega> M"

  1393       using \<open>A \<in> dynkin \<Omega> M\<close> sets_into_space by auto

  1394   qed

  1395   from sigma_algebra.sigma_sets_subset[OF this, of "M"]

  1396   have "sigma_sets (\<Omega>) (M) \<subseteq> dynkin \<Omega> M" by auto

  1397   ultimately have "sigma_sets (\<Omega>) (M) = dynkin \<Omega> M" by auto

  1398   then show ?thesis

  1399     by (auto simp: dynkin_def)

  1400 qed

  1401

  1402 lemma (in dynkin_system) dynkin_idem:

  1403   "dynkin \<Omega> M = M"

  1404 proof -

  1405   have "dynkin \<Omega> M = M"

  1406   proof

  1407     show "M \<subseteq> dynkin \<Omega> M"

  1408       using dynkin_Basic by auto

  1409     show "dynkin \<Omega> M \<subseteq> M"

  1410       by (intro dynkin_subset) auto

  1411   qed

  1412   then show ?thesis

  1413     by (auto simp: dynkin_def)

  1414 qed

  1415

  1416 lemma (in dynkin_system) dynkin_lemma:

  1417   assumes "Int_stable E"

  1418   and E: "E \<subseteq> M" "M \<subseteq> sigma_sets \<Omega> E"

  1419   shows "sigma_sets \<Omega> E = M"

  1420 proof -

  1421   have "E \<subseteq> Pow \<Omega>"

  1422     using E sets_into_space by force

  1423   then have *: "sigma_sets \<Omega> E = dynkin \<Omega> E"

  1424     using \<open>Int_stable E\<close> by (rule sigma_eq_dynkin)

  1425   then have "dynkin \<Omega> E = M"

  1426     using assms dynkin_subset[OF E(1)] by simp

  1427   with * show ?thesis

  1428     using assms by (auto simp: dynkin_def)

  1429 qed

  1430

  1431 subsubsection \<open>Induction rule for intersection-stable generators\<close>

  1432

  1433 text%important \<open>The reason to introduce Dynkin-systems is the following induction rules for $\sigma$-algebras

  1434 generated by a generator closed under intersection.\<close>

  1435

  1436 proposition sigma_sets_induct_disjoint[consumes 3, case_names basic empty compl union]:

  1437   assumes "Int_stable G"

  1438     and closed: "G \<subseteq> Pow \<Omega>"

  1439     and A: "A \<in> sigma_sets \<Omega> G"

  1440   assumes basic: "\<And>A. A \<in> G \<Longrightarrow> P A"

  1441     and empty: "P {}"

  1442     and compl: "\<And>A. A \<in> sigma_sets \<Omega> G \<Longrightarrow> P A \<Longrightarrow> P (\<Omega> - A)"

  1443     and union: "\<And>A. disjoint_family A \<Longrightarrow> range A \<subseteq> sigma_sets \<Omega> G \<Longrightarrow> (\<And>i. P (A i)) \<Longrightarrow> P (\<Union>i::nat. A i)"

  1444   shows "P A"

  1445 proof -

  1446   let ?D = "{ A \<in> sigma_sets \<Omega> G. P A }"

  1447   interpret sigma_algebra \<Omega> "sigma_sets \<Omega> G"

  1448     using closed by (rule sigma_algebra_sigma_sets)

  1449   from compl[OF _ empty] closed have space: "P \<Omega>" by simp

  1450   interpret dynkin_system \<Omega> ?D

  1451     by standard (auto dest: sets_into_space intro!: space compl union)

  1452   have "sigma_sets \<Omega> G = ?D"

  1453     by (rule dynkin_lemma) (auto simp: basic \<open>Int_stable G\<close>)

  1454   with A show ?thesis by auto

  1455 qed

  1456

  1457 subsection \<open>Measure type\<close>

  1458

  1459 definition%important positive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ennreal) \<Rightarrow> bool" where

  1460   "positive M \<mu> \<longleftrightarrow> \<mu> {} = 0"

  1461

  1462 definition%important countably_additive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ennreal) \<Rightarrow> bool" where

  1463   "countably_additive M f \<longleftrightarrow> (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow>

  1464     (\<Sum>i. f (A i)) = f (\<Union>i. A i))"

  1465

  1466 definition%important measure_space :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ennreal) \<Rightarrow> bool" where

  1467   "measure_space \<Omega> A \<mu> \<longleftrightarrow> sigma_algebra \<Omega> A \<and> positive A \<mu> \<and> countably_additive A \<mu>"

  1468

  1469 typedef%important 'a measure = "{(\<Omega>::'a set, A, \<mu>). (\<forall>a\<in>-A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu> }"

  1470 proof%unimportant

  1471   have "sigma_algebra UNIV {{}, UNIV}"

  1472     by (auto simp: sigma_algebra_iff2)

  1473   then show "(UNIV, {{}, UNIV}, \<lambda>A. 0) \<in> {(\<Omega>, A, \<mu>). (\<forall>a\<in>-A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu>} "

  1474     by (auto simp: measure_space_def positive_def countably_additive_def)

  1475 qed

  1476

  1477 definition%important space :: "'a measure \<Rightarrow> 'a set" where

  1478   "space M = fst (Rep_measure M)"

  1479

  1480 definition%important sets :: "'a measure \<Rightarrow> 'a set set" where

  1481   "sets M = fst (snd (Rep_measure M))"

  1482

  1483 definition%important emeasure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> ennreal" where

  1484   "emeasure M = snd (snd (Rep_measure M))"

  1485

  1486 definition%important measure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> real" where

  1487   "measure M A = enn2real (emeasure M A)"

  1488

  1489 declare [[coercion sets]]

  1490

  1491 declare [[coercion measure]]

  1492

  1493 declare [[coercion emeasure]]

  1494

  1495 lemma measure_space: "measure_space (space M) (sets M) (emeasure M)"

  1496   by (cases M) (auto simp: space_def sets_def emeasure_def Abs_measure_inverse)

  1497

  1498 interpretation sets: sigma_algebra "space M" "sets M" for M :: "'a measure"

  1499   using measure_space[of M] by (auto simp: measure_space_def)

  1500

  1501 definition%important measure_of :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ennreal) \<Rightarrow> 'a measure" where

  1502   "measure_of \<Omega> A \<mu> = Abs_measure (\<Omega>, if A \<subseteq> Pow \<Omega> then sigma_sets \<Omega> A else {{}, \<Omega>},

  1503     \<lambda>a. if a \<in> sigma_sets \<Omega> A \<and> measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> then \<mu> a else 0)"

  1504

  1505 abbreviation "sigma \<Omega> A \<equiv> measure_of \<Omega> A (\<lambda>x. 0)"

  1506

  1507 lemma measure_space_0: "A \<subseteq> Pow \<Omega> \<Longrightarrow> measure_space \<Omega> (sigma_sets \<Omega> A) (\<lambda>x. 0)"

  1508   unfolding measure_space_def

  1509   by (auto intro!: sigma_algebra_sigma_sets simp: positive_def countably_additive_def)

  1510

  1511 lemma sigma_algebra_trivial: "sigma_algebra \<Omega> {{}, \<Omega>}"

  1512 by unfold_locales(fastforce intro: exI[where x="{{}}"] exI[where x="{\<Omega>}"])+

