src/HOL/Probability/Sigma_Algebra.thy
 author haftmann Sat Jul 05 11:01:53 2014 +0200 (2014-07-05) changeset 57514 bdc2c6b40bf2 parent 57512 cc97b347b301 child 58588 93d87fd1583d permissions -rw-r--r--
prefer ac_simps collections over separate name bindings for add and mult
     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 header {* Describing measurable sets *}

     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 begin

    18

    19 text {* Sigma algebras are an elementary concept in measure

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

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

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

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

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

    25   three very natural and desirable properties. *}

    26

    27 subsection {* Families of sets *}

    28

    29 locale subset_class =

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

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

    32

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

    34   by (metis PowD contra_subsetD space_closed)

    35

    36 subsubsection {* Semiring of sets *}

    37

    38 definition "disjoint A \<longleftrightarrow> (\<forall>a\<in>A. \<forall>b\<in>A. a \<noteq> b \<longrightarrow> a \<inter> b = {})"

    39

    40 lemma disjointI:

    41   "(\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> a \<inter> b = {}) \<Longrightarrow> disjoint A"

    42   unfolding disjoint_def by auto

    43

    44 lemma disjointD:

    45   "disjoint A \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> a \<inter> b = {}"

    46   unfolding disjoint_def by auto

    47

    48 lemma disjoint_empty[iff]: "disjoint {}"

    49   by (auto simp: disjoint_def)

    50

    51 lemma disjoint_union:

    52   assumes C: "disjoint C" and B: "disjoint B" and disj: "\<Union>C \<inter> \<Union>B = {}"

    53   shows "disjoint (C \<union> B)"

    54 proof (rule disjointI)

    55   fix c d assume sets: "c \<in> C \<union> B" "d \<in> C \<union> B" and "c \<noteq> d"

    56   show "c \<inter> d = {}"

    57   proof cases

    58     assume "(c \<in> C \<and> d \<in> C) \<or> (c \<in> B \<and> d \<in> B)"

    59     then show ?thesis

    60     proof

    61       assume "c \<in> C \<and> d \<in> C" with c \<noteq> d C show "c \<inter> d = {}"

    62         by (auto simp: disjoint_def)

    63     next

    64       assume "c \<in> B \<and> d \<in> B" with c \<noteq> d B show "c \<inter> d = {}"

    65         by (auto simp: disjoint_def)

    66     qed

    67   next

    68     assume "\<not> ((c \<in> C \<and> d \<in> C) \<or> (c \<in> B \<and> d \<in> B))"

    69     with sets have "(c \<subseteq> \<Union>C \<and> d \<subseteq> \<Union>B) \<or> (c \<subseteq> \<Union>B \<and> d \<subseteq> \<Union>C)"

    70       by auto

    71     with disj show "c \<inter> d = {}" by auto

    72   qed

    73 qed

    74

    75 lemma disjoint_singleton [simp]: "disjoint {A}"

    76 by(simp add: disjoint_def)

    77

    78 locale semiring_of_sets = subset_class +

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

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

    81   assumes Diff_cover:

    82     "\<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"

    83

    84 lemma (in semiring_of_sets) finite_INT[intro]:

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

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

    87   using assms by (induct rule: finite_ne_induct) auto

    88

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

    90   by (metis Int_absorb1 sets_into_space)

    91

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

    93   by (metis Int_absorb2 sets_into_space)

    94

    95 lemma (in semiring_of_sets) sets_Collect_conj:

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

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

    98 proof -

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

   100     by auto

   101   with assms show ?thesis by auto

   102 qed

   103

   104 lemma (in semiring_of_sets) sets_Collect_finite_All':

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

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

   107 proof -

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

   109     using S \<noteq> {} by auto

   110   with assms show ?thesis by auto

   111 qed

   112

   113 locale ring_of_sets = semiring_of_sets +

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

   115

   116 lemma (in ring_of_sets) finite_Union [intro]:

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

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

   119

   120 lemma (in ring_of_sets) finite_UN[intro]:

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

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

   123   using assms by induct auto

   124

   125 lemma (in ring_of_sets) Diff [intro]:

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

   127   using Diff_cover[OF assms] by auto

   128

   129 lemma ring_of_setsI:

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

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

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

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

   134   shows "ring_of_sets \<Omega> M"

   135 proof

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

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

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

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

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

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

   142 qed fact+

   143

   144 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)"

   145 proof

   146   assume "ring_of_sets \<Omega> M"

   147   then interpret ring_of_sets \<Omega> M .

   148   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)"

   149     using space_closed by auto

   150 qed (auto intro!: ring_of_setsI)

   151

   152 lemma (in ring_of_sets) insert_in_sets:

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

   154 proof -

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

   156   thus ?thesis by auto

   157 qed

   158

   159 lemma (in ring_of_sets) sets_Collect_disj:

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

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

   162 proof -

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

   164     by auto

   165   with assms show ?thesis by auto

   166 qed

   167

   168 lemma (in ring_of_sets) sets_Collect_finite_Ex:

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

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

   171 proof -

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

   173     by auto

   174   with assms show ?thesis by auto

   175 qed

   176

   177 locale algebra = ring_of_sets +

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

   179

   180 lemma (in algebra) compl_sets [intro]:

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

   182   by auto

   183

   184 lemma algebra_iff_Un:

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

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

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

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

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

   190 proof

   191   assume "algebra \<Omega> M"

   192   then interpret algebra \<Omega> M .

   193   show ?Un using sets_into_space by auto

   194 next

   195   assume ?Un

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

   197   interpret ring_of_sets \<Omega> M

   198   proof (rule ring_of_setsI)

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

   200       using ?Un by auto

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

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

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

   204       using \<Omega> a b by auto

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

   206       using a b  ?Un by auto

   207   qed

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

   209 qed

   210

   211 lemma algebra_iff_Int:

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

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

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

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

   216 proof

   217   assume "algebra \<Omega> M"

   218   then interpret algebra \<Omega> M .

   219   show ?Int using sets_into_space by auto

   220 next

   221   assume ?Int

   222   show "algebra \<Omega> M"

   223   proof (unfold algebra_iff_Un, intro conjI ballI)

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

   225       using ?Int by auto

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

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

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

   229       using \<Omega> by blast

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

   231       using M ?Int by auto

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

   233   qed

   234 qed

   235

   236 lemma (in algebra) sets_Collect_neg:

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

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

   239 proof -

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

   241   with assms show ?thesis by auto

   242 qed

   243

   244 lemma (in algebra) sets_Collect_imp:

   245   "{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"

   246   unfolding imp_conv_disj by (intro sets_Collect_disj sets_Collect_neg)

   247

   248 lemma (in algebra) sets_Collect_const:

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

   250   by (cases P) auto

   251

   252 lemma algebra_single_set:

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

   254   by (auto simp: algebra_iff_Int)

   255

   256 subsubsection {* Restricted algebras *}

   257

   258 abbreviation (in algebra)

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

   260

   261 lemma (in algebra) restricted_algebra:

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

   263   using assms by (auto simp: algebra_iff_Int)

   264

   265 subsubsection {* Sigma Algebras *}

   266

   267 locale sigma_algebra = algebra +

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

   269

   270 lemma (in algebra) is_sigma_algebra:

   271   assumes "finite M"

   272   shows "sigma_algebra \<Omega> M"

   273 proof

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

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

   276     by auto

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

   278     using finite M by auto

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

   280 qed

   281

   282 lemma countable_UN_eq:

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

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

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

   286 proof -

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

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

   289   proof safe

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

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

   292   next

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

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

   295   qed

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

   297     using surj_from_nat

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

   299   show ?thesis unfolding * ** ..

   300 qed

   301

   302 lemma (in sigma_algebra) countable_Union [intro]:

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

   304 proof cases

   305   assume "X \<noteq> {}"

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

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

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

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

   310   finally show ?thesis .

   311 qed simp

   312

   313 lemma (in sigma_algebra) countable_UN[intro]:

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

   315   assumes "AX \<subseteq> M"

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

   317 proof -

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

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

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

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

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

   323   ultimately show ?thesis by simp

   324 qed

   325

   326 lemma (in sigma_algebra) countable_UN':

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

   328   assumes X: "countable X"

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

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

   331 proof -

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

   333     using X by auto

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

   335     using A X

   336     by (intro countable_UN) auto

   337   finally show ?thesis .

   338 qed

   339

   340 lemma (in sigma_algebra) countable_INT [intro]:

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

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

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

   344 proof -

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

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

   347   moreover

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

   349     by blast

   350   ultimately show ?thesis by metis

   351 qed

   352

   353 lemma (in sigma_algebra) countable_INT':

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

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

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

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

   358 proof -

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

   360     using X by auto

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

   362     using A X

   363     by (intro countable_INT) auto

   364   finally show ?thesis .

   365 qed

   366

   367

   368 lemma (in sigma_algebra) countable:

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

   370   shows "A \<in> M"

   371 proof -

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

   373     using assms by (intro countable_UN') auto

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

   375   finally show ?thesis by auto

   376 qed

   377

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

   379   by (auto simp: ring_of_sets_iff)

   380

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

   382   by (auto simp: algebra_iff_Un)

   383

   384 lemma sigma_algebra_iff:

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

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

   387   by (simp add: sigma_algebra_def sigma_algebra_axioms_def)

   388

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

   390   by (auto simp: sigma_algebra_iff algebra_iff_Int)

   391

   392 lemma (in sigma_algebra) sets_Collect_countable_All:

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

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

   395 proof -

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

   397   with assms show ?thesis by auto

   398 qed

   399

   400 lemma (in sigma_algebra) sets_Collect_countable_Ex:

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

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

   403 proof -

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

   405   with assms show ?thesis by auto

   406 qed

   407

   408 lemma (in sigma_algebra) sets_Collect_countable_Ex':

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

   410   assumes "countable I"

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

   412 proof -

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

   414   with assms show ?thesis

   415     by (auto intro!: countable_UN')

   416 qed

   417

   418 lemma (in sigma_algebra) sets_Collect_countable_All':

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

   420   assumes "countable I"

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

   422 proof -

   423   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

   424   with assms show ?thesis

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

   426 qed

   427

   428 lemma (in sigma_algebra) sets_Collect_countable_Ex1':

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

   430   assumes "countable I"

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

   432 proof -

   433   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)}"

   434     by auto

   435   with assms show ?thesis

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

   437 qed

   438

   439 lemmas (in sigma_algebra) sets_Collect =

   440   sets_Collect_imp sets_Collect_disj sets_Collect_conj sets_Collect_neg sets_Collect_const

   441   sets_Collect_countable_All sets_Collect_countable_Ex sets_Collect_countable_All

   442

   443 lemma (in sigma_algebra) sets_Collect_countable_Ball:

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

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

   446   unfolding Ball_def by (intro sets_Collect assms)

   447

   448 lemma (in sigma_algebra) sets_Collect_countable_Bex:

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

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

   451   unfolding Bex_def by (intro sets_Collect assms)

   452

   453 lemma sigma_algebra_single_set:

   454   assumes "X \<subseteq> S"

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

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

   457

   458 subsubsection {* Binary Unions *}

   459

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

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

   462

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

   464   by (auto simp add: binary_def)

   465

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

   467   by (simp add: SUP_def range_binary_eq)

   468

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

   470   by (simp add: INF_def range_binary_eq)

   471

   472 lemma sigma_algebra_iff2:

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

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

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

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

   477   by (auto simp add: range_binary_eq sigma_algebra_def sigma_algebra_axioms_def

   478          algebra_iff_Un Un_range_binary)

   479

   480 subsubsection {* Initial Sigma Algebra *}

   481

   482 text {*Sigma algebras can naturally be created as the closure of any set of

   483   M with regard to the properties just postulated.  *}

   484

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

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

   487   where

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

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

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

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

   492

   493 lemma (in sigma_algebra) sigma_sets_subset:

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

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

   496 proof

   497   fix x

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

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

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

   501 qed

   502

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

   504   by (erule sigma_sets.induct, auto)

   505

   506 lemma sigma_algebra_sigma_sets:

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

   508   by (auto simp add: sigma_algebra_iff2 dest: sigma_sets_into_sp

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

   510

   511 lemma sigma_sets_least_sigma_algebra:

   512   assumes "A \<subseteq> Pow S"

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

   514 proof safe

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

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

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

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

   519 next

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

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

   522      by simp

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

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

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

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

   527 qed

   528

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

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

   531

   532 lemma sigma_sets_Un:

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

   534 apply (simp add: Un_range_binary range_binary_eq)

   535 apply (rule Union, simp add: binary_def)

   536 done

   537

   538 lemma sigma_sets_Inter:

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

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

   541 proof -

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

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

   544     by (rule sigma_sets.Compl)

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

   546     by (rule sigma_sets.Union)

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

   548     by (rule sigma_sets.Compl)

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

   550     by auto

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

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

   553   finally show ?thesis .

   554 qed

   555

   556 lemma sigma_sets_INTER:

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

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

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

   560 proof -

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

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

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

   564     by (rule sigma_sets_Inter [OF Asb])

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

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

   567   finally show ?thesis .

