src/HOL/Probability/Sigma_Algebra.thy
 author Andreas Lochbihler Tue Apr 14 14:11:01 2015 +0200 (2015-04-14) changeset 60063 81835db730e8 parent 59415 854fe701c984 child 60727 53697011b03a permissions -rw-r--r--
     1 (*  Title:      HOL/Probability/Sigma_Algebra.thy

     2     Author:     Stefan Richter, Markus Wenzel, TU München

     3     Author:     Johannes Hölzl, TU München

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

     5     translated by Lawrence Paulson.

     6 *)

     7

     8 section {* 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 lemma (in sigma_algebra) countable_INT'':

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

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

   370

   371 lemma (in sigma_algebra) countable:

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

   373   shows "A \<in> M"

   374 proof -

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

   376     using assms by (intro countable_UN') auto

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

   378   finally show ?thesis by auto

   379 qed

   380

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

   382   by (auto simp: ring_of_sets_iff)

   383

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

   385   by (auto simp: algebra_iff_Un)

   386

   387 lemma sigma_algebra_iff:

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

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

   390   by (simp add: sigma_algebra_def sigma_algebra_axioms_def)

   391

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

   393   by (auto simp: sigma_algebra_iff algebra_iff_Int)

   394

   395 lemma (in sigma_algebra) sets_Collect_countable_All:

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

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

   398 proof -

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

   400   with assms show ?thesis by auto

   401 qed

   402

   403 lemma (in sigma_algebra) sets_Collect_countable_Ex:

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

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

   406 proof -

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

   408   with assms show ?thesis by auto

   409 qed

   410

   411 lemma (in sigma_algebra) sets_Collect_countable_Ex':

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

   413   assumes "countable I"

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

   415 proof -

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

   417   with assms show ?thesis

   418     by (auto intro!: countable_UN')

   419 qed

   420

   421 lemma (in sigma_algebra) sets_Collect_countable_All':

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

   423   assumes "countable I"

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

   425 proof -

   426   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

   427   with assms show ?thesis

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

   429 qed

   430

   431 lemma (in sigma_algebra) sets_Collect_countable_Ex1':

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

   433   assumes "countable I"

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

   435 proof -

   436   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)}"

   437     by auto

   438   with assms show ?thesis

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

   440 qed

   441

   442 lemmas (in sigma_algebra) sets_Collect =

   443   sets_Collect_imp sets_Collect_disj sets_Collect_conj sets_Collect_neg sets_Collect_const

   444   sets_Collect_countable_All sets_Collect_countable_Ex sets_Collect_countable_All

   445

   446 lemma (in sigma_algebra) sets_Collect_countable_Ball:

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

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

   449   unfolding Ball_def by (intro sets_Collect assms)

   450

   451 lemma (in sigma_algebra) sets_Collect_countable_Bex:

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

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

   454   unfolding Bex_def by (intro sets_Collect assms)

   455

   456 lemma sigma_algebra_single_set:

   457   assumes "X \<subseteq> S"

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

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

   460

   461 subsubsection {* Binary Unions *}

   462

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

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

   465

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

   467   by (auto simp add: binary_def)

   468

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

   470   by (simp add: SUP_def range_binary_eq)

   471

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

   473   by (simp add: INF_def range_binary_eq)

   474

   475 lemma sigma_algebra_iff2:

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

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

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

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

   480   by (auto simp add: range_binary_eq sigma_algebra_def sigma_algebra_axioms_def

   481          algebra_iff_Un Un_range_binary)

   482

   483 subsubsection {* Initial Sigma Algebra *}

   484

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

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

   487

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

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

   490   where

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

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

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

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

   495

   496 lemma (in sigma_algebra) sigma_sets_subset:

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

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

   499 proof

   500   fix x

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

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

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

   504 qed

   505

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

   507   by (erule sigma_sets.induct, auto)

   508

   509 lemma sigma_algebra_sigma_sets:

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

   511   by (auto simp add: sigma_algebra_iff2 dest: sigma_sets_into_sp

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

   513

   514 lemma sigma_sets_least_sigma_algebra:

   515   assumes "A \<subseteq> Pow S"

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

   517 proof safe

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

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

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

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

   522 next

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

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

   525      by simp

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

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

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

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

   530 qed

   531

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

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

   534

   535 lemma sigma_sets_Un:

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

   537 apply (simp add: Un_range_binary range_binary_eq)

   538 apply (rule Union, simp add: binary_def)

   539 done

   540

   541 lemma sigma_sets_Inter:

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

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

   544 proof -

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

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

   547     by (rule sigma_sets.Compl)

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

   549     by (rule sigma_sets.Union)

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

   551     by (rule sigma_sets.Compl)

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

   553     by auto

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

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

   556   finally show ?thesis .

   557 qed

   558

   559 lemma sigma_sets_INTER:

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

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

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

   563 proof -

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

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

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

   567     by (rule sigma_sets_Inter [OF Asb])

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

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

   570   finally show ?thesis .

   571 qed

   572

   573 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"

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

   575   apply (cases "B = {}")

   576   apply (simp add: sigma_sets.Empty)

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

   578   done

   579

   580 lemma (in sigma_algebra) sigma_sets_eq:

   581      "sigma_sets \<Omega> M = M"

   582 proof

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

   584     by (metis Set.subsetI sigma_sets.Basic)

   585   next

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

   587     by (metis sigma_sets_subset subset_refl)

   588 qed

   589

   590 lemma sigma_sets_eqI:

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

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

   593   shows "sigma_sets M A = sigma_sets M B"

   594 proof (intro set_eqI iffI)

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

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

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

   598 next

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

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

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

   602 qed

   603

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

   605 proof

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

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

   608 qed

   609

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

   611 proof

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

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

   614 qed

   615

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

   617 proof

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

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

   620 qed

   621

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

   623   by (auto intro: sigma_sets.Basic)

   624

   625 lemma (in sigma_algebra) restriction_in_sets:

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

   627   assumes "S \<in> M"

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

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

   630 proof -

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

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

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

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

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

   636 qed

   637

   638 lemma (in sigma_algebra) restricted_sigma_algebra:

   639   assumes "S \<in> M"

   640   shows "sigma_algebra S (restricted_space S)"

   641   unfolding sigma_algebra_def sigma_algebra_axioms_def

   642 proof safe

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

   644 next

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

   646   from restriction_in_sets[OF assms this[simplified]]

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

   648 qed

   649

   650 lemma sigma_sets_Int:

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

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

   653 proof (intro equalityI subsetI)

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

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

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

   657   proof (induct arbitrary: x)

   658     case (Compl a)

   659     then show ?case

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

   661   next

   662     case (Union a)

   663     then show ?case

   664       by (auto intro!: sigma_sets.Union

   665                simp add: UN_extend_simps simp del: UN_simps)

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

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

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

   669 next

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

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

   672   proof induct

   673     case (Compl a)

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

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

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

   677   next

   678     case (Union a)

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

   680       by (auto simp: image_iff Bex_def)

   681     from choice[OF this] guess f ..