  1513

  1514 lemma measure_space_0': "measure_space \<Omega> {{}, \<Omega>} (\<lambda>x. 0)"

  1515 by(simp add: measure_space_def positive_def countably_additive_def sigma_algebra_trivial)

  1516

  1517 lemma measure_space_closed:

  1518   assumes "measure_space \<Omega> M \<mu>"

  1519   shows "M \<subseteq> Pow \<Omega>"

  1520 proof -

  1521   interpret sigma_algebra \<Omega> M using assms by(simp add: measure_space_def)

  1522   show ?thesis by(rule space_closed)

  1523 qed

  1524

  1525 lemma (in ring_of_sets) positive_cong_eq:

  1526   "(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> positive M \<mu>' = positive M \<mu>"

  1527   by (auto simp add: positive_def)

  1528

  1529 lemma (in sigma_algebra) countably_additive_eq:

  1530   "(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> countably_additive M \<mu>' = countably_additive M \<mu>"

  1531   unfolding countably_additive_def

  1532   by (intro arg_cong[where f=All] ext) (auto simp add: countably_additive_def subset_eq)

  1533

  1534 lemma measure_space_eq:

  1535   assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a"

  1536   shows "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'"

  1537 proof -

  1538   interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" using closed by (rule sigma_algebra_sigma_sets)

  1539   from positive_cong_eq[OF eq, of "\<lambda>i. i"] countably_additive_eq[OF eq, of "\<lambda>i. i"] show ?thesis

  1540     by (auto simp: measure_space_def)

  1541 qed

  1542

  1543 lemma measure_of_eq:

  1544   assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "(\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a)"

  1545   shows "measure_of \<Omega> A \<mu> = measure_of \<Omega> A \<mu>'"

  1546 proof -

  1547   have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'"

  1548     using assms by (rule measure_space_eq)

  1549   with eq show ?thesis

  1550     by (auto simp add: measure_of_def intro!: arg_cong[where f=Abs_measure])

  1551 qed

  1552

  1553 lemma

  1554   shows space_measure_of_conv: "space (measure_of \<Omega> A \<mu>) = \<Omega>" (is ?space)

  1555   and sets_measure_of_conv:

  1556   "sets (measure_of \<Omega> A \<mu>) = (if A \<subseteq> Pow \<Omega> then sigma_sets \<Omega> A else {{}, \<Omega>})" (is ?sets)

  1557   and emeasure_measure_of_conv:

  1558   "emeasure (measure_of \<Omega> A \<mu>) =

  1559   (\<lambda>B. if B \<in> sigma_sets \<Omega> A \<and> measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> then \<mu> B else 0)" (is ?emeasure)

  1560 proof -

  1561   have "?space \<and> ?sets \<and> ?emeasure"

  1562   proof(cases "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>")

  1563     case True

  1564     from measure_space_closed[OF this] sigma_sets_superset_generator[of A \<Omega>]

  1565     have "A \<subseteq> Pow \<Omega>" by simp

  1566     hence "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A)

  1567       (\<lambda>a. if a \<in> sigma_sets \<Omega> A then \<mu> a else 0)"

  1568       by(rule measure_space_eq) auto

  1569     with True \<open>A \<subseteq> Pow \<Omega>\<close> show ?thesis

  1570       by(simp add: measure_of_def space_def sets_def emeasure_def Abs_measure_inverse)

  1571   next

  1572     case False thus ?thesis

  1573       by(cases "A \<subseteq> Pow \<Omega>")(simp_all add: Abs_measure_inverse measure_of_def sets_def space_def emeasure_def measure_space_0 measure_space_0')

  1574   qed

  1575   thus ?space ?sets ?emeasure by simp_all

  1576 qed

  1577

  1578 lemma [simp]:

  1579   assumes A: "A \<subseteq> Pow \<Omega>"

  1580   shows sets_measure_of: "sets (measure_of \<Omega> A \<mu>) = sigma_sets \<Omega> A"

  1581     and space_measure_of: "space (measure_of \<Omega> A \<mu>) = \<Omega>"

  1582 using assms

  1583 by(simp_all add: sets_measure_of_conv space_measure_of_conv)

  1584

  1585 lemma space_in_measure_of[simp]: "\<Omega> \<in> sets (measure_of \<Omega> M \<mu>)"

  1586   by (subst sets_measure_of_conv) (auto simp: sigma_sets_top)

  1587

  1588 lemma (in sigma_algebra) sets_measure_of_eq[simp]: "sets (measure_of \<Omega> M \<mu>) = M"

  1589   using space_closed by (auto intro!: sigma_sets_eq)

  1590

  1591 lemma (in sigma_algebra) space_measure_of_eq[simp]: "space (measure_of \<Omega> M \<mu>) = \<Omega>"

  1592   by (rule space_measure_of_conv)

  1593

  1594 lemma measure_of_subset: "M \<subseteq> Pow \<Omega> \<Longrightarrow> M' \<subseteq> M \<Longrightarrow> sets (measure_of \<Omega> M' \<mu>) \<subseteq> sets (measure_of \<Omega> M \<mu>')"

  1595   by (auto intro!: sigma_sets_subseteq)

  1596

  1597 lemma emeasure_sigma: "emeasure (sigma \<Omega> A) = (\<lambda>x. 0)"

  1598   unfolding measure_of_def emeasure_def

  1599   by (subst Abs_measure_inverse)

  1600      (auto simp: measure_space_def positive_def countably_additive_def

  1601            intro!: sigma_algebra_sigma_sets sigma_algebra_trivial)

  1602

  1603 lemma sigma_sets_mono'':

  1604   assumes "A \<in> sigma_sets C D"

  1605   assumes "B \<subseteq> D"

  1606   assumes "D \<subseteq> Pow C"

  1607   shows "sigma_sets A B \<subseteq> sigma_sets C D"

  1608 proof

  1609   fix x assume "x \<in> sigma_sets A B"

  1610   thus "x \<in> sigma_sets C D"

  1611   proof induct

  1612     case (Basic a) with assms have "a \<in> D" by auto

  1613     thus ?case ..

  1614   next

  1615     case Empty show ?case by (rule sigma_sets.Empty)

  1616   next

  1617     from assms have "A \<in> sets (sigma C D)" by (subst sets_measure_of[OF \<open>D \<subseteq> Pow C\<close>])

  1618     moreover case (Compl a) hence "a \<in> sets (sigma C D)" by (subst sets_measure_of[OF \<open>D \<subseteq> Pow C\<close>])

  1619     ultimately have "A - a \<in> sets (sigma C D)" ..