   568 qed

   569

   570 lemma sigma_sets_UNION: "countable B \<Longrightarrow> (\<And>b. b \<in> B \<Longrightarrow> b \<in> sigma_sets X A) \<Longrightarrow> (\<Union>B) \<in> sigma_sets X A"

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

   572   apply (cases "B = {}")

   573   apply (simp add: sigma_sets.Empty)

   574   apply (simp del: Union_image_eq add: Union_image_eq[symmetric])

   575   done

   576

   577 lemma (in sigma_algebra) sigma_sets_eq:

   578      "sigma_sets \<Omega> M = M"

   579 proof

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

   581     by (metis Set.subsetI sigma_sets.Basic)

   582   next

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

   584     by (metis sigma_sets_subset subset_refl)

   585 qed

   586

   587 lemma sigma_sets_eqI:

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

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

   590   shows "sigma_sets M A = sigma_sets M B"

   591 proof (intro set_eqI iffI)

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

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

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

   595 next

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

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

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

   599 qed

   600

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

   602 proof

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

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

   605 qed

   606

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

   608 proof

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

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

   611 qed

   612

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

   614 proof

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

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

   617 qed

   618

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

   620   by (auto intro: sigma_sets.Basic)

   621

   622 lemma (in sigma_algebra) restriction_in_sets:

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

   624   assumes "S \<in> M"

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

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

   627 proof -

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

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

   630     hence "A i \<subseteq> S" "A i \<in> M" using S \<in> M by auto }

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

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

   633 qed

   634

   635 lemma (in sigma_algebra) restricted_sigma_algebra:

   636   assumes "S \<in> M"

   637   shows "sigma_algebra S (restricted_space S)"

   638   unfolding sigma_algebra_def sigma_algebra_axioms_def

   639 proof safe

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

   641 next

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

   643   from restriction_in_sets[OF assms this[simplified]]

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

   645 qed

   646

   647 lemma sigma_sets_Int:

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

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

   650 proof (intro equalityI subsetI)

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

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

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

   654   proof (induct arbitrary: x)

   655     case (Compl a)

   656     then show ?case

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

   658   next

   659     case (Union a)

   660     then show ?case

   661       by (auto intro!: sigma_sets.Union

   662                simp add: UN_extend_simps simp del: UN_simps)

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

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

   665     using A \<subseteq> sp by (simp add: Int_absorb2)

   666 next

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

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

   669   proof induct

   670     case (Compl a)

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

   672     then show ?case using A \<subseteq> sp

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

   674   next

   675     case (Union a)

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

   677       by (auto simp: image_iff Bex_def)

   678     from choice[OF this] guess f ..

   679     then show ?case

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

   681                simp add: image_iff)

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

   683 qed

   684

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

   686 proof (intro set_eqI iffI)

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

   688     by induct blast+

   689 qed (auto intro: sigma_sets.Empty sigma_sets_top)

   690

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

   692 proof (intro set_eqI iffI)

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

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

   695     by induct blast+

   696 next

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

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

   699     by (auto intro: sigma_sets.Empty sigma_sets_top)

   700 qed

   701

   702 lemma sigma_sets_sigma_sets_eq:

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

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

   705

   706 lemma sigma_sets_singleton:

   707   assumes "X \<subseteq> S"

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

   709 proof -

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

   711     by (rule sigma_algebra_single_set) fact

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

   713     by (rule sigma_sets_subseteq) simp

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

   715     using sigma_sets_eq by simp

   716   moreover

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

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

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

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

   721     by (intro antisym) auto

   722   with sigma_sets_eq show ?thesis by simp

   723 qed

   724

   725 lemma restricted_sigma:

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

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

   728     sigma_sets S (algebra.restricted_space M S)"

   729 proof -

   730   from S sigma_sets_into_sp[OF M]

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

   732   from sigma_sets_Int[OF this]

   733   show ?thesis by simp

   734 qed

   735

   736 lemma sigma_sets_vimage_commute:

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

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

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

   740 proof

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

   742   proof clarify

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

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

   745     proof induct

   746       case Empty then show ?case

   747         by (auto intro!: sigma_sets.Empty)

   748     next

   749       case (Compl B)

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

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

   752       with Compl show ?case

   753         by (auto intro!: sigma_sets.Compl)

   754     next

   755       case (Union F)

   756       then show ?case

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

   758                  intro!: sigma_sets.Union)

   759     qed auto

   760   qed

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

   762   proof clarify

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

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

   765     proof induct

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

   767     next

   768       case Empty then show ?case

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

   770     next

   771       case (Compl B)

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

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

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

   775       with A(2) show ?case

   776         by (auto intro: sigma_sets.Compl)

   777     next

   778       case (Union F)

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

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

   781       with A show ?case

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

   783     qed

   784   qed

   785 qed

   786

   787 subsubsection "Disjoint families"

   788

   789 definition

   790   disjoint_family_on  where

   791   "disjoint_family_on A S \<longleftrightarrow> (\<forall>m\<in>S. \<forall>n\<in>S. m \<noteq> n \<longrightarrow> A m \<inter> A n = {})"

   792

   793 abbreviation

   794   "disjoint_family A \<equiv> disjoint_family_on A UNIV"

   795

   796 lemma range_subsetD: "range f \<subseteq> B \<Longrightarrow> f i \<in> B"

   797   by blast

   798

   799 lemma disjoint_family_onD: "disjoint_family_on A I \<Longrightarrow> i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> i \<noteq> j \<Longrightarrow> A i \<inter> A j = {}"

   800   by (auto simp: disjoint_family_on_def)

   801

   802 lemma Int_Diff_disjoint: "A \<inter> B \<inter> (A - B) = {}"

   803   by blast

   804

   805 lemma Int_Diff_Un: "A \<inter> B \<union> (A - B) = A"

   806   by blast

   807

   808 lemma disjoint_family_subset:

   809      "disjoint_family A \<Longrightarrow> (!!x. B x \<subseteq> A x) \<Longrightarrow> disjoint_family B"

   810   by (force simp add: disjoint_family_on_def)

   811

   812 lemma disjoint_family_on_bisimulation:

   813   assumes "disjoint_family_on f S"

   814   and "\<And>n m. n \<in> S \<Longrightarrow> m \<in> S \<Longrightarrow> n \<noteq> m \<Longrightarrow> f n \<inter> f m = {} \<Longrightarrow> g n \<inter> g m = {}"

   815   shows "disjoint_family_on g S"

   816   using assms unfolding disjoint_family_on_def by auto

   817

   818 lemma disjoint_family_on_mono:

   819   "A \<subseteq> B \<Longrightarrow> disjoint_family_on f B \<Longrightarrow> disjoint_family_on f A"

   820   unfolding disjoint_family_on_def by auto

   821

   822 lemma disjoint_family_Suc:

   823   assumes Suc: "!!n. A n \<subseteq> A (Suc n)"

   824   shows "disjoint_family (\<lambda>i. A (Suc i) - A i)"

   825 proof -

   826   {

   827     fix m

   828     have "!!n. A n \<subseteq> A (m+n)"

   829     proof (induct m)

   830       case 0 show ?case by simp

   831     next

   832       case (Suc m) thus ?case

   833         by (metis Suc_eq_plus1 assms add.commute add.left_commute subset_trans)

   834     qed

   835   }

   836   hence "!!m n. m < n \<Longrightarrow> A m \<subseteq> A n"

   837     by (metis add.commute le_add_diff_inverse nat_less_le)

   838   thus ?thesis

   839     by (auto simp add: disjoint_family_on_def)

   840       (metis insert_absorb insert_subset le_SucE le_antisym not_leE)

   841 qed

   842

   843 lemma setsum_indicator_disjoint_family:

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

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

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

   847 proof -

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

   849     using d x \<in> A j j \<in> P unfolding disjoint_family_on_def

   850     by auto

   851   thus ?thesis

   852     unfolding indicator_def

   853     by (simp add: if_distrib setsum.If_cases[OF finite P])

   854 qed

   855

   856 definition disjointed :: "(nat \<Rightarrow> 'a set) \<Rightarrow> nat \<Rightarrow> 'a set "

   857   where "disjointed A n = A n - (\<Union>i\<in>{0..<n}. A i)"

   858

   859 lemma finite_UN_disjointed_eq: "(\<Union>i\<in>{0..<n}. disjointed A i) = (\<Union>i\<in>{0..<n}. A i)"

   860 proof (induct n)

   861   case 0 show ?case by simp

   862 next

   863   case (Suc n)

   864   thus ?case by (simp add: atLeastLessThanSuc disjointed_def)

   865 qed

   866

   867 lemma UN_disjointed_eq: "(\<Union>i. disjointed A i) = (\<Union>i. A i)"

   868   apply (rule UN_finite2_eq [where k=0])

   869   apply (simp add: finite_UN_disjointed_eq)

   870   done

   871

   872 lemma less_disjoint_disjointed: "m<n \<Longrightarrow> disjointed A m \<inter> disjointed A n = {}"

   873   by (auto simp add: disjointed_def)

   874

   875 lemma disjoint_family_disjointed: "disjoint_family (disjointed A)"

   876   by (simp add: disjoint_family_on_def)

   877      (metis neq_iff Int_commute less_disjoint_disjointed)

   878

   879 lemma disjointed_subset: "disjointed A n \<subseteq> A n"

   880   by (auto simp add: disjointed_def)

   881

   882 lemma (in ring_of_sets) UNION_in_sets:

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

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

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

   886 proof (induct n)

   887   case 0 show ?case by simp

   888 next

   889   case (Suc n)

   890   thus ?case

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

   892 qed

   893

   894 lemma (in ring_of_sets) range_disjointed_sets:

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

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

   897 proof (auto simp add: disjointed_def)

   898   fix n

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

   900     by (metis A Diff UNIV_I image_subset_iff)

   901 qed

   902

   903 lemma (in algebra) range_disjointed_sets':

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

   905   using range_disjointed_sets .

   906

   907 lemma disjointed_0[simp]: "disjointed A 0 = A 0"

   908   by (simp add: disjointed_def)

   909

   910 lemma incseq_Un:

   911   "incseq A \<Longrightarrow> (\<Union>i\<le>n. A i) = A n"

   912   unfolding incseq_def by auto

   913

   914 lemma disjointed_incseq:

   915   "incseq A \<Longrightarrow> disjointed A (Suc n) = A (Suc n) - A n"

   916   using incseq_Un[of A]

   917   by (simp add: disjointed_def atLeastLessThanSuc_atLeastAtMost atLeast0AtMost)

   918

   919 lemma sigma_algebra_disjoint_iff:

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

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

   922 proof (auto simp add: sigma_algebra_iff)

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

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

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

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

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

   928          disjoint_family (disjointed A) \<longrightarrow>

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

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

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

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

   933 qed

   934

   935 lemma disjoint_family_on_disjoint_image:

   936   "disjoint_family_on A I \<Longrightarrow> disjoint (A  I)"

   937   unfolding disjoint_family_on_def disjoint_def by force

   938

   939 lemma disjoint_image_disjoint_family_on:

   940   assumes d: "disjoint (A  I)" and i: "inj_on A I"

   941   shows "disjoint_family_on A I"

   942   unfolding disjoint_family_on_def

   943 proof (intro ballI impI)

   944   fix n m assume nm: "m \<in> I" "n \<in> I" and "n \<noteq> m"

   945   with i[THEN inj_onD, of n m] show "A n \<inter> A m = {}"

   946     by (intro disjointD[OF d]) auto

   947 qed

   948

   949 subsubsection {* Ring generated by a semiring *}

   950

   951 definition (in semiring_of_sets)

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

   953

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

   955   assumes "a \<in> generated_ring"

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

   957   using assms unfolding generated_ring_def by auto

   958

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

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

   961   shows "a \<in> generated_ring"

   962   using assms unfolding generated_ring_def by auto

   963

   964 lemma (in semiring_of_sets) generated_ringI_Basic:

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

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

   967

   968 lemma (in semiring_of_sets) generated_ring_disjoint_Un[intro]:

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

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

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

   972 proof -

   973   from a guess Ca .. note Ca = this

   974   from b guess Cb .. note Cb = this

   975   show ?thesis

   976   proof

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

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

   979   qed (insert Ca Cb, auto)

   980 qed

   981

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

   983   by (auto simp: generated_ring_def disjoint_def)

   984

   985 lemma (in semiring_of_sets) generated_ring_disjoint_Union:

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

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

   988

   989 lemma (in semiring_of_sets) generated_ring_disjoint_UNION:

   990   "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"

   991   unfolding SUP_def by (intro generated_ring_disjoint_Union) auto

   992

   993 lemma (in semiring_of_sets) generated_ring_Int:

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

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

   996 proof -

   997   from a guess Ca .. note Ca = this

   998   from b guess Cb .. note Cb = this

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

  1000   show ?thesis

  1001   proof

  1002     show "disjoint C"

  1003     proof (simp add: disjoint_def C_def, intro ballI impI)

  1004       fix a1 b1 a2 b2 assume sets: "a1 \<in> Ca" "b1 \<in> Cb" "a2 \<in> Ca" "b2 \<in> Cb"

  1005       assume "a1 \<inter> b1 \<noteq> a2 \<inter> b2"

  1006       then have "a1 \<noteq> a2 \<or> b1 \<noteq> b2" by auto

  1007       then show "(a1 \<inter> b1) \<inter> (a2 \<inter> b2) = {}"

  1008       proof

  1009         assume "a1 \<noteq> a2"

  1010         with sets Ca have "a1 \<inter> a2 = {}"

  1011           by (auto simp: disjoint_def)

  1012         then show ?thesis by auto

  1013       next

  1014         assume "b1 \<noteq> b2"

  1015         with sets Cb have "b1 \<inter> b2 = {}"

  1016           by (auto simp: disjoint_def)

  1017         then show ?thesis by auto

  1018       qed

  1019     qed

  1020   qed (insert Ca Cb, auto simp: C_def)

  1021 qed

  1022

  1023 lemma (in semiring_of_sets) generated_ring_Inter:

  1024   assumes "finite A" "A \<noteq> {}" shows "A \<subseteq> generated_ring \<Longrightarrow> \<Inter>A \<in> generated_ring"

