src/HOL/Probability/Sigma_Algebra.thy
 author hoelzl Fri Feb 19 13:40:50 2016 +0100 (2016-02-19) changeset 62378 85ed00c1fe7c parent 62343 24106dc44def child 62390 842917225d56 permissions -rw-r--r--
generalize more theorems to support enat and ennreal
     1 (*  Title:      HOL/Probability/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>Describing measurable sets\<close>

     9

    10 theory Sigma_Algebra

    11 imports

    12   Complex_Main

    13   "~~/src/HOL/Library/Countable_Set"

    14   "~~/src/HOL/Library/FuncSet"

    15   "~~/src/HOL/Library/Indicator_Function"

    16   "~~/src/HOL/Library/Extended_Real"

    17   "~~/src/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 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 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 locale ring_of_sets = semiring_of_sets +

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

    76

    77 lemma (in ring_of_sets) finite_Union [intro]:

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

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

    80

    81 lemma (in ring_of_sets) finite_UN[intro]:

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

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

    84   using assms by induct auto

    85

    86 lemma (in ring_of_sets) Diff [intro]:

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

    88   using Diff_cover[OF assms] by auto

    89

    90 lemma ring_of_setsI:

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

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

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

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

    95   shows "ring_of_sets \<Omega> M"

    96 proof

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

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

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

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

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

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

   103 qed fact+

   104

   105 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)"

   106 proof

   107   assume "ring_of_sets \<Omega> M"

   108   then interpret ring_of_sets \<Omega> M .

   109   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)"

   110     using space_closed by auto

   111 qed (auto intro!: ring_of_setsI)

   112

   113 lemma (in ring_of_sets) insert_in_sets:

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

   115 proof -

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

   117   thus ?thesis by auto

   118 qed

   119

   120 lemma (in ring_of_sets) sets_Collect_disj:

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

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

   123 proof -

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

   125     by auto

   126   with assms show ?thesis by auto

   127 qed

   128

   129 lemma (in ring_of_sets) sets_Collect_finite_Ex:

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

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

   132 proof -

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

   134     by auto

   135   with assms show ?thesis by auto

   136 qed

   137

   138 locale algebra = ring_of_sets +

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

   140

   141 lemma (in algebra) compl_sets [intro]:

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

   143   by auto

   144

   145 lemma algebra_iff_Un:

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

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

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

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

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

   151 proof

   152   assume "algebra \<Omega> M"

   153   then interpret algebra \<Omega> M .

   154   show ?Un using sets_into_space by auto

   155 next

   156   assume ?Un

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

   158   interpret ring_of_sets \<Omega> M

   159   proof (rule ring_of_setsI)

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

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

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

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

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

   165       using \<Omega> a b by auto

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

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

   168   qed

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

   170 qed

   171

   172 lemma algebra_iff_Int:

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

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

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

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

   177 proof

   178   assume "algebra \<Omega> M"

   179   then interpret algebra \<Omega> M .

   180   show ?Int using sets_into_space by auto

   181 next

   182   assume ?Int

   183   show "algebra \<Omega> M"

   184   proof (unfold algebra_iff_Un, intro conjI ballI)

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

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

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

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

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

   190       using \<Omega> by blast

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

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

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

   194   qed

   195 qed

   196

   197 lemma (in algebra) sets_Collect_neg:

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

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

   200 proof -

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

   202   with assms show ?thesis by auto

   203 qed

   204

   205 lemma (in algebra) sets_Collect_imp:

   206   "{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"

   207   unfolding imp_conv_disj by (intro sets_Collect_disj sets_Collect_neg)

   208

   209 lemma (in algebra) sets_Collect_const:

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

   211   by (cases P) auto

   212

   213 lemma algebra_single_set:

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

   215   by (auto simp: algebra_iff_Int)

   216

   217 subsubsection \<open>Restricted algebras\<close>

   218

   219 abbreviation (in algebra)

   220   "restricted_space A \<equiv> (op \<inter> A)  M"

   221

   222 lemma (in algebra) restricted_algebra:

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

   224   using assms by (auto simp: algebra_iff_Int)

   225

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

   227

   228 locale sigma_algebra = algebra +

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

   230

   231 lemma (in algebra) is_sigma_algebra:

   232   assumes "finite M"

   233   shows "sigma_algebra \<Omega> M"

   234 proof

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

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

   237     by auto

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

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

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

   241 qed

   242

   243 lemma countable_UN_eq:

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

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

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

   247 proof -

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

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

   250   proof safe

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

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

   253   next

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

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

   256   qed

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

   258     using surj_from_nat

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

   260   show ?thesis unfolding * ** ..

   261 qed

   262

   263 lemma (in sigma_algebra) countable_Union [intro]:

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

   265 proof cases

   266   assume "X \<noteq> {}"

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

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

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

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

   271   finally show ?thesis .

   272 qed simp

   273

   274 lemma (in sigma_algebra) countable_UN[intro]:

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

   276   assumes "AX \<subseteq> M"

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

   278 proof -

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

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

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

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

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

   284   ultimately show ?thesis by simp

   285 qed

   286

   287 lemma (in sigma_algebra) countable_UN':

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

   289   assumes X: "countable X"

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

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

   292 proof -

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

   294     using X by auto

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

   296     using A X

   297     by (intro countable_UN) auto

   298   finally show ?thesis .

   299 qed

   300

   301 lemma (in sigma_algebra) countable_UN'':

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

   303 by(erule countable_UN')(auto)

   304

   305 lemma (in sigma_algebra) countable_INT [intro]:

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

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

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

   309 proof -

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

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

   312   moreover

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

   314     by blast

   315   ultimately show ?thesis by metis

   316 qed

   317

   318 lemma (in sigma_algebra) countable_INT':

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

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

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

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

   323 proof -

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

   325     using X by auto

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

   327     using A X

   328     by (intro countable_INT) auto

   329   finally show ?thesis .

   330 qed

   331

   332 lemma (in sigma_algebra) countable_INT'':

   333   "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"

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

   335

   336 lemma (in sigma_algebra) countable:

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

   338   shows "A \<in> M"

   339 proof -

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

   341     using assms by (intro countable_UN') auto

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

   343   finally show ?thesis by auto

   344 qed

   345

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

   347   by (auto simp: ring_of_sets_iff)

   348

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

   350   by (auto simp: algebra_iff_Un)

   351

   352 lemma sigma_algebra_iff:

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

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

   355   by (simp add: sigma_algebra_def sigma_algebra_axioms_def)

   356

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

   358   by (auto simp: sigma_algebra_iff algebra_iff_Int)

   359

   360 lemma (in sigma_algebra) sets_Collect_countable_All:

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

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

   363 proof -

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

   365   with assms show ?thesis by auto

   366 qed

   367

   368 lemma (in sigma_algebra) sets_Collect_countable_Ex:

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

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

   371 proof -

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

   373   with assms show ?thesis by auto

   374 qed

   375

   376 lemma (in sigma_algebra) sets_Collect_countable_Ex':

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

   378   assumes "countable I"

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

   380 proof -

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

   382   with assms show ?thesis

   383     by (auto intro!: countable_UN')

   384 qed

   385

   386 lemma (in sigma_algebra) sets_Collect_countable_All':

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

   388   assumes "countable I"

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

   390 proof -

   391   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

   392   with assms show ?thesis

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

   394 qed

   395

   396 lemma (in sigma_algebra) sets_Collect_countable_Ex1':

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

   398   assumes "countable I"

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

   400 proof -

   401   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)}"

   402     by auto

   403   with assms show ?thesis

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

   405 qed

   406

   407 lemmas (in sigma_algebra) sets_Collect =

   408   sets_Collect_imp sets_Collect_disj sets_Collect_conj sets_Collect_neg sets_Collect_const

   409   sets_Collect_countable_All sets_Collect_countable_Ex sets_Collect_countable_All

   410

   411 lemma (in sigma_algebra) sets_Collect_countable_Ball:

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

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

   414   unfolding Ball_def by (intro sets_Collect assms)

   415

   416 lemma (in sigma_algebra) sets_Collect_countable_Bex:

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

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

   419   unfolding Bex_def by (intro sets_Collect assms)

   420

   421 lemma sigma_algebra_single_set:

   422   assumes "X \<subseteq> S"

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

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

   425

   426 subsubsection \<open>Binary Unions\<close>

   427

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

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

   430

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

   432   by (auto simp add: binary_def)

   433

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

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

   436

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

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

   439

   440 lemma sigma_algebra_iff2:

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

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

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

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

   445   by (auto simp add: range_binary_eq sigma_algebra_def sigma_algebra_axioms_def

   446          algebra_iff_Un Un_range_binary)

   447

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

   449

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

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

   452

   453 inductive_set sigma_sets :: "'a set \<Rightarrow> 'a set set \<Rightarrow> 'a set set"

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

   455   where

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

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

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

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

   460

   461 lemma (in sigma_algebra) sigma_sets_subset:

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

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

   464 proof

   465   fix x

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

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

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

   469 qed

   470

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

   472   by (erule sigma_sets.induct, auto)

   473

   474 lemma sigma_algebra_sigma_sets:

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

   476   by (auto simp add: sigma_algebra_iff2 dest: sigma_sets_into_sp

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

   478

   479 lemma sigma_sets_least_sigma_algebra:

   480   assumes "A \<subseteq> Pow S"

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

   482 proof safe

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

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

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

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

   487 next

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

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

   490      by simp

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

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

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

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

   495 qed

   496

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

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

   499

   500 lemma sigma_sets_Un:

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

   502 apply (simp add: Un_range_binary range_binary_eq)

   503 apply (rule Union, simp add: binary_def)

   504 done

   505

   506 lemma sigma_sets_Inter:

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

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

   509 proof -

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

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

   512     by (rule sigma_sets.Compl)

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

   514     by (rule sigma_sets.Union)

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

   516     by (rule sigma_sets.Compl)

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

   518     by auto

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

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

   521   finally show ?thesis .

   522 qed

   523

   524 lemma sigma_sets_INTER:

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

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

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

   528 proof -

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

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

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

   532     by (rule sigma_sets_Inter [OF Asb])

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

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

   535   finally show ?thesis .

   536 qed

   537

   538 lemma sigma_sets_UNION:

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

   540   apply (cases "B = {}")

   541   apply (simp add: sigma_sets.Empty)

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

   543   apply simp

   544   apply auto

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

   546   done

   547

   548 lemma (in sigma_algebra) sigma_sets_eq:

   549      "sigma_sets \<Omega> M = M"

   550 proof

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

   552     by (metis Set.subsetI sigma_sets.Basic)

   553   next

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

   555     by (metis sigma_sets_subset subset_refl)

   556 qed

   557

   558 lemma sigma_sets_eqI:

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

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

   561   shows "sigma_sets M A = sigma_sets M B"

   562 proof (intro set_eqI iffI)

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

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

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

   566 next

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

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

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

   570 qed

   571

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

   573 proof

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

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

   576 qed

   577

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

   579 proof

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

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

   582 qed

   583

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

   585 proof

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

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

   588 qed

   589

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

   591   by (auto intro: sigma_sets.Basic)

   592

   593 lemma (in sigma_algebra) restriction_in_sets:

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

   595   assumes "S \<in> M"

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

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

   598 proof -

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

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

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

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

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

   604 qed

   605

   606 lemma (in sigma_algebra) restricted_sigma_algebra:

   607   assumes "S \<in> M"

   608   shows "sigma_algebra S (restricted_space S)"

   609   unfolding sigma_algebra_def sigma_algebra_axioms_def

   610 proof safe

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

   612 next

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

   614   from restriction_in_sets[OF assms this[simplified]]

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

   616 qed

   617

   618 lemma sigma_sets_Int:

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

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

   621 proof (intro equalityI subsetI)

   622   fix x assume "x \<in> op \<inter> A  sigma_sets sp st"

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

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

   625   proof (induct arbitrary: x)

   626     case (Compl a)

   627     then show ?case

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

   629   next

   630     case (Union a)

   631     then show ?case

   632       by (auto intro!: sigma_sets.Union

   633                simp add: UN_extend_simps simp del: UN_simps)

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

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

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

   637 next

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

   639   then show "x \<in> op \<inter> A  sigma_sets sp st"

   640   proof induct

   641     case (Compl a)

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

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

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

   645   next

   646     case (Union a)

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

   648       by (auto simp: image_iff Bex_def)

   649     from choice[OF this] guess f ..