  1620     thus ?case by (subst (asm) sets_measure_of[OF \<open>D \<subseteq> Pow C\<close>])

  1621   next

  1622     case (Union a)

  1623     thus ?case by (intro sigma_sets.Union)

  1624   qed

  1625 qed

  1626

  1627 lemma in_measure_of[intro, simp]: "M \<subseteq> Pow \<Omega> \<Longrightarrow> A \<in> M \<Longrightarrow> A \<in> sets (measure_of \<Omega> M \<mu>)"

  1628   by auto

  1629

  1630 lemma space_empty_iff: "space N = {} \<longleftrightarrow> sets N = {{}}"

  1631   by (metis Pow_empty Sup_bot_conv(1) cSup_singleton empty_iff

  1632             sets.sigma_sets_eq sets.space_closed sigma_sets_top subset_singletonD)

  1633

  1634 subsubsection \<open>Constructing simple @{typ "'a measure"}\<close>

  1635

  1636 proposition emeasure_measure_of:

  1637   assumes M: "M = measure_of \<Omega> A \<mu>"

  1638   assumes ms: "A \<subseteq> Pow \<Omega>" "positive (sets M) \<mu>" "countably_additive (sets M) \<mu>"

  1639   assumes X: "X \<in> sets M"

  1640   shows "emeasure M X = \<mu> X"

  1641 proof -

  1642   interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" by (rule sigma_algebra_sigma_sets) fact

  1643   have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>"

  1644     using ms M by (simp add: measure_space_def sigma_algebra_sigma_sets)

  1645   thus ?thesis using X ms

  1646     by(simp add: M emeasure_measure_of_conv sets_measure_of_conv)

  1647 qed

  1648

  1649 lemma emeasure_measure_of_sigma:

  1650   assumes ms: "sigma_algebra \<Omega> M" "positive M \<mu>" "countably_additive M \<mu>"

  1651   assumes A: "A \<in> M"

  1652   shows "emeasure (measure_of \<Omega> M \<mu>) A = \<mu> A"

  1653 proof -

  1654   interpret sigma_algebra \<Omega> M by fact

  1655   have "measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"

  1656     using ms sigma_sets_eq by (simp add: measure_space_def)

  1657   thus ?thesis by(simp add: emeasure_measure_of_conv A)

  1658 qed

  1659

  1660 lemma measure_cases[cases type: measure]:

  1661   obtains (measure) \<Omega> A \<mu> where "x = Abs_measure (\<Omega>, A, \<mu>)" "\<forall>a\<in>-A. \<mu> a = 0" "measure_space \<Omega> A \<mu>"

  1662   by atomize_elim (cases x, auto)

  1663

  1664 lemma sets_le_imp_space_le: "sets A \<subseteq> sets B \<Longrightarrow> space A \<subseteq> space B"

  1665   by (auto dest: sets.sets_into_space)

  1666

  1667 lemma sets_eq_imp_space_eq: "sets M = sets M' \<Longrightarrow> space M = space M'"

  1668   by (auto intro!: antisym sets_le_imp_space_le)

  1669

  1670 lemma emeasure_notin_sets: "A \<notin> sets M \<Longrightarrow> emeasure M A = 0"

  1671   by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)

  1672

  1673 lemma emeasure_neq_0_sets: "emeasure M A \<noteq> 0 \<Longrightarrow> A \<in> sets M"

  1674   using emeasure_notin_sets[of A M] by blast

  1675

  1676 lemma measure_notin_sets: "A \<notin> sets M \<Longrightarrow> measure M A = 0"

  1677   by (simp add: measure_def emeasure_notin_sets zero_ennreal.rep_eq)

  1678

  1679 lemma measure_eqI:

  1680   fixes M N :: "'a measure"

  1681   assumes "sets M = sets N" and eq: "\<And>A. A \<in> sets M \<Longrightarrow> emeasure M A = emeasure N A"

  1682   shows "M = N"

  1683 proof (cases M N rule: measure_cases[case_product measure_cases])

  1684   case (measure_measure \<Omega> A \<mu> \<Omega>' A' \<mu>')

  1685   interpret M: sigma_algebra \<Omega> A using measure_measure by (auto simp: measure_space_def)

  1686   interpret N: sigma_algebra \<Omega>' A' using measure_measure by (auto simp: measure_space_def)

  1687   have "A = sets M" "A' = sets N"

  1688     using measure_measure by (simp_all add: sets_def Abs_measure_inverse)

  1689   with \<open>sets M = sets N\<close> have AA': "A = A'" by simp

  1690   moreover from M.top N.top M.space_closed N.space_closed AA' have "\<Omega> = \<Omega>'" by auto

  1691   moreover { fix B have "\<mu> B = \<mu>' B"

  1692     proof cases

  1693       assume "B \<in> A"

  1694       with eq \<open>A = sets M\<close> have "emeasure M B = emeasure N B" by simp

  1695       with measure_measure show "\<mu> B = \<mu>' B"

  1696         by (simp add: emeasure_def Abs_measure_inverse)

  1697     next

  1698       assume "B \<notin> A"

  1699       with \<open>A = sets M\<close> \<open>A' = sets N\<close> \<open>A = A'\<close> have "B \<notin> sets M" "B \<notin> sets N"

  1700         by auto

  1701       then have "emeasure M B = 0" "emeasure N B = 0"

  1702         by (simp_all add: emeasure_notin_sets)

  1703       with measure_measure show "\<mu> B = \<mu>' B"

  1704         by (simp add: emeasure_def Abs_measure_inverse)

  1705     qed }

  1706   then have "\<mu> = \<mu>'" by auto

  1707   ultimately show "M = N"

  1708     by (simp add: measure_measure)

  1709 qed

  1710

  1711 lemma sigma_eqI:

  1712   assumes [simp]: "M \<subseteq> Pow \<Omega>" "N \<subseteq> Pow \<Omega>" "sigma_sets \<Omega> M = sigma_sets \<Omega> N"

  1713   shows "sigma \<Omega> M = sigma \<Omega> N"

  1714   by (rule measure_eqI) (simp_all add: emeasure_sigma)

  1715

  1716 subsubsection \<open>Measurable functions\<close>

  1717

  1718 definition%important measurable :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) set" (infixr "\<rightarrow>\<^sub>M" 60) where

  1719   "measurable A B = {f \<in> space A \<rightarrow> space B. \<forall>y \<in> sets B. f - y \<inter> space A \<in> sets A}"

  1720

  1721 lemma measurableI:

  1722   "(\<And>x. x \<in> space M \<Longrightarrow> f x \<in> space N) \<Longrightarrow> (\<And>A. A \<in> sets N \<Longrightarrow> f - A \<inter> space M \<in> sets M) \<Longrightarrow>

  1723     f \<in> measurable M N"

  1724   by (auto simp: measurable_def)

  1725

  1726 lemma measurable_space:

  1727   "f \<in> measurable M A \<Longrightarrow> x \<in> space M \<Longrightarrow> f x \<in> space A"

  1728    unfolding measurable_def by auto

  1729

  1730 lemma measurable_sets:

  1731   "f \<in> measurable M A \<Longrightarrow> S \<in> sets A \<Longrightarrow> f - S \<inter> space M \<in> sets M"

  1732    unfolding measurable_def by auto

  1733

  1734 lemma measurable_sets_Collect:

  1735   assumes f: "f \<in> measurable M N" and P: "{x\<in>space N. P x} \<in> sets N" shows "{x\<in>space M. P (f x)} \<in> sets M"

  1736 proof -

  1737   have "f - {x \<in> space N. P x} \<inter> space M = {x\<in>space M. P (f x)}"

  1738     using measurable_space[OF f] by auto

  1739   with measurable_sets[OF f P] show ?thesis

  1740     by simp

  1741 qed

  1742

  1743 lemma measurable_sigma_sets:

  1744   assumes B: "sets N = sigma_sets \<Omega> A" "A \<subseteq> Pow \<Omega>"

  1745       and f: "f \<in> space M \<rightarrow> \<Omega>"

  1746       and ba: "\<And>y. y \<in> A \<Longrightarrow> (f - y) \<inter> space M \<in> sets M"