  1025   using assms by (induct A rule: finite_ne_induct) (auto intro: generated_ring_Int)

  1026

  1027 lemma (in semiring_of_sets) generated_ring_INTER:

  1028   "finite I \<Longrightarrow> I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> A i \<in> generated_ring) \<Longrightarrow> INTER I A \<in> generated_ring"

  1029   unfolding INF_def by (intro generated_ring_Inter) auto

  1030

  1031 lemma (in semiring_of_sets) generating_ring:

  1032   "ring_of_sets \<Omega> generated_ring"

  1033 proof (rule ring_of_setsI)

  1034   let ?R = generated_ring

  1035   show "?R \<subseteq> Pow \<Omega>"

  1036     using sets_into_space by (auto simp: generated_ring_def generated_ring_empty)

  1037   show "{} \<in> ?R" by (rule generated_ring_empty)

  1038

  1039   { fix a assume a: "a \<in> ?R" then guess Ca .. note Ca = this

  1040     fix b assume b: "b \<in> ?R" then guess Cb .. note Cb = this

  1041

  1042     show "a - b \<in> ?R"

  1043     proof cases

  1044       assume "Cb = {}" with Cb a \<in> ?R show ?thesis

  1045         by simp

  1046     next

  1047       assume "Cb \<noteq> {}"

  1048       with Ca Cb have "a - b = (\<Union>a'\<in>Ca. \<Inter>b'\<in>Cb. a' - b')" by auto

  1049       also have "\<dots> \<in> ?R"

  1050       proof (intro generated_ring_INTER generated_ring_disjoint_UNION)

  1051         fix a b assume "a \<in> Ca" "b \<in> Cb"

  1052         with Ca Cb Diff_cover[of a b] show "a - b \<in> ?R"

  1053           by (auto simp add: generated_ring_def)

  1054       next

  1055         show "disjoint ((\<lambda>a'. \<Inter>b'\<in>Cb. a' - b')Ca)"

  1056           using Ca by (auto simp add: disjoint_def Cb \<noteq> {})

  1057       next

  1058         show "finite Ca" "finite Cb" "Cb \<noteq> {}" by fact+

  1059       qed

  1060       finally show "a - b \<in> ?R" .

  1061     qed }

  1062   note Diff = this

  1063

  1064   fix a b assume sets: "a \<in> ?R" "b \<in> ?R"

  1065   have "a \<union> b = (a - b) \<union> (a \<inter> b) \<union> (b - a)" by auto

  1066   also have "\<dots> \<in> ?R"

  1067     by (intro sets generated_ring_disjoint_Un generated_ring_Int Diff) auto

  1068   finally show "a \<union> b \<in> ?R" .

  1069 qed

  1070

  1071 lemma (in semiring_of_sets) sigma_sets_generated_ring_eq: "sigma_sets \<Omega> generated_ring = sigma_sets \<Omega> M"

  1072 proof

  1073   interpret M: sigma_algebra \<Omega> "sigma_sets \<Omega> M"

  1074     using space_closed by (rule sigma_algebra_sigma_sets)

  1075   show "sigma_sets \<Omega> generated_ring \<subseteq> sigma_sets \<Omega> M"

  1076     by (blast intro!: sigma_sets_mono elim: generated_ringE)

  1077 qed (auto intro!: generated_ringI_Basic sigma_sets_mono)

  1078

  1079 subsubsection {* A Two-Element Series *}

  1080

  1081 definition binaryset :: "'a set \<Rightarrow> 'a set \<Rightarrow> nat \<Rightarrow> 'a set "

  1082   where "binaryset A B = (\<lambda>x. {})(0 := A, Suc 0 := B)"

  1083

  1084 lemma range_binaryset_eq: "range(binaryset A B) = {A,B,{}}"

  1085   apply (simp add: binaryset_def)

  1086   apply (rule set_eqI)

  1087   apply (auto simp add: image_iff)

  1088   done

  1089

  1090 lemma UN_binaryset_eq: "(\<Union>i. binaryset A B i) = A \<union> B"

  1091   by (simp add: SUP_def range_binaryset_eq)

  1092

  1093 subsubsection {* Closed CDI *}

  1094

  1095 definition closed_cdi where

  1096   "closed_cdi \<Omega> M \<longleftrightarrow>

  1097    M \<subseteq> Pow \<Omega> &

  1098    (\<forall>s \<in> M. \<Omega> - s \<in> M) &

  1099    (\<forall>A. (range A \<subseteq> M) & (A 0 = {}) & (\<forall>n. A n \<subseteq> A (Suc n)) \<longrightarrow>

  1100         (\<Union>i. A i) \<in> M) &

  1101    (\<forall>A. (range A \<subseteq> M) & disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"

  1102

  1103 inductive_set

  1104   smallest_ccdi_sets :: "'a set \<Rightarrow> 'a set set \<Rightarrow> 'a set set"

  1105   for \<Omega> M

  1106   where

  1107     Basic [intro]:

  1108       "a \<in> M \<Longrightarrow> a \<in> smallest_ccdi_sets \<Omega> M"

  1109   | Compl [intro]:

  1110       "a \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> \<Omega> - a \<in> smallest_ccdi_sets \<Omega> M"

  1111   | Inc:

  1112       "range A \<in> Pow(smallest_ccdi_sets \<Omega> M) \<Longrightarrow> A 0 = {} \<Longrightarrow> (\<And>n. A n \<subseteq> A (Suc n))

  1113        \<Longrightarrow> (\<Union>i. A i) \<in> smallest_ccdi_sets \<Omega> M"

  1114   | Disj:

  1115       "range A \<in> Pow(smallest_ccdi_sets \<Omega> M) \<Longrightarrow> disjoint_family A

  1116        \<Longrightarrow> (\<Union>i::nat. A i) \<in> smallest_ccdi_sets \<Omega> M"

  1117

  1118 lemma (in subset_class) smallest_closed_cdi1: "M \<subseteq> smallest_ccdi_sets \<Omega> M"

  1119   by auto

  1120

  1121 lemma (in subset_class) smallest_ccdi_sets: "smallest_ccdi_sets \<Omega> M \<subseteq> Pow \<Omega>"

  1122   apply (rule subsetI)

  1123   apply (erule smallest_ccdi_sets.induct)

  1124   apply (auto intro: range_subsetD dest: sets_into_space)

  1125   done

  1126

  1127 lemma (in subset_class) smallest_closed_cdi2: "closed_cdi \<Omega> (smallest_ccdi_sets \<Omega> M)"

  1128   apply (auto simp add: closed_cdi_def smallest_ccdi_sets)

  1129   apply (blast intro: smallest_ccdi_sets.Inc smallest_ccdi_sets.Disj) +

  1130   done

  1131

  1132 lemma closed_cdi_subset: "closed_cdi \<Omega> M \<Longrightarrow> M \<subseteq> Pow \<Omega>"

  1133   by (simp add: closed_cdi_def)

  1134

  1135 lemma closed_cdi_Compl: "closed_cdi \<Omega> M \<Longrightarrow> s \<in> M \<Longrightarrow> \<Omega> - s \<in> M"

  1136   by (simp add: closed_cdi_def)

  1137

  1138 lemma closed_cdi_Inc:

  1139   "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"

  1140   by (simp add: closed_cdi_def)

  1141

  1142 lemma closed_cdi_Disj:

  1143   "closed_cdi \<Omega> M \<Longrightarrow> range A \<subseteq> M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"

  1144   by (simp add: closed_cdi_def)

  1145

  1146 lemma closed_cdi_Un:

  1147   assumes cdi: "closed_cdi \<Omega> M" and empty: "{} \<in> M"

  1148       and A: "A \<in> M" and B: "B \<in> M"

  1149       and disj: "A \<inter> B = {}"

  1150     shows "A \<union> B \<in> M"

  1151 proof -

  1152   have ra: "range (binaryset A B) \<subseteq> M"

  1153    by (simp add: range_binaryset_eq empty A B)

  1154  have di:  "disjoint_family (binaryset A B)" using disj

  1155    by (simp add: disjoint_family_on_def binaryset_def Int_commute)

  1156  from closed_cdi_Disj [OF cdi ra di]

  1157  show ?thesis

  1158    by (simp add: UN_binaryset_eq)

  1159 qed

  1160

  1161 lemma (in algebra) smallest_ccdi_sets_Un:

  1162   assumes A: "A \<in> smallest_ccdi_sets \<Omega> M" and B: "B \<in> smallest_ccdi_sets \<Omega> M"

  1163       and disj: "A \<inter> B = {}"

  1164     shows "A \<union> B \<in> smallest_ccdi_sets \<Omega> M"

  1165 proof -

  1166   have ra: "range (binaryset A B) \<in> Pow (smallest_ccdi_sets \<Omega> M)"

  1167     by (simp add: range_binaryset_eq  A B smallest_ccdi_sets.Basic)

  1168   have di:  "disjoint_family (binaryset A B)" using disj

  1169     by (simp add: disjoint_family_on_def binaryset_def Int_commute)

  1170   from Disj [OF ra di]

  1171   show ?thesis

  1172     by (simp add: UN_binaryset_eq)

  1173 qed

  1174

  1175 lemma (in algebra) smallest_ccdi_sets_Int1:

  1176   assumes a: "a \<in> M"

  1177   shows "b \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets \<Omega> M"

  1178 proof (induct rule: smallest_ccdi_sets.induct)

  1179   case (Basic x)

  1180   thus ?case

  1181     by (metis a Int smallest_ccdi_sets.Basic)

  1182 next

  1183   case (Compl x)

  1184   have "a \<inter> (\<Omega> - x) = \<Omega> - ((\<Omega> - a) \<union> (a \<inter> x))"

  1185     by blast

  1186   also have "... \<in> smallest_ccdi_sets \<Omega> M"

  1187     by (metis smallest_ccdi_sets.Compl a Compl(2) Diff_Int2 Diff_Int_distrib2

  1188            Diff_disjoint Int_Diff Int_empty_right smallest_ccdi_sets_Un

  1189            smallest_ccdi_sets.Basic smallest_ccdi_sets.Compl)

  1190   finally show ?case .

  1191 next

  1192   case (Inc A)

  1193   have 1: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) = a \<inter> (\<Union>i. A i)"

  1194     by blast

  1195   have "range (\<lambda>i. a \<inter> A i) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Inc

  1196     by blast

  1197   moreover have "(\<lambda>i. a \<inter> A i) 0 = {}"

  1198     by (simp add: Inc)

  1199   moreover have "!!n. (\<lambda>i. a \<inter> A i) n \<subseteq> (\<lambda>i. a \<inter> A i) (Suc n)" using Inc

  1200     by blast

  1201   ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets \<Omega> M"

  1202     by (rule smallest_ccdi_sets.Inc)

  1203   show ?case

  1204     by (metis 1 2)

  1205 next

  1206   case (Disj A)

  1207   have 1: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) = a \<inter> (\<Union>i. A i)"

  1208     by blast

  1209   have "range (\<lambda>i. a \<inter> A i) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Disj

  1210     by blast

  1211   moreover have "disjoint_family (\<lambda>i. a \<inter> A i)" using Disj

  1212     by (auto simp add: disjoint_family_on_def)

  1213   ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets \<Omega> M"

  1214     by (rule smallest_ccdi_sets.Disj)

  1215   show ?case

  1216     by (metis 1 2)

  1217 qed

  1218

  1219

  1220 lemma (in algebra) smallest_ccdi_sets_Int:

  1221   assumes b: "b \<in> smallest_ccdi_sets \<Omega> M"

  1222   shows "a \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets \<Omega> M"

  1223 proof (induct rule: smallest_ccdi_sets.induct)

  1224   case (Basic x)

  1225   thus ?case

  1226     by (metis b smallest_ccdi_sets_Int1)

  1227 next

  1228   case (Compl x)

  1229   have "(\<Omega> - x) \<inter> b = \<Omega> - (x \<inter> b \<union> (\<Omega> - b))"

  1230     by blast

  1231   also have "... \<in> smallest_ccdi_sets \<Omega> M"

  1232     by (metis Compl(2) Diff_disjoint Int_Diff Int_commute Int_empty_right b

  1233            smallest_ccdi_sets.Compl smallest_ccdi_sets_Un)

  1234   finally show ?case .