   682     then show ?case

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

   684                simp add: image_iff)

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

   686 qed

   687

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

   689 proof (intro set_eqI iffI)

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

   691     by induct blast+

   692 qed (auto intro: sigma_sets.Empty sigma_sets_top)

   693

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

   695 proof (intro set_eqI iffI)

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

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

   698     by induct blast+

   699 next

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

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

   702     by (auto intro: sigma_sets.Empty sigma_sets_top)

   703 qed

   704

   705 lemma sigma_sets_sigma_sets_eq:

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

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

   708

   709 lemma sigma_sets_singleton:

   710   assumes "X \<subseteq> S"

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

   712 proof -

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

   714     by (rule sigma_algebra_single_set) fact

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

   716     by (rule sigma_sets_subseteq) simp

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

   718     using sigma_sets_eq by simp

   719   moreover

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

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

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

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

   724     by (intro antisym) auto

   725   with sigma_sets_eq show ?thesis by simp

   726 qed

   727

   728 lemma restricted_sigma:

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

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

   731     sigma_sets S (algebra.restricted_space M S)"

   732 proof -

   733   from S sigma_sets_into_sp[OF M]

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

   735   from sigma_sets_Int[OF this]

   736   show ?thesis by simp

   737 qed

   738

   739 lemma sigma_sets_vimage_commute:

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

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

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

   743 proof

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

   745   proof clarify

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

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

   748     proof induct

   749       case Empty then show ?case

   750         by (auto intro!: sigma_sets.Empty)

   751     next

   752       case (Compl B)

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

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

   755       with Compl show ?case

   756         by (auto intro!: sigma_sets.Compl)

   757     next

   758       case (Union F)

   759       then show ?case

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

   761                  intro!: sigma_sets.Union)

   762     qed auto

   763   qed

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

   765   proof clarify

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

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

   768     proof induct

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

   770     next

   771       case Empty then show ?case

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

   773     next

   774       case (Compl B)

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

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

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

   778       with A(2) show ?case

   779         by (auto intro: sigma_sets.Compl)

   780     next

   781       case (Union F)

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

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

   784       with A show ?case

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

   786     qed

   787   qed

   788 qed

   789

   790 subsubsection "Disjoint families"

   791

   792 definition

   793   disjoint_family_on  where

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

   795

   796 abbreviation

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

   798

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

   800   by blast

   801

   802 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 = {}"

   803   by (auto simp: disjoint_family_on_def)

   804

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

   806   by blast

   807

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

   809   by blast

   810

   811 lemma disjoint_family_subset:

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

   813   by (force simp add: disjoint_family_on_def)

   814

   815 lemma disjoint_family_on_bisimulation:

   816   assumes "disjoint_family_on f S"

   817   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 = {}"

   818   shows "disjoint_family_on g S"

   819   using assms unfolding disjoint_family_on_def by auto

   820

   821 lemma disjoint_family_on_mono:

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

   823   unfolding disjoint_family_on_def by auto

   824

   825 lemma disjoint_family_Suc:

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

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

   828 proof -

   829   {

   830     fix m

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

   832     proof (induct m)

   833       case 0 show ?case by simp

   834     next

   835       case (Suc m) thus ?case

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

   837     qed

   838   }

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

   840     by (metis add.commute le_add_diff_inverse nat_less_le)

   841   thus ?thesis

   842     by (auto simp add: disjoint_family_on_def)

   843       (metis insert_absorb insert_subset le_SucE le_antisym not_leE)

   844 qed

   845

   846 lemma setsum_indicator_disjoint_family:

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

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

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

   850 proof -

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

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

   853     by auto

   854   thus ?thesis

   855     unfolding indicator_def

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

   857 qed

   858

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

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

   861

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

   863 proof (induct n)

   864   case 0 show ?case by simp

   865 next

   866   case (Suc n)

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

   868 qed

   869

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

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

   872   apply (simp add: finite_UN_disjointed_eq)

   873   done

   874

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

   876   by (auto simp add: disjointed_def)

   877

   878 lemma disjoint_family_disjointed: "disjoint_family (disjointed A)"

   879   by (simp add: disjoint_family_on_def)

   880      (metis neq_iff Int_commute less_disjoint_disjointed)

   881

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

   883   by (auto simp add: disjointed_def)

   884

   885 lemma (in ring_of_sets) UNION_in_sets:

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

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

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

   889 proof (induct n)

   890   case 0 show ?case by simp

   891 next

   892   case (Suc n)

   893   thus ?case

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

   895 qed

   896

   897 lemma (in ring_of_sets) range_disjointed_sets:

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

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

   900 proof (auto simp add: disjointed_def)

   901   fix n

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

   903     by (metis A Diff UNIV_I image_subset_iff)

   904 qed

   905

   906 lemma (in algebra) range_disjointed_sets':

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

   908   using range_disjointed_sets .

   909

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

   911   by (simp add: disjointed_def)

   912

   913 lemma incseq_Un:

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

   915   unfolding incseq_def by auto

   916

   917 lemma disjointed_incseq:

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

   919   using incseq_Un[of A]

   920   by (simp add: disjointed_def atLeastLessThanSuc_atLeastAtMost atLeast0AtMost)

   921

   922 lemma sigma_algebra_disjoint_iff:

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

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

   925 proof (auto simp add: sigma_algebra_iff)

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

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

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

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

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

   931          disjoint_family (disjointed A) \<longrightarrow>

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

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

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

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

   936 qed

   937

   938 lemma disjoint_family_on_disjoint_image:

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

   940   unfolding disjoint_family_on_def disjoint_def by force

   941

   942 lemma disjoint_image_disjoint_family_on:

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

   944   shows "disjoint_family_on A I"

   945   unfolding disjoint_family_on_def

   946 proof (intro ballI impI)

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

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

   949     by (intro disjointD[OF d]) auto

   950 qed

   951

   952 subsubsection {* Ring generated by a semiring *}

   953

   954 definition (in semiring_of_sets)

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

   956

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

   958   assumes "a \<in> generated_ring"

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

   960   using assms unfolding generated_ring_def by auto

   961

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

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

   964   shows "a \<in> generated_ring"

   965   using assms unfolding generated_ring_def by auto

   966

   967 lemma (in semiring_of_sets) generated_ringI_Basic:

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

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

   970

   971 lemma (in semiring_of_sets) generated_ring_disjoint_Un[intro]:

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

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

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

   975 proof -

   976   from a guess Ca .. note Ca = this

   977   from b guess Cb .. note Cb = this

   978   show ?thesis

   979   proof

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

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

   982   qed (insert Ca Cb, auto)

   983 qed

   984

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

   986   by (auto simp: generated_ring_def disjoint_def)

   987

   988 lemma (in semiring_of_sets) generated_ring_disjoint_Union:

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

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

   991

   992 lemma (in semiring_of_sets) generated_ring_disjoint_UNION:

   993   "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"

   994   unfolding SUP_def by (intro generated_ring_disjoint_Union) auto

   995

   996 lemma (in semiring_of_sets) generated_ring_Int:

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

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

   999 proof -

  1000   from a guess Ca .. note Ca = this

  1001   from b guess Cb .. note Cb = this

  1002   def C \<equiv> "(\<lambda>(a,b). a \<inter> b) (Ca\<times>Cb)"

  1003   show ?thesis

  1004   proof

  1005     show "disjoint C"

  1006     proof (simp add: disjoint_def C_def, intro ballI impI)

  1007       fix a1 b1 a2 b2 assume sets: "a1 \<in> Ca" "b1 \<in> Cb" "a2 \<in> Ca" "b2 \<in> Cb"

  1008       assume "a1 \<inter> b1 \<noteq> a2 \<inter> b2"

  1009       then have "a1 \<noteq> a2 \<or> b1 \<noteq> b2" by auto

  1010       then show "(a1 \<inter> b1) \<inter> (a2 \<inter> b2) = {}"

  1011       proof

  1012         assume "a1 \<noteq> a2"

  1013         with sets Ca have "a1 \<inter> a2 = {}"

  1014           by (auto simp: disjoint_def)

  1015         then show ?thesis by auto

  1016       next

  1017         assume "b1 \<noteq> b2"

  1018         with sets Cb have "b1 \<inter> b2 = {}"

  1019           by (auto simp: disjoint_def)

  1020         then show ?thesis by auto

  1021       qed

  1022     qed

  1023   qed (insert Ca Cb, auto simp: C_def)

  1024 qed

  1025

  1026 lemma (in semiring_of_sets) generated_ring_Inter:

  1027   assumes "finite A" "A \<noteq> {}" shows "A \<subseteq> generated_ring \<Longrightarrow> \<Inter>A \<in> generated_ring"

  1028   using assms by (induct A rule: finite_ne_induct) (auto intro: generated_ring_Int)

  1029

  1030 lemma (in semiring_of_sets) generated_ring_INTER:

  1031   "finite I \<Longrightarrow> I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> A i \<in> generated_ring) \<Longrightarrow> INTER I A \<in> generated_ring"

  1032   unfolding INF_def by (intro generated_ring_Inter) auto

  1033

  1034 lemma (in semiring_of_sets) generating_ring:

  1035   "ring_of_sets \<Omega> generated_ring"

  1036 proof (rule ring_of_setsI)

  1037   let ?R = generated_ring

  1038   show "?R \<subseteq> Pow \<Omega>"

  1039     using sets_into_space by (auto simp: generated_ring_def generated_ring_empty)

  1040   show "{} \<in> ?R" by (rule generated_ring_empty)

  1041

  1042   { fix a assume a: "a \<in> ?R" then guess Ca .. note Ca = this

  1043     fix b assume b: "b \<in> ?R" then guess Cb .. note Cb = this

  1044

  1045     show "a - b \<in> ?R"

  1046     proof cases

  1047       assume "Cb = {}" with Cb a \<in> ?R show ?thesis

  1048         by simp

  1049     next

  1050       assume "Cb \<noteq> {}"

  1051       with Ca Cb have "a - b = (\<Union>a'\<in>Ca. \<Inter>b'\<in>Cb. a' - b')" by auto

  1052       also have "\<dots> \<in> ?R"

  1053       proof (intro generated_ring_INTER generated_ring_disjoint_UNION)

  1054         fix a b assume "a \<in> Ca" "b \<in> Cb"

  1055         with Ca Cb Diff_cover[of a b] show "a - b \<in> ?R"

  1056           by (auto simp add: generated_ring_def)

  1057       next

  1058         show "disjoint ((\<lambda>a'. \<Inter>b'\<in>Cb. a' - b')Ca)"

  1059           using Ca by (auto simp add: disjoint_def Cb \<noteq> {})

  1060       next

  1061         show "finite Ca" "finite Cb" "Cb \<noteq> {}" by fact+

  1062       qed

  1063       finally show "a - b \<in> ?R" .

  1064     qed }

  1065   note Diff = this

  1066

  1067   fix a b assume sets: "a \<in> ?R" "b \<in> ?R"

  1068   have "a \<union> b = (a - b) \<union> (a \<inter> b) \<union> (b - a)" by auto

  1069   also have "\<dots> \<in> ?R"

  1070     by (intro sets generated_ring_disjoint_Un generated_ring_Int Diff) auto

  1071   finally show "a \<union> b \<in> ?R" .

  1072 qed

  1073

  1074 lemma (in semiring_of_sets) sigma_sets_generated_ring_eq: "sigma_sets \<Omega> generated_ring = sigma_sets \<Omega> M"

  1075 proof

  1076   interpret M: sigma_algebra \<Omega> "sigma_sets \<Omega> M"

  1077     using space_closed by (rule sigma_algebra_sigma_sets)

  1078   show "sigma_sets \<Omega> generated_ring \<subseteq> sigma_sets \<Omega> M"

  1079     by (blast intro!: sigma_sets_mono elim: generated_ringE)

  1080 qed (auto intro!: generated_ringI_Basic sigma_sets_mono)

  1081

  1082 subsubsection {* A Two-Element Series *}

  1083

  1084 definition binaryset :: "'a set \<Rightarrow> 'a set \<Rightarrow> nat \<Rightarrow> 'a set "

  1085   where "binaryset A B = (\<lambda>x. {})(0 := A, Suc 0 := B)"

  1086

  1087 lemma range_binaryset_eq: "range(binaryset A B) = {A,B,{}}"

  1088   apply (simp add: binaryset_def)

  1089   apply (rule set_eqI)

  1090   apply (auto simp add: image_iff)

  1091   done

  1092

  1093 lemma UN_binaryset_eq: "(\<Union>i. binaryset A B i) = A \<union> B"

  1094   by (simp add: SUP_def range_binaryset_eq)

  1095

  1096 subsubsection {* Closed CDI *}

  1097

  1098 definition closed_cdi where

  1099   "closed_cdi \<Omega> M \<longleftrightarrow>

  1100    M \<subseteq> Pow \<Omega> &

  1101    (\<forall>s \<in> M. \<Omega> - s \<in> M) &

  1102    (\<forall>A. (range A \<subseteq> M) & (A 0 = {}) & (\<forall>n. A n \<subseteq> A (Suc n)) \<longrightarrow>