  1747   shows "f \<in> measurable M N"

  1748 proof -

  1749   interpret A: sigma_algebra \<Omega> "sigma_sets \<Omega> A" using B(2) by (rule sigma_algebra_sigma_sets)

  1750   from B sets.top[of N] A.top sets.space_closed[of N] A.space_closed have \<Omega>: "\<Omega> = space N" by force

  1751

  1752   { fix X assume "X \<in> sigma_sets \<Omega> A"

  1753     then have "f - X \<inter> space M \<in> sets M \<and> X \<subseteq> \<Omega>"

  1754       proof induct

  1755         case (Basic a) then show ?case

  1756           by (auto simp add: ba) (metis B(2) subsetD PowD)

  1757       next

  1758         case (Compl a)

  1759         have [simp]: "f - \<Omega> \<inter> space M = space M"

  1760           by (auto simp add: funcset_mem [OF f])

  1761         then show ?case

  1762           by (auto simp add: vimage_Diff Diff_Int_distrib2 sets.compl_sets Compl)

  1763       next

  1764         case (Union a)

  1765         then show ?case

  1766           by (simp add: vimage_UN, simp only: UN_extend_simps(4)) blast

  1767       qed auto }

  1768   with f show ?thesis

  1769     by (auto simp add: measurable_def B \<Omega>)

  1770 qed

  1771

  1772 lemma measurable_measure_of:

  1773   assumes B: "N \<subseteq> Pow \<Omega>"

  1774       and f: "f \<in> space M \<rightarrow> \<Omega>"

  1775       and ba: "\<And>y. y \<in> N \<Longrightarrow> (f - y) \<inter> space M \<in> sets M"

  1776   shows "f \<in> measurable M (measure_of \<Omega> N \<mu>)"

  1777 proof -

  1778   have "sets (measure_of \<Omega> N \<mu>) = sigma_sets \<Omega> N"

  1779     using B by (rule sets_measure_of)

  1780   from this assms show ?thesis by (rule measurable_sigma_sets)

  1781 qed

  1782

  1783 lemma measurable_iff_measure_of:

  1784   assumes "N \<subseteq> Pow \<Omega>" "f \<in> space M \<rightarrow> \<Omega>"

  1785   shows "f \<in> measurable M (measure_of \<Omega> N \<mu>) \<longleftrightarrow> (\<forall>A\<in>N. f - A \<inter> space M \<in> sets M)"

  1786   by (metis assms in_measure_of measurable_measure_of assms measurable_sets)

  1787

  1788 lemma measurable_cong_sets:

  1789   assumes sets: "sets M = sets M'" "sets N = sets N'"

  1790   shows "measurable M N = measurable M' N'"

  1791   using sets[THEN sets_eq_imp_space_eq] sets by (simp add: measurable_def)

  1792

  1793 lemma measurable_cong:

  1794   assumes "\<And>w. w \<in> space M \<Longrightarrow> f w = g w"

  1795   shows "f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable M M'"

  1796   unfolding measurable_def using assms

  1797   by (simp cong: vimage_inter_cong Pi_cong)

  1798

  1799 lemma measurable_cong':

  1800   assumes "\<And>w. w \<in> space M =simp=> f w = g w"

  1801   shows "f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable M M'"

  1802   unfolding measurable_def using assms

  1803   by (simp cong: vimage_inter_cong Pi_cong add: simp_implies_def)

  1804

  1805 lemma measurable_cong_strong:

  1806   "M = N \<Longrightarrow> M' = N' \<Longrightarrow> (\<And>w. w \<in> space M \<Longrightarrow> f w = g w) \<Longrightarrow>

  1807     f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable N N'"

  1808   by (metis measurable_cong)

  1809

  1810 lemma measurable_compose:

  1811   assumes f: "f \<in> measurable M N" and g: "g \<in> measurable N L"

  1812   shows "(\<lambda>x. g (f x)) \<in> measurable M L"

  1813 proof -

  1814   have "\<And>A. (\<lambda>x. g (f x)) - A \<inter> space M = f - (g - A \<inter> space N) \<inter> space M"

  1815     using measurable_space[OF f] by auto

  1816   with measurable_space[OF f] measurable_space[OF g] show ?thesis

  1817     by (auto intro: measurable_sets[OF f] measurable_sets[OF g]

  1818              simp del: vimage_Int simp add: measurable_def)

  1819 qed

  1820

  1821 lemma measurable_comp:

  1822   "f \<in> measurable M N \<Longrightarrow> g \<in> measurable N L \<Longrightarrow> g \<circ> f \<in> measurable M L"

  1823   using measurable_compose[of f M N g L] by (simp add: comp_def)

  1824

  1825 lemma measurable_const:

  1826   "c \<in> space M' \<Longrightarrow> (\<lambda>x. c) \<in> measurable M M'"

  1827   by (auto simp add: measurable_def)

  1828

  1829 lemma measurable_ident: "id \<in> measurable M M"

  1830   by (auto simp add: measurable_def)

  1831

  1832 lemma measurable_id: "(\<lambda>x. x) \<in> measurable M M"

  1833   by (simp add: measurable_def)

  1834

  1835 lemma measurable_ident_sets:

  1836   assumes eq: "sets M = sets M'" shows "(\<lambda>x. x) \<in> measurable M M'"

  1837   using measurable_ident[of M]

  1838   unfolding id_def measurable_def eq sets_eq_imp_space_eq[OF eq] .

  1839

  1840 lemma sets_Least:

  1841   assumes meas: "\<And>i::nat. {x\<in>space M. P i x} \<in> M"

  1842   shows "(\<lambda>x. LEAST j. P j x) - A \<inter> space M \<in> sets M"

  1843 proof -

  1844   { fix i have "(\<lambda>x. LEAST j. P j x) - {i} \<inter> space M \<in> sets M"

  1845     proof cases

  1846       assume i: "(LEAST j. False) = i"

  1847       have "(\<lambda>x. LEAST j. P j x) - {i} \<inter> space M =

  1848         {x\<in>space M. P i x} \<inter> (space M - (\<Union>j<i. {x\<in>space M. P j x})) \<union> (space M - (\<Union>i. {x\<in>space M. P i x}))"

  1849         by (simp add: set_eq_iff, safe)

  1850            (insert i, auto dest: Least_le intro: LeastI intro!: Least_equality)

  1851       with meas show ?thesis

  1852         by (auto intro!: sets.Int)

  1853     next

  1854       assume i: "(LEAST j. False) \<noteq> i"

  1855       then have "(\<lambda>x. LEAST j. P j x) - {i} \<inter> space M =

  1856         {x\<in>space M. P i x} \<inter> (space M - (\<Union>j<i. {x\<in>space M. P j x}))"

  1857       proof (simp add: set_eq_iff, safe)

  1858         fix x assume neq: "(LEAST j. False) \<noteq> (LEAST j. P j x)"

  1859         have "\<exists>j. P j x"

  1860           by (rule ccontr) (insert neq, auto)

  1861         then show "P (LEAST j. P j x) x" by (rule LeastI_ex)

  1862       qed (auto dest: Least_le intro!: Least_equality)

  1863       with meas show ?thesis

  1864         by auto

  1865     qed }

  1866   then have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) - {i} \<inter> space M) \<in> sets M"