  1235 next

  1236   case (Inc A)

  1237   have 1: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) = (\<Union>i. A i) \<inter> b"

  1238     by blast

  1239   have "range (\<lambda>i. A i \<inter> b) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Inc

  1240     by blast

  1241   moreover have "(\<lambda>i. A i \<inter> b) 0 = {}"

  1242     by (simp add: Inc)

  1243   moreover have "!!n. (\<lambda>i. A i \<inter> b) n \<subseteq> (\<lambda>i. A i \<inter> b) (Suc n)" using Inc

  1244     by blast

  1245   ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets \<Omega> M"

  1246     by (rule smallest_ccdi_sets.Inc)

  1247   show ?case

  1248     by (metis 1 2)

  1249 next

  1250   case (Disj A)

  1251   have 1: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) = (\<Union>i. A i) \<inter> b"

  1252     by blast

  1253   have "range (\<lambda>i. A i \<inter> b) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Disj

  1254     by blast

  1255   moreover have "disjoint_family (\<lambda>i. A i \<inter> b)" using Disj

  1256     by (auto simp add: disjoint_family_on_def)

  1257   ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets \<Omega> M"

  1258     by (rule smallest_ccdi_sets.Disj)

  1259   show ?case

  1260     by (metis 1 2)

  1261 qed

  1262

  1263 lemma (in algebra) sigma_property_disjoint_lemma:

  1264   assumes sbC: "M \<subseteq> C"

  1265       and ccdi: "closed_cdi \<Omega> C"

  1266   shows "sigma_sets \<Omega> M \<subseteq> C"

  1267 proof -

  1268   have "smallest_ccdi_sets \<Omega> M \<in> {B . M \<subseteq> B \<and> sigma_algebra \<Omega> B}"

  1269     apply (auto simp add: sigma_algebra_disjoint_iff algebra_iff_Int

  1270             smallest_ccdi_sets_Int)

  1271     apply (metis Union_Pow_eq Union_upper subsetD smallest_ccdi_sets)

  1272     apply (blast intro: smallest_ccdi_sets.Disj)

  1273     done

  1274   hence "sigma_sets (\<Omega>) (M) \<subseteq> smallest_ccdi_sets \<Omega> M"

  1275     by clarsimp

  1276        (drule sigma_algebra.sigma_sets_subset [where a="M"], auto)

  1277   also have "...  \<subseteq> C"

  1278     proof

  1279       fix x

  1280       assume x: "x \<in> smallest_ccdi_sets \<Omega> M"

  1281       thus "x \<in> C"

  1282         proof (induct rule: smallest_ccdi_sets.induct)

  1283           case (Basic x)

  1284           thus ?case

  1285             by (metis Basic subsetD sbC)

  1286         next

  1287           case (Compl x)

  1288           thus ?case

  1289             by (blast intro: closed_cdi_Compl [OF ccdi, simplified])

  1290         next

  1291           case (Inc A)

  1292           thus ?case

  1293                by (auto intro: closed_cdi_Inc [OF ccdi, simplified])

  1294         next

  1295           case (Disj A)

  1296           thus ?case

  1297                by (auto intro: closed_cdi_Disj [OF ccdi, simplified])

  1298         qed

  1299     qed

  1300   finally show ?thesis .

  1301 qed

  1302

  1303 lemma (in algebra) sigma_property_disjoint:

  1304   assumes sbC: "M \<subseteq> C"

  1305       and compl: "!!s. s \<in> C \<inter> sigma_sets (\<Omega>) (M) \<Longrightarrow> \<Omega> - s \<in> C"

  1306       and inc: "!!A. range A \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)

  1307                      \<Longrightarrow> A 0 = {} \<Longrightarrow> (!!n. A n \<subseteq> A (Suc n))

  1308                      \<Longrightarrow> (\<Union>i. A i) \<in> C"

  1309       and disj: "!!A. range A \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)

  1310                       \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> C"

  1311   shows "sigma_sets (\<Omega>) (M) \<subseteq> C"

  1312 proof -

  1313   have "sigma_sets (\<Omega>) (M) \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)"

  1314     proof (rule sigma_property_disjoint_lemma)

  1315       show "M \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)"

  1316         by (metis Int_greatest Set.subsetI sbC sigma_sets.Basic)

  1317     next

  1318       show "closed_cdi \<Omega> (C \<inter> sigma_sets (\<Omega>) (M))"

  1319         by (simp add: closed_cdi_def compl inc disj)

  1320            (metis PowI Set.subsetI le_infI2 sigma_sets_into_sp space_closed

  1321              IntE sigma_sets.Compl range_subsetD sigma_sets.Union)

  1322     qed

  1323   thus ?thesis

  1324     by blast

  1325 qed

  1326

  1327 subsubsection {* Dynkin systems *}

  1328

  1329 locale dynkin_system = subset_class +

  1330   assumes space: "\<Omega> \<in> M"

  1331     and   compl[intro!]: "\<And>A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"

  1332     and   UN[intro!]: "\<And>A. disjoint_family A \<Longrightarrow> range A \<subseteq> M

  1333                            \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"

  1334

  1335 lemma (in dynkin_system) empty[intro, simp]: "{} \<in> M"

  1336   using space compl[of "\<Omega>"] by simp

  1337

  1338 lemma (in dynkin_system) diff:

  1339   assumes sets: "D \<in> M" "E \<in> M" and "D \<subseteq> E"

  1340   shows "E - D \<in> M"

  1341 proof -

  1342   let ?f = "\<lambda>x. if x = 0 then D else if x = Suc 0 then \<Omega> - E else {}"

  1343   have "range ?f = {D, \<Omega> - E, {}}"

  1344     by (auto simp: image_iff)

  1345   moreover have "D \<union> (\<Omega> - E) = (\<Union>i. ?f i)"

  1346     by (auto simp: image_iff split: split_if_asm)

  1347   moreover

  1348   have "disjoint_family ?f" unfolding disjoint_family_on_def

  1349     using D \<in> M[THEN sets_into_space] D \<subseteq> E by auto

  1350   ultimately have "\<Omega> - (D \<union> (\<Omega> - E)) \<in> M"

  1351     using sets by auto

  1352   also have "\<Omega> - (D \<union> (\<Omega> - E)) = E - D"

  1353     using assms sets_into_space by auto

  1354   finally show ?thesis .

  1355 qed

  1356

  1357 lemma dynkin_systemI:

  1358   assumes "\<And> A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>" "\<Omega> \<in> M"

  1359   assumes "\<And> A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"

  1360   assumes "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> M

  1361           \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"

  1362   shows "dynkin_system \<Omega> M"

  1363   using assms by (auto simp: dynkin_system_def dynkin_system_axioms_def subset_class_def)

  1364

  1365 lemma dynkin_systemI':

  1366   assumes 1: "\<And> A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>"

  1367   assumes empty: "{} \<in> M"

  1368   assumes Diff: "\<And> A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"

  1369   assumes 2: "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> M

  1370           \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"

  1371   shows "dynkin_system \<Omega> M"

  1372 proof -

  1373   from Diff[OF empty] have "\<Omega> \<in> M" by auto

  1374   from 1 this Diff 2 show ?thesis

  1375     by (intro dynkin_systemI) auto

  1376 qed

  1377

  1378 lemma dynkin_system_trivial:

  1379   shows "dynkin_system A (Pow A)"

  1380   by (rule dynkin_systemI) auto

  1381

  1382 lemma sigma_algebra_imp_dynkin_system:

  1383   assumes "sigma_algebra \<Omega> M" shows "dynkin_system \<Omega> M"

  1384 proof -

  1385   interpret sigma_algebra \<Omega> M by fact

  1386   show ?thesis using sets_into_space by (fastforce intro!: dynkin_systemI)

  1387 qed

  1388

  1389 subsubsection "Intersection sets systems"

  1390

  1391 definition "Int_stable M \<longleftrightarrow> (\<forall> a \<in> M. \<forall> b \<in> M. a \<inter> b \<in> M)"

  1392

  1393 lemma (in algebra) Int_stable: "Int_stable M"

  1394   unfolding Int_stable_def by auto

  1395

  1396 lemma Int_stableI:

  1397   "(\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<inter> b \<in> A) \<Longrightarrow> Int_stable A"

  1398   unfolding Int_stable_def by auto

  1399

  1400 lemma Int_stableD:

  1401   "Int_stable M \<Longrightarrow> a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<inter> b \<in> M"

  1402   unfolding Int_stable_def by auto

  1403

  1404 lemma (in dynkin_system) sigma_algebra_eq_Int_stable:

  1405   "sigma_algebra \<Omega> M \<longleftrightarrow> Int_stable M"

  1406 proof

  1407   assume "sigma_algebra \<Omega> M" then show "Int_stable M"

  1408     unfolding sigma_algebra_def using algebra.Int_stable by auto

  1409 next

  1410   assume "Int_stable M"

  1411   show "sigma_algebra \<Omega> M"

  1412     unfolding sigma_algebra_disjoint_iff algebra_iff_Un

  1413   proof (intro conjI ballI allI impI)

  1414     show "M \<subseteq> Pow (\<Omega>)" using sets_into_space by auto

  1415   next

  1416     fix A B assume "A \<in> M" "B \<in> M"

  1417     then have "A \<union> B = \<Omega> - ((\<Omega> - A) \<inter> (\<Omega> - B))"

  1418               "\<Omega> - A \<in> M" "\<Omega> - B \<in> M"

  1419       using sets_into_space by auto

  1420     then show "A \<union> B \<in> M"

  1421       using Int_stable M unfolding Int_stable_def by auto

  1422   qed auto

  1423 qed

  1424

  1425 subsubsection "Smallest Dynkin systems"

  1426

  1427 definition dynkin where

  1428   "dynkin \<Omega> M =  (\<Inter>{D. dynkin_system \<Omega> D \<and> M \<subseteq> D})"

  1429

  1430 lemma dynkin_system_dynkin:

  1431   assumes "M \<subseteq> Pow (\<Omega>)"

  1432   shows "dynkin_system \<Omega> (dynkin \<Omega> M)"

  1433 proof (rule dynkin_systemI)

  1434   fix A assume "A \<in> dynkin \<Omega> M"

  1435   moreover

  1436   { fix D assume "A \<in> D" and d: "dynkin_system \<Omega> D"

  1437     then have "A \<subseteq> \<Omega>" by (auto simp: dynkin_system_def subset_class_def) }

  1438   moreover have "{D. dynkin_system \<Omega> D \<and> M \<subseteq> D} \<noteq> {}"

  1439     using assms dynkin_system_trivial by fastforce

  1440   ultimately show "A \<subseteq> \<Omega>"

  1441     unfolding dynkin_def using assms

  1442     by auto

  1443 next

  1444   show "\<Omega> \<in> dynkin \<Omega> M"

  1445     unfolding dynkin_def using dynkin_system.space by fastforce

  1446 next

  1447   fix A assume "A \<in> dynkin \<Omega> M"

  1448   then show "\<Omega> - A \<in> dynkin \<Omega> M"

  1449     unfolding dynkin_def using dynkin_system.compl by force

  1450 next

  1451   fix A :: "nat \<Rightarrow> 'a set"

  1452   assume A: "disjoint_family A" "range A \<subseteq> dynkin \<Omega> M"

  1453   show "(\<Union>i. A i) \<in> dynkin \<Omega> M" unfolding dynkin_def

  1454   proof (simp, safe)

  1455     fix D assume "dynkin_system \<Omega> D" "M \<subseteq> D"

  1456     with A have "(\<Union>i. A i) \<in> D"

  1457       by (intro dynkin_system.UN) (auto simp: dynkin_def)

  1458     then show "(\<Union>i. A i) \<in> D" by auto

  1459   qed

  1460 qed

  1461

  1462 lemma dynkin_Basic[intro]: "A \<in> M \<Longrightarrow> A \<in> dynkin \<Omega> M"

  1463   unfolding dynkin_def by auto

  1464

  1465 lemma (in dynkin_system) restricted_dynkin_system:

  1466   assumes "D \<in> M"

  1467   shows "dynkin_system \<Omega> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}"

  1468 proof (rule dynkin_systemI, simp_all)

  1469   have "\<Omega> \<inter> D = D"

  1470     using D \<in> M sets_into_space by auto

  1471   then show "\<Omega> \<inter> D \<in> M"

  1472     using D \<in> M by auto

  1473 next

  1474   fix A assume "A \<subseteq> \<Omega> \<and> A \<inter> D \<in> M"

  1475   moreover have "(\<Omega> - A) \<inter> D = (\<Omega> - (A \<inter> D)) - (\<Omega> - D)"

  1476     by auto

  1477   ultimately show "\<Omega> - A \<subseteq> \<Omega> \<and> (\<Omega> - A) \<inter> D \<in> M"

  1478     using  D \<in> M by (auto intro: diff)

  1479 next

  1480   fix A :: "nat \<Rightarrow> 'a set"

  1481   assume "disjoint_family A" "range A \<subseteq> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}"

  1482   then have "\<And>i. A i \<subseteq> \<Omega>" "disjoint_family (\<lambda>i. A i \<inter> D)"

  1483     "range (\<lambda>i. A i \<inter> D) \<subseteq> M" "(\<Union>x. A x) \<inter> D = (\<Union>x. A x \<inter> D)"

  1484     by ((fastforce simp: disjoint_family_on_def)+)

  1485   then show "(\<Union>x. A x) \<subseteq> \<Omega> \<and> (\<Union>x. A x) \<inter> D \<in> M"

  1486     by (auto simp del: UN_simps)

  1487 qed

  1488

  1489 lemma (in dynkin_system) dynkin_subset:

  1490   assumes "N \<subseteq> M"

  1491   shows "dynkin \<Omega> N \<subseteq> M"

  1492 proof -

  1493   have "dynkin_system \<Omega> M" by default

  1494   then have "dynkin_system \<Omega> M"

  1495     using assms unfolding dynkin_system_def dynkin_system_axioms_def subset_class_def by simp

  1496   with N \<subseteq> M show ?thesis by (auto simp add: dynkin_def)

  1497 qed

  1498

  1499 lemma sigma_eq_dynkin:

  1500   assumes sets: "M \<subseteq> Pow \<Omega>"

  1501   assumes "Int_stable M"

  1502   shows "sigma_sets \<Omega> M = dynkin \<Omega> M"

  1503 proof -

  1504   have "dynkin \<Omega> M \<subseteq> sigma_sets (\<Omega>) (M)"

  1505     using sigma_algebra_imp_dynkin_system

  1506     unfolding dynkin_def sigma_sets_least_sigma_algebra[OF sets] by auto

  1507   moreover

  1508   interpret dynkin_system \<Omega> "dynkin \<Omega> M"

  1509     using dynkin_system_dynkin[OF sets] .