   650     then show ?case

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

   652                simp add: image_iff)

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

   654 qed

   655

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

   657 proof (intro set_eqI iffI)

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

   659     by induct blast+

   660 qed (auto intro: sigma_sets.Empty sigma_sets_top)

   661

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

   663 proof (intro set_eqI iffI)

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

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

   666     by induct blast+

   667 next

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

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

   670     by (auto intro: sigma_sets.Empty sigma_sets_top)

   671 qed

   672

   673 lemma sigma_sets_sigma_sets_eq:

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

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

   676

   677 lemma sigma_sets_singleton:

   678   assumes "X \<subseteq> S"

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

   680 proof -

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

   682     by (rule sigma_algebra_single_set) fact

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

   684     by (rule sigma_sets_subseteq) simp

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

   686     using sigma_sets_eq by simp

   687   moreover

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

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

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

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

   692     by (intro antisym) auto

   693   with sigma_sets_eq show ?thesis by simp

   694 qed

   695

   696 lemma restricted_sigma:

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

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

   699     sigma_sets S (algebra.restricted_space M S)"

   700 proof -

   701   from S sigma_sets_into_sp[OF M]

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

   703   from sigma_sets_Int[OF this]

   704   show ?thesis by simp

   705 qed

   706

   707 lemma sigma_sets_vimage_commute:

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

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

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

   711 proof

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

   713   proof clarify

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

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

   716     proof induct

   717       case Empty then show ?case

   718         by (auto intro!: sigma_sets.Empty)

   719     next

   720       case (Compl B)

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

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

   723       with Compl show ?case

   724         by (auto intro!: sigma_sets.Compl)

   725     next

   726       case (Union F)

   727       then show ?case

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

   729                  intro!: sigma_sets.Union)

   730     qed auto

   731   qed

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

   733   proof clarify

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

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

   736     proof induct

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

   738     next

   739       case Empty then show ?case

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

   741     next

   742       case (Compl B)

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

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

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

   746       with A(2) show ?case

   747         by (auto intro: sigma_sets.Compl)

   748     next

   749       case (Union F)

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

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

   752       with A show ?case

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

   754     qed

   755   qed

   756 qed

   757

   758 lemma (in ring_of_sets) UNION_in_sets:

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

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

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

   762 proof (induct n)

   763   case 0 show ?case by simp

   764 next

   765   case (Suc n)

   766   thus ?case

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

   768 qed

   769

   770 lemma (in ring_of_sets) range_disjointed_sets:

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

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

   773 proof (auto simp add: disjointed_def)

   774   fix n

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

   776     by (metis A Diff UNIV_I image_subset_iff)

   777 qed

   778

   779 lemma (in algebra) range_disjointed_sets':

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

   781   using range_disjointed_sets .

   782

   783 lemma sigma_algebra_disjoint_iff:

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

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

   786 proof (auto simp add: sigma_algebra_iff)

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

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

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

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

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

   792          disjoint_family (disjointed A) \<longrightarrow>

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

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

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

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

   797 qed

   798

   799 subsubsection \<open>Ring generated by a semiring\<close>

   800

   801 definition (in semiring_of_sets)

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

   803

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

   805   assumes "a \<in> generated_ring"

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

   807   using assms unfolding generated_ring_def by auto

   808

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

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

   811   shows "a \<in> generated_ring"

   812   using assms unfolding generated_ring_def by auto

   813

   814 lemma (in semiring_of_sets) generated_ringI_Basic:

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

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

   817

   818 lemma (in semiring_of_sets) generated_ring_disjoint_Un[intro]:

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

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

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

   822 proof -

   823   from a guess Ca .. note Ca = this

   824   from b guess Cb .. note Cb = this

   825   show ?thesis

   826   proof

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

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

   829   qed (insert Ca Cb, auto)

   830 qed

   831

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

   833   by (auto simp: generated_ring_def disjoint_def)

   834

   835 lemma (in semiring_of_sets) generated_ring_disjoint_Union:

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

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

   838

   839 lemma (in semiring_of_sets) generated_ring_disjoint_UNION:

   840   "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"

   841   by (intro generated_ring_disjoint_Union) auto

   842

   843 lemma (in semiring_of_sets) generated_ring_Int:

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

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

   846 proof -

   847   from a guess Ca .. note Ca = this

   848   from b guess Cb .. note Cb = this

   849   def C \<equiv> "(\<lambda>(a,b). a \<inter> b) (Ca\<times>Cb)"

   850   show ?thesis

   851   proof

   852     show "disjoint C"

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

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

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

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

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

   858       proof

   859         assume "a1 \<noteq> a2"

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

   861           by (auto simp: disjoint_def)

   862         then show ?thesis by auto

   863       next

   864         assume "b1 \<noteq> b2"

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

   866           by (auto simp: disjoint_def)

   867         then show ?thesis by auto

   868       qed

   869     qed

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

   871 qed

   872

   873 lemma (in semiring_of_sets) generated_ring_Inter:

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

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

   876

   877 lemma (in semiring_of_sets) generated_ring_INTER:

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

   879   by (intro generated_ring_Inter) auto

   880

   881 lemma (in semiring_of_sets) generating_ring:

   882   "ring_of_sets \<Omega> generated_ring"

   883 proof (rule ring_of_setsI)

   884   let ?R = generated_ring

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

   886     using sets_into_space by (auto simp: generated_ring_def generated_ring_empty)

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

   888

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

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

   891

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

   893     proof cases

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

   895         by simp

   896     next

   897       assume "Cb \<noteq> {}"

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

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

   900       proof (intro generated_ring_INTER generated_ring_disjoint_UNION)

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

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

   903           by (auto simp add: generated_ring_def)

   904             (metis DiffI Diff_eq_empty_iff empty_iff)

   905       next

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

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

   908       next

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

   910       qed

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

   912     qed }

   913   note Diff = this

   914

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

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

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

   918     by (intro sets generated_ring_disjoint_Un generated_ring_Int Diff) auto

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

   920 qed

   921

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

   923 proof

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

   925     using space_closed by (rule sigma_algebra_sigma_sets)

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

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

   928 qed (auto intro!: generated_ringI_Basic sigma_sets_mono)

   929

   930 subsubsection \<open>A Two-Element Series\<close>

   931

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

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

   934

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

   936   apply (simp add: binaryset_def)

   937   apply (rule set_eqI)

   938   apply (auto simp add: image_iff)

   939   done

   940

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

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

   943

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

   945

   946 definition closed_cdi where

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

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

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

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

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

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

   953

   954 inductive_set

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

   956   for \<Omega> M

   957   where

   958     Basic [intro]:

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

   960   | Compl [intro]:

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

   962   | Inc:

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

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

   965   | Disj:

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

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

   968

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

   970   by auto

   971

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

   973   apply (rule subsetI)

   974   apply (erule smallest_ccdi_sets.induct)

   975   apply (auto intro: range_subsetD dest: sets_into_space)

   976   done

   977

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

   979   apply (auto simp add: closed_cdi_def smallest_ccdi_sets)

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

   981   done

   982

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

   984   by (simp add: closed_cdi_def)

   985

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

   987   by (simp add: closed_cdi_def)

   988

   989 lemma closed_cdi_Inc:

   990   "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"

   991   by (simp add: closed_cdi_def)

   992

   993 lemma closed_cdi_Disj:

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

   995   by (simp add: closed_cdi_def)

   996

   997 lemma closed_cdi_Un:

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

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

  1000       and disj: "A \<inter> B = {}"

  1001     shows "A \<union> B \<in> M"

  1002 proof -

  1003   have ra: "range (binaryset A B) \<subseteq> M"

  1004    by (simp add: range_binaryset_eq empty A B)

  1005  have di:  "disjoint_family (binaryset A B)" using disj

  1006    by (simp add: disjoint_family_on_def binaryset_def Int_commute)

  1007  from closed_cdi_Disj [OF cdi ra di]

  1008  show ?thesis

  1009    by (simp add: UN_binaryset_eq)

  1010 qed

  1011

  1012 lemma (in algebra) smallest_ccdi_sets_Un:

  1013   assumes A: "A \<in> smallest_ccdi_sets \<Omega> M" and B: "B \<in> smallest_ccdi_sets \<Omega> M"

  1014       and disj: "A \<inter> B = {}"

  1015     shows "A \<union> B \<in> smallest_ccdi_sets \<Omega> M"

  1016 proof -

  1017   have ra: "range (binaryset A B) \<in> Pow (smallest_ccdi_sets \<Omega> M)"

  1018     by (simp add: range_binaryset_eq  A B smallest_ccdi_sets.Basic)

  1019   have di:  "disjoint_family (binaryset A B)" using disj

  1020     by (simp add: disjoint_family_on_def binaryset_def Int_commute)

  1021   from Disj [OF ra di]

  1022   show ?thesis

  1023     by (simp add: UN_binaryset_eq)

  1024 qed

  1025

  1026 lemma (in algebra) smallest_ccdi_sets_Int1:

  1027   assumes a: "a \<in> M"

  1028   shows "b \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets \<Omega> M"

  1029 proof (induct rule: smallest_ccdi_sets.induct)

  1030   case (Basic x)

  1031   thus ?case

  1032     by (metis a Int smallest_ccdi_sets.Basic)

  1033 next

  1034   case (Compl x)

  1035   have "a \<inter> (\<Omega> - x) = \<Omega> - ((\<Omega> - a) \<union> (a \<inter> x))"

  1036     by blast

  1037   also have "... \<in> smallest_ccdi_sets \<Omega> M"

  1038     by (metis smallest_ccdi_sets.Compl a Compl(2) Diff_Int2 Diff_Int_distrib2

  1039            Diff_disjoint Int_Diff Int_empty_right smallest_ccdi_sets_Un

  1040            smallest_ccdi_sets.Basic smallest_ccdi_sets.Compl)

  1041   finally show ?case .

  1042 next

  1043   case (Inc A)

  1044   have 1: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) = a \<inter> (\<Union>i. A i)"

  1045     by blast

  1046   have "range (\<lambda>i. a \<inter> A i) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Inc

  1047     by blast

  1048   moreover have "(\<lambda>i. a \<inter> A i) 0 = {}"

  1049     by (simp add: Inc)

  1050   moreover have "!!n. (\<lambda>i. a \<inter> A i) n \<subseteq> (\<lambda>i. a \<inter> A i) (Suc n)" using Inc

  1051     by blast

  1052   ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets \<Omega> M"

  1053     by (rule smallest_ccdi_sets.Inc)

  1054   show ?case

  1055     by (metis 1 2)

  1056 next

  1057   case (Disj A)

  1058   have 1: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) = a \<inter> (\<Union>i. A i)"

  1059     by blast

  1060   have "range (\<lambda>i. a \<inter> A i) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Disj

  1061     by blast

  1062   moreover have "disjoint_family (\<lambda>i. a \<inter> A i)" using Disj

  1063     by (auto simp add: disjoint_family_on_def)

  1064   ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets \<Omega> M"

  1065     by (rule smallest_ccdi_sets.Disj)

  1066   show ?case

  1067     by (metis 1 2)

  1068 qed

  1069

  1070

  1071 lemma (in algebra) smallest_ccdi_sets_Int:

  1072   assumes b: "b \<in> smallest_ccdi_sets \<Omega> M"

  1073   shows "a \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets \<Omega> M"

  1074 proof (induct rule: smallest_ccdi_sets.induct)

  1075   case (Basic x)

  1076   thus ?case

  1077     by (metis b smallest_ccdi_sets_Int1)

  1078 next

  1079   case (Compl x)

  1080   have "(\<Omega> - x) \<inter> b = \<Omega> - (x \<inter> b \<union> (\<Omega> - b))"

  1081     by blast

  1082   also have "... \<in> smallest_ccdi_sets \<Omega> M"

  1083     by (metis Compl(2) Diff_disjoint Int_Diff Int_commute Int_empty_right b

  1084            smallest_ccdi_sets.Compl smallest_ccdi_sets_Un)

  1085   finally show ?case .