  1867     by (intro sets.countable_UN) auto

  1868   moreover have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) - {i} \<inter> space M) =

  1869     (\<lambda>x. LEAST j. P j x) - A \<inter> space M" by auto

  1870   ultimately show ?thesis by auto

  1871 qed

  1872

  1873 lemma measurable_mono1:

  1874   "M' \<subseteq> Pow \<Omega> \<Longrightarrow> M \<subseteq> M' \<Longrightarrow>

  1875     measurable (measure_of \<Omega> M \<mu>) N \<subseteq> measurable (measure_of \<Omega> M' \<mu>') N"

  1876   using measure_of_subset[of M' \<Omega> M] by (auto simp add: measurable_def)

  1877

  1878 subsubsection \<open>Counting space\<close>

  1879

  1880 definition%important count_space :: "'a set \<Rightarrow> 'a measure" where

  1881   "count_space \<Omega> = measure_of \<Omega> (Pow \<Omega>) (\<lambda>A. if finite A then of_nat (card A) else \<infinity>)"

  1882

  1883 lemma

  1884   shows space_count_space[simp]: "space (count_space \<Omega>) = \<Omega>"

  1885     and sets_count_space[simp]: "sets (count_space \<Omega>) = Pow \<Omega>"

  1886   using sigma_sets_into_sp[of "Pow \<Omega>" \<Omega>]

  1887   by (auto simp: count_space_def)

  1888

  1889 lemma measurable_count_space_eq1[simp]:

  1890   "f \<in> measurable (count_space A) M \<longleftrightarrow> f \<in> A \<rightarrow> space M"

  1891  unfolding measurable_def by simp

  1892

  1893 lemma measurable_compose_countable':

  1894   assumes f: "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. f i x) \<in> measurable M N"

  1895   and g: "g \<in> measurable M (count_space I)" and I: "countable I"

  1896   shows "(\<lambda>x. f (g x) x) \<in> measurable M N"

  1897   unfolding measurable_def

  1898 proof safe

  1899   fix x assume "x \<in> space M" then show "f (g x) x \<in> space N"

  1900     using measurable_space[OF f] g[THEN measurable_space] by auto

  1901 next

  1902   fix A assume A: "A \<in> sets N"

  1903   have "(\<lambda>x. f (g x) x) - A \<inter> space M = (\<Union>i\<in>I. (g - {i} \<inter> space M) \<inter> (f i - A \<inter> space M))"

  1904     using measurable_space[OF g] by auto

  1905   also have "\<dots> \<in> sets M"

  1906     using f[THEN measurable_sets, OF _ A] g[THEN measurable_sets]

  1907     by (auto intro!: sets.countable_UN' I intro: sets.Int[OF measurable_sets measurable_sets])

  1908   finally show "(\<lambda>x. f (g x) x) - A \<inter> space M \<in> sets M" .

  1909 qed

  1910

  1911 lemma measurable_count_space_eq_countable:

  1912   assumes "countable A"

  1913   shows "f \<in> measurable M (count_space A) \<longleftrightarrow> (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f - {a} \<inter> space M \<in> sets M))"

  1914 proof -

  1915   { fix X assume "X \<subseteq> A" "f \<in> space M \<rightarrow> A"

  1916     with \<open>countable A\<close> have "f - X \<inter> space M = (\<Union>a\<in>X. f - {a} \<inter> space M)" "countable X"

  1917       by (auto dest: countable_subset)

  1918     moreover assume "\<forall>a\<in>A. f - {a} \<inter> space M \<in> sets M"

  1919     ultimately have "f - X \<inter> space M \<in> sets M"

  1920       using \<open>X \<subseteq> A\<close> by (auto intro!: sets.countable_UN' simp del: UN_simps) }

  1921   then show ?thesis

  1922     unfolding measurable_def by auto

  1923 qed

  1924

  1925 lemma measurable_count_space_eq2:

  1926   "finite A \<Longrightarrow> f \<in> measurable M (count_space A) \<longleftrightarrow> (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f - {a} \<inter> space M \<in> sets M))"

  1927   by (intro measurable_count_space_eq_countable countable_finite)

  1928

  1929 lemma measurable_count_space_eq2_countable:

  1930   fixes f :: "'a => 'c::countable"

  1931   shows "f \<in> measurable M (count_space A) \<longleftrightarrow> (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f - {a} \<inter> space M \<in> sets M))"

  1932   by (intro measurable_count_space_eq_countable countableI_type)

  1933

  1934 lemma measurable_compose_countable:

  1935   assumes f: "\<And>i::'i::countable. (\<lambda>x. f i x) \<in> measurable M N" and g: "g \<in> measurable M (count_space UNIV)"

  1936   shows "(\<lambda>x. f (g x) x) \<in> measurable M N"

  1937   by (rule measurable_compose_countable'[OF assms]) auto

  1938

  1939 lemma measurable_count_space_const:

  1940   "(\<lambda>x. c) \<in> measurable M (count_space UNIV)"

  1941   by (simp add: measurable_const)

  1942

  1943 lemma measurable_count_space:

  1944   "f \<in> measurable (count_space A) (count_space UNIV)"

  1945   by simp

  1946

  1947 lemma measurable_compose_rev:

  1948   assumes f: "f \<in> measurable L N" and g: "g \<in> measurable M L"

  1949   shows "(\<lambda>x. f (g x)) \<in> measurable M N"

  1950   using measurable_compose[OF g f] .

  1951

  1952 lemma measurable_empty_iff:

  1953   "space N = {} \<Longrightarrow> f \<in> measurable M N \<longleftrightarrow> space M = {}"

  1954   by (auto simp add: measurable_def Pi_iff)

  1955

  1956 subsubsection%unimportant \<open>Extend measure\<close>

  1957

  1958 definition "extend_measure \<Omega> I G \<mu> =

  1959   (if (\<exists>\<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (GI)) \<mu>') \<and> \<not> (\<forall>i\<in>I. \<mu> i = 0)

  1960       then measure_of \<Omega> (GI) (SOME \<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (GI)) \<mu>')

  1961       else measure_of \<Omega> (GI) (\<lambda>_. 0))"

  1962

  1963 lemma space_extend_measure: "G  I \<subseteq> Pow \<Omega> \<Longrightarrow> space (extend_measure \<Omega> I G \<mu>) = \<Omega>"

  1964   unfolding extend_measure_def by simp

  1965

  1966 lemma sets_extend_measure: "G  I \<subseteq> Pow \<Omega> \<Longrightarrow> sets (extend_measure \<Omega> I G \<mu>) = sigma_sets \<Omega> (GI)"

  1967   unfolding extend_measure_def by simp

  1968

  1969 lemma emeasure_extend_measure:

  1970   assumes M: "M = extend_measure \<Omega> I G \<mu>"

  1971     and eq: "\<And>i. i \<in> I \<Longrightarrow> \<mu>' (G i) = \<mu> i"

  1972     and ms: "G  I \<subseteq> Pow \<Omega>" "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'"

  1973     and "i \<in> I"

  1974   shows "emeasure M (G i) = \<mu> i"

  1975 proof cases

  1976   assume *: "(\<forall>i\<in>I. \<mu> i = 0)"

  1977   with M have M_eq: "M = measure_of \<Omega> (GI) (\<lambda>_. 0)"

  1978    by (simp add: extend_measure_def)