  1103         (\<Union>i. A i) \<in> M) &

  1104    (\<forall>A. (range A \<subseteq> M) & disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"

  1105

  1106 inductive_set

  1107   smallest_ccdi_sets :: "'a set \<Rightarrow> 'a set set \<Rightarrow> 'a set set"

  1108   for \<Omega> M

  1109   where

  1110     Basic [intro]:

  1111       "a \<in> M \<Longrightarrow> a \<in> smallest_ccdi_sets \<Omega> M"

  1112   | Compl [intro]:

  1113       "a \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> \<Omega> - a \<in> smallest_ccdi_sets \<Omega> M"

  1114   | Inc:

  1115       "range A \<in> Pow(smallest_ccdi_sets \<Omega> M) \<Longrightarrow> A 0 = {} \<Longrightarrow> (\<And>n. A n \<subseteq> A (Suc n))

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

  1117   | Disj:

  1118       "range A \<in> Pow(smallest_ccdi_sets \<Omega> M) \<Longrightarrow> disjoint_family A

  1119        \<Longrightarrow> (\<Union>i::nat. A i) \<in> smallest_ccdi_sets \<Omega> M"

  1120

  1121 lemma (in subset_class) smallest_closed_cdi1: "M \<subseteq> smallest_ccdi_sets \<Omega> M"

  1122   by auto

  1123

  1124 lemma (in subset_class) smallest_ccdi_sets: "smallest_ccdi_sets \<Omega> M \<subseteq> Pow \<Omega>"

  1125   apply (rule subsetI)

  1126   apply (erule smallest_ccdi_sets.induct)

  1127   apply (auto intro: range_subsetD dest: sets_into_space)

  1128   done

  1129

  1130 lemma (in subset_class) smallest_closed_cdi2: "closed_cdi \<Omega> (smallest_ccdi_sets \<Omega> M)"

  1131   apply (auto simp add: closed_cdi_def smallest_ccdi_sets)

  1132   apply (blast intro: smallest_ccdi_sets.Inc smallest_ccdi_sets.Disj) +

  1133   done

  1134

  1135 lemma closed_cdi_subset: "closed_cdi \<Omega> M \<Longrightarrow> M \<subseteq> Pow \<Omega>"

  1136   by (simp add: closed_cdi_def)

  1137

  1138 lemma closed_cdi_Compl: "closed_cdi \<Omega> M \<Longrightarrow> s \<in> M \<Longrightarrow> \<Omega> - s \<in> M"

  1139   by (simp add: closed_cdi_def)

  1140

  1141 lemma closed_cdi_Inc:

  1142   "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"

  1143   by (simp add: closed_cdi_def)

  1144

  1145 lemma closed_cdi_Disj:

  1146   "closed_cdi \<Omega> M \<Longrightarrow> range A \<subseteq> M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"

  1147   by (simp add: closed_cdi_def)

  1148

  1149 lemma closed_cdi_Un:

  1150   assumes cdi: "closed_cdi \<Omega> M" and empty: "{} \<in> M"

  1151       and A: "A \<in> M" and B: "B \<in> M"

  1152       and disj: "A \<inter> B = {}"

  1153     shows "A \<union> B \<in> M"

  1154 proof -

  1155   have ra: "range (binaryset A B) \<subseteq> M"

  1156    by (simp add: range_binaryset_eq empty A B)

  1157  have di:  "disjoint_family (binaryset A B)" using disj

  1158    by (simp add: disjoint_family_on_def binaryset_def Int_commute)

  1159  from closed_cdi_Disj [OF cdi ra di]

  1160  show ?thesis

  1161    by (simp add: UN_binaryset_eq)

  1162 qed

  1163

  1164 lemma (in algebra) smallest_ccdi_sets_Un:

  1165   assumes A: "A \<in> smallest_ccdi_sets \<Omega> M" and B: "B \<in> smallest_ccdi_sets \<Omega> M"

  1166       and disj: "A \<inter> B = {}"

  1167     shows "A \<union> B \<in> smallest_ccdi_sets \<Omega> M"

  1168 proof -

  1169   have ra: "range (binaryset A B) \<in> Pow (smallest_ccdi_sets \<Omega> M)"

  1170     by (simp add: range_binaryset_eq  A B smallest_ccdi_sets.Basic)

  1171   have di:  "disjoint_family (binaryset A B)" using disj

  1172     by (simp add: disjoint_family_on_def binaryset_def Int_commute)

  1173   from Disj [OF ra di]

  1174   show ?thesis

  1175     by (simp add: UN_binaryset_eq)

  1176 qed

  1177

  1178 lemma (in algebra) smallest_ccdi_sets_Int1:

  1179   assumes a: "a \<in> M"

  1180   shows "b \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets \<Omega> M"

  1181 proof (induct rule: smallest_ccdi_sets.induct)

  1182   case (Basic x)

  1183   thus ?case

  1184     by (metis a Int smallest_ccdi_sets.Basic)

  1185 next

  1186   case (Compl x)

  1187   have "a \<inter> (\<Omega> - x) = \<Omega> - ((\<Omega> - a) \<union> (a \<inter> x))"

  1188     by blast

  1189   also have "... \<in> smallest_ccdi_sets \<Omega> M"

  1190     by (metis smallest_ccdi_sets.Compl a Compl(2) Diff_Int2 Diff_Int_distrib2

  1191            Diff_disjoint Int_Diff Int_empty_right smallest_ccdi_sets_Un

  1192            smallest_ccdi_sets.Basic smallest_ccdi_sets.Compl)

  1193   finally show ?case .