  1086 next

  1087   case (Inc A)

  1088   have 1: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) = (\<Union>i. A i) \<inter> b"

  1089     by blast

  1090   have "range (\<lambda>i. A i \<inter> b) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Inc

  1091     by blast

  1092   moreover have "(\<lambda>i. A i \<inter> b) 0 = {}"

  1093     by (simp add: Inc)

  1094   moreover have "!!n. (\<lambda>i. A i \<inter> b) n \<subseteq> (\<lambda>i. A i \<inter> b) (Suc n)" using Inc

  1095     by blast

  1096   ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets \<Omega> M"

  1097     by (rule smallest_ccdi_sets.Inc)

  1098   show ?case

  1099     by (metis 1 2)

  1100 next

  1101   case (Disj A)

  1102   have 1: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) = (\<Union>i. A i) \<inter> b"

  1103     by blast

  1104   have "range (\<lambda>i. A i \<inter> b) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Disj

  1105     by blast

  1106   moreover have "disjoint_family (\<lambda>i. A i \<inter> b)" using Disj

  1107     by (auto simp add: disjoint_family_on_def)

  1108   ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets \<Omega> M"

  1109     by (rule smallest_ccdi_sets.Disj)

  1110   show ?case

  1111     by (metis 1 2)

  1112 qed

  1113

  1114 lemma (in algebra) sigma_property_disjoint_lemma:

  1115   assumes sbC: "M \<subseteq> C"

  1116       and ccdi: "closed_cdi \<Omega> C"

  1117   shows "sigma_sets \<Omega> M \<subseteq> C"

  1118 proof -

  1119   have "smallest_ccdi_sets \<Omega> M \<in> {B . M \<subseteq> B \<and> sigma_algebra \<Omega> B}"

  1120     apply (auto simp add: sigma_algebra_disjoint_iff algebra_iff_Int

  1121             smallest_ccdi_sets_Int)

  1122     apply (metis Union_Pow_eq Union_upper subsetD smallest_ccdi_sets)

  1123     apply (blast intro: smallest_ccdi_sets.Disj)

  1124     done

  1125   hence "sigma_sets (\<Omega>) (M) \<subseteq> smallest_ccdi_sets \<Omega> M"

  1126     by clarsimp

  1127        (drule sigma_algebra.sigma_sets_subset [where a="M"], auto)

  1128   also have "...  \<subseteq> C"

  1129     proof

  1130       fix x

  1131       assume x: "x \<in> smallest_ccdi_sets \<Omega> M"

  1132       thus "x \<in> C"

  1133         proof (induct rule: smallest_ccdi_sets.induct)

  1134           case (Basic x)

  1135           thus ?case

  1136             by (metis Basic subsetD sbC)

  1137         next

  1138           case (Compl x)

  1139           thus ?case

  1140             by (blast intro: closed_cdi_Compl [OF ccdi, simplified])

  1141         next

  1142           case (Inc A)

  1143           thus ?case

  1144                by (auto intro: closed_cdi_Inc [OF ccdi, simplified])

  1145         next

  1146           case (Disj A)

  1147           thus ?case

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

  1149         qed

  1150     qed

  1151   finally show ?thesis .

  1152 qed

  1153

  1154 lemma (in algebra) sigma_property_disjoint:

  1155   assumes sbC: "M \<subseteq> C"

  1156       and compl: "!!s. s \<in> C \<inter> sigma_sets (\<Omega>) (M) \<Longrightarrow> \<Omega> - s \<in> C"

  1157       and inc: "!!A. range A \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)

  1158                      \<Longrightarrow> A 0 = {} \<Longrightarrow> (!!n. A n \<subseteq> A (Suc n))

  1159                      \<Longrightarrow> (\<Union>i. A i) \<in> C"

  1160       and disj: "!!A. range A \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)

  1161                       \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> C"

  1162   shows "sigma_sets (\<Omega>) (M) \<subseteq> C"

  1163 proof -

  1164   have "sigma_sets (\<Omega>) (M) \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)"

  1165     proof (rule sigma_property_disjoint_lemma)

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

  1167         by (metis Int_greatest Set.subsetI sbC sigma_sets.Basic)

  1168     next

  1169       show "closed_cdi \<Omega> (C \<inter> sigma_sets (\<Omega>) (M))"

  1170         by (simp add: closed_cdi_def compl inc disj)

  1171            (metis PowI Set.subsetI le_infI2 sigma_sets_into_sp space_closed

  1172              IntE sigma_sets.Compl range_subsetD sigma_sets.Union)

  1173     qed

  1174   thus ?thesis

  1175     by blast

  1176 qed

  1177

  1178 subsubsection \<open>Dynkin systems\<close>

  1179

  1180 locale dynkin_system = subset_class +

  1181   assumes space: "\<Omega> \<in> M"

  1182     and   compl[intro!]: "\<And>A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"

  1183     and   UN[intro!]: "\<And>A. disjoint_family A \<Longrightarrow> range A \<subseteq> M

  1184                            \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"

  1185

  1186 lemma (in dynkin_system) empty[intro, simp]: "{} \<in> M"

  1187   using space compl[of "\<Omega>"] by simp

  1188

  1189 lemma (in dynkin_system) diff:

  1190   assumes sets: "D \<in> M" "E \<in> M" and "D \<subseteq> E"

  1191   shows "E - D \<in> M"

  1192 proof -

  1193   let ?f = "\<lambda>x. if x = 0 then D else if x = Suc 0 then \<Omega> - E else {}"

  1194   have "range ?f = {D, \<Omega> - E, {}}"

  1195     by (auto simp: image_iff)

  1196   moreover have "D \<union> (\<Omega> - E) = (\<Union>i. ?f i)"

  1197     by (auto simp: image_iff split: split_if_asm)

  1198   moreover

  1199   have "disjoint_family ?f" unfolding disjoint_family_on_def

  1200     using \<open>D \<in> M\<close>[THEN sets_into_space] \<open>D \<subseteq> E\<close> by auto

  1201   ultimately have "\<Omega> - (D \<union> (\<Omega> - E)) \<in> M"

  1202     using sets by auto

  1203   also have "\<Omega> - (D \<union> (\<Omega> - E)) = E - D"

  1204     using assms sets_into_space by auto

  1205   finally show ?thesis .

  1206 qed

  1207

  1208 lemma dynkin_systemI:

  1209   assumes "\<And> A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>" "\<Omega> \<in> M"

  1210   assumes "\<And> A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"

  1211   assumes "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> M

  1212           \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"

  1213   shows "dynkin_system \<Omega> M"

  1214   using assms by (auto simp: dynkin_system_def dynkin_system_axioms_def subset_class_def)

  1215

  1216 lemma dynkin_systemI':

  1217   assumes 1: "\<And> A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>"

  1218   assumes empty: "{} \<in> M"

  1219   assumes Diff: "\<And> A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"

  1220   assumes 2: "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> M

  1221           \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"

  1222   shows "dynkin_system \<Omega> M"

  1223 proof -

  1224   from Diff[OF empty] have "\<Omega> \<in> M" by auto

  1225   from 1 this Diff 2 show ?thesis

  1226     by (intro dynkin_systemI) auto

  1227 qed

  1228

  1229 lemma dynkin_system_trivial:

  1230   shows "dynkin_system A (Pow A)"

  1231   by (rule dynkin_systemI) auto

  1232

  1233 lemma sigma_algebra_imp_dynkin_system:

  1234   assumes "sigma_algebra \<Omega> M" shows "dynkin_system \<Omega> M"

  1235 proof -

  1236   interpret sigma_algebra \<Omega> M by fact

  1237   show ?thesis using sets_into_space by (fastforce intro!: dynkin_systemI)

  1238 qed

  1239

  1240 subsubsection "Intersection sets systems"

  1241

  1242 definition "Int_stable M \<longleftrightarrow> (\<forall> a \<in> M. \<forall> b \<in> M. a \<inter> b \<in> M)"

  1243

  1244 lemma (in algebra) Int_stable: "Int_stable M"

  1245   unfolding Int_stable_def by auto

  1246

  1247 lemma Int_stableI:

  1248   "(\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<inter> b \<in> A) \<Longrightarrow> Int_stable A"

  1249   unfolding Int_stable_def by auto

  1250

  1251 lemma Int_stableD:

  1252   "Int_stable M \<Longrightarrow> a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<inter> b \<in> M"

  1253   unfolding Int_stable_def by auto

  1254

  1255 lemma (in dynkin_system) sigma_algebra_eq_Int_stable:

  1256   "sigma_algebra \<Omega> M \<longleftrightarrow> Int_stable M"

  1257 proof

  1258   assume "sigma_algebra \<Omega> M" then show "Int_stable M"

  1259     unfolding sigma_algebra_def using algebra.Int_stable by auto

  1260 next

  1261   assume "Int_stable M"

  1262   show "sigma_algebra \<Omega> M"

  1263     unfolding sigma_algebra_disjoint_iff algebra_iff_Un

  1264   proof (intro conjI ballI allI impI)

  1265     show "M \<subseteq> Pow (\<Omega>)" using sets_into_space by auto

  1266   next

  1267     fix A B assume "A \<in> M" "B \<in> M"

  1268     then have "A \<union> B = \<Omega> - ((\<Omega> - A) \<inter> (\<Omega> - B))"

  1269               "\<Omega> - A \<in> M" "\<Omega> - B \<in> M"

  1270       using sets_into_space by auto

  1271     then show "A \<union> B \<in> M"

  1272       using \<open>Int_stable M\<close> unfolding Int_stable_def by auto

  1273   qed auto

  1274 qed

  1275

  1276 subsubsection "Smallest Dynkin systems"

  1277

  1278 definition dynkin where

  1279   "dynkin \<Omega> M =  (\<Inter>{D. dynkin_system \<Omega> D \<and> M \<subseteq> D})"

  1280

  1281 lemma dynkin_system_dynkin:

  1282   assumes "M \<subseteq> Pow (\<Omega>)"

  1283   shows "dynkin_system \<Omega> (dynkin \<Omega> M)"

  1284 proof (rule dynkin_systemI)

  1285   fix A assume "A \<in> dynkin \<Omega> M"

  1286   moreover

  1287   { fix D assume "A \<in> D" and d: "dynkin_system \<Omega> D"

  1288     then have "A \<subseteq> \<Omega>" by (auto simp: dynkin_system_def subset_class_def) }

  1289   moreover have "{D. dynkin_system \<Omega> D \<and> M \<subseteq> D} \<noteq> {}"

  1290     using assms dynkin_system_trivial by fastforce

  1291   ultimately show "A \<subseteq> \<Omega>"

  1292     unfolding dynkin_def using assms

  1293     by auto

  1294 next

  1295   show "\<Omega> \<in> dynkin \<Omega> M"

  1296     unfolding dynkin_def using dynkin_system.space by fastforce

  1297 next

  1298   fix A assume "A \<in> dynkin \<Omega> M"

  1299   then show "\<Omega> - A \<in> dynkin \<Omega> M"

  1300     unfolding dynkin_def using dynkin_system.compl by force

  1301 next

  1302   fix A :: "nat \<Rightarrow> 'a set"

  1303   assume A: "disjoint_family A" "range A \<subseteq> dynkin \<Omega> M"

  1304   show "(\<Union>i. A i) \<in> dynkin \<Omega> M" unfolding dynkin_def

  1305   proof (simp, safe)

  1306     fix D assume "dynkin_system \<Omega> D" "M \<subseteq> D"

  1307     with A have "(\<Union>i. A i) \<in> D"

  1308       by (intro dynkin_system.UN) (auto simp: dynkin_def)

  1309     then show "(\<Union>i. A i) \<in> D" by auto

  1310   qed

  1311 qed

  1312

  1313 lemma dynkin_Basic[intro]: "A \<in> M \<Longrightarrow> A \<in> dynkin \<Omega> M"

  1314   unfolding dynkin_def by auto

  1315

  1316 lemma (in dynkin_system) restricted_dynkin_system:

  1317   assumes "D \<in> M"

  1318   shows "dynkin_system \<Omega> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}"

  1319 proof (rule dynkin_systemI, simp_all)

  1320   have "\<Omega> \<inter> D = D"

  1321     using \<open>D \<in> M\<close> sets_into_space by auto

  1322   then show "\<Omega> \<inter> D \<in> M"

  1323     using \<open>D \<in> M\<close> by auto

  1324 next

  1325   fix A assume "A \<subseteq> \<Omega> \<and> A \<inter> D \<in> M"

  1326   moreover have "(\<Omega> - A) \<inter> D = (\<Omega> - (A \<inter> D)) - (\<Omega> - D)"

  1327     by auto

  1328   ultimately show "\<Omega> - A \<subseteq> \<Omega> \<and> (\<Omega> - A) \<inter> D \<in> M"

  1329     using  \<open>D \<in> M\<close> by (auto intro: diff)

  1330 next

  1331   fix A :: "nat \<Rightarrow> 'a set"

  1332   assume "disjoint_family A" "range A \<subseteq> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}"

  1333   then have "\<And>i. A i \<subseteq> \<Omega>" "disjoint_family (\<lambda>i. A i \<inter> D)"

  1334     "range (\<lambda>i. A i \<inter> D) \<subseteq> M" "(\<Union>x. A x) \<inter> D = (\<Union>x. A x \<inter> D)"

  1335     by ((fastforce simp: disjoint_family_on_def)+)

  1336   then show "(\<Union>x. A x) \<subseteq> \<Omega> \<and> (\<Union>x. A x) \<inter> D \<in> M"

  1337     by (auto simp del: UN_simps)

  1338 qed

  1339

  1340 lemma (in dynkin_system) dynkin_subset:

  1341   assumes "N \<subseteq> M"

  1342   shows "dynkin \<Omega> N \<subseteq> M"

  1343 proof -

  1344   have "dynkin_system \<Omega> M" ..