  1979   from measure_space_0[OF ms(1)] ms \<open>i\<in>I\<close>

  1980   have "emeasure M (G i) = 0"

  1981     by (intro emeasure_measure_of[OF M_eq]) (auto simp add: M measure_space_def sets_extend_measure)

  1982   with \<open>i\<in>I\<close> * show ?thesis

  1983     by simp

  1984 next

  1985   define P where "P \<mu>' \<longleftrightarrow> (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (GI)) \<mu>'" for \<mu>'

  1986   assume "\<not> (\<forall>i\<in>I. \<mu> i = 0)"

  1987   moreover

  1988   have "measure_space (space M) (sets M) \<mu>'"

  1989     using ms unfolding measure_space_def by auto standard

  1990   with ms eq have "\<exists>\<mu>'. P \<mu>'"

  1991     unfolding P_def

  1992     by (intro exI[of _ \<mu>']) (auto simp add: M space_extend_measure sets_extend_measure)

  1993   ultimately have M_eq: "M = measure_of \<Omega> (GI) (Eps P)"

  1994     by (simp add: M extend_measure_def P_def[symmetric])

  1995

  1996   from \<open>\<exists>\<mu>'. P \<mu>'\<close> have P: "P (Eps P)" by (rule someI_ex)

  1997   show "emeasure M (G i) = \<mu> i"

  1998   proof (subst emeasure_measure_of[OF M_eq])

  1999     have sets_M: "sets M = sigma_sets \<Omega> (GI)"

  2000       using M_eq ms by (auto simp: sets_extend_measure)

  2001     then show "G i \<in> sets M" using \<open>i \<in> I\<close> by auto

  2002     show "positive (sets M) (Eps P)" "countably_additive (sets M) (Eps P)" "Eps P (G i) = \<mu> i"

  2003       using P \<open>i\<in>I\<close> by (auto simp add: sets_M measure_space_def P_def)

  2004   qed fact

  2005 qed

  2006

  2007 lemma emeasure_extend_measure_Pair:

  2008   assumes M: "M = extend_measure \<Omega> {(i, j). I i j} (\<lambda>(i, j). G i j) (\<lambda>(i, j). \<mu> i j)"

  2009     and eq: "\<And>i j. I i j \<Longrightarrow> \<mu>' (G i j) = \<mu> i j"

  2010     and ms: "\<And>i j. I i j \<Longrightarrow> G i j \<in> Pow \<Omega>" "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'"

  2011     and "I i j"

  2012   shows "emeasure M (G i j) = \<mu> i j"

  2013   using emeasure_extend_measure[OF M _ _ ms(2,3), of "(i,j)"] eq ms(1) \<open>I i j\<close>

  2014   by (auto simp: subset_eq)

  2015

  2016 subsection \<open>The smallest $\sigma$-algebra regarding a function\<close>

  2017

  2018 definition%important

  2019   "vimage_algebra X f M = sigma X {f - A \<inter> X | A. A \<in> sets M}"

  2020

  2021 lemma space_vimage_algebra[simp]: "space (vimage_algebra X f M) = X"

  2022   unfolding vimage_algebra_def by (rule space_measure_of) auto

  2023

  2024 lemma sets_vimage_algebra: "sets (vimage_algebra X f M) = sigma_sets X {f - A \<inter> X | A. A \<in> sets M}"

  2025   unfolding vimage_algebra_def by (rule sets_measure_of) auto

  2026

  2027 lemma sets_vimage_algebra2:

  2028   "f \<in> X \<rightarrow> space M \<Longrightarrow> sets (vimage_algebra X f M) = {f - A \<inter> X | A. A \<in> sets M}"

  2029   using sigma_sets_vimage_commute[of f X "space M" "sets M"]

  2030   unfolding sets_vimage_algebra sets.sigma_sets_eq by simp

  2031

  2032 lemma sets_vimage_algebra_cong: "sets M = sets N \<Longrightarrow> sets (vimage_algebra X f M) = sets (vimage_algebra X f N)"

  2033   by (simp add: sets_vimage_algebra)

  2034

  2035 lemma vimage_algebra_cong:

  2036   assumes "X = Y"

  2037   assumes "\<And>x. x \<in> Y \<Longrightarrow> f x = g x"

  2038   assumes "sets M = sets N"

  2039   shows "vimage_algebra X f M = vimage_algebra Y g N"

  2040   by (auto simp: vimage_algebra_def assms intro!: arg_cong2[where f=sigma])

  2041

  2042 lemma in_vimage_algebra: "A \<in> sets M \<Longrightarrow> f - A \<inter> X \<in> sets (vimage_algebra X f M)"

  2043   by (auto simp: vimage_algebra_def)

  2044

  2045 lemma sets_image_in_sets:

  2046   assumes N: "space N = X"

  2047   assumes f: "f \<in> measurable N M"

  2048   shows "sets (vimage_algebra X f M) \<subseteq> sets N"

  2049   unfolding sets_vimage_algebra N[symmetric]

  2050   by (rule sets.sigma_sets_subset) (auto intro!: measurable_sets f)

  2051

  2052 lemma measurable_vimage_algebra1: "f \<in> X \<rightarrow> space M \<Longrightarrow> f \<in> measurable (vimage_algebra X f M) M"

  2053   unfolding measurable_def by (auto intro: in_vimage_algebra)

  2054

  2055 lemma measurable_vimage_algebra2:

  2056   assumes g: "g \<in> space N \<rightarrow> X" and f: "(\<lambda>x. f (g x)) \<in> measurable N M"

  2057   shows "g \<in> measurable N (vimage_algebra X f M)"

  2058   unfolding vimage_algebra_def

  2059 proof (rule measurable_measure_of)

  2060   fix A assume "A \<in> {f - A \<inter> X | A. A \<in> sets M}"

  2061   then obtain Y where Y: "Y \<in> sets M" and A: "A = f - Y \<inter> X"

  2062     by auto

  2063   then have "g - A \<inter> space N = (\<lambda>x. f (g x)) - Y \<inter> space N"

  2064     using g by auto

  2065   also have "\<dots> \<in> sets N"

  2066     using f Y by (rule measurable_sets)

  2067   finally show "g - A \<inter> space N \<in> sets N" .

  2068 qed (insert g, auto)

  2069

  2070 lemma vimage_algebra_sigma:

  2071   assumes X: "X \<subseteq> Pow \<Omega>'" and f: "f \<in> \<Omega> \<rightarrow> \<Omega>'"

  2072   shows "vimage_algebra \<Omega> f (sigma \<Omega>' X) = sigma \<Omega> {f - A \<inter> \<Omega> | A. A \<in> X }" (is "?V = ?S")

  2073 proof (rule measure_eqI)

  2074   have \<Omega>: "{f - A \<inter> \<Omega> |A. A \<in> X} \<subseteq> Pow \<Omega>" by auto

  2075   show "sets ?V = sets ?S"

  2076     using sigma_sets_vimage_commute[OF f, of X]

  2077     by (simp add: space_measure_of_conv f sets_vimage_algebra2 \<Omega> X)

  2078 qed (simp add: vimage_algebra_def emeasure_sigma)

  2079

  2080 lemma vimage_algebra_vimage_algebra_eq:

  2081   assumes *: "f \<in> X \<rightarrow> Y" "g \<in> Y \<rightarrow> space M"