  1194 next

  1195   case (Inc A)

  1196   have 1: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) = a \<inter> (\<Union>i. A i)"

  1197     by blast

  1198   have "range (\<lambda>i. a \<inter> A i) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Inc

  1199     by blast

  1200   moreover have "(\<lambda>i. a \<inter> A i) 0 = {}"

  1201     by (simp add: Inc)

  1202   moreover have "!!n. (\<lambda>i. a \<inter> A i) n \<subseteq> (\<lambda>i. a \<inter> A i) (Suc n)" using Inc

  1203     by blast

  1204   ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets \<Omega> M"

  1205     by (rule smallest_ccdi_sets.Inc)

  1206   show ?case

  1207     by (metis 1 2)

  1208 next

  1209   case (Disj A)

  1210   have 1: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) = a \<inter> (\<Union>i. A i)"

  1211     by blast

  1212   have "range (\<lambda>i. a \<inter> A i) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Disj

  1213     by blast

  1214   moreover have "disjoint_family (\<lambda>i. a \<inter> A i)" using Disj

  1215     by (auto simp add: disjoint_family_on_def)

  1216   ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets \<Omega> M"

  1217     by (rule smallest_ccdi_sets.Disj)

  1218   show ?case

  1219     by (metis 1 2)

  1220 qed

  1221

  1222

  1223 lemma (in algebra) smallest_ccdi_sets_Int:

  1224   assumes b: "b \<in> smallest_ccdi_sets \<Omega> M"

  1225   shows "a \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets \<Omega> M"

  1226 proof (induct rule: smallest_ccdi_sets.induct)

  1227   case (Basic x)

  1228   thus ?case

  1229     by (metis b smallest_ccdi_sets_Int1)

  1230 next

  1231   case (Compl x)

  1232   have "(\<Omega> - x) \<inter> b = \<Omega> - (x \<inter> b \<union> (\<Omega> - b))"

  1233     by blast

  1234   also have "... \<in> smallest_ccdi_sets \<Omega> M"

  1235     by (metis Compl(2) Diff_disjoint Int_Diff Int_commute Int_empty_right b

  1236            smallest_ccdi_sets.Compl smallest_ccdi_sets_Un)

  1237   finally show ?case .

  1238 next

  1239   case (Inc A)

  1240   have 1: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) = (\<Union>i. A i) \<inter> b"

  1241     by blast

  1242   have "range (\<lambda>i. A i \<inter> b) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Inc

  1243     by blast

  1244   moreover have "(\<lambda>i. A i \<inter> b) 0 = {}"

  1245     by (simp add: Inc)

  1246   moreover have "!!n. (\<lambda>i. A i \<inter> b) n \<subseteq> (\<lambda>i. A i \<inter> b) (Suc n)" using Inc

  1247     by blast

  1248   ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets \<Omega> M"

  1249     by (rule smallest_ccdi_sets.Inc)

  1250   show ?case

  1251     by (metis 1 2)

  1252 next

  1253   case (Disj A)

  1254   have 1: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) = (\<Union>i. A i) \<inter> b"

  1255     by blast

  1256   have "range (\<lambda>i. A i \<inter> b) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Disj

  1257     by blast

  1258   moreover have "disjoint_family (\<lambda>i. A i \<inter> b)" using Disj

  1259     by (auto simp add: disjoint_family_on_def)

  1260   ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets \<Omega> M"

  1261     by (rule smallest_ccdi_sets.Disj)

  1262   show ?case

  1263     by (metis 1 2)

  1264 qed

  1265

  1266 lemma (in algebra) sigma_property_disjoint_lemma:

  1267   assumes sbC: "M \<subseteq> C"

  1268       and ccdi: "closed_cdi \<Omega> C"

  1269   shows "sigma_sets \<Omega> M \<subseteq> C"

  1270 proof -

  1271   have "smallest_ccdi_sets \<Omega> M \<in> {B . M \<subseteq> B \<and> sigma_algebra \<Omega> B}"

  1272     apply (auto simp add: sigma_algebra_disjoint_iff algebra_iff_Int

  1273             smallest_ccdi_sets_Int)

  1274     apply (metis Union_Pow_eq Union_upper subsetD smallest_ccdi_sets)

  1275     apply (blast intro: smallest_ccdi_sets.Disj)

  1276     done

  1277   hence "sigma_sets (\<Omega>) (M) \<subseteq> smallest_ccdi_sets \<Omega> M"

  1278     by clarsimp

  1279        (drule sigma_algebra.sigma_sets_subset [where a="M"], auto)

  1280   also have "...  \<subseteq> C"

  1281     proof

  1282       fix x

  1283       assume x: "x \<in> smallest_ccdi_sets \<Omega> M"

  1284       thus "x \<in> C"

  1285         proof (induct rule: smallest_ccdi_sets.induct)

  1286           case (Basic x)

  1287           thus ?case

  1288             by (metis Basic subsetD sbC)

  1289         next

  1290           case (Compl x)

  1291           thus ?case

  1292             by (blast intro: closed_cdi_Compl [OF ccdi, simplified])

  1293         next

  1294           case (Inc A)

  1295           thus ?case

  1296                by (auto intro: closed_cdi_Inc [OF ccdi, simplified])

  1297         next

  1298           case (Disj A)

  1299           thus ?case

  1300                by (auto intro: closed_cdi_Disj [OF ccdi, simplified])

  1301         qed

  1302     qed

  1303   finally show ?thesis .

  1304 qed

  1305

  1306 lemma (in algebra) sigma_property_disjoint:

  1307   assumes sbC: "M \<subseteq> C"

  1308       and compl: "!!s. s \<in> C \<inter> sigma_sets (\<Omega>) (M) \<Longrightarrow> \<Omega> - s \<in> C"

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

  1310                      \<Longrightarrow> A 0 = {} \<Longrightarrow> (!!n. A n \<subseteq> A (Suc n))

  1311                      \<Longrightarrow> (\<Union>i. A i) \<in> C"

  1312       and disj: "!!A. range A \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)

  1313                       \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> C"

  1314   shows "sigma_sets (\<Omega>) (M) \<subseteq> C"

  1315 proof -

  1316   have "sigma_sets (\<Omega>) (M) \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)"

  1317     proof (rule sigma_property_disjoint_lemma)

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

  1319         by (metis Int_greatest Set.subsetI sbC sigma_sets.Basic)

  1320     next

  1321       show "closed_cdi \<Omega> (C \<inter> sigma_sets (\<Omega>) (M))"

  1322         by (simp add: closed_cdi_def compl inc disj)

  1323            (metis PowI Set.subsetI le_infI2 sigma_sets_into_sp space_closed

  1324              IntE sigma_sets.Compl range_subsetD sigma_sets.Union)

  1325     qed

  1326   thus ?thesis

  1327     by blast

  1328 qed

  1329

  1330 subsubsection {* Dynkin systems *}

  1331

  1332 locale dynkin_system = subset_class +

  1333   assumes space: "\<Omega> \<in> M"

  1334     and   compl[intro!]: "\<And>A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"

  1335     and   UN[intro!]: "\<And>A. disjoint_family A \<Longrightarrow> range A \<subseteq> M

  1336                            \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"

  1337

  1338 lemma (in dynkin_system) empty[intro, simp]: "{} \<in> M"

  1339   using space compl[of "\<Omega>"] by simp

  1340

  1341 lemma (in dynkin_system) diff:

  1342   assumes sets: "D \<in> M" "E \<in> M" and "D \<subseteq> E"

  1343   shows "E - D \<in> M"

  1344 proof -

  1345   let ?f = "\<lambda>x. if x = 0 then D else if x = Suc 0 then \<Omega> - E else {}"

  1346   have "range ?f = {D, \<Omega> - E, {}}"

  1347     by (auto simp: image_iff)

  1348   moreover have "D \<union> (\<Omega> - E) = (\<Union>i. ?f i)"

  1349     by (auto simp: image_iff split: split_if_asm)

  1350   moreover

  1351   have "disjoint_family ?f" unfolding disjoint_family_on_def

  1352     using D \<in> M[THEN sets_into_space] D \<subseteq> E by auto

  1353   ultimately have "\<Omega> - (D \<union> (\<Omega> - E)) \<in> M"

  1354     using sets by auto

  1355   also have "\<Omega> - (D \<union> (\<Omega> - E)) = E - D"

  1356     using assms sets_into_space by auto

  1357   finally show ?thesis .