  1345   then have "dynkin_system \<Omega> M"

  1346     using assms unfolding dynkin_system_def dynkin_system_axioms_def subset_class_def by simp

  1347   with \<open>N \<subseteq> M\<close> show ?thesis by (auto simp add: dynkin_def)

  1348 qed

  1349

  1350 lemma sigma_eq_dynkin:

  1351   assumes sets: "M \<subseteq> Pow \<Omega>"

  1352   assumes "Int_stable M"

  1353   shows "sigma_sets \<Omega> M = dynkin \<Omega> M"

  1354 proof -

  1355   have "dynkin \<Omega> M \<subseteq> sigma_sets (\<Omega>) (M)"

  1356     using sigma_algebra_imp_dynkin_system

  1357     unfolding dynkin_def sigma_sets_least_sigma_algebra[OF sets] by auto

  1358   moreover

  1359   interpret dynkin_system \<Omega> "dynkin \<Omega> M"

  1360     using dynkin_system_dynkin[OF sets] .

  1361   have "sigma_algebra \<Omega> (dynkin \<Omega> M)"

  1362     unfolding sigma_algebra_eq_Int_stable Int_stable_def

  1363   proof (intro ballI)

  1364     fix A B assume "A \<in> dynkin \<Omega> M" "B \<in> dynkin \<Omega> M"

  1365     let ?D = "\<lambda>E. {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> E \<in> dynkin \<Omega> M}"

  1366     have "M \<subseteq> ?D B"

  1367     proof

  1368       fix E assume "E \<in> M"

  1369       then have "M \<subseteq> ?D E" "E \<in> dynkin \<Omega> M"

  1370         using sets_into_space \<open>Int_stable M\<close> by (auto simp: Int_stable_def)

  1371       then have "dynkin \<Omega> M \<subseteq> ?D E"

  1372         using restricted_dynkin_system \<open>E \<in> dynkin \<Omega> M\<close>

  1373         by (intro dynkin_system.dynkin_subset) simp_all

  1374       then have "B \<in> ?D E"

  1375         using \<open>B \<in> dynkin \<Omega> M\<close> by auto

  1376       then have "E \<inter> B \<in> dynkin \<Omega> M"

  1377         by (subst Int_commute) simp

  1378       then show "E \<in> ?D B"

  1379         using sets \<open>E \<in> M\<close> by auto

  1380     qed

  1381     then have "dynkin \<Omega> M \<subseteq> ?D B"

  1382       using restricted_dynkin_system \<open>B \<in> dynkin \<Omega> M\<close>

  1383       by (intro dynkin_system.dynkin_subset) simp_all

  1384     then show "A \<inter> B \<in> dynkin \<Omega> M"

  1385       using \<open>A \<in> dynkin \<Omega> M\<close> sets_into_space by auto

  1386   qed

  1387   from sigma_algebra.sigma_sets_subset[OF this, of "M"]

  1388   have "sigma_sets (\<Omega>) (M) \<subseteq> dynkin \<Omega> M" by auto

  1389   ultimately have "sigma_sets (\<Omega>) (M) = dynkin \<Omega> M" by auto

  1390   then show ?thesis

  1391     by (auto simp: dynkin_def)

  1392 qed

  1393

  1394 lemma (in dynkin_system) dynkin_idem:

  1395   "dynkin \<Omega> M = M"

  1396 proof -

  1397   have "dynkin \<Omega> M = M"

  1398   proof

  1399     show "M \<subseteq> dynkin \<Omega> M"

  1400       using dynkin_Basic by auto

  1401     show "dynkin \<Omega> M \<subseteq> M"

  1402       by (intro dynkin_subset) auto

  1403   qed

  1404   then show ?thesis

  1405     by (auto simp: dynkin_def)

  1406 qed

  1407

  1408 lemma (in dynkin_system) dynkin_lemma:

  1409   assumes "Int_stable E"

  1410   and E: "E \<subseteq> M" "M \<subseteq> sigma_sets \<Omega> E"

  1411   shows "sigma_sets \<Omega> E = M"

  1412 proof -

  1413   have "E \<subseteq> Pow \<Omega>"

  1414     using E sets_into_space by force

  1415   then have *: "sigma_sets \<Omega> E = dynkin \<Omega> E"

  1416     using \<open>Int_stable E\<close> by (rule sigma_eq_dynkin)

  1417   then have "dynkin \<Omega> E = M"

  1418     using assms dynkin_subset[OF E(1)] by simp

  1419   with * show ?thesis

  1420     using assms by (auto simp: dynkin_def)

  1421 qed

  1422

  1423 subsubsection \<open>Induction rule for intersection-stable generators\<close>

  1424

  1425 text \<open>The reason to introduce Dynkin-systems is the following induction rules for $\sigma$-algebras

  1426 generated by a generator closed under intersection.\<close>

  1427

  1428 lemma sigma_sets_induct_disjoint[consumes 3, case_names basic empty compl union]:

  1429   assumes "Int_stable G"

  1430     and closed: "G \<subseteq> Pow \<Omega>"

  1431     and A: "A \<in> sigma_sets \<Omega> G"

  1432   assumes basic: "\<And>A. A \<in> G \<Longrightarrow> P A"

  1433     and empty: "P {}"

  1434     and compl: "\<And>A. A \<in> sigma_sets \<Omega> G \<Longrightarrow> P A \<Longrightarrow> P (\<Omega> - A)"

  1435     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)"

  1436   shows "P A"

  1437 proof -

  1438   let ?D = "{ A \<in> sigma_sets \<Omega> G. P A }"

  1439   interpret sigma_algebra \<Omega> "sigma_sets \<Omega> G"

  1440     using closed by (rule sigma_algebra_sigma_sets)

  1441   from compl[OF _ empty] closed have space: "P \<Omega>" by simp

  1442   interpret dynkin_system \<Omega> ?D

  1443     by standard (auto dest: sets_into_space intro!: space compl union)

  1444   have "sigma_sets \<Omega> G = ?D"

  1445     by (rule dynkin_lemma) (auto simp: basic \<open>Int_stable G\<close>)

  1446   with A show ?thesis by auto

  1447 qed

  1448

  1449 subsection \<open>Measure type\<close>

  1450

  1451 definition positive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where

  1452   "positive M \<mu> \<longleftrightarrow> \<mu> {} = 0 \<and> (\<forall>A\<in>M. 0 \<le> \<mu> A)"

  1453

  1454 definition countably_additive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where

  1455   "countably_additive M f \<longleftrightarrow> (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow>

  1456     (\<Sum>i. f (A i)) = f (\<Union>i. A i))"

  1457

  1458 definition measure_space :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where

  1459   "measure_space \<Omega> A \<mu> \<longleftrightarrow> sigma_algebra \<Omega> A \<and> positive A \<mu> \<and> countably_additive A \<mu>"

  1460

  1461 typedef 'a measure = "{(\<Omega>::'a set, A, \<mu>). (\<forall>a\<in>-A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu> }"

  1462 proof

  1463   have "sigma_algebra UNIV {{}, UNIV}"

  1464     by (auto simp: sigma_algebra_iff2)

  1465   then show "(UNIV, {{}, UNIV}, \<lambda>A. 0) \<in> {(\<Omega>, A, \<mu>). (\<forall>a\<in>-A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu>} "

  1466     by (auto simp: measure_space_def positive_def countably_additive_def)

  1467 qed

  1468

  1469 definition space :: "'a measure \<Rightarrow> 'a set" where

  1470   "space M = fst (Rep_measure M)"

  1471

  1472 definition sets :: "'a measure \<Rightarrow> 'a set set" where

  1473   "sets M = fst (snd (Rep_measure M))"

  1474

  1475 definition emeasure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> ereal" where

  1476   "emeasure M = snd (snd (Rep_measure M))"

  1477

  1478 definition measure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> real" where

  1479   "measure M A = real_of_ereal (emeasure M A)"

  1480

  1481 declare [[coercion sets]]

  1482

  1483 declare [[coercion measure]]

  1484

  1485 declare [[coercion emeasure]]

  1486

  1487 lemma measure_space: "measure_space (space M) (sets M) (emeasure M)"

  1488   by (cases M) (auto simp: space_def sets_def emeasure_def Abs_measure_inverse)

  1489

  1490 interpretation sets: sigma_algebra "space M" "sets M" for M :: "'a measure"

  1491   using measure_space[of M] by (auto simp: measure_space_def)

  1492

  1493 definition measure_of :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> 'a measure" where

  1494   "measure_of \<Omega> A \<mu> = Abs_measure (\<Omega>, if A \<subseteq> Pow \<Omega> then sigma_sets \<Omega> A else {{}, \<Omega>},

  1495     \<lambda>a. if a \<in> sigma_sets \<Omega> A \<and> measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> then \<mu> a else 0)"

  1496

  1497 abbreviation "sigma \<Omega> A \<equiv> measure_of \<Omega> A (\<lambda>x. 0)"

  1498

  1499 lemma measure_space_0: "A \<subseteq> Pow \<Omega> \<Longrightarrow> measure_space \<Omega> (sigma_sets \<Omega> A) (\<lambda>x. 0)"

  1500   unfolding measure_space_def

  1501   by (auto intro!: sigma_algebra_sigma_sets simp: positive_def countably_additive_def)

  1502

  1503 lemma sigma_algebra_trivial: "sigma_algebra \<Omega> {{}, \<Omega>}"

  1504 by unfold_locales(fastforce intro: exI[where x="{{}}"] exI[where x="{\<Omega>}"])+

  1505

  1506 lemma measure_space_0': "measure_space \<Omega> {{}, \<Omega>} (\<lambda>x. 0)"

  1507 by(simp add: measure_space_def positive_def countably_additive_def sigma_algebra_trivial)

  1508

  1509 lemma measure_space_closed:

  1510   assumes "measure_space \<Omega> M \<mu>"

  1511   shows "M \<subseteq> Pow \<Omega>"

  1512 proof -

  1513   interpret sigma_algebra \<Omega> M using assms by(simp add: measure_space_def)

  1514   show ?thesis by(rule space_closed)

  1515 qed

  1516

  1517 lemma (in ring_of_sets) positive_cong_eq:

  1518   "(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> positive M \<mu>' = positive M \<mu>"

  1519   by (auto simp add: positive_def)

  1520

  1521 lemma (in sigma_algebra) countably_additive_eq:

  1522   "(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> countably_additive M \<mu>' = countably_additive M \<mu>"

  1523   unfolding countably_additive_def

  1524   by (intro arg_cong[where f=All] ext) (auto simp add: countably_additive_def subset_eq)

  1525

  1526 lemma measure_space_eq:

  1527   assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a"

  1528   shows "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'"

  1529 proof -

  1530   interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" using closed by (rule sigma_algebra_sigma_sets)

  1531   from positive_cong_eq[OF eq, of "\<lambda>i. i"] countably_additive_eq[OF eq, of "\<lambda>i. i"] show ?thesis

  1532     by (auto simp: measure_space_def)

  1533 qed

  1534

  1535 lemma measure_of_eq:

  1536   assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "(\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a)"

  1537   shows "measure_of \<Omega> A \<mu> = measure_of \<Omega> A \<mu>'"

  1538 proof -

  1539   have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'"

  1540     using assms by (rule measure_space_eq)

  1541   with eq show ?thesis

  1542     by (auto simp add: measure_of_def intro!: arg_cong[where f=Abs_measure])

  1543 qed

  1544

  1545 lemma

  1546   shows space_measure_of_conv: "space (measure_of \<Omega> A \<mu>) = \<Omega>" (is ?space)

  1547   and sets_measure_of_conv:

  1548   "sets (measure_of \<Omega> A \<mu>) = (if A \<subseteq> Pow \<Omega> then sigma_sets \<Omega> A else {{}, \<Omega>})" (is ?sets)

  1549   and emeasure_measure_of_conv:

  1550   "emeasure (measure_of \<Omega> A \<mu>) =

  1551   (\<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)

  1552 proof -

  1553   have "?space \<and> ?sets \<and> ?emeasure"

  1554   proof(cases "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>")

  1555     case True

  1556     from measure_space_closed[OF this] sigma_sets_superset_generator[of A \<Omega>]

  1557     have "A \<subseteq> Pow \<Omega>" by simp

  1558     hence "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A)

  1559       (\<lambda>a. if a \<in> sigma_sets \<Omega> A then \<mu> a else 0)"

  1560       by(rule measure_space_eq) auto

  1561     with True \<open>A \<subseteq> Pow \<Omega>\<close> show ?thesis

  1562       by(simp add: measure_of_def space_def sets_def emeasure_def Abs_measure_inverse)

  1563   next

  1564     case False thus ?thesis

  1565       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')

  1566   qed

  1567   thus ?space ?sets ?emeasure by simp_all

  1568 qed

  1569

  1570 lemma [simp]:

  1571   assumes A: "A \<subseteq> Pow \<Omega>"

  1572   shows sets_measure_of: "sets (measure_of \<Omega> A \<mu>) = sigma_sets \<Omega> A"

  1573     and space_measure_of: "space (measure_of \<Omega> A \<mu>) = \<Omega>"

  1574 using assms

  1575 by(simp_all add: sets_measure_of_conv space_measure_of_conv)

  1576

  1577 lemma (in sigma_algebra) sets_measure_of_eq[simp]: "sets (measure_of \<Omega> M \<mu>) = M"

  1578   using space_closed by (auto intro!: sigma_sets_eq)

  1579

  1580 lemma (in sigma_algebra) space_measure_of_eq[simp]: "space (measure_of \<Omega> M \<mu>) = \<Omega>"

  1581   by (rule space_measure_of_conv)

  1582

  1583 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>')"

  1584   by (auto intro!: sigma_sets_subseteq)

  1585

  1586 lemma emeasure_sigma: "emeasure (sigma \<Omega> A) = (\<lambda>x. 0)"

  1587   unfolding measure_of_def emeasure_def

  1588   by (subst Abs_measure_inverse)

  1589      (auto simp: measure_space_def positive_def countably_additive_def

  1590            intro!: sigma_algebra_sigma_sets sigma_algebra_trivial)

  1591

  1592 lemma sigma_sets_mono'':

  1593   assumes "A \<in> sigma_sets C D"

  1594   assumes "B \<subseteq> D"

  1595   assumes "D \<subseteq> Pow C"

  1596   shows "sigma_sets A B \<subseteq> sigma_sets C D"

  1597 proof

  1598   fix x assume "x \<in> sigma_sets A B"

  1599   thus "x \<in> sigma_sets C D"

  1600   proof induct

  1601     case (Basic a) with assms have "a \<in> D" by auto

  1602     thus ?case ..