  1510   have "sigma_algebra \<Omega> (dynkin \<Omega> M)"

  1511     unfolding sigma_algebra_eq_Int_stable Int_stable_def

  1512   proof (intro ballI)

  1513     fix A B assume "A \<in> dynkin \<Omega> M" "B \<in> dynkin \<Omega> M"

  1514     let ?D = "\<lambda>E. {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> E \<in> dynkin \<Omega> M}"

  1515     have "M \<subseteq> ?D B"

  1516     proof

  1517       fix E assume "E \<in> M"

  1518       then have "M \<subseteq> ?D E" "E \<in> dynkin \<Omega> M"

  1519         using sets_into_space Int_stable M by (auto simp: Int_stable_def)

  1520       then have "dynkin \<Omega> M \<subseteq> ?D E"

  1521         using restricted_dynkin_system E \<in> dynkin \<Omega> M

  1522         by (intro dynkin_system.dynkin_subset) simp_all

  1523       then have "B \<in> ?D E"

  1524         using B \<in> dynkin \<Omega> M by auto

  1525       then have "E \<inter> B \<in> dynkin \<Omega> M"

  1526         by (subst Int_commute) simp

  1527       then show "E \<in> ?D B"

  1528         using sets E \<in> M by auto

  1529     qed

  1530     then have "dynkin \<Omega> M \<subseteq> ?D B"

  1531       using restricted_dynkin_system B \<in> dynkin \<Omega> M

  1532       by (intro dynkin_system.dynkin_subset) simp_all

  1533     then show "A \<inter> B \<in> dynkin \<Omega> M"

  1534       using A \<in> dynkin \<Omega> M sets_into_space by auto

  1535   qed

  1536   from sigma_algebra.sigma_sets_subset[OF this, of "M"]

  1537   have "sigma_sets (\<Omega>) (M) \<subseteq> dynkin \<Omega> M" by auto

  1538   ultimately have "sigma_sets (\<Omega>) (M) = dynkin \<Omega> M" by auto

  1539   then show ?thesis

  1540     by (auto simp: dynkin_def)

  1541 qed

  1542

  1543 lemma (in dynkin_system) dynkin_idem:

  1544   "dynkin \<Omega> M = M"

  1545 proof -

  1546   have "dynkin \<Omega> M = M"

  1547   proof

  1548     show "M \<subseteq> dynkin \<Omega> M"

  1549       using dynkin_Basic by auto

  1550     show "dynkin \<Omega> M \<subseteq> M"

  1551       by (intro dynkin_subset) auto

  1552   qed

  1553   then show ?thesis

  1554     by (auto simp: dynkin_def)

  1555 qed

  1556

  1557 lemma (in dynkin_system) dynkin_lemma:

  1558   assumes "Int_stable E"

  1559   and E: "E \<subseteq> M" "M \<subseteq> sigma_sets \<Omega> E"

  1560   shows "sigma_sets \<Omega> E = M"

  1561 proof -

  1562   have "E \<subseteq> Pow \<Omega>"

  1563     using E sets_into_space by force

  1564   then have *: "sigma_sets \<Omega> E = dynkin \<Omega> E"

  1565     using Int_stable E by (rule sigma_eq_dynkin)

  1566   then have "dynkin \<Omega> E = M"

  1567     using assms dynkin_subset[OF E(1)] by simp

  1568   with * show ?thesis

  1569     using assms by (auto simp: dynkin_def)

  1570 qed

  1571

  1572 subsubsection {* Induction rule for intersection-stable generators *}

  1573

  1574 text {* The reason to introduce Dynkin-systems is the following induction rules for $\sigma$-algebras

  1575 generated by a generator closed under intersection. *}

  1576

  1577 lemma sigma_sets_induct_disjoint[consumes 3, case_names basic empty compl union]:

  1578   assumes "Int_stable G"

  1579     and closed: "G \<subseteq> Pow \<Omega>"

  1580     and A: "A \<in> sigma_sets \<Omega> G"

  1581   assumes basic: "\<And>A. A \<in> G \<Longrightarrow> P A"

  1582     and empty: "P {}"

  1583     and compl: "\<And>A. A \<in> sigma_sets \<Omega> G \<Longrightarrow> P A \<Longrightarrow> P (\<Omega> - A)"

  1584     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)"

  1585   shows "P A"

  1586 proof -

  1587   let ?D = "{ A \<in> sigma_sets \<Omega> G. P A }"

  1588   interpret sigma_algebra \<Omega> "sigma_sets \<Omega> G"

  1589     using closed by (rule sigma_algebra_sigma_sets)

  1590   from compl[OF _ empty] closed have space: "P \<Omega>" by simp

  1591   interpret dynkin_system \<Omega> ?D

  1592     by default (auto dest: sets_into_space intro!: space compl union)

  1593   have "sigma_sets \<Omega> G = ?D"

  1594     by (rule dynkin_lemma) (auto simp: basic Int_stable G)

  1595   with A show ?thesis by auto

  1596 qed

  1597

  1598 subsection {* Measure type *}

  1599

  1600 definition positive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where

  1601   "positive M \<mu> \<longleftrightarrow> \<mu> {} = 0 \<and> (\<forall>A\<in>M. 0 \<le> \<mu> A)"

  1602

  1603 definition countably_additive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where

  1604   "countably_additive M f \<longleftrightarrow> (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow>

  1605     (\<Sum>i. f (A i)) = f (\<Union>i. A i))"

  1606

  1607 definition measure_space :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where

  1608   "measure_space \<Omega> A \<mu> \<longleftrightarrow> sigma_algebra \<Omega> A \<and> positive A \<mu> \<and> countably_additive A \<mu>"

  1609

  1610 typedef 'a measure = "{(\<Omega>::'a set, A, \<mu>). (\<forall>a\<in>-A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu> }"

  1611 proof

  1612   have "sigma_algebra UNIV {{}, UNIV}"

  1613     by (auto simp: sigma_algebra_iff2)

  1614   then show "(UNIV, {{}, UNIV}, \<lambda>A. 0) \<in> {(\<Omega>, A, \<mu>). (\<forall>a\<in>-A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu>} "

  1615     by (auto simp: measure_space_def positive_def countably_additive_def)

  1616 qed

  1617

  1618 definition space :: "'a measure \<Rightarrow> 'a set" where

  1619   "space M = fst (Rep_measure M)"

  1620

  1621 definition sets :: "'a measure \<Rightarrow> 'a set set" where

  1622   "sets M = fst (snd (Rep_measure M))"

  1623

  1624 definition emeasure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> ereal" where

  1625   "emeasure M = snd (snd (Rep_measure M))"

  1626

  1627 definition measure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> real" where

  1628   "measure M A = real (emeasure M A)"

  1629

  1630 declare [[coercion sets]]

  1631

  1632 declare [[coercion measure]]

  1633

  1634 declare [[coercion emeasure]]

  1635

  1636 lemma measure_space: "measure_space (space M) (sets M) (emeasure M)"

  1637   by (cases M) (auto simp: space_def sets_def emeasure_def Abs_measure_inverse)

  1638

  1639 interpretation sets!: sigma_algebra "space M" "sets M" for M :: "'a measure"

  1640   using measure_space[of M] by (auto simp: measure_space_def)

  1641

  1642 definition measure_of :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> 'a measure" where

  1643   "measure_of \<Omega> A \<mu> = Abs_measure (\<Omega>, if A \<subseteq> Pow \<Omega> then sigma_sets \<Omega> A else {{}, \<Omega>},

  1644     \<lambda>a. if a \<in> sigma_sets \<Omega> A \<and> measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> then \<mu> a else 0)"

  1645

  1646 abbreviation "sigma \<Omega> A \<equiv> measure_of \<Omega> A (\<lambda>x. 0)"

  1647

  1648 lemma measure_space_0: "A \<subseteq> Pow \<Omega> \<Longrightarrow> measure_space \<Omega> (sigma_sets \<Omega> A) (\<lambda>x. 0)"

  1649   unfolding measure_space_def

  1650   by (auto intro!: sigma_algebra_sigma_sets simp: positive_def countably_additive_def)

  1651

  1652 lemma sigma_algebra_trivial: "sigma_algebra \<Omega> {{}, \<Omega>}"

  1653 by unfold_locales(fastforce intro: exI[where x="{{}}"] exI[where x="{\<Omega>}"])+

  1654

  1655 lemma measure_space_0': "measure_space \<Omega> {{}, \<Omega>} (\<lambda>x. 0)"

  1656 by(simp add: measure_space_def positive_def countably_additive_def sigma_algebra_trivial)

  1657

  1658 lemma measure_space_closed:

  1659   assumes "measure_space \<Omega> M \<mu>"

  1660   shows "M \<subseteq> Pow \<Omega>"

  1661 proof -

  1662   interpret sigma_algebra \<Omega> M using assms by(simp add: measure_space_def)

  1663   show ?thesis by(rule space_closed)

  1664 qed

  1665

  1666 lemma (in ring_of_sets) positive_cong_eq:

  1667   "(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> positive M \<mu>' = positive M \<mu>"

  1668   by (auto simp add: positive_def)

  1669

  1670 lemma (in sigma_algebra) countably_additive_eq:

  1671   "(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> countably_additive M \<mu>' = countably_additive M \<mu>"

  1672   unfolding countably_additive_def

  1673   by (intro arg_cong[where f=All] ext) (auto simp add: countably_additive_def subset_eq)

  1674

  1675 lemma measure_space_eq:

  1676   assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a"

  1677   shows "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'"

  1678 proof -

  1679   interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" using closed by (rule sigma_algebra_sigma_sets)

  1680   from positive_cong_eq[OF eq, of "\<lambda>i. i"] countably_additive_eq[OF eq, of "\<lambda>i. i"] show ?thesis

  1681     by (auto simp: measure_space_def)

  1682 qed

  1683

  1684 lemma measure_of_eq:

  1685   assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "(\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a)"

  1686   shows "measure_of \<Omega> A \<mu> = measure_of \<Omega> A \<mu>'"

  1687 proof -

  1688   have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'"

  1689     using assms by (rule measure_space_eq)

  1690   with eq show ?thesis

  1691     by (auto simp add: measure_of_def intro!: arg_cong[where f=Abs_measure])

  1692 qed

  1693

  1694 lemma

  1695   shows space_measure_of_conv: "space (measure_of \<Omega> A \<mu>) = \<Omega>" (is ?space)

  1696   and sets_measure_of_conv:

  1697   "sets (measure_of \<Omega> A \<mu>) = (if A \<subseteq> Pow \<Omega> then sigma_sets \<Omega> A else {{}, \<Omega>})" (is ?sets)

  1698   and emeasure_measure_of_conv:

  1699   "emeasure (measure_of \<Omega> A \<mu>) =

  1700   (\<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)

  1701 proof -

  1702   have "?space \<and> ?sets \<and> ?emeasure"

  1703   proof(cases "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>")

  1704     case True

  1705     from measure_space_closed[OF this] sigma_sets_superset_generator[of A \<Omega>]

  1706     have "A \<subseteq> Pow \<Omega>" by simp

  1707     hence "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A)

  1708       (\<lambda>a. if a \<in> sigma_sets \<Omega> A then \<mu> a else 0)"

  1709       by(rule measure_space_eq) auto

  1710     with True A \<subseteq> Pow \<Omega> show ?thesis

  1711       by(simp add: measure_of_def space_def sets_def emeasure_def Abs_measure_inverse)

  1712   next

  1713     case False thus ?thesis

  1714       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')

  1715   qed

  1716   thus ?space ?sets ?emeasure by simp_all

  1717 qed

  1718

  1719 lemma [simp]:

  1720   assumes A: "A \<subseteq> Pow \<Omega>"

  1721   shows sets_measure_of: "sets (measure_of \<Omega> A \<mu>) = sigma_sets \<Omega> A"

  1722     and space_measure_of: "space (measure_of \<Omega> A \<mu>) = \<Omega>"

  1723 using assms

  1724 by(simp_all add: sets_measure_of_conv space_measure_of_conv)

  1725

  1726 lemma (in sigma_algebra) sets_measure_of_eq[simp]: "sets (measure_of \<Omega> M \<mu>) = M"

  1727   using space_closed by (auto intro!: sigma_sets_eq)

  1728

  1729 lemma (in sigma_algebra) space_measure_of_eq[simp]: "space (measure_of \<Omega> M \<mu>) = \<Omega>"

  1730   by (rule space_measure_of_conv)

  1731

  1732 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>')"

  1733   by (auto intro!: sigma_sets_subseteq)

  1734

  1735 lemma sigma_sets_mono'':

  1736   assumes "A \<in> sigma_sets C D"

  1737   assumes "B \<subseteq> D"

  1738   assumes "D \<subseteq> Pow C"

  1739   shows "sigma_sets A B \<subseteq> sigma_sets C D"

  1740 proof

  1741   fix x assume "x \<in> sigma_sets A B"

  1742   thus "x \<in> sigma_sets C D"

  1743   proof induct

  1744     case (Basic a) with assms have "a \<in> D" by auto

  1745     thus ?case ..

  1746   next

  1747     case Empty show ?case by (rule sigma_sets.Empty)

  1748   next

  1749     from assms have "A \<in> sets (sigma C D)" by (subst sets_measure_of[OF D \<subseteq> Pow C])

  1750     moreover case (Compl a) hence "a \<in> sets (sigma C D)" by (subst sets_measure_of[OF D \<subseteq> Pow C])

  1751     ultimately have "A - a \<in> sets (sigma C D)" ..