  1358 qed

  1359

  1360 lemma dynkin_systemI:

  1361   assumes "\<And> A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>" "\<Omega> \<in> M"

  1362   assumes "\<And> A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"

  1363   assumes "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> M

  1364           \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"

  1365   shows "dynkin_system \<Omega> M"

  1366   using assms by (auto simp: dynkin_system_def dynkin_system_axioms_def subset_class_def)

  1367

  1368 lemma dynkin_systemI':

  1369   assumes 1: "\<And> A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>"

  1370   assumes empty: "{} \<in> M"

  1371   assumes Diff: "\<And> A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"

  1372   assumes 2: "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> M

  1373           \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"

  1374   shows "dynkin_system \<Omega> M"

  1375 proof -

  1376   from Diff[OF empty] have "\<Omega> \<in> M" by auto

  1377   from 1 this Diff 2 show ?thesis

  1378     by (intro dynkin_systemI) auto

  1379 qed

  1380

  1381 lemma dynkin_system_trivial:

  1382   shows "dynkin_system A (Pow A)"

  1383   by (rule dynkin_systemI) auto

  1384

  1385 lemma sigma_algebra_imp_dynkin_system:

  1386   assumes "sigma_algebra \<Omega> M" shows "dynkin_system \<Omega> M"

  1387 proof -

  1388   interpret sigma_algebra \<Omega> M by fact

  1389   show ?thesis using sets_into_space by (fastforce intro!: dynkin_systemI)

  1390 qed

  1391

  1392 subsubsection "Intersection sets systems"

  1393

  1394 definition "Int_stable M \<longleftrightarrow> (\<forall> a \<in> M. \<forall> b \<in> M. a \<inter> b \<in> M)"

  1395

  1396 lemma (in algebra) Int_stable: "Int_stable M"

  1397   unfolding Int_stable_def by auto

  1398

  1399 lemma Int_stableI:

  1400   "(\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<inter> b \<in> A) \<Longrightarrow> Int_stable A"

  1401   unfolding Int_stable_def by auto

  1402

  1403 lemma Int_stableD:

  1404   "Int_stable M \<Longrightarrow> a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<inter> b \<in> M"

  1405   unfolding Int_stable_def by auto

  1406

  1407 lemma (in dynkin_system) sigma_algebra_eq_Int_stable:

  1408   "sigma_algebra \<Omega> M \<longleftrightarrow> Int_stable M"

  1409 proof

  1410   assume "sigma_algebra \<Omega> M" then show "Int_stable M"

  1411     unfolding sigma_algebra_def using algebra.Int_stable by auto

  1412 next

  1413   assume "Int_stable M"

  1414   show "sigma_algebra \<Omega> M"

  1415     unfolding sigma_algebra_disjoint_iff algebra_iff_Un

  1416   proof (intro conjI ballI allI impI)

  1417     show "M \<subseteq> Pow (\<Omega>)" using sets_into_space by auto

  1418   next

  1419     fix A B assume "A \<in> M" "B \<in> M"

  1420     then have "A \<union> B = \<Omega> - ((\<Omega> - A) \<inter> (\<Omega> - B))"

  1421               "\<Omega> - A \<in> M" "\<Omega> - B \<in> M"

  1422       using sets_into_space by auto

  1423     then show "A \<union> B \<in> M"

  1424       using Int_stable M unfolding Int_stable_def by auto

  1425   qed auto

  1426 qed

  1427

  1428 subsubsection "Smallest Dynkin systems"

  1429

  1430 definition dynkin where

  1431   "dynkin \<Omega> M =  (\<Inter>{D. dynkin_system \<Omega> D \<and> M \<subseteq> D})"

  1432

  1433 lemma dynkin_system_dynkin:

  1434   assumes "M \<subseteq> Pow (\<Omega>)"

  1435   shows "dynkin_system \<Omega> (dynkin \<Omega> M)"

  1436 proof (rule dynkin_systemI)

  1437   fix A assume "A \<in> dynkin \<Omega> M"

  1438   moreover

  1439   { fix D assume "A \<in> D" and d: "dynkin_system \<Omega> D"

  1440     then have "A \<subseteq> \<Omega>" by (auto simp: dynkin_system_def subset_class_def) }

  1441   moreover have "{D. dynkin_system \<Omega> D \<and> M \<subseteq> D} \<noteq> {}"

  1442     using assms dynkin_system_trivial by fastforce

  1443   ultimately show "A \<subseteq> \<Omega>"

  1444     unfolding dynkin_def using assms

  1445     by auto

  1446 next

  1447   show "\<Omega> \<in> dynkin \<Omega> M"

  1448     unfolding dynkin_def using dynkin_system.space by fastforce

  1449 next

  1450   fix A assume "A \<in> dynkin \<Omega> M"

  1451   then show "\<Omega> - A \<in> dynkin \<Omega> M"

  1452     unfolding dynkin_def using dynkin_system.compl by force

  1453 next

  1454   fix A :: "nat \<Rightarrow> 'a set"

  1455   assume A: "disjoint_family A" "range A \<subseteq> dynkin \<Omega> M"

  1456   show "(\<Union>i. A i) \<in> dynkin \<Omega> M" unfolding dynkin_def

  1457   proof (simp, safe)

  1458     fix D assume "dynkin_system \<Omega> D" "M \<subseteq> D"

  1459     with A have "(\<Union>i. A i) \<in> D"

  1460       by (intro dynkin_system.UN) (auto simp: dynkin_def)

  1461     then show "(\<Union>i. A i) \<in> D" by auto

  1462   qed

  1463 qed

  1464

  1465 lemma dynkin_Basic[intro]: "A \<in> M \<Longrightarrow> A \<in> dynkin \<Omega> M"

  1466   unfolding dynkin_def by auto

  1467

  1468 lemma (in dynkin_system) restricted_dynkin_system:

  1469   assumes "D \<in> M"