  1603   next

  1604     case Empty show ?case by (rule sigma_sets.Empty)

  1605   next

  1606     from assms have "A \<in> sets (sigma C D)" by (subst sets_measure_of[OF \<open>D \<subseteq> Pow C\<close>])

  1607     moreover case (Compl a) hence "a \<in> sets (sigma C D)" by (subst sets_measure_of[OF \<open>D \<subseteq> Pow C\<close>])

  1608     ultimately have "A - a \<in> sets (sigma C D)" ..

  1609     thus ?case by (subst (asm) sets_measure_of[OF \<open>D \<subseteq> Pow C\<close>])

  1610   next

  1611     case (Union a)

  1612     thus ?case by (intro sigma_sets.Union)

  1613   qed

  1614 qed

  1615

  1616 lemma in_measure_of[intro, simp]: "M \<subseteq> Pow \<Omega> \<Longrightarrow> A \<in> M \<Longrightarrow> A \<in> sets (measure_of \<Omega> M \<mu>)"

  1617   by auto

  1618

  1619 lemma space_empty_iff: "space N = {} \<longleftrightarrow> sets N = {{}}"

  1620   by (metis Pow_empty Sup_bot_conv(1) cSup_singleton empty_iff

  1621             sets.sigma_sets_eq sets.space_closed sigma_sets_top subset_singletonD)

  1622

  1623 subsubsection \<open>Constructing simple @{typ "'a measure"}\<close>

  1624

  1625 lemma emeasure_measure_of:

  1626   assumes M: "M = measure_of \<Omega> A \<mu>"

  1627   assumes ms: "A \<subseteq> Pow \<Omega>" "positive (sets M) \<mu>" "countably_additive (sets M) \<mu>"

  1628   assumes X: "X \<in> sets M"

  1629   shows "emeasure M X = \<mu> X"

  1630 proof -

  1631   interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" by (rule sigma_algebra_sigma_sets) fact

  1632   have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>"

  1633     using ms M by (simp add: measure_space_def sigma_algebra_sigma_sets)

  1634   thus ?thesis using X ms

  1635     by(simp add: M emeasure_measure_of_conv sets_measure_of_conv)

  1636 qed

  1637

  1638 lemma emeasure_measure_of_sigma:

  1639   assumes ms: "sigma_algebra \<Omega> M" "positive M \<mu>" "countably_additive M \<mu>"

  1640   assumes A: "A \<in> M"

  1641   shows "emeasure (measure_of \<Omega> M \<mu>) A = \<mu> A"

  1642 proof -

  1643   interpret sigma_algebra \<Omega> M by fact

  1644   have "measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"

  1645     using ms sigma_sets_eq by (simp add: measure_space_def)

  1646   thus ?thesis by(simp add: emeasure_measure_of_conv A)

  1647 qed

  1648

  1649 lemma measure_cases[cases type: measure]:

  1650   obtains (measure) \<Omega> A \<mu> where "x = Abs_measure (\<Omega>, A, \<mu>)" "\<forall>a\<in>-A. \<mu> a = 0" "measure_space \<Omega> A \<mu>"

  1651   by atomize_elim (cases x, auto)

  1652

  1653 lemma sets_le_imp_space_le: "sets A \<subseteq> sets B \<Longrightarrow> space A \<subseteq> space B"

  1654   by (auto dest: sets.sets_into_space)

  1655

  1656 lemma sets_eq_imp_space_eq: "sets M = sets M' \<Longrightarrow> space M = space M'"

  1657   by (auto intro!: antisym sets_le_imp_space_le)

  1658

  1659 lemma emeasure_notin_sets: "A \<notin> sets M \<Longrightarrow> emeasure M A = 0"

  1660   by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)

  1661

  1662 lemma emeasure_neq_0_sets: "emeasure M A \<noteq> 0 \<Longrightarrow> A \<in> sets M"

  1663   using emeasure_notin_sets[of A M] by blast

  1664

  1665 lemma measure_notin_sets: "A \<notin> sets M \<Longrightarrow> measure M A = 0"

  1666   by (simp add: measure_def emeasure_notin_sets)

  1667

  1668 lemma measure_eqI:

  1669   fixes M N :: "'a measure"

  1670   assumes "sets M = sets N" and eq: "\<And>A. A \<in> sets M \<Longrightarrow> emeasure M A = emeasure N A"

  1671   shows "M = N"

  1672 proof (cases M N rule: measure_cases[case_product measure_cases])

  1673   case (measure_measure \<Omega> A \<mu> \<Omega>' A' \<mu>')

  1674   interpret M: sigma_algebra \<Omega> A using measure_measure by (auto simp: measure_space_def)

  1675   interpret N: sigma_algebra \<Omega>' A' using measure_measure by (auto simp: measure_space_def)

  1676   have "A = sets M" "A' = sets N"

  1677     using measure_measure by (simp_all add: sets_def Abs_measure_inverse)

  1678   with \<open>sets M = sets N\<close> have AA': "A = A'" by simp

  1679   moreover from M.top N.top M.space_closed N.space_closed AA' have "\<Omega> = \<Omega>'" by auto

  1680   moreover { fix B have "\<mu> B = \<mu>' B"

  1681     proof cases

  1682       assume "B \<in> A"

  1683       with eq \<open>A = sets M\<close> have "emeasure M B = emeasure N B" by simp

  1684       with measure_measure show "\<mu> B = \<mu>' B"

  1685         by (simp add: emeasure_def Abs_measure_inverse)

  1686     next

  1687       assume "B \<notin> A"

  1688       with \<open>A = sets M\<close> \<open>A' = sets N\<close> \<open>A = A'\<close> have "B \<notin> sets M" "B \<notin> sets N"

  1689         by auto

  1690       then have "emeasure M B = 0" "emeasure N B = 0"

  1691         by (simp_all add: emeasure_notin_sets)

  1692       with measure_measure show "\<mu> B = \<mu>' B"

  1693         by (simp add: emeasure_def Abs_measure_inverse)

  1694     qed }

  1695   then have "\<mu> = \<mu>'" by auto

  1696   ultimately show "M = N"

  1697     by (simp add: measure_measure)

  1698 qed

  1699

  1700 lemma sigma_eqI:

  1701   assumes [simp]: "M \<subseteq> Pow \<Omega>" "N \<subseteq> Pow \<Omega>" "sigma_sets \<Omega> M = sigma_sets \<Omega> N"

  1702   shows "sigma \<Omega> M = sigma \<Omega> N"

  1703   by (rule measure_eqI) (simp_all add: emeasure_sigma)

  1704

  1705 subsubsection \<open>Measurable functions\<close>

  1706

  1707 definition measurable :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) set" (infixr "\<rightarrow>\<^sub>M" 60) where

  1708   "measurable A B = {f \<in> space A \<rightarrow> space B. \<forall>y \<in> sets B. f - y \<inter> space A \<in> sets A}"

  1709

  1710 lemma measurableI:

  1711   "(\<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>

  1712     f \<in> measurable M N"

  1713   by (auto simp: measurable_def)

  1714

  1715 lemma measurable_space:

  1716   "f \<in> measurable M A \<Longrightarrow> x \<in> space M \<Longrightarrow> f x \<in> space A"

  1717    unfolding measurable_def by auto

  1718

  1719 lemma measurable_sets:

  1720   "f \<in> measurable M A \<Longrightarrow> S \<in> sets A \<Longrightarrow> f - S \<inter> space M \<in> sets M"

  1721    unfolding measurable_def by auto

  1722

  1723 lemma measurable_sets_Collect:

  1724   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"

  1725 proof -

  1726   have "f - {x \<in> space N. P x} \<inter> space M = {x\<in>space M. P (f x)}"

  1727     using measurable_space[OF f] by auto

  1728   with measurable_sets[OF f P] show ?thesis

  1729     by simp

  1730 qed

  1731

  1732 lemma measurable_sigma_sets:

  1733   assumes B: "sets N = sigma_sets \<Omega> A" "A \<subseteq> Pow \<Omega>"

  1734       and f: "f \<in> space M \<rightarrow> \<Omega>"

  1735       and ba: "\<And>y. y \<in> A \<Longrightarrow> (f - y) \<inter> space M \<in> sets M"

  1736   shows "f \<in> measurable M N"

  1737 proof -

  1738   interpret A: sigma_algebra \<Omega> "sigma_sets \<Omega> A" using B(2) by (rule sigma_algebra_sigma_sets)

  1739   from B sets.top[of N] A.top sets.space_closed[of N] A.space_closed have \<Omega>: "\<Omega> = space N" by force

  1740

  1741   { fix X assume "X \<in> sigma_sets \<Omega> A"

  1742     then have "f - X \<inter> space M \<in> sets M \<and> X \<subseteq> \<Omega>"

  1743       proof induct

  1744         case (Basic a) then show ?case

  1745           by (auto simp add: ba) (metis B(2) subsetD PowD)

  1746       next

  1747         case (Compl a)

  1748         have [simp]: "f - \<Omega> \<inter> space M = space M"

  1749           by (auto simp add: funcset_mem [OF f])

  1750         then show ?case

  1751           by (auto simp add: vimage_Diff Diff_Int_distrib2 sets.compl_sets Compl)

  1752       next

  1753         case (Union a)

  1754         then show ?case

  1755           by (simp add: vimage_UN, simp only: UN_extend_simps(4)) blast

  1756       qed auto }

  1757   with f show ?thesis

  1758     by (auto simp add: measurable_def B \<Omega>)

  1759 qed

  1760

  1761 lemma measurable_measure_of:

  1762   assumes B: "N \<subseteq> Pow \<Omega>"

  1763       and f: "f \<in> space M \<rightarrow> \<Omega>"

  1764       and ba: "\<And>y. y \<in> N \<Longrightarrow> (f - y) \<inter> space M \<in> sets M"

  1765   shows "f \<in> measurable M (measure_of \<Omega> N \<mu>)"

  1766 proof -

  1767   have "sets (measure_of \<Omega> N \<mu>) = sigma_sets \<Omega> N"

  1768     using B by (rule sets_measure_of)

  1769   from this assms show ?thesis by (rule measurable_sigma_sets)

  1770 qed

  1771

  1772 lemma measurable_iff_measure_of:

  1773   assumes "N \<subseteq> Pow \<Omega>" "f \<in> space M \<rightarrow> \<Omega>"

  1774   shows "f \<in> measurable M (measure_of \<Omega> N \<mu>) \<longleftrightarrow> (\<forall>A\<in>N. f - A \<inter> space M \<in> sets M)"

  1775   by (metis assms in_measure_of measurable_measure_of assms measurable_sets)

  1776

  1777 lemma measurable_cong_sets:

  1778   assumes sets: "sets M = sets M'" "sets N = sets N'"

  1779   shows "measurable M N = measurable M' N'"

  1780   using sets[THEN sets_eq_imp_space_eq] sets by (simp add: measurable_def)

  1781

  1782 lemma measurable_cong:

  1783   assumes "\<And>w. w \<in> space M \<Longrightarrow> f w = g w"

  1784   shows "f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable M M'"

  1785   unfolding measurable_def using assms

  1786   by (simp cong: vimage_inter_cong Pi_cong)

  1787

  1788 lemma measurable_cong':

  1789   assumes "\<And>w. w \<in> space M =simp=> f w = g w"