  2082   shows "vimage_algebra X f (vimage_algebra Y g M) = vimage_algebra X (\<lambda>x. g (f x)) M"

  2083     (is "?VV = ?V")

  2084 proof (rule measure_eqI)

  2085   have "(\<lambda>x. g (f x)) \<in> X \<rightarrow> space M" "\<And>A. A \<inter> f - Y \<inter> X = A \<inter> X"

  2086     using * by auto

  2087   with * show "sets ?VV = sets ?V"

  2088     by (simp add: sets_vimage_algebra2 vimage_comp comp_def flip: ex_simps)

  2089 qed (simp add: vimage_algebra_def emeasure_sigma)

  2090

  2091 subsubsection \<open>Restricted Space Sigma Algebra\<close>

  2092

  2093 definition restrict_space where

  2094   "restrict_space M \<Omega> = measure_of (\<Omega> \<inter> space M) (((\<inter>) \<Omega>)  sets M) (emeasure M)"

  2095

  2096 lemma space_restrict_space: "space (restrict_space M \<Omega>) = \<Omega> \<inter> space M"

  2097   using sets.sets_into_space unfolding restrict_space_def by (subst space_measure_of) auto

  2098

  2099 lemma space_restrict_space2 [simp]: "\<Omega> \<in> sets M \<Longrightarrow> space (restrict_space M \<Omega>) = \<Omega>"

  2100   by (simp add: space_restrict_space sets.sets_into_space)

  2101

  2102 lemma sets_restrict_space: "sets (restrict_space M \<Omega>) = ((\<inter>) \<Omega>)  sets M"

  2103   unfolding restrict_space_def

  2104 proof (subst sets_measure_of)

  2105   show "(\<inter>) \<Omega>  sets M \<subseteq> Pow (\<Omega> \<inter> space M)"

  2106     by (auto dest: sets.sets_into_space)

  2107   have "sigma_sets (\<Omega> \<inter> space M) {((\<lambda>x. x) - X) \<inter> (\<Omega> \<inter> space M) | X. X \<in> sets M} =

  2108     (\<lambda>X. X \<inter> (\<Omega> \<inter> space M))  sets M"

  2109     by (subst sigma_sets_vimage_commute[symmetric, where \<Omega>' = "space M"])

  2110        (auto simp add: sets.sigma_sets_eq)

  2111   moreover have "{((\<lambda>x. x) - X) \<inter> (\<Omega> \<inter> space M) | X. X \<in> sets M} = (\<lambda>X. X \<inter> (\<Omega> \<inter> space M))   sets M"

  2112     by auto

  2113   moreover have "(\<lambda>X. X \<inter> (\<Omega> \<inter> space M))   sets M = ((\<inter>) \<Omega>)  sets M"

  2114     by (intro image_cong) (auto dest: sets.sets_into_space)

  2115   ultimately show "sigma_sets (\<Omega> \<inter> space M) ((\<inter>) \<Omega>  sets M) = (\<inter>) \<Omega>  sets M"

  2116     by simp

  2117 qed

  2118

  2119 lemma restrict_space_sets_cong:

  2120   "A = B \<Longrightarrow> sets M = sets N \<Longrightarrow> sets (restrict_space M A) = sets (restrict_space N B)"

  2121   by (auto simp: sets_restrict_space)

  2122

  2123 lemma sets_restrict_space_count_space :

  2124   "sets (restrict_space (count_space A) B) = sets (count_space (A \<inter> B))"

  2125 by(auto simp add: sets_restrict_space)

  2126

  2127 lemma sets_restrict_UNIV[simp]: "sets (restrict_space M UNIV) = sets M"

  2128   by (auto simp add: sets_restrict_space)

  2129

  2130 lemma sets_restrict_restrict_space:

  2131   "sets (restrict_space (restrict_space M A) B) = sets (restrict_space M (A \<inter> B))"

  2132   unfolding sets_restrict_space image_comp by (intro image_cong) auto

  2133

  2134 lemma sets_restrict_space_iff:

  2135   "\<Omega> \<inter> space M \<in> sets M \<Longrightarrow> A \<in> sets (restrict_space M \<Omega>) \<longleftrightarrow> (A \<subseteq> \<Omega> \<and> A \<in> sets M)"

  2136 proof (subst sets_restrict_space, safe)

  2137   fix A assume "\<Omega> \<inter> space M \<in> sets M" and A: "A \<in> sets M"

  2138   then have "(\<Omega> \<inter> space M) \<inter> A \<in> sets M"

  2139     by rule

  2140   also have "(\<Omega> \<inter> space M) \<inter> A = \<Omega> \<inter> A"

  2141     using sets.sets_into_space[OF A] by auto

  2142   finally show "\<Omega> \<inter> A \<in> sets M"

  2143     by auto

  2144 qed auto

  2145

  2146 lemma sets_restrict_space_cong: "sets M = sets N \<Longrightarrow> sets (restrict_space M \<Omega>) = sets (restrict_space N \<Omega>)"

  2147   by (simp add: sets_restrict_space)

  2148

  2149 lemma restrict_space_eq_vimage_algebra:

  2150   "\<Omega> \<subseteq> space M \<Longrightarrow> sets (restrict_space M \<Omega>) = sets (vimage_algebra \<Omega> (\<lambda>x. x) M)"

  2151   unfolding restrict_space_def

  2152   apply (subst sets_measure_of)

  2153   apply (auto simp add: image_subset_iff dest: sets.sets_into_space) []

  2154   apply (auto simp add: sets_vimage_algebra intro!: arg_cong2[where f=sigma_sets])

  2155   done

  2156

  2157 lemma sets_Collect_restrict_space_iff:

  2158   assumes "S \<in> sets M"

  2159   shows "{x\<in>space (restrict_space M S). P x} \<in> sets (restrict_space M S) \<longleftrightarrow> {x\<in>space M. x \<in> S \<and> P x} \<in> sets M"

  2160 proof -

  2161   have "{x\<in>S. P x} = {x\<in>space M. x \<in> S \<and> P x}"

  2162     using sets.sets_into_space[OF assms] by auto

  2163   then show ?thesis

  2164     by (subst sets_restrict_space_iff) (auto simp add: space_restrict_space assms)

  2165 qed

  2166

  2167 lemma measurable_restrict_space1:

  2168   assumes f: "f \<in> measurable M N"

  2169   shows "f \<in> measurable (restrict_space M \<Omega>) N"

  2170   unfolding measurable_def

  2171 proof (intro CollectI conjI ballI)

  2172   show sp: "f \<in> space (restrict_space M \<Omega>) \<rightarrow> space N"

  2173     using measurable_space[OF f] by (auto simp: space_restrict_space)

  2174

  2175   fix A assume "A \<in> sets N"

  2176   have "f - A \<inter> space (restrict_space M \<Omega>) = (f - A \<inter> space M) \<inter> (\<Omega> \<inter> space M)"

  2177     by (auto simp: space_restrict_space)

  2178   also have "\<dots> \<in> sets (restrict_space M \<Omega>)"

  2179     unfolding sets_restrict_space

  2180     using measurable_sets[OF f \<open>A \<in> sets N\<close>] by blast

  2181   finally show "f - A \<inter> space (restrict_space M \<Omega>) \<in> sets (restrict_space M \<Omega>)" .