  1752     thus ?case by (subst (asm) sets_measure_of[OF D \<subseteq> Pow C])

  1753   next

  1754     case (Union a)

  1755     thus ?case by (intro sigma_sets.Union)

  1756   qed

  1757 qed

  1758

  1759 lemma in_measure_of[intro, simp]: "M \<subseteq> Pow \<Omega> \<Longrightarrow> A \<in> M \<Longrightarrow> A \<in> sets (measure_of \<Omega> M \<mu>)"

  1760   by auto

  1761

  1762 subsubsection {* Constructing simple @{typ "'a measure"} *}

  1763

  1764 lemma emeasure_measure_of:

  1765   assumes M: "M = measure_of \<Omega> A \<mu>"

  1766   assumes ms: "A \<subseteq> Pow \<Omega>" "positive (sets M) \<mu>" "countably_additive (sets M) \<mu>"

  1767   assumes X: "X \<in> sets M"

  1768   shows "emeasure M X = \<mu> X"

  1769 proof -

  1770   interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" by (rule sigma_algebra_sigma_sets) fact

  1771   have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>"

  1772     using ms M by (simp add: measure_space_def sigma_algebra_sigma_sets)

  1773   thus ?thesis using X ms

  1774     by(simp add: M emeasure_measure_of_conv sets_measure_of_conv)

  1775 qed

  1776

  1777 lemma emeasure_measure_of_sigma:

  1778   assumes ms: "sigma_algebra \<Omega> M" "positive M \<mu>" "countably_additive M \<mu>"

  1779   assumes A: "A \<in> M"

  1780   shows "emeasure (measure_of \<Omega> M \<mu>) A = \<mu> A"

  1781 proof -

  1782   interpret sigma_algebra \<Omega> M by fact

  1783   have "measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"

  1784     using ms sigma_sets_eq by (simp add: measure_space_def)

  1785   thus ?thesis by(simp add: emeasure_measure_of_conv A)

  1786 qed

  1787

  1788 lemma measure_cases[cases type: measure]:

  1789   obtains (measure) \<Omega> A \<mu> where "x = Abs_measure (\<Omega>, A, \<mu>)" "\<forall>a\<in>-A. \<mu> a = 0" "measure_space \<Omega> A \<mu>"

  1790   by atomize_elim (cases x, auto)

  1791

  1792 lemma sets_eq_imp_space_eq:

  1793   "sets M = sets M' \<Longrightarrow> space M = space M'"

  1794   using sets.top[of M] sets.top[of M'] sets.space_closed[of M] sets.space_closed[of M']

  1795   by blast

  1796

  1797 lemma emeasure_notin_sets: "A \<notin> sets M \<Longrightarrow> emeasure M A = 0"

  1798   by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)

  1799

  1800 lemma emeasure_neq_0_sets: "emeasure M A \<noteq> 0 \<Longrightarrow> A \<in> sets M"

  1801   using emeasure_notin_sets[of A M] by blast

  1802

  1803 lemma measure_notin_sets: "A \<notin> sets M \<Longrightarrow> measure M A = 0"

  1804   by (simp add: measure_def emeasure_notin_sets)

  1805

  1806 lemma measure_eqI:

  1807   fixes M N :: "'a measure"

  1808   assumes "sets M = sets N" and eq: "\<And>A. A \<in> sets M \<Longrightarrow> emeasure M A = emeasure N A"

  1809   shows "M = N"

  1810 proof (cases M N rule: measure_cases[case_product measure_cases])

  1811   case (measure_measure \<Omega> A \<mu> \<Omega>' A' \<mu>')

  1812   interpret M: sigma_algebra \<Omega> A using measure_measure by (auto simp: measure_space_def)

  1813   interpret N: sigma_algebra \<Omega>' A' using measure_measure by (auto simp: measure_space_def)

  1814   have "A = sets M" "A' = sets N"

  1815     using measure_measure by (simp_all add: sets_def Abs_measure_inverse)

  1816   with sets M = sets N have AA': "A = A'" by simp

  1817   moreover from M.top N.top M.space_closed N.space_closed AA' have "\<Omega> = \<Omega>'" by auto

  1818   moreover { fix B have "\<mu> B = \<mu>' B"

  1819     proof cases

  1820       assume "B \<in> A"

  1821       with eq A = sets M have "emeasure M B = emeasure N B" by simp

  1822       with measure_measure show "\<mu> B = \<mu>' B"

  1823         by (simp add: emeasure_def Abs_measure_inverse)

  1824     next

  1825       assume "B \<notin> A"

  1826       with A = sets M A' = sets N A = A' have "B \<notin> sets M" "B \<notin> sets N"

  1827         by auto

  1828       then have "emeasure M B = 0" "emeasure N B = 0"

  1829         by (simp_all add: emeasure_notin_sets)

  1830       with measure_measure show "\<mu> B = \<mu>' B"

  1831         by (simp add: emeasure_def Abs_measure_inverse)

  1832     qed }

  1833   then have "\<mu> = \<mu>'" by auto

  1834   ultimately show "M = N"

  1835     by (simp add: measure_measure)

  1836 qed

  1837

  1838 lemma emeasure_sigma: "A \<subseteq> Pow \<Omega> \<Longrightarrow> emeasure (sigma \<Omega> A) = (\<lambda>_. 0)"

  1839   using measure_space_0[of A \<Omega>]

  1840   by (simp add: measure_of_def emeasure_def Abs_measure_inverse)

  1841

  1842 lemma sigma_eqI:

  1843   assumes [simp]: "M \<subseteq> Pow \<Omega>" "N \<subseteq> Pow \<Omega>" "sigma_sets \<Omega> M = sigma_sets \<Omega> N"

  1844   shows "sigma \<Omega> M = sigma \<Omega> N"

  1845   by (rule measure_eqI) (simp_all add: emeasure_sigma)

  1846

  1847 subsubsection {* Measurable functions *}

  1848

  1849 definition measurable :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) set" where

  1850   "measurable A B = {f \<in> space A -> space B. \<forall>y \<in> sets B. f - y \<inter> space A \<in> sets A}"

  1851

  1852 lemma measurable_space:

  1853   "f \<in> measurable M A \<Longrightarrow> x \<in> space M \<Longrightarrow> f x \<in> space A"

  1854    unfolding measurable_def by auto

  1855

  1856 lemma measurable_sets:

  1857   "f \<in> measurable M A \<Longrightarrow> S \<in> sets A \<Longrightarrow> f - S \<inter> space M \<in> sets M"

  1858    unfolding measurable_def by auto

  1859

  1860 lemma measurable_sets_Collect:

  1861   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"

  1862 proof -

  1863   have "f - {x \<in> space N. P x} \<inter> space M = {x\<in>space M. P (f x)}"

  1864     using measurable_space[OF f] by auto

  1865   with measurable_sets[OF f P] show ?thesis

  1866     by simp

  1867 qed

  1868

  1869 lemma measurable_sigma_sets:

  1870   assumes B: "sets N = sigma_sets \<Omega> A" "A \<subseteq> Pow \<Omega>"

  1871       and f: "f \<in> space M \<rightarrow> \<Omega>"

  1872       and ba: "\<And>y. y \<in> A \<Longrightarrow> (f - y) \<inter> space M \<in> sets M"

  1873   shows "f \<in> measurable M N"

  1874 proof -

  1875   interpret A: sigma_algebra \<Omega> "sigma_sets \<Omega> A" using B(2) by (rule sigma_algebra_sigma_sets)

  1876   from B sets.top[of N] A.top sets.space_closed[of N] A.space_closed have \<Omega>: "\<Omega> = space N" by force

  1877

  1878   { fix X assume "X \<in> sigma_sets \<Omega> A"

  1879     then have "f - X \<inter> space M \<in> sets M \<and> X \<subseteq> \<Omega>"

  1880       proof induct

  1881         case (Basic a) then show ?case

  1882           by (auto simp add: ba) (metis B(2) subsetD PowD)

  1883       next

  1884         case (Compl a)

  1885         have [simp]: "f - \<Omega> \<inter> space M = space M"

  1886           by (auto simp add: funcset_mem [OF f])

  1887         then show ?case

  1888           by (auto simp add: vimage_Diff Diff_Int_distrib2 sets.compl_sets Compl)

  1889       next

  1890         case (Union a)

  1891         then show ?case

  1892           by (simp add: vimage_UN, simp only: UN_extend_simps(4)) blast

  1893       qed auto }

  1894   with f show ?thesis

  1895     by (auto simp add: measurable_def B \<Omega>)

  1896 qed

  1897

  1898 lemma measurable_measure_of:

  1899   assumes B: "N \<subseteq> Pow \<Omega>"

  1900       and f: "f \<in> space M \<rightarrow> \<Omega>"

  1901       and ba: "\<And>y. y \<in> N \<Longrightarrow> (f - y) \<inter> space M \<in> sets M"

  1902   shows "f \<in> measurable M (measure_of \<Omega> N \<mu>)"

  1903 proof -

  1904   have "sets (measure_of \<Omega> N \<mu>) = sigma_sets \<Omega> N"

  1905     using B by (rule sets_measure_of)

  1906   from this assms show ?thesis by (rule measurable_sigma_sets)

  1907 qed

  1908

  1909 lemma measurable_iff_measure_of:

  1910   assumes "N \<subseteq> Pow \<Omega>" "f \<in> space M \<rightarrow> \<Omega>"

  1911   shows "f \<in> measurable M (measure_of \<Omega> N \<mu>) \<longleftrightarrow> (\<forall>A\<in>N. f - A \<inter> space M \<in> sets M)"

  1912   by (metis assms in_measure_of measurable_measure_of assms measurable_sets)

  1913

  1914 lemma measurable_cong_sets:

  1915   assumes sets: "sets M = sets M'" "sets N = sets N'"

  1916   shows "measurable M N = measurable M' N'"

  1917   using sets[THEN sets_eq_imp_space_eq] sets by (simp add: measurable_def)

  1918

  1919 lemma measurable_cong:

  1920   assumes "\<And> w. w \<in> space M \<Longrightarrow> f w = g w"

  1921   shows "f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable M M'"

  1922   unfolding measurable_def using assms

  1923   by (simp cong: vimage_inter_cong Pi_cong)

  1924

  1925 lemma measurable_cong_strong:

  1926   "M = N \<Longrightarrow> M' = N' \<Longrightarrow> (\<And>w. w \<in> space M \<Longrightarrow> f w = g w) \<Longrightarrow>

  1927     f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable N N'"

  1928   by (metis measurable_cong)

  1929

  1930 lemma measurable_eqI:

  1931      "\<lbrakk> space m1 = space m1' ; space m2 = space m2' ;

  1932         sets m1 = sets m1' ; sets m2 = sets m2' \<rbrakk>

  1933       \<Longrightarrow> measurable m1 m2 = measurable m1' m2'"

  1934   by (simp add: measurable_def sigma_algebra_iff2)

  1935

  1936 lemma measurable_compose:

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

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

  1939 proof -

  1940   have "\<And>A. (\<lambda>x. g (f x)) - A \<inter> space M = f - (g - A \<inter> space N) \<inter> space M"

  1941     using measurable_space[OF f] by auto

  1942   with measurable_space[OF f] measurable_space[OF g] show ?thesis

  1943     by (auto intro: measurable_sets[OF f] measurable_sets[OF g]

  1944              simp del: vimage_Int simp add: measurable_def)

  1945 qed

  1946

  1947 lemma measurable_comp:

  1948   "f \<in> measurable M N \<Longrightarrow> g \<in> measurable N L \<Longrightarrow> g \<circ> f \<in> measurable M L"

  1949   using measurable_compose[of f M N g L] by (simp add: comp_def)

  1950

  1951 lemma measurable_const:

  1952   "c \<in> space M' \<Longrightarrow> (\<lambda>x. c) \<in> measurable M M'"

  1953   by (auto simp add: measurable_def)

  1954

  1955 lemma measurable_If:

  1956   assumes measure: "f \<in> measurable M M'" "g \<in> measurable M M'"

  1957   assumes P: "{x\<in>space M. P x} \<in> sets M"

  1958   shows "(\<lambda>x. if P x then f x else g x) \<in> measurable M M'"

  1959   unfolding measurable_def

  1960 proof safe

  1961   fix x assume "x \<in> space M"

  1962   thus "(if P x then f x else g x) \<in> space M'"

  1963     using measure unfolding measurable_def by auto

  1964 next

  1965   fix A assume "A \<in> sets M'"

  1966   hence *: "(\<lambda>x. if P x then f x else g x) - A \<inter> space M =

  1967     ((f - A \<inter> space M) \<inter> {x\<in>space M. P x}) \<union>

  1968     ((g - A \<inter> space M) \<inter> (space M - {x\<in>space M. P x}))"

  1969     using measure unfolding measurable_def by (auto split: split_if_asm)

  1970   show "(\<lambda>x. if P x then f x else g x) - A \<inter> space M \<in> sets M"

  1971     using A \<in> sets M' measure P unfolding * measurable_def

  1972     by (auto intro!: sets.Un)

  1973 qed

  1974

  1975 lemma measurable_If_set:

  1976   assumes measure: "f \<in> measurable M M'" "g \<in> measurable M M'"

  1977   assumes P: "A \<inter> space M \<in> sets M"

  1978   shows "(\<lambda>x. if x \<in> A then f x else g x) \<in> measurable M M'"

  1979 proof (rule measurable_If[OF measure])

  1980   have "{x \<in> space M. x \<in> A} = A \<inter> space M" by auto

  1981   thus "{x \<in> space M. x \<in> A} \<in> sets M" using A \<inter> space M \<in> sets M by auto

  1982 qed

  1983

  1984 lemma measurable_ident: "id \<in> measurable M M"

  1985   by (auto simp add: measurable_def)

  1986

  1987 lemma measurable_ident_sets:

  1988   assumes eq: "sets M = sets M'" shows "(\<lambda>x. x) \<in> measurable M M'"

  1989   using measurable_ident[of M]

  1990   unfolding id_def measurable_def eq sets_eq_imp_space_eq[OF eq] .