  1470   shows "dynkin_system \<Omega> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}"

  1471 proof (rule dynkin_systemI, simp_all)

  1472   have "\<Omega> \<inter> D = D"

  1473     using D \<in> M sets_into_space by auto

  1474   then show "\<Omega> \<inter> D \<in> M"

  1475     using D \<in> M by auto

  1476 next

  1477   fix A assume "A \<subseteq> \<Omega> \<and> A \<inter> D \<in> M"

  1478   moreover have "(\<Omega> - A) \<inter> D = (\<Omega> - (A \<inter> D)) - (\<Omega> - D)"

  1479     by auto

  1480   ultimately show "\<Omega> - A \<subseteq> \<Omega> \<and> (\<Omega> - A) \<inter> D \<in> M"

  1481     using  D \<in> M by (auto intro: diff)

  1482 next

  1483   fix A :: "nat \<Rightarrow> 'a set"

  1484   assume "disjoint_family A" "range A \<subseteq> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}"

  1485   then have "\<And>i. A i \<subseteq> \<Omega>" "disjoint_family (\<lambda>i. A i \<inter> D)"

  1486     "range (\<lambda>i. A i \<inter> D) \<subseteq> M" "(\<Union>x. A x) \<inter> D = (\<Union>x. A x \<inter> D)"

  1487     by ((fastforce simp: disjoint_family_on_def)+)

  1488   then show "(\<Union>x. A x) \<subseteq> \<Omega> \<and> (\<Union>x. A x) \<inter> D \<in> M"

  1489     by (auto simp del: UN_simps)

  1490 qed

  1491

  1492 lemma (in dynkin_system) dynkin_subset:

  1493   assumes "N \<subseteq> M"

  1494   shows "dynkin \<Omega> N \<subseteq> M"

  1495 proof -

  1496   have "dynkin_system \<Omega> M" by default

  1497   then have "dynkin_system \<Omega> M"

  1498     using assms unfolding dynkin_system_def dynkin_system_axioms_def subset_class_def by simp

  1499   with N \<subseteq> M show ?thesis by (auto simp add: dynkin_def)

  1500 qed

  1501

  1502 lemma sigma_eq_dynkin:

  1503   assumes sets: "M \<subseteq> Pow \<Omega>"

  1504   assumes "Int_stable M"

  1505   shows "sigma_sets \<Omega> M = dynkin \<Omega> M"

  1506 proof -

  1507   have "dynkin \<Omega> M \<subseteq> sigma_sets (\<Omega>) (M)"

  1508     using sigma_algebra_imp_dynkin_system

  1509     unfolding dynkin_def sigma_sets_least_sigma_algebra[OF sets] by auto

  1510   moreover

  1511   interpret dynkin_system \<Omega> "dynkin \<Omega> M"

  1512     using dynkin_system_dynkin[OF sets] .

  1513   have "sigma_algebra \<Omega> (dynkin \<Omega> M)"

  1514     unfolding sigma_algebra_eq_Int_stable Int_stable_def

  1515   proof (intro ballI)

  1516     fix A B assume "A \<in> dynkin \<Omega> M" "B \<in> dynkin \<Omega> M"

  1517     let ?D = "\<lambda>E. {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> E \<in> dynkin \<Omega> M}"

  1518     have "M \<subseteq> ?D B"

  1519     proof

  1520       fix E assume "E \<in> M"

  1521       then have "M \<subseteq> ?D E" "E \<in> dynkin \<Omega> M"

  1522         using sets_into_space Int_stable M by (auto simp: Int_stable_def)

  1523       then have "dynkin \<Omega> M \<subseteq> ?D E"

  1524         using restricted_dynkin_system E \<in> dynkin \<Omega> M

  1525         by (intro dynkin_system.dynkin_subset) simp_all

  1526       then have "B \<in> ?D E"

  1527         using B \<in> dynkin \<Omega> M by auto

  1528       then have "E \<inter> B \<in> dynkin \<Omega> M"

  1529         by (subst Int_commute) simp

  1530       then show "E \<in> ?D B"

  1531         using sets E \<in> M by auto

  1532     qed

  1533     then have "dynkin \<Omega> M \<subseteq> ?D B"

  1534       using restricted_dynkin_system B \<in> dynkin \<Omega> M

  1535       by (intro dynkin_system.dynkin_subset) simp_all

  1536     then show "A \<inter> B \<in> dynkin \<Omega> M"

  1537       using A \<in> dynkin \<Omega> M sets_into_space by auto

  1538   qed

  1539   from sigma_algebra.sigma_sets_subset[OF this, of "M"]

  1540   have "sigma_sets (\<Omega>) (M) \<subseteq> dynkin \<Omega> M" by auto

  1541   ultimately have "sigma_sets (\<Omega>) (M) = dynkin \<Omega> M" by auto

  1542   then show ?thesis

  1543     by (auto simp: dynkin_def)

  1544 qed

  1545

  1546 lemma (in dynkin_system) dynkin_idem:

  1547   "dynkin \<Omega> M = M"

  1548 proof -

  1549   have "dynkin \<Omega> M = M"

  1550   proof

  1551     show "M \<subseteq> dynkin \<Omega> M"

  1552       using dynkin_Basic by auto

  1553     show "dynkin \<Omega> M \<subseteq> M"

  1554       by (intro dynkin_subset) auto

  1555   qed

  1556   then show ?thesis

  1557     by (auto simp: dynkin_def)

  1558 qed

  1559

  1560 lemma (in dynkin_system) dynkin_lemma:

  1561   assumes "Int_stable E"

  1562   and E: "E \<subseteq> M" "M \<subseteq> sigma_sets \<Omega> E"

  1563   shows "sigma_sets \<Omega> E = M"

  1564 proof -

  1565   have "E \<subseteq> Pow \<Omega>"

  1566     using E sets_into_space by force

  1567   then have *: "sigma_sets \<Omega> E = dynkin \<Omega> E"

  1568     using Int_stable E by (rule sigma_eq_dynkin)

  1569   then have "dynkin \<Omega> E = M"

  1570     using assms dynkin_subset[OF E(1)] by simp

  1571   with * show ?thesis

  1572     using assms by (auto simp: dynkin_def)

  1573 qed

  1574

  1575 subsubsection {* Induction rule for intersection-stable generators *}

  1576

  1577 text {* The reason to introduce Dynkin-systems is the following induction rules for $\sigma$-algebras

  1578 generated by a generator closed under intersection. *}

  1579

  1580 lemma sigma_sets_induct_disjoint[consumes 3, case_names basic empty compl union]:

  1581   assumes "Int_stable G"

  1582     and closed: "G \<subseteq> Pow \<Omega>"

  1583     and A: "A \<in> sigma_sets \<Omega> G"

  1584   assumes basic: "\<And>A. A \<in> G \<Longrightarrow> P A"

  1585     and empty: "P {}"

  1586     and compl: "\<And>A. A \<in> sigma_sets \<Omega> G \<Longrightarrow> P A \<Longrightarrow> P (\<Omega> - A)"

  1587     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)"

  1588   shows "P A"

  1589 proof -

  1590   let ?D = "{ A \<in> sigma_sets \<Omega> G. P A }"

  1591   interpret sigma_algebra \<Omega> "sigma_sets \<Omega> G"

  1592     using closed by (rule sigma_algebra_sigma_sets)

  1593   from compl[OF _ empty] closed have space: "P \<Omega>" by simp

  1594   interpret dynkin_system \<Omega> ?D

  1595     by default (auto dest: sets_into_space intro!: space compl union)

  1596   have "sigma_sets \<Omega> G = ?D"

  1597     by (rule dynkin_lemma) (auto simp: basic Int_stable G)

  1598   with A show ?thesis by auto

  1599 qed

  1600

  1601 subsection {* Measure type *}

  1602

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

  1604   "positive M \<mu> \<longleftrightarrow> \<mu> {} = 0 \<and> (\<forall>A\<in>M. 0 \<le> \<mu> A)"

  1605

  1606 definition countably_additive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where

  1607   "countably_additive M f \<longleftrightarrow> (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow>

  1608     (\<Sum>i. f (A i)) = f (\<Union>i. A i))"

  1609

  1610 definition measure_space :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where

  1611   "measure_space \<Omega> A \<mu> \<longleftrightarrow> sigma_algebra \<Omega> A \<and> positive A \<mu> \<and> countably_additive A \<mu>"

  1612

  1613 typedef 'a measure = "{(\<Omega>::'a set, A, \<mu>). (\<forall>a\<in>-A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu> }"

  1614 proof

  1615   have "sigma_algebra UNIV {{}, UNIV}"

  1616     by (auto simp: sigma_algebra_iff2)

  1617   then show "(UNIV, {{}, UNIV}, \<lambda>A. 0) \<in> {(\<Omega>, A, \<mu>). (\<forall>a\<in>-A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu>} "

  1618     by (auto simp: measure_space_def positive_def countably_additive_def)

  1619 qed

  1620

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

  1622   "space M = fst (Rep_measure M)"

  1623

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

  1625   "sets M = fst (snd (Rep_measure M))"

  1626

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

  1628   "emeasure M = snd (snd (Rep_measure M))"

  1629

  1630 definition measure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> real" where

  1631   "measure M A = real (emeasure M A)"

  1632

  1633 declare [[coercion sets]]

  1634

  1635 declare [[coercion measure]]

  1636

  1637 declare [[coercion emeasure]]

  1638

  1639 lemma measure_space: "measure_space (space M) (sets M) (emeasure M)"

  1640   by (cases M) (auto simp: space_def sets_def emeasure_def Abs_measure_inverse)

  1641

  1642 interpretation sets!: sigma_algebra "space M" "sets M" for M :: "'a measure"

  1643   using measure_space[of M] by (auto simp: measure_space_def)

  1644

  1645 definition measure_of :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> 'a measure" where

  1646   "measure_of \<Omega> A \<mu> = Abs_measure (\<Omega>, if A \<subseteq> Pow \<Omega> then sigma_sets \<Omega> A else {{}, \<Omega>},

  1647     \<lambda>a. if a \<in> sigma_sets \<Omega> A \<and> measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> then \<mu> a else 0)"

  1648

  1649 abbreviation "sigma \<Omega> A \<equiv> measure_of \<Omega> A (\<lambda>x. 0)"

  1650

  1651 lemma measure_space_0: "A \<subseteq> Pow \<Omega> \<Longrightarrow> measure_space \<Omega> (sigma_sets \<Omega> A) (\<lambda>x. 0)"

  1652   unfolding measure_space_def

  1653   by (auto intro!: sigma_algebra_sigma_sets simp: positive_def countably_additive_def)

  1654

  1655 lemma sigma_algebra_trivial: "sigma_algebra \<Omega> {{}, \<Omega>}"

  1656 by unfold_locales(fastforce intro: exI[where x="{{}}"] exI[where x="{\<Omega>}"])+

  1657

  1658 lemma measure_space_0': "measure_space \<Omega> {{}, \<Omega>} (\<lambda>x. 0)"

  1659 by(simp add: measure_space_def positive_def countably_additive_def sigma_algebra_trivial)

  1660

  1661 lemma measure_space_closed:

  1662   assumes "measure_space \<Omega> M \<mu>"

  1663   shows "M \<subseteq> Pow \<Omega>"

  1664 proof -

  1665   interpret sigma_algebra \<Omega> M using assms by(simp add: measure_space_def)

  1666   show ?thesis by(rule space_closed)

  1667 qed

  1668

  1669 lemma (in ring_of_sets) positive_cong_eq:

  1670   "(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> positive M \<mu>' = positive M \<mu>"

  1671   by (auto simp add: positive_def)

  1672

  1673 lemma (in sigma_algebra) countably_additive_eq:

  1674   "(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> countably_additive M \<mu>' = countably_additive M \<mu>"

  1675   unfolding countably_additive_def

  1676   by (intro arg_cong[where f=All] ext) (auto simp add: countably_additive_def subset_eq)

  1677

  1678 lemma measure_space_eq:

  1679   assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a"

  1680   shows "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'"

  1681 proof -

  1682   interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" using closed by (rule sigma_algebra_sigma_sets)

  1683   from positive_cong_eq[OF eq, of "\<lambda>i. i"] countably_additive_eq[OF eq, of "\<lambda>i. i"] show ?thesis

  1684     by (auto simp: measure_space_def)

  1685 qed

  1686

  1687 lemma measure_of_eq:

  1688   assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "(\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a)"

  1689   shows "measure_of \<Omega> A \<mu> = measure_of \<Omega> A \<mu>'"

  1690 proof -

  1691   have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'"

  1692     using assms by (rule measure_space_eq)

  1693   with eq show ?thesis

  1694     by (auto simp add: measure_of_def intro!: arg_cong[where f=Abs_measure])

  1695 qed

  1696

  1697 lemma

  1698   shows space_measure_of_conv: "space (measure_of \<Omega> A \<mu>) = \<Omega>" (is ?space)

  1699   and sets_measure_of_conv:

  1700   "sets (measure_of \<Omega> A \<mu>) = (if A \<subseteq> Pow \<Omega> then sigma_sets \<Omega> A else {{}, \<Omega>})" (is ?sets)

  1701   and emeasure_measure_of_conv:

  1702   "emeasure (measure_of