  1790   shows "f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable M M'"

  1791   unfolding measurable_def using assms

  1792   by (simp cong: vimage_inter_cong Pi_cong add: simp_implies_def)

  1793

  1794 lemma measurable_cong_strong:

  1795   "M = N \<Longrightarrow> M' = N' \<Longrightarrow> (\<And>w. w \<in> space M \<Longrightarrow> f w = g w) \<Longrightarrow>

  1796     f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable N N'"

  1797   by (metis measurable_cong)

  1798

  1799 lemma measurable_compose:

  1800   assumes f: "f \<in> measurable M N" and g: "g \<in> measurable N L"

  1801   shows "(\<lambda>x. g (f x)) \<in> measurable M L"

  1802 proof -

  1803   have "\<And>A. (\<lambda>x. g (f x)) - A \<inter> space M = f - (g - A \<inter> space N) \<inter> space M"

  1804     using measurable_space[OF f] by auto

  1805   with measurable_space[OF f] measurable_space[OF g] show ?thesis

  1806     by (auto intro: measurable_sets[OF f] measurable_sets[OF g]

  1807              simp del: vimage_Int simp add: measurable_def)

  1808 qed

  1809

  1810 lemma measurable_comp:

  1811   "f \<in> measurable M N \<Longrightarrow> g \<in> measurable N L \<Longrightarrow> g \<circ> f \<in> measurable M L"

  1812   using measurable_compose[of f M N g L] by (simp add: comp_def)

  1813

  1814 lemma measurable_const:

  1815   "c \<in> space M' \<Longrightarrow> (\<lambda>x. c) \<in> measurable M M'"

  1816   by (auto simp add: measurable_def)

  1817

  1818 lemma measurable_ident: "id \<in> measurable M M"

  1819   by (auto simp add: measurable_def)

  1820

  1821 lemma measurable_id: "(\<lambda>x. x) \<in> measurable M M"

  1822   by (simp add: measurable_def)

  1823

  1824 lemma measurable_ident_sets:

  1825   assumes eq: "sets M = sets M'" shows "(\<lambda>x. x) \<in> measurable M M'"

  1826   using measurable_ident[of M]

  1827   unfolding id_def measurable_def eq sets_eq_imp_space_eq[OF eq] .

  1828

  1829 lemma sets_Least:

  1830   assumes meas: "\<And>i::nat. {x\<in>space M. P i x} \<in> M"

  1831   shows "(\<lambda>x. LEAST j. P j x) - A \<inter> space M \<in> sets M"

  1832 proof -

  1833   { fix i have "(\<lambda>x. LEAST j. P j x) - {i} \<inter> space M \<in> sets M"

  1834     proof cases

  1835       assume i: "(LEAST j. False) = i"

  1836       have "(\<lambda>x. LEAST j. P j x) - {i} \<inter> space M =

  1837         {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}))"

  1838         by (simp add: set_eq_iff, safe)

  1839            (insert i, auto dest: Least_le intro: LeastI intro!: Least_equality)

  1840       with meas show ?thesis

  1841         by (auto intro!: sets.Int)

  1842     next

  1843       assume i: "(LEAST j. False) \<noteq> i"

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

  1845         {x\<in>space M. P i x} \<inter> (space M - (\<Union>j<i. {x\<in>space M. P j x}))"

  1846       proof (simp add: set_eq_iff, safe)

  1847         fix x assume neq: "(LEAST j. False) \<noteq> (LEAST j. P j x)"

  1848         have "\<exists>j. P j x"

  1849           by (rule ccontr) (insert neq, auto)

  1850         then show "P (LEAST j. P j x) x" by (rule LeastI_ex)

  1851       qed (auto dest: Least_le intro!: Least_equality)

  1852       with meas show ?thesis

  1853         by auto

  1854     qed }

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

  1856     by (intro sets.countable_UN) auto

  1857   moreover have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) - {i} \<inter> space M) =

  1858     (\<lambda>x. LEAST j. P j x) - A \<inter> space M" by auto

  1859   ultimately show ?thesis by auto

  1860 qed

  1861

  1862 lemma measurable_mono1:

  1863   "M' \<subseteq> Pow \<Omega> \<Longrightarrow> M \<subseteq> M' \<Longrightarrow>

  1864     measurable (measure_of \<Omega> M \<mu>) N \<subseteq> measurable (measure_of \<Omega> M' \<mu>') N"

  1865   using measure_of_subset[of M' \<Omega> M] by (auto simp add: measurable_def)

  1866

  1867 subsubsection \<open>Counting space\<close>

  1868

  1869 definition count_space :: "'a set \<Rightarrow> 'a measure" where

  1870   "count_space \<Omega> = measure_of \<Omega> (Pow \<Omega>) (\<lambda>A. if finite A then ereal (card A) else \<infinity>)"

  1871

  1872 lemma

  1873   shows space_count_space[simp]: "space (count_space \<Omega>) = \<Omega>"

  1874     and sets_count_space[simp]: "sets (count_space \<Omega>) = Pow \<Omega>"

  1875   using sigma_sets_into_sp[of "Pow \<Omega>" \<Omega>]

  1876   by (auto simp: count_space_def)

  1877

  1878 lemma measurable_count_space_eq1[simp]:

  1879   "f \<in> measurable (count_space A) M \<longleftrightarrow> f \<in> A \<rightarrow> space M"

  1880  unfolding measurable_def by simp

  1881

  1882 lemma measurable_compose_countable':

  1883   assumes f: "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. f i x) \<in> measurable M N"

  1884   and g: "g \<in> measurable M (count_space I)" and I: "countable I"

  1885   shows "(\<lambda>x. f (g x) x) \<in> measurable M N"

  1886   unfolding measurable_def

  1887 proof safe

  1888   fix x assume "x \<in> space M" then show "f (g x) x \<in> space N"

  1889     using measurable_space[OF f] g[THEN measurable_space] by auto

  1890 next

  1891   fix A assume A: "A \<in> sets N"

  1892   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))"

  1893     using measurable_space[OF g] by auto

  1894   also have "\<dots> \<in> sets M"

  1895     using f[THEN measurable_sets, OF _ A] g[THEN measurable_sets]

  1896     by (auto intro!: sets.countable_UN' I intro: sets.Int[OF measurable_sets measurable_sets])

  1897   finally show "(\<lambda>x. f (g x) x) - A \<inter> space M \<in> sets M" .

  1898 qed

  1899

  1900 lemma measurable_count_space_eq_countable:

  1901   assumes "countable A"

  1902   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))"

  1903 proof -

  1904   { fix X assume "X \<subseteq> A" "f \<in> space M \<rightarrow> A"

  1905     with \<open>countable A\<close> have "f - X \<inter> space M = (\<Union>a\<in>X. f - {a} \<inter> space M)" "countable X"

  1906       by (auto dest: countable_subset)

  1907     moreover assume "\<forall>a\<in>A. f - {a} \<inter> space M \<in> sets M"

  1908     ultimately have "f - X \<inter> space M \<in> sets M"

  1909       using \<open>X \<subseteq> A\<close> by (auto intro!: sets.countable_UN' simp del: UN_simps) }

  1910   then show ?thesis

  1911     unfolding measurable_def by auto

  1912 qed

  1913

  1914 lemma measurable_count_space_eq2:

  1915   "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))"

  1916   by (intro measurable_count_space_eq_countable countable_finite)

  1917

  1918 lemma measurable_count_space_eq2_countable:

  1919   fixes f :: "'a => 'c::countable"

  1920   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))"

  1921   by (intro measurable_count_space_eq_countable countableI_type)

  1922

  1923 lemma measurable_compose_countable:

  1924   assumes f: "\<And>i::'i::countable. (\<lambda>x. f i x) \<in> measurable M N" and g: "g \<in> measurable M (count_space UNIV)"

  1925   shows "(\<lambda>x. f (g x) x) \<in> measurable M N"

  1926   by (rule measurable_compose_countable'[OF assms]) auto

  1927

  1928 lemma measurable_count_space_const:

  1929   "(\<lambda>x. c) \<in> measurable M (count_space UNIV)"

  1930   by (simp add: measurable_const)

  1931

  1932 lemma measurable_count_space:

  1933   "f \<in> measurable (count_space A) (count_space UNIV)"

  1934   by simp

  1935

  1936 lemma measurable_compose_rev:

  1937   assumes f: "f \<in> measurable L N" and g: "g \<in> measurable M L"

  1938   shows "(\<lambda>x. f (g x)) \<in> measurable M N"

  1939   using measurable_compose[OF g f] .

  1940

  1941 lemma measurable_empty_iff:

  1942   "space N = {} \<Longrightarrow> f \<in> measurable M N \<longleftrightarrow> space M = {}"

  1943   by (auto simp add: measurable_def Pi_iff)

  1944

  1945 subsubsection \<open>Extend measure\<close>

  1946

  1947 definition "extend_measure \<Omega> I G \<mu> =

  1948   (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)

  1949       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>')

  1950       else measure_of \<Omega> (GI) (\<lambda>_. 0))"

  1951

  1952 lemma space_extend_measure: "G  I \<subseteq> Pow \<Omega> \<Longrightarrow> space (extend_measure \<Omega> I G \<mu>) = \<Omega>"

  1953   unfolding extend_measure_def by simp

  1954

  1955 lemma sets_extend_measure: "G  I \<subseteq> Pow \<Omega> \<Longrightarrow> sets (extend_measure \<Omega> I G \<mu>) = sigma_sets \<Omega> (GI)"

  1956   unfolding extend_measure_def by simp

  1957

  1958 lemma emeasure_extend_measure:

  1959   assumes M: "M = extend_measure \<Omega> I G \<mu>"

  1960     and eq: "\<And>i. i \<in> I \<Longrightarrow> \<mu>' (G i) = \<mu> i"

  1961     and ms: "G  I \<subseteq> Pow \<Omega>" "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'"

  1962     and "i \<in> I"

  1963   shows "emeasure M (G i) = \<mu> i"

  1964 proof cases

  1965   assume *: "(\<forall>i\<in>I. \<mu> i = 0)"

  1966   with M have M_eq: "M = measure_of \<Omega> (GI) (\<lambda>_. 0)"

  1967    by (simp add: extend_measure_def)

  1968   from measure_space_0[OF ms(1)] ms \<open>i\<in>I\<close>

  1969   have "emeasure M (G i) = 0"

  1970     by (intro emeasure_measure_of[OF M_eq]) (auto simp add: M measure_space_def sets_extend_measure)

  1971   with \<open>i\<in>I\<close> * show ?thesis

  1972     by simp

  1973 next

  1974   def P \<equiv> "\<lambda>\<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (GI)) \<mu>'"

  1975   assume "\<not> (\<forall>i\<in>I. \<mu> i = 0)"

  1976   moreover

  1977   have "measure_space (space M) (sets M) \<mu>'"

  1978     using ms unfolding measure_space_def by auto standard

  1979   with ms eq have "\<exists>\<mu>'. P \<mu>'"

  1980     unfolding P_def

  1981     by (intro exI[of _ \<mu>']) (auto simp add: M space_extend_measure sets_extend_measure)

  1982   ultimately have M_eq: "M = measure_of \<Omega> (GI) (Eps P)"

  1983     by (simp add: M extend_measure_def P_def[symmetric])

  1984

  1985   from \<open>\<exists>\<mu>'. P \<mu>'\<close> have P: "P (Eps P)" by (rule someI_ex)

  1986   show "emeasure M (G i) = \<mu> i"

  1987   proof (subst emeasure_measure_of[OF M_eq])

  1988     have sets_M: "sets M = sigma_sets \<Omega> (GI)"

  1989       using M_eq ms by (auto simp: sets_extend_measure)

  1990     then show "G i \<in> sets M" using \<open>i \<in> I\<close> by auto

  1991     show "positive (sets M) (Eps P)" "countably_additive (sets M) (Eps P)" "Eps P (G i) = \<mu> i"

  1992       using P \<open>i\<in>I\<close> by (auto simp add: sets_M measure_space_def P_def)

  1993   qed fact

  1994 qed

  1995

  1996 lemma emeasure_extend_measure_Pair:

  1997   assumes M: "M = extend_measure \<Omega> {(i, j). I i j} (\<lambda>(i, j). G i j) (\<lambda>(i, j). \<mu> i j)"

  1998     and eq: "\<And>i j. I i j \<Longrightarrow> \<mu>' (G i j) = \<mu> i j"

  1999     and ms: "\<And>i j. I i j \<Longrightarrow> G i j \<in> Pow \<Omega>" "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'"

  2000     and "I i j"

  2001   shows "emeasure M (G i j) = \<mu> i j"

  2002   using emeasure_extend_measure[OF M _ _ ms(2,3), of "(i,j)"] eq ms(1) \<open>I i j\<close>

  2003   by (auto simp: subset_eq)