  1991

  1992 lemma sets_Least:

  1993   assumes meas: "\<And>i::nat. {x\<in>space M. P i x} \<in> M"

  1994   shows "(\<lambda>x. LEAST j. P j x) - A \<inter> space M \<in> sets M"

  1995 proof -

  1996   { fix i have "(\<lambda>x. LEAST j. P j x) - {i} \<inter> space M \<in> sets M"

  1997     proof cases

  1998       assume i: "(LEAST j. False) = i"

  1999       have "(\<lambda>x. LEAST j. P j x) - {i} \<inter> space M =

  2000         {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}))"

  2001         by (simp add: set_eq_iff, safe)

  2002            (insert i, auto dest: Least_le intro: LeastI intro!: Least_equality)

  2003       with meas show ?thesis

  2004         by (auto intro!: sets.Int)

  2005     next

  2006       assume i: "(LEAST j. False) \<noteq> i"

  2007       then have "(\<lambda>x. LEAST j. P j x) - {i} \<inter> space M =

  2008         {x\<in>space M. P i x} \<inter> (space M - (\<Union>j<i. {x\<in>space M. P j x}))"

  2009       proof (simp add: set_eq_iff, safe)

  2010         fix x assume neq: "(LEAST j. False) \<noteq> (LEAST j. P j x)"

  2011         have "\<exists>j. P j x"

  2012           by (rule ccontr) (insert neq, auto)

  2013         then show "P (LEAST j. P j x) x" by (rule LeastI_ex)

  2014       qed (auto dest: Least_le intro!: Least_equality)

  2015       with meas show ?thesis

  2016         by auto

  2017     qed }

  2018   then have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) - {i} \<inter> space M) \<in> sets M"

  2019     by (intro sets.countable_UN) auto

  2020   moreover have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) - {i} \<inter> space M) =

  2021     (\<lambda>x. LEAST j. P j x) - A \<inter> space M" by auto

  2022   ultimately show ?thesis by auto

  2023 qed

  2024

  2025 lemma measurable_strong:

  2026   fixes f :: "'a \<Rightarrow> 'b" and g :: "'b \<Rightarrow> 'c"

  2027   assumes f: "f \<in> measurable a b" and g: "g \<in> space b \<rightarrow> space c"

  2028       and t: "f  (space a) \<subseteq> t"

  2029       and cb: "\<And>s. s \<in> sets c \<Longrightarrow> (g - s) \<inter> t \<in> sets b"

  2030   shows "(g o f) \<in> measurable a c"

  2031 proof -

  2032   have fab: "f \<in> (space a -> space b)"

  2033    and ba: "\<And>y. y \<in> sets b \<Longrightarrow> (f - y) \<inter> (space a) \<in> sets a" using f

  2034      by (auto simp add: measurable_def)

  2035   have eq: "\<And>y. (g \<circ> f) - y \<inter> space a = f - (g - y \<inter> t) \<inter> space a" using t

  2036     by force

  2037   show ?thesis

  2038     apply (auto simp add: measurable_def vimage_comp)

  2039     apply (metis funcset_mem fab g)

  2040     apply (subst eq)

  2041     apply (metis ba cb)

  2042     done

  2043 qed

  2044

  2045 lemma measurable_discrete_difference:

  2046   assumes f: "f \<in> measurable M N"

  2047   assumes X: "countable X"

  2048   assumes sets: "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"

  2049   assumes space: "\<And>x. x \<in> X \<Longrightarrow> g x \<in> space N"

  2050   assumes eq: "\<And>x. x \<in> space M \<Longrightarrow> x \<notin> X \<Longrightarrow> f x = g x"

  2051   shows "g \<in> measurable M N"

  2052   unfolding measurable_def

  2053 proof safe

  2054   fix x assume "x \<in> space M" then show "g x \<in> space N"

  2055     using measurable_space[OF f, of x] eq[of x] space[of x] by (cases "x \<in> X") auto

  2056 next

  2057   fix S assume S: "S \<in> sets N"

  2058   have "g - S \<inter> space M = (f - S \<inter> space M) - (\<Union>x\<in>X. {x}) \<union> (\<Union>x\<in>{x\<in>X. g x \<in> S}. {x})"

  2059     using sets.sets_into_space[OF sets] eq by auto

  2060   also have "\<dots> \<in> sets M"

  2061     by (safe intro!: sets.Diff sets.Un measurable_sets[OF f] S sets.countable_UN' X countable_Collect sets)

  2062   finally show "g - S \<inter> space M \<in> sets M" .

  2063 qed

  2064

  2065 lemma measurable_mono1:

  2066   "M' \<subseteq> Pow \<Omega> \<Longrightarrow> M \<subseteq> M' \<Longrightarrow>

  2067     measurable (measure_of \<Omega> M \<mu>) N \<subseteq> measurable (measure_of \<Omega> M' \<mu>') N"

  2068   using measure_of_subset[of M' \<Omega> M] by (auto simp add: measurable_def)

  2069

  2070 subsubsection {* Counting space *}

  2071

  2072 definition count_space :: "'a set \<Rightarrow> 'a measure" where

  2073   "count_space \<Omega> = measure_of \<Omega> (Pow \<Omega>) (\<lambda>A. if finite A then ereal (card A) else \<infinity>)"

  2074

  2075 lemma

  2076   shows space_count_space[simp]: "space (count_space \<Omega>) = \<Omega>"

  2077     and sets_count_space[simp]: "sets (count_space \<Omega>) = Pow \<Omega>"

  2078   using sigma_sets_into_sp[of "Pow \<Omega>" \<Omega>]

  2079   by (auto simp: count_space_def)

  2080

  2081 lemma measurable_count_space_eq1[simp]:

  2082   "f \<in> measurable (count_space A) M \<longleftrightarrow> f \<in> A \<rightarrow> space M"

  2083  unfolding measurable_def by simp

  2084

  2085 lemma measurable_count_space_eq2:

  2086   assumes "finite A"

  2087   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))"

  2088 proof -

  2089   { fix X assume "X \<subseteq> A" "f \<in> space M \<rightarrow> A"

  2090     with finite A have "f - X \<inter> space M = (\<Union>a\<in>X. f - {a} \<inter> space M)" "finite X"

  2091       by (auto dest: finite_subset)

  2092     moreover assume "\<forall>a\<in>A. f - {a} \<inter> space M \<in> sets M"

  2093     ultimately have "f - X \<inter> space M \<in> sets M"

  2094       using X \<subseteq> A by (auto intro!: sets.finite_UN simp del: UN_simps) }

  2095   then show ?thesis

  2096     unfolding measurable_def by auto

  2097 qed

  2098

  2099 lemma measurable_count_space_eq2_countable:

  2100   fixes f :: "'a => 'c::countable"

  2101   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))"

  2102 proof -

  2103   { fix X assume "X \<subseteq> A" "f \<in> space M \<rightarrow> A"

  2104     assume *: "\<And>a. a\<in>A \<Longrightarrow> f - {a} \<inter> space M \<in> sets M"

  2105     have "f - X \<inter> space M = (\<Union>a\<in>X. f - {a} \<inter> space M)"

  2106       by auto

  2107     also have "\<dots> \<in> sets M"

  2108       using * X \<subseteq> A by (intro sets.countable_UN) auto

  2109     finally have "f - X \<inter> space M \<in> sets M" . }

  2110   then show ?thesis

  2111     unfolding measurable_def by auto

  2112 qed

  2113

  2114 lemma measurable_compose_countable:

  2115   assumes f: "\<And>i::'i::countable. (\<lambda>x. f i x) \<in> measurable M N" and g: "g \<in> measurable M (count_space UNIV)"

  2116   shows "(\<lambda>x. f (g x) x) \<in> measurable M N"

  2117   unfolding measurable_def

  2118 proof safe

  2119   fix x assume "x \<in> space M" then show "f (g x) x \<in> space N"

  2120     using f[THEN measurable_space] g[THEN measurable_space] by auto

  2121 next

  2122   fix A assume A: "A \<in> sets N"

  2123   have "(\<lambda>x. f (g x) x) - A \<inter> space M = (\<Union>i. (g - {i} \<inter> space M) \<inter> (f i - A \<inter> space M))"

  2124     by auto

  2125   also have "\<dots> \<in> sets M" using f[THEN measurable_sets, OF A] g[THEN measurable_sets]

  2126     by (auto intro!: sets.countable_UN measurable_sets)

  2127   finally show "(\<lambda>x. f (g x) x) - A \<inter> space M \<in> sets M" .

  2128 qed

  2129

  2130 lemma measurable_count_space_const:

  2131   "(\<lambda>x. c) \<in> measurable M (count_space UNIV)"

  2132   by (simp add: measurable_const)

  2133

  2134 lemma measurable_count_space:

  2135   "f \<in> measurable (count_space A) (count_space UNIV)"

  2136   by simp

  2137

  2138 lemma measurable_compose_rev:

  2139   assumes f: "f \<in> measurable L N" and g: "g \<in> measurable M L"

  2140   shows "(\<lambda>x. f (g x)) \<in> measurable M N"

  2141   using measurable_compose[OF g f] .

  2142

  2143 lemma measurable_count_space_eq_countable:

  2144   assumes "countable A"

  2145   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))"

  2146 proof -

  2147   { fix X assume "X \<subseteq> A" "f \<in> space M \<rightarrow> A"

  2148     with countable A have "f - X \<inter> space M = (\<Union>a\<in>X. f - {a} \<inter> space M)" "countable X"

  2149       by (auto dest: countable_subset)

  2150     moreover assume "\<forall>a\<in>A. f - {a} \<inter> space M \<in> sets M"

  2151     ultimately have "f - X \<inter> space M \<in> sets M"

  2152       using X \<subseteq> A by (auto intro!: sets.countable_UN' simp del: UN_simps) }

  2153   then show ?thesis

  2154     unfolding measurable_def by auto

  2155 qed

  2156

  2157 subsubsection {* Extend measure *}

  2158

  2159 definition "extend_measure \<Omega> I G \<mu> =

  2160   (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)

  2161       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>')

  2162       else measure_of \<Omega> (GI) (\<lambda>_. 0))"

  2163

  2164 lemma space_extend_measure: "G  I \<subseteq> Pow \<Omega> \<Longrightarrow> space (extend_measure \<Omega> I G \<mu>) = \<Omega>"

  2165   unfolding extend_measure_def by simp

  2166

  2167 lemma sets_extend_measure: "G  I \<subseteq> Pow \<Omega> \<Longrightarrow> sets (extend_measure \<Omega> I G \<mu>) = sigma_sets \<Omega> (GI)"

  2168   unfolding extend_measure_def by simp

  2169

  2170 lemma emeasure_extend_measure:

  2171   assumes M: "M = extend_measure \<Omega> I G \<mu>"

  2172     and eq: "\<And>i. i \<in> I \<Longrightarrow> \<mu>' (G i) = \<mu> i"

  2173     and ms: "G  I \<subseteq> Pow \<Omega>" "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'"

  2174     and "i \<in> I"

  2175   shows "emeasure M (G i) = \<mu> i"

  2176 proof cases

  2177   assume *: "(\<forall>i\<in>I. \<mu> i = 0)"

  2178   with M have M_eq: "M = measure_of \<Omega> (GI) (\<lambda>_. 0)"

  2179    by (simp add: extend_measure_def)

  2180   from measure_space_0[OF ms(1)] ms i\<in>I

  2181   have "emeasure M (G i) = 0"

  2182     by (intro emeasure_measure_of[OF M_eq]) (auto simp add: M measure_space_def sets_extend_measure)

  2183   with i\<in>I * show ?thesis

  2184     by simp

  2185 next

  2186   def P \<equiv> "\<lambda>\<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (GI)) \<mu>'"

  2187   assume "\<not> (\<forall>i\<in>I. \<mu> i = 0)"

  2188   moreover

  2189   have "measure_space (space M) (sets M) \<mu>'"

  2190     using ms unfolding measure_space_def by auto default

  2191   with ms eq have "\<exists>\<mu>'. P \<mu>'"

  2192     unfolding P_def

  2193     by (intro exI[of _ \<mu>']) (auto simp add: M space_extend_measure sets_extend_measure)

  2194   ultimately have M_eq: "M = measure_of \<Omega> (GI) (Eps P)"

  2195     by (simp add: M extend_measure_def P_def[symmetric])

  2196

  2197   from \<exists>\<mu>'. P \<mu>' have P: "P (Eps P)" by (rule someI_ex)

  2198   show "emeasure M (G i) = \<mu> i"

  2199   proof (subst emeasure_measure_of[OF M_eq])

  2200     have sets_M: "sets M = sigma_sets \<Omega> (GI)"

  2201       using M_eq ms by (auto simp: sets_extend_measure)

  2202     then show "G i \<in> sets M" using i \<in> I by auto

  2203     show "positive (sets M) (Eps P)" "countably_additive (sets M) (Eps P)" "Eps P (G i) = \<mu> i"

  2204       using P i\<in>I by (auto simp add: sets_M measure_space_def P_def)

  2205   qed fact

  2206 qed

  2207

  2208 lemma emeasure_extend_measure_Pair:

  2209   assumes M: "M = extend_measure \<Omega> {(i, j). I i j} (\<lambda>(i, j). G i j) (\<lambda>(i, j). \<mu> i j)"

  2210     and eq: "\<And>i j. I i j \<Longrightarrow> \<mu>' (G i j) = \<mu> i j"

  2211     and ms: "\<And>i j. I i j \<Longrightarrow> G i j \<in> Pow \<Omega>" "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'"

  2212     and "I i j"

  2213   shows "emeasure M (G i j) = \<mu> i j"