  2182 qed

  2183

  2184 lemma measurable_restrict_space2_iff:

  2185   "f \<in> measurable M (restrict_space N \<Omega>) \<longleftrightarrow> (f \<in> measurable M N \<and> f \<in> space M \<rightarrow> \<Omega>)"

  2186 proof -

  2187   have "\<And>A. f \<in> space M \<rightarrow> \<Omega> \<Longrightarrow> f - \<Omega> \<inter> f - A \<inter> space M = f - A \<inter> space M"

  2188     by auto

  2189   then show ?thesis

  2190     by (auto simp: measurable_def space_restrict_space Pi_Int[symmetric] sets_restrict_space)

  2191 qed

  2192

  2193 lemma measurable_restrict_space2:

  2194   "f \<in> space M \<rightarrow> \<Omega> \<Longrightarrow> f \<in> measurable M N \<Longrightarrow> f \<in> measurable M (restrict_space N \<Omega>)"

  2195   by (simp add: measurable_restrict_space2_iff)

  2196

  2197 lemma measurable_piecewise_restrict:

  2198   assumes I: "countable C"

  2199     and X: "\<And>\<Omega>. \<Omega> \<in> C \<Longrightarrow> \<Omega> \<inter> space M \<in> sets M" "space M \<subseteq> \<Union>C"

  2200     and f: "\<And>\<Omega>. \<Omega> \<in> C \<Longrightarrow> f \<in> measurable (restrict_space M \<Omega>) N"

  2201   shows "f \<in> measurable M N"

  2202 proof (rule measurableI)

  2203   fix x assume "x \<in> space M"

  2204   with X obtain \<Omega> where "\<Omega> \<in> C" "x \<in> \<Omega>" "x \<in> space M" by auto

  2205   then show "f x \<in> space N"

  2206     by (auto simp: space_restrict_space intro: f measurable_space)

  2207 next

  2208   fix A assume A: "A \<in> sets N"

  2209   have "f - A \<inter> space M = (\<Union>\<Omega>\<in>C. (f - A \<inter> (\<Omega> \<inter> space M)))"

  2210     using X by (auto simp: subset_eq)

  2211   also have "\<dots> \<in> sets M"

  2212     using measurable_sets[OF f A] X I

  2213     by (intro sets.countable_UN') (auto simp: sets_restrict_space_iff space_restrict_space)

  2214   finally show "f - A \<inter> space M \<in> sets M" .

  2215 qed

  2216

  2217 lemma measurable_piecewise_restrict_iff:

  2218   "countable C \<Longrightarrow> (\<And>\<Omega>. \<Omega> \<in> C \<Longrightarrow> \<Omega> \<inter> space M \<in> sets M) \<Longrightarrow> space M \<subseteq> (\<Union>C) \<Longrightarrow>

  2219     f \<in> measurable M N \<longleftrightarrow> (\<forall>\<Omega>\<in>C. f \<in> measurable (restrict_space M \<Omega>) N)"

  2220   by (auto intro: measurable_piecewise_restrict measurable_restrict_space1)

  2221

  2222 lemma measurable_If_restrict_space_iff:

  2223   "{x\<in>space M. P x} \<in> sets M \<Longrightarrow>

  2224     (\<lambda>x. if P x then f x else g x) \<in> measurable M N \<longleftrightarrow>

  2225     (f \<in> measurable (restrict_space M {x. P x}) N \<and> g \<in> measurable (restrict_space M {x. \<not> P x}) N)"

  2226   by (subst measurable_piecewise_restrict_iff[where C="{{x. P x}, {x. \<not> P x}}"])

  2227      (auto simp: Int_def sets.sets_Collect_neg space_restrict_space conj_commute[of _ "x \<in> space M" for x]

  2228            cong: measurable_cong')

  2229

  2230 lemma measurable_If:

  2231   "f \<in> measurable M M' \<Longrightarrow> g \<in> measurable M M' \<Longrightarrow> {x\<in>space M. P x} \<in> sets M \<Longrightarrow>

  2232     (\<lambda>x. if P x then f x else g x) \<in> measurable M M'"

  2233   unfolding measurable_If_restrict_space_iff by (auto intro: measurable_restrict_space1)

  2234

  2235 lemma measurable_If_set:

  2236   assumes measure: "f \<in> measurable M M'" "g \<in> measurable M M'"

  2237   assumes P: "A \<inter> space M \<in> sets M"

  2238   shows "(\<lambda>x. if x \<in> A then f x else g x) \<in> measurable M M'"

  2239 proof (rule measurable_If[OF measure])

  2240   have "{x \<in> space M. x \<in> A} = A \<inter> space M" by auto

  2241   thus "{x \<in> space M. x \<in> A} \<in> sets M" using \<open>A \<inter> space M \<in> sets M\<close> by auto

  2242 qed

  2243

  2244 lemma measurable_restrict_space_iff:

  2245   "\<Omega> \<inter> space M \<in> sets M \<Longrightarrow> c \<in> space N \<Longrightarrow>

  2246     f \<in> measurable (restrict_space M \<Omega>) N \<longleftrightarrow> (\<lambda>x. if x \<in> \<Omega> then f x else c) \<in> measurable M N"

  2247   by (subst measurable_If_restrict_space_iff)

  2248      (simp_all add: Int_def conj_commute measurable_const)

  2249

  2250 lemma restrict_space_singleton: "{x} \<in> sets M \<Longrightarrow> sets (restrict_space M {x}) = sets (count_space {x})"

  2251   using sets_restrict_space_iff[of "{x}" M]

  2252   by (auto simp add: sets_restrict_space_iff dest!: subset_singletonD)

  2253

  2254 lemma measurable_restrict_countable:

  2255   assumes X[intro]: "countable X"

  2256   assumes sets[simp]: "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"

  2257   assumes space[simp]: "\<And>x. x \<in> X \<Longrightarrow> f x \<in> space N"

  2258   assumes f: "f \<in> measurable (restrict_space M (- X)) N"

  2259   shows "f \<in> measurable M N"

  2260   using f sets.countable[OF sets X]

  2261   by (intro measurable_piecewise_restrict[where M=M and C="{- X} \<union> ((\<lambda>x. {x})  X)"])

  2262      (auto simp: Diff_Int_distrib2 Compl_eq_Diff_UNIV Int_insert_left sets.Diff restrict_space_singleton

  2263            simp del: sets_count_space  cong: measurable_cong_sets)

  2264

  2265 lemma measurable_discrete_difference:

  2266   assumes f: "f \<in> measurable M N"

  2267   assumes X: "countable X" "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M" "\<And>x. x \<in> X \<Longrightarrow> g x \<in> space N"

  2268   assumes eq: "\<And>x. x \<in> space M \<Longrightarrow> x \<notin> X \<Longrightarrow> f x = g x"

  2269   shows "g \<in> measurable M N"

  2270   by (rule measurable_restrict_countable[OF X])

  2271      (auto simp: eq[symmetric] space_restrict_space cong: measurable_cong' intro: f measurable_restrict_space1)

  2272

  2273 lemma measurable_count_space_extend: "A \<subseteq> B \<Longrightarrow> f \<in> space M \<rightarrow> A \<Longrightarrow> f \<in> M \<rightarrow>\<^sub>M count_space B \<Longrightarrow> f \<in> M \<rightarrow>\<^sub>M count_space A"

  2274   by (auto simp: measurable_def)

  2275

  2276 end
`