  2004

  2005 subsubsection \<open>Supremum of a set of $\sigma$-algebras\<close>

  2006

  2007 definition "Sup_sigma M = sigma (\<Union>x\<in>M. space x) (\<Union>x\<in>M. sets x)"

  2008

  2009 syntax

  2010   "_SUP_sigma"   :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>\<^sub>\<sigma> _\<in>_./ _)" [0, 0, 10] 10)

  2011

  2012 translations

  2013   "\<Squnion>\<^sub>\<sigma> x\<in>A. B"   == "CONST Sup_sigma ((\<lambda>x. B)  A)"

  2014

  2015 lemma space_Sup_sigma: "space (Sup_sigma M) = (\<Union>x\<in>M. space x)"

  2016   unfolding Sup_sigma_def by (rule space_measure_of) (auto dest: sets.sets_into_space)

  2017

  2018 lemma sets_Sup_sigma: "sets (Sup_sigma M) = sigma_sets (\<Union>x\<in>M. space x) (\<Union>x\<in>M. sets x)"

  2019   unfolding Sup_sigma_def by (rule sets_measure_of) (auto dest: sets.sets_into_space)

  2020

  2021 lemma in_Sup_sigma: "m \<in> M \<Longrightarrow> A \<in> sets m \<Longrightarrow> A \<in> sets (Sup_sigma M)"

  2022   unfolding sets_Sup_sigma by auto

  2023

  2024 lemma SUP_sigma_cong:

  2025   assumes *: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets (N i)" shows "sets (\<Squnion>\<^sub>\<sigma> i\<in>I. M i) = sets (\<Squnion>\<^sub>\<sigma> i\<in>I. N i)"

  2026   using * sets_eq_imp_space_eq[OF *] by (simp add: Sup_sigma_def)

  2027

  2028 lemma sets_Sup_in_sets:

  2029   assumes "M \<noteq> {}"

  2030   assumes "\<And>m. m \<in> M \<Longrightarrow> space m = space N"

  2031   assumes "\<And>m. m \<in> M \<Longrightarrow> sets m \<subseteq> sets N"

  2032   shows "sets (Sup_sigma M) \<subseteq> sets N"

  2033 proof -

  2034   have *: "UNION M space = space N"

  2035     using assms by auto

  2036   show ?thesis

  2037     unfolding sets_Sup_sigma * using assms by (auto intro!: sets.sigma_sets_subset)

  2038 qed

  2039

  2040 lemma measurable_Sup_sigma1:

  2041   assumes m: "m \<in> M" and f: "f \<in> measurable m N"

  2042     and const_space: "\<And>m n. m \<in> M \<Longrightarrow> n \<in> M \<Longrightarrow> space m = space n"

  2043   shows "f \<in> measurable (Sup_sigma M) N"

  2044 proof -

  2045   have "space (Sup_sigma M) = space m"

  2046     using m by (auto simp add: space_Sup_sigma dest: const_space)

  2047   then show ?thesis

  2048     using m f unfolding measurable_def by (auto intro: in_Sup_sigma)

  2049 qed

  2050

  2051 lemma measurable_Sup_sigma2:

  2052   assumes M: "M \<noteq> {}"

  2053   assumes f: "\<And>m. m \<in> M \<Longrightarrow> f \<in> measurable N m"

  2054   shows "f \<in> measurable N (Sup_sigma M)"

  2055   unfolding Sup_sigma_def

  2056 proof (rule measurable_measure_of)

  2057   show "f \<in> space N \<rightarrow> UNION M space"

  2058     using measurable_space[OF f] M by auto

  2059 qed (auto intro: measurable_sets f dest: sets.sets_into_space)

  2060

  2061 lemma Sup_sigma_sigma:

  2062   assumes [simp]: "M \<noteq> {}" and M: "\<And>m. m \<in> M \<Longrightarrow> m \<subseteq> Pow \<Omega>"

  2063   shows "(\<Squnion>\<^sub>\<sigma> m\<in>M. sigma \<Omega> m) = sigma \<Omega> (\<Union>M)"

  2064 proof (rule measure_eqI)

  2065   { fix a m assume "a \<in> sigma_sets \<Omega> m" "m \<in> M"

  2066     then have "a \<in> sigma_sets \<Omega> (\<Union>M)"

  2067      by induction (auto intro: sigma_sets.intros) }

  2068   then show "sets (\<Squnion>\<^sub>\<sigma> m\<in>M. sigma \<Omega> m) = sets (sigma \<Omega> (\<Union>M))"

  2069     apply (simp add: sets_Sup_sigma space_measure_of_conv M Union_least)

  2070     apply (rule sigma_sets_eqI)

  2071     apply auto

  2072     done

  2073 qed (simp add: Sup_sigma_def emeasure_sigma)

  2074

  2075 lemma SUP_sigma_sigma:

  2076   assumes M: "M \<noteq> {}" "\<And>m. m \<in> M \<Longrightarrow> f m \<subseteq> Pow \<Omega>"

  2077   shows "(\<Squnion>\<^sub>\<sigma> m\<in>M. sigma \<Omega> (f m)) = sigma \<Omega> (\<Union>m\<in>M. f m)"

  2078 proof -

  2079   have "Sup_sigma (sigma \<Omega>  f  M) = sigma \<Omega> (\<Union>(f  M))"

  2080     using M by (intro Sup_sigma_sigma) auto

  2081   then show ?thesis

  2082     by (simp add: image_image)

  2083 qed

  2084

  2085 subsection \<open>The smallest $\sigma$-algebra regarding a function\<close>

  2086

  2087 definition

  2088   "vimage_algebra X f M = sigma X {f - A \<inter> X | A. A \<in> sets M}"

  2089

  2090 lemma space_vimage_algebra[simp]: "space (vimage_algebra X f M) = X"

  2091   unfolding vimage_algebra_def by (rule space_measure_of) auto

  2092

  2093 lemma sets_vimage_algebra: "sets (vimage_algebra X f M) = sigma_sets X {f - A \<inter> X | A. A \<in> sets M}"

  2094   unfolding vimage_algebra_def by (rule sets_measure_of) auto

  2095

  2096 lemma sets_vimage_algebra2:

  2097   "f \<in> X \<rightarrow> space M \<Longrightarrow> sets (vimage_algebra X f M) = {f - A \<inter> X | A. A \<in> sets M}"

  2098   using sigma_sets_vimage_commute[of f X "space M" "sets M"]

  2099   unfolding sets_vimage_algebra sets.sigma_sets_eq by simp

  2100

  2101 lemma sets_vimage_algebra_cong: "sets M = sets N \<Longrightarrow> sets (vimage_algebra X f M) = sets (vimage_algebra X f N)"

  2102   by (simp add: sets_vimage_algebra)

  2103

  2104 lemma vimage_algebra_cong:

  2105   assumes "X = Y"

  2106   assumes "\<And>x. x \<in> Y \<Longrightarrow> f x = g x"

  2107   assumes "sets M = sets N"

  2108   shows "vimage_algebra X f M = vimage_algebra Y g N"

  2109   by (auto simp: vimage_algebra_def assms intro!: arg_cong2[where f=sigma])

  2110

  2111 lemma in_vimage_algebra: "A \<in> sets M \<Longrightarrow> f - A \<inter> X \<in> sets (vimage_algebra X f M)"

  2112   by (auto simp: vimage_algebra_def)

  2113

  2114 lemma sets_image_in_sets:

  2115   assumes N: "space N = X"

  2116   assumes f: "f \<in> measurable N M"

  2117   shows "sets (vimage_algebra X f M) \<subseteq> sets N"

  2118   unfolding sets_vimage_algebra N[symmetric]

  2119   by (rule sets.sigma_sets_subset) (auto intro!: measurable_sets f)

  2120

  2121 lemma measurable_vimage_algebra1: "f \<in> X \<rightarrow> space M \<Longrightarrow> f \<in> measurable (vimage_algebra X f M) M"

  2122   unfolding measurable_def by (auto intro: in_vimage_algebra)

  2123

  2124 lemma measurable_vimage_algebra2:

  2125   assumes g: "g \<in> space N \<rightarrow> X" and f: "(\<lambda>x. f (g x)) \<in> measurable N M"

  2126   shows "g \<in> measurable N (vimage_algebra X f M)"

  2127   unfolding vimage_algebra_def

  2128 proof (rule measurable_measure_of)

  2129   fix A assume "A \<in> {f - A \<inter> X | A. A \<in> sets M}"

  2130   then obtain Y where Y: "Y \<in> sets M" and A: "A = f - Y \<inter> X"

  2131     by auto

  2132   then have "g - A \<inter> space N = (\<lambda>x. f (g x)) - Y \<inter> space N"

  2133     using g by auto

  2134   also have "\<dots> \<in> sets N"

  2135     using f Y by (rule measurable_sets)

  2136   finally show "g - A \<inter> space N \<in> sets N" .

  2137 qed (insert g, auto)

  2138

  2139 lemma vimage_algebra_sigma:

  2140   assumes X: "X \<subseteq> Pow \<Omega>'" and f: "f \<in> \<Omega> \<rightarrow> \<Omega>'"

  2141   shows "vimage_algebra \<Omega> f (sigma \<Omega>' X) = sigma \<Omega> {f - A \<inter> \<Omega> | A. A \<in> X }" (is "?V = ?S")

  2142 proof (rule measure_eqI)

  2143   have \<Omega>: "{f - A \<inter> \<Omega> |A. A \<in> X} \<subseteq> Pow \<Omega>" by auto

  2144   show "sets ?V = sets ?S"

  2145     using sigma_sets_vimage_commute[OF f, of X]

  2146     by (simp add: space_measure_of_conv f sets_vimage_algebra2 \<Omega> X)

  2147 qed (simp add: vimage_algebra_def emeasure_sigma)

  2148

  2149 lemma vimage_algebra_vimage_algebra_eq:

  2150   assumes *: "f \<in> X \<rightarrow> Y" "g \<in> Y \<rightarrow> space M"

  2151   shows "vimage_algebra X f (vimage_algebra Y g M) = vimage_algebra X (\<lambda>x. g (f x)) M"

  2152     (is "?VV = ?V")

  2153 proof (rule measure_eqI)

  2154   have "(\<lambda>x. g (f x)) \<in> X \<rightarrow> space M" "\<And>A. A \<inter> f - Y \<inter> X = A \<inter> X"

  2155     using * by auto

  2156   with * show "sets ?VV = sets ?V"

  2157     by (simp add: sets_vimage_algebra2 ex_simps[symmetric] vimage_comp comp_def del: ex_simps)

  2158 qed (simp add: vimage_algebra_def emeasure_sigma)

  2159

  2160 lemma sets_vimage_Sup_eq:

  2161   assumes *: "M \<noteq> {}" "\<And>m. m \<in> M \<Longrightarrow> f \<in> X \<rightarrow> space m"

  2162   shows "sets (vimage_algebra X f (Sup_sigma M)) = sets (\<Squnion>\<^sub>\<sigma> m \<in> M. vimage_algebra X f m)"

  2163   (is "?IS = ?SI")

  2164 proof

  2165   show "?IS \<subseteq> ?SI"

  2166     by (intro sets_image_in_sets measurable_Sup_sigma2 measurable_Sup_sigma1)

  2167        (auto simp: space_Sup_sigma measurable_vimage_algebra1 *)

  2168   { fix m assume "m \<in> M"

  2169     moreover then have "f \<in> X \<rightarrow> space (Sup_sigma M)" "f \<in> X \<rightarrow> space m"

  2170       using * by (auto simp: space_Sup_sigma)

  2171     ultimately have "f \<in> measurable (vimage_algebra X f (Sup_sigma M)) m"

  2172       by (auto simp add: measurable_def sets_vimage_algebra2 intro: in_Sup_sigma) }

  2173   then show "?SI \<subseteq> ?IS"

  2174     by (auto intro!: sets_image_in_sets sets_Sup_in_sets del: subsetI simp: *)

  2175 qed

  2176

  2177 lemma vimage_algebra_Sup_sigma:

  2178   assumes [simp]: "MM \<noteq> {}" and "\<And>M. M \<in> MM \<Longrightarrow> f \<in> X \<rightarrow> space M"

  2179   shows "vimage_algebra X f (Sup_sigma MM) = Sup_sigma (vimage_algebra X f  MM)"

  2180 proof (rule measure_eqI)

  2181   show "sets (vimage_algebra X f (Sup_sigma MM)) = sets (Sup_sigma (vimage_algebra X f  MM))"

  2182     using assms by (rule sets_vimage_Sup_eq)

  2183 qed (simp add: vimage_algebra_def Sup_sigma_def emeasure_sigma)

  2184

  2185 subsubsection \<open>Restricted Space Sigma Algebra\<close>

  2186

  2187 definition restrict_space where

  2188   "restrict_space M \<Omega> = measure_of (\<Omega> \<inter> space M) ((op \<inter> \<Omega>)  sets M) (emeasure M)"