  2214   using emeasure_extend_measure[OF M _ _ ms(2,3), of "(i,j)"] eq ms(1) I i j

  2215   by (auto simp: subset_eq)

  2216

  2217 subsubsection {* Sigma algebra generated by function preimages *}

  2218

  2219 definition

  2220   "vimage_algebra M S X = sigma S ((\<lambda>A. X - A \<inter> S)  sets M)"

  2221

  2222 lemma sigma_algebra_preimages:

  2223   fixes f :: "'x \<Rightarrow> 'a"

  2224   assumes "f \<in> S \<rightarrow> space M"

  2225   shows "sigma_algebra S ((\<lambda>A. f - A \<inter> S)  sets M)"

  2226     (is "sigma_algebra _ (?F  sets M)")

  2227 proof (simp add: sigma_algebra_iff2, safe)

  2228   show "{} \<in> ?F  sets M" by blast

  2229 next

  2230   fix A assume "A \<in> sets M"

  2231   moreover have "S - ?F A = ?F (space M - A)"

  2232     using assms by auto

  2233   ultimately show "S - ?F A \<in> ?F  sets M"

  2234     by blast

  2235 next

  2236   fix A :: "nat \<Rightarrow> 'x set" assume *: "range A \<subseteq> ?F  M"

  2237   have "\<forall>i. \<exists>b. b \<in> M \<and> A i = ?F b"

  2238   proof safe

  2239     fix i

  2240     have "A i \<in> ?F  M" using * by auto

  2241     then show "\<exists>b. b \<in> M \<and> A i = ?F b" by auto

  2242   qed

  2243   from choice[OF this] obtain b where b: "range b \<subseteq> M" "\<And>i. A i = ?F (b i)"

  2244     by auto

  2245   then have "(\<Union>i. A i) = ?F (\<Union>i. b i)" by auto

  2246   then show "(\<Union>i. A i) \<in> ?F  M" using b(1) by blast

  2247 qed

  2248

  2249 lemma sets_vimage_algebra[simp]:

  2250   "f \<in> S \<rightarrow> space M \<Longrightarrow> sets (vimage_algebra M S f) = (\<lambda>A. f - A \<inter> S)  sets M"

  2251   using sigma_algebra.sets_measure_of_eq[OF sigma_algebra_preimages, of f S M]

  2252   by (simp add: vimage_algebra_def)

  2253

  2254 lemma space_vimage_algebra[simp]:

  2255   "f \<in> S \<rightarrow> space M \<Longrightarrow> space (vimage_algebra M S f) = S"

  2256   using sigma_algebra.space_measure_of_eq[OF sigma_algebra_preimages, of f S M]

  2257   by (simp add: vimage_algebra_def)

  2258

  2259 lemma in_vimage_algebra[simp]:

  2260   "f \<in> S \<rightarrow> space M \<Longrightarrow> A \<in> sets (vimage_algebra M S f) \<longleftrightarrow> (\<exists>B\<in>sets M. A = f - B \<inter> S)"

  2261   by (simp add: image_iff)

  2262

  2263 lemma measurable_vimage_algebra:

  2264   fixes S :: "'c set" assumes "f \<in> S \<rightarrow> space M"

  2265   shows "f \<in> measurable (vimage_algebra M S f) M"

  2266   unfolding measurable_def using assms by force

  2267

  2268 lemma measurable_vimage:

  2269   fixes g :: "'a \<Rightarrow> 'c" and f :: "'d \<Rightarrow> 'a"

  2270   assumes "g \<in> measurable M M2" "f \<in> S \<rightarrow> space M"

  2271   shows "(\<lambda>x. g (f x)) \<in> measurable (vimage_algebra M S f) M2"

  2272 proof -

  2273   note measurable_vimage_algebra[OF assms(2)]

  2274   from measurable_comp[OF this assms(1)]

  2275   show ?thesis by (simp add: comp_def)

  2276 qed

  2277

  2278 lemma sigma_sets_vimage:

  2279   assumes "f \<in> S' \<rightarrow> S" and "A \<subseteq> Pow S"

  2280   shows "sigma_sets S' ((\<lambda>X. f - X \<inter> S')  A) = (\<lambda>X. f - X \<inter> S')  sigma_sets S A"

  2281 proof (intro set_eqI iffI)

  2282   let ?F = "\<lambda>X. f - X \<inter> S'"

  2283   fix X assume "X \<in> sigma_sets S' (?F  A)"

  2284   then show "X \<in> ?F  sigma_sets S A"

  2285   proof induct

  2286     case (Basic X) then obtain X' where "X = ?F X'" "X' \<in> A"

  2287       by auto

  2288     then show ?case by auto

  2289   next

  2290     case Empty then show ?case

  2291       by (auto intro!: image_eqI[of _ _ "{}"] sigma_sets.Empty)

  2292   next

  2293     case (Compl X) then obtain X' where X: "X = ?F X'" and "X' \<in> sigma_sets S A"

  2294       by auto

  2295     then have "S - X' \<in> sigma_sets S A"

  2296       by (auto intro!: sigma_sets.Compl)

  2297     then show ?case

  2298       using X assms by (auto intro!: image_eqI[where x="S - X'"])

  2299   next

  2300     case (Union F)

  2301     then have "\<forall>i. \<exists>F'.  F' \<in> sigma_sets S A \<and> F i = f - F' \<inter> S'"

  2302       by (auto simp: image_iff Bex_def)

  2303     from choice[OF this] obtain F' where

  2304       "\<And>i. F' i \<in> sigma_sets S A" and "\<And>i. F i = f - F' i \<inter> S'"

  2305       by auto

  2306     then show ?case

  2307       by (auto intro!: sigma_sets.Union image_eqI[where x="\<Union>i. F' i"])

  2308   qed

  2309 next

  2310   let ?F = "\<lambda>X. f - X \<inter> S'"

  2311   fix X assume "X \<in> ?F  sigma_sets S A"

  2312   then obtain X' where "X' \<in> sigma_sets S A" "X = ?F X'" by auto

  2313   then show "X \<in> sigma_sets S' (?F  A)"

  2314   proof (induct arbitrary: X)

  2315     case Empty then show ?case by (auto intro: sigma_sets.Empty)

  2316   next

  2317     case (Compl X')

  2318     have "S' - (S' - X) \<in> sigma_sets S' (?F  A)"

  2319       apply (rule sigma_sets.Compl)

  2320       using assms by (auto intro!: Compl.hyps simp: Compl.prems)

  2321     also have "S' - (S' - X) = X"

  2322       using assms Compl by auto

  2323     finally show ?case .

  2324   next

  2325     case (Union F)

  2326     have "(\<Union>i. f - F i \<inter> S') \<in> sigma_sets S' (?F  A)"

  2327       by (intro sigma_sets.Union Union.hyps) simp

  2328     also have "(\<Union>i. f - F i \<inter> S') = X"

  2329       using assms Union by auto

  2330     finally show ?case .

  2331   qed auto

  2332 qed

  2333

  2334 subsubsection {* Restricted Space Sigma Algebra *}

  2335

  2336 definition restrict_space where

  2337   "restrict_space M \<Omega> = measure_of (\<Omega> \<inter> space M) ((op \<inter> \<Omega>)  sets M) (emeasure M)"

  2338

  2339 lemma space_restrict_space: "space (restrict_space M \<Omega>) = \<Omega> \<inter> space M"

  2340   using sets.sets_into_space unfolding restrict_space_def by (subst space_measure_of) auto

  2341

  2342 lemma space_restrict_space2: "\<Omega> \<in> sets M \<Longrightarrow> space (restrict_space M \<Omega>) = \<Omega>"

  2343   by (simp add: space_restrict_space sets.sets_into_space)

  2344

  2345 lemma sets_restrict_space: "sets (restrict_space M \<Omega>) = (op \<inter> \<Omega>)  sets M"

  2346 proof -

  2347   have "sigma_sets (\<Omega> \<inter> space M) ((\<lambda>X. X \<inter> (\<Omega> \<inter> space M))  sets M) =

  2348     (\<lambda>X. X \<inter> (\<Omega> \<inter> space M))  sets M"

  2349     using sigma_sets_vimage[of "\<lambda>x. x" "\<Omega> \<inter> space M" "space M" "sets M"] sets.space_closed[of M]

  2350     by (simp add: sets.sigma_sets_eq)

  2351   moreover have "(\<lambda>X. X \<inter> (\<Omega> \<inter> space M))  sets M = (op \<inter> \<Omega>)  sets M"

  2352     using sets.sets_into_space by (intro image_cong) auto

  2353   ultimately show ?thesis

  2354     using sets.sets_into_space[of _ M] unfolding restrict_space_def

  2355     by (subst sets_measure_of) fastforce+

  2356 qed

  2357

  2358 lemma sets_restrict_space_iff:

  2359   "\<Omega> \<inter> space M \<in> sets M \<Longrightarrow> A \<in> sets (restrict_space M \<Omega>) \<longleftrightarrow> (A \<subseteq> \<Omega> \<and> A \<in> sets M)"

  2360 proof (subst sets_restrict_space, safe)

  2361   fix A assume "\<Omega> \<inter> space M \<in> sets M" and A: "A \<in> sets M"

  2362   then have "(\<Omega> \<inter> space M) \<inter> A \<in> sets M"

  2363     by rule

  2364   also have "(\<Omega> \<inter> space M) \<inter> A = \<Omega> \<inter> A"

  2365     using sets.sets_into_space[OF A] by auto

  2366   finally show "\<Omega> \<inter> A \<in> sets M"

  2367     by auto

  2368 qed auto

  2369

  2370 lemma measurable_restrict_space1:

  2371   assumes \<Omega>: "\<Omega> \<inter> space M \<in> sets M" and f: "f \<in> measurable M N"

  2372   shows "f \<in> measurable (restrict_space M \<Omega>) N"

  2373   unfolding measurable_def

  2374 proof (intro CollectI conjI ballI)

  2375   show sp: "f \<in> space (restrict_space M \<Omega>) \<rightarrow> space N"

  2376     using measurable_space[OF f] sets.sets_into_space[OF \<Omega>] by (auto simp: space_restrict_space)

  2377

  2378   fix A assume "A \<in> sets N"

  2379   have "f - A \<inter> space (restrict_space M \<Omega>) = (f - A \<inter> space M) \<inter> (\<Omega> \<inter> space M)"

  2380     using sets.sets_into_space[OF \<Omega>] by (auto simp: space_restrict_space)

  2381   also have "\<dots> \<in> sets (restrict_space M \<Omega>)"

  2382     unfolding sets_restrict_space_iff[OF \<Omega>]

  2383     using measurable_sets[OF f A \<in> sets N] \<Omega> by blast

  2384   finally show "f - A \<inter> space (restrict_space M \<Omega>) \<in> sets (restrict_space M \<Omega>)" .

  2385 qed

  2386

  2387 lemma measurable_restrict_space2:

  2388   "\<Omega> \<inter> space N \<in> sets N \<Longrightarrow> f \<in> space M \<rightarrow> \<Omega> \<Longrightarrow> f \<in> measurable M N \<Longrightarrow>

  2389     f \<in> measurable M (restrict_space N \<Omega>)"

  2390   by (simp add: measurable_def space_restrict_space sets_restrict_space_iff Pi_Int[symmetric])

  2391

  2392 lemma measurable_restrict_space_iff:

  2393   assumes \<Omega>[simp, intro]: "\<Omega> \<inter> space M \<in> sets M" "c \<in> space N"

  2394   shows "f \<in> measurable (restrict_space M \<Omega>) N \<longleftrightarrow>

  2395     (\<lambda>x. if x \<in> \<Omega> then f x else c) \<in> measurable M N" (is "f \<in> measurable ?R N \<longleftrightarrow> ?f \<in> measurable M N")

  2396   unfolding measurable_def

  2397 proof safe

  2398   fix x assume "f \<in> space ?R \<rightarrow> space N" "x \<in> space M" then show "?f x \<in> space N"

  2399     using c\<in>space N by (auto simp: space_restrict_space)

  2400 next

  2401   fix x assume "?f \<in> space M \<rightarrow> space N" "x \<in> space ?R" then show "f x \<in> space N"

  2402     using c\<in>space N by (auto simp: space_restrict_space Pi_iff)

  2403 next

  2404   fix X assume X: "X \<in> sets N"

  2405   assume *[THEN bspec]: "\<forall>y\<in>sets N. f - y \<inter> space ?R \<in> sets ?R"

  2406   have "?f - X \<inter> space M = (f - X \<inter> (\<Omega> \<inter> space M)) \<union> (if c \<in> X then (space M - (\<Omega> \<inter> space M)) else {})"

  2407     by (auto split: split_if_asm)

  2408   also have "\<dots> \<in> sets M"

  2409     using *[OF X] by (auto simp add: space_restrict_space sets_restrict_space_iff)

  2410   finally show "?f - X \<inter> space M \<in> sets M" .

  2411 next

  2412   assume *[THEN bspec]: "\<forall>y\<in>sets N. ?f - y \<inter> space M \<in> sets M"

  2413   fix X :: "'b set" assume X: "X \<in> sets N"

  2414   have "f - X \<inter> (\<Omega> \<inter> space M) = (?f - X \<inter> space M) \<inter> (\<Omega> \<inter> space M)"

  2415     by (auto simp: space_restrict_space)

  2416   also have "\<dots> \<in> sets M"

  2417     using *[OF X] by auto

  2418   finally show "f - X \<inter> space ?R \<in> sets ?R"

  2419     by (auto simp add: sets_restrict_space_iff space_restrict_space)

  2420 qed

  2421

  2422 end

  2423
`