  2189

  2190 lemma space_restrict_space: "space (restrict_space M \<Omega>) = \<Omega> \<inter> space M"

  2191   using sets.sets_into_space unfolding restrict_space_def by (subst space_measure_of) auto

  2192

  2193 lemma space_restrict_space2: "\<Omega> \<in> sets M \<Longrightarrow> space (restrict_space M \<Omega>) = \<Omega>"

  2194   by (simp add: space_restrict_space sets.sets_into_space)

  2195

  2196 lemma sets_restrict_space: "sets (restrict_space M \<Omega>) = (op \<inter> \<Omega>)  sets M"

  2197   unfolding restrict_space_def

  2198 proof (subst sets_measure_of)

  2199   show "op \<inter> \<Omega>  sets M \<subseteq> Pow (\<Omega> \<inter> space M)"

  2200     by (auto dest: sets.sets_into_space)

  2201   have "sigma_sets (\<Omega> \<inter> space M) {((\<lambda>x. x) - X) \<inter> (\<Omega> \<inter> space M) | X. X \<in> sets M} =

  2202     (\<lambda>X. X \<inter> (\<Omega> \<inter> space M))  sets M"

  2203     by (subst sigma_sets_vimage_commute[symmetric, where \<Omega>' = "space M"])

  2204        (auto simp add: sets.sigma_sets_eq)

  2205   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"

  2206     by auto

  2207   moreover have "(\<lambda>X. X \<inter> (\<Omega> \<inter> space M))   sets M = (op \<inter> \<Omega>)  sets M"

  2208     by (intro image_cong) (auto dest: sets.sets_into_space)

  2209   ultimately show "sigma_sets (\<Omega> \<inter> space M) (op \<inter> \<Omega>  sets M) = op \<inter> \<Omega>  sets M"

  2210     by simp

  2211 qed

  2212

  2213 lemma restrict_space_sets_cong:

  2214   "A = B \<Longrightarrow> sets M = sets N \<Longrightarrow> sets (restrict_space M A) = sets (restrict_space N B)"

  2215   by (auto simp: sets_restrict_space)

  2216

  2217 lemma sets_restrict_space_count_space :

  2218   "sets (restrict_space (count_space A) B) = sets (count_space (A \<inter> B))"

  2219 by(auto simp add: sets_restrict_space)

  2220

  2221 lemma sets_restrict_UNIV[simp]: "sets (restrict_space M UNIV) = sets M"

  2222   by (auto simp add: sets_restrict_space)

  2223

  2224 lemma sets_restrict_restrict_space:

  2225   "sets (restrict_space (restrict_space M A) B) = sets (restrict_space M (A \<inter> B))"

  2226   unfolding sets_restrict_space image_comp by (intro image_cong) auto

  2227

  2228 lemma sets_restrict_space_iff:

  2229   "\<Omega> \<inter> space M \<in> sets M \<Longrightarrow> A \<in> sets (restrict_space M \<Omega>) \<longleftrightarrow> (A \<subseteq> \<Omega> \<and> A \<in> sets M)"

  2230 proof (subst sets_restrict_space, safe)

  2231   fix A assume "\<Omega> \<inter> space M \<in> sets M" and A: "A \<in> sets M"

  2232   then have "(\<Omega> \<inter> space M) \<inter> A \<in> sets M"

  2233     by rule

  2234   also have "(\<Omega> \<inter> space M) \<inter> A = \<Omega> \<inter> A"

  2235     using sets.sets_into_space[OF A] by auto

  2236   finally show "\<Omega> \<inter> A \<in> sets M"

  2237     by auto

  2238 qed auto

  2239

  2240 lemma sets_restrict_space_cong: "sets M = sets N \<Longrightarrow> sets (restrict_space M \<Omega>) = sets (restrict_space N \<Omega>)"

  2241   by (simp add: sets_restrict_space)

  2242

  2243 lemma restrict_space_eq_vimage_algebra:

  2244   "\<Omega> \<subseteq> space M \<Longrightarrow> sets (restrict_space M \<Omega>) = sets (vimage_algebra \<Omega> (\<lambda>x. x) M)"

  2245   unfolding restrict_space_def

  2246   apply (subst sets_measure_of)

  2247   apply (auto simp add: image_subset_iff dest: sets.sets_into_space) []

  2248   apply (auto simp add: sets_vimage_algebra intro!: arg_cong2[where f=sigma_sets])

  2249   done

  2250

  2251 lemma sets_Collect_restrict_space_iff:

  2252   assumes "S \<in> sets M"

  2253   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"

  2254 proof -

  2255   have "{x\<in>S. P x} = {x\<in>space M. x \<in> S \<and> P x}"

  2256     using sets.sets_into_space[OF assms] by auto

  2257   then show ?thesis

  2258     by (subst sets_restrict_space_iff) (auto simp add: space_restrict_space assms)

  2259 qed

  2260

  2261 lemma measurable_restrict_space1:

  2262   assumes f: "f \<in> measurable M N"

  2263   shows "f \<in> measurable (restrict_space M \<Omega>) N"

  2264   unfolding measurable_def

  2265 proof (intro CollectI conjI ballI)

  2266   show sp: "f \<in> space (restrict_space M \<Omega>) \<rightarrow> space N"

  2267     using measurable_space[OF f] by (auto simp: space_restrict_space)

  2268

  2269   fix A assume "A \<in> sets N"

  2270   have "f - A \<inter> space (restrict_space M \<Omega>) = (f - A \<inter> space M) \<inter> (\<Omega> \<inter> space M)"

  2271     by (auto simp: space_restrict_space)

  2272   also have "\<dots> \<in> sets (restrict_space M \<Omega>)"

  2273     unfolding sets_restrict_space

  2274     using measurable_sets[OF f \<open>A \<in> sets N\<close>] by blast

  2275   finally show "f - A \<inter> space (restrict_space M \<Omega>) \<in> sets (restrict_space M \<Omega>)" .

  2276 qed

  2277

  2278 lemma measurable_restrict_space2_iff:

  2279   "f \<in> measurable M (restrict_space N \<Omega>) \<longleftrightarrow> (f \<in> measurable M N \<and> f \<in> space M \<rightarrow> \<Omega>)"

  2280 proof -

  2281   have "\<And>A. f \<in> space M \<rightarrow> \<Omega> \<Longrightarrow> f - \<Omega> \<inter> f - A \<inter> space M = f - A \<inter> space M"

  2282     by auto

  2283   then show ?thesis

  2284     by (auto simp: measurable_def space_restrict_space Pi_Int[symmetric] sets_restrict_space)

  2285 qed

  2286

  2287 lemma measurable_restrict_space2:

  2288   "f \<in> space M \<rightarrow> \<Omega> \<Longrightarrow> f \<in> measurable M N \<Longrightarrow> f \<in> measurable M (restrict_space N \<Omega>)"

  2289   by (simp add: measurable_restrict_space2_iff)

  2290

  2291 lemma measurable_piecewise_restrict:

  2292   assumes I: "countable C"

  2293     and X: "\<And>\<Omega>. \<Omega> \<in> C \<Longrightarrow> \<Omega> \<inter> space M \<in> sets M" "space M \<subseteq> \<Union>C"

  2294     and f: "\<And>\<Omega>. \<Omega> \<in> C \<Longrightarrow> f \<in> measurable (restrict_space M \<Omega>) N"

  2295   shows "f \<in> measurable M N"

  2296 proof (rule measurableI)

  2297   fix x assume "x \<in> space M"

  2298   with X obtain \<Omega> where "\<Omega> \<in> C" "x \<in> \<Omega>" "x \<in> space M" by auto

  2299   then show "f x \<in> space N"

  2300     by (auto simp: space_restrict_space intro: f measurable_space)

  2301 next

  2302   fix A assume A: "A \<in> sets N"

  2303   have "f - A \<inter> space M = (\<Union>\<Omega>\<in>C. (f - A \<inter> (\<Omega> \<inter> space M)))"

  2304     using X by (auto simp: subset_eq)

  2305   also have "\<dots> \<in> sets M"

  2306     using measurable_sets[OF f A] X I

  2307     by (intro sets.countable_UN') (auto simp: sets_restrict_space_iff space_restrict_space)

  2308   finally show "f - A \<inter> space M \<in> sets M" .

  2309 qed

  2310

  2311 lemma measurable_piecewise_restrict_iff:

  2312   "countable C \<Longrightarrow> (\<And>\<Omega>. \<Omega> \<in> C \<Longrightarrow> \<Omega> \<inter> space M \<in> sets M) \<Longrightarrow> space M \<subseteq> (\<Union>C) \<Longrightarrow>

  2313     f \<in> measurable M N \<longleftrightarrow> (\<forall>\<Omega>\<in>C. f \<in> measurable (restrict_space M \<Omega>) N)"

  2314   by (auto intro: measurable_piecewise_restrict measurable_restrict_space1)

  2315

  2316 lemma measurable_If_restrict_space_iff:

  2317   "{x\<in>space M. P x} \<in> sets M \<Longrightarrow>

  2318     (\<lambda>x. if P x then f x else g x) \<in> measurable M N \<longleftrightarrow>

  2319     (f \<in> measurable (restrict_space M {x. P x}) N \<and> g \<in> measurable (restrict_space M {x. \<not> P x}) N)"

  2320   by (subst measurable_piecewise_restrict_iff[where C="{{x. P x}, {x. \<not> P x}}"])

  2321      (auto simp: Int_def sets.sets_Collect_neg space_restrict_space conj_commute[of _ "x \<in> space M" for x]

  2322            cong: measurable_cong')

  2323

  2324 lemma measurable_If:

  2325   "f \<in> measurable M M' \<Longrightarrow> g \<in> measurable M M' \<Longrightarrow> {x\<in>space M. P x} \<in> sets M \<Longrightarrow>

  2326     (\<lambda>x. if P x then f x else g x) \<in> measurable M M'"

  2327   unfolding measurable_If_restrict_space_iff by (auto intro: measurable_restrict_space1)

  2328

  2329 lemma measurable_If_set:

  2330   assumes measure: "f \<in> measurable M M'" "g \<in> measurable M M'"

  2331   assumes P: "A \<inter> space M \<in> sets M"

  2332   shows "(\<lambda>x. if x \<in> A then f x else g x) \<in> measurable M M'"

  2333 proof (rule measurable_If[OF measure])

  2334   have "{x \<in> space M. x \<in> A} = A \<inter> space M" by auto

  2335   thus "{x \<in> space M. x \<in> A} \<in> sets M" using \<open>A \<inter> space M \<in> sets M\<close> by auto

  2336 qed

  2337

  2338 lemma measurable_restrict_space_iff:

  2339   "\<Omega> \<inter> space M \<in> sets M \<Longrightarrow> c \<in> space N \<Longrightarrow>

  2340     f \<in> measurable (restrict_space M \<Omega>) N \<longleftrightarrow> (\<lambda>x. if x \<in> \<Omega> then f x else c) \<in> measurable M N"

  2341   by (subst measurable_If_restrict_space_iff)

  2342      (simp_all add: Int_def conj_commute measurable_const)

  2343

  2344 lemma restrict_space_singleton: "{x} \<in> sets M \<Longrightarrow> sets (restrict_space M {x}) = sets (count_space {x})"

  2345   using sets_restrict_space_iff[of "{x}" M]

  2346   by (auto simp add: sets_restrict_space_iff dest!: subset_singletonD)

  2347

  2348 lemma measurable_restrict_countable:

  2349   assumes X[intro]: "countable X"

  2350   assumes sets[simp]: "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"

  2351   assumes space[simp]: "\<And>x. x \<in> X \<Longrightarrow> f x \<in> space N"

  2352   assumes f: "f \<in> measurable (restrict_space M (- X)) N"

  2353   shows "f \<in> measurable M N"

  2354   using f sets.countable[OF sets X]

  2355   by (intro measurable_piecewise_restrict[where M=M and C="{- X} \<union> ((\<lambda>x. {x})  X)"])

  2356      (auto simp: Diff_Int_distrib2 Compl_eq_Diff_UNIV Int_insert_left sets.Diff restrict_space_singleton

  2357            simp del: sets_count_space  cong: measurable_cong_sets)

  2358

  2359 lemma measurable_discrete_difference:

  2360   assumes f: "f \<in> measurable M N"

  2361   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"

  2362   assumes eq: "\<And>x. x \<in> space M \<Longrightarrow> x \<notin> X \<Longrightarrow> f x = g x"

  2363   shows "g \<in> measurable M N"

  2364   by (rule measurable_restrict_countable[OF X])

  2365      (auto simp: eq[symmetric] space_restrict_space cong: measurable_cong' intro: f measurable_restrict_space1)

  2366

  2367 end

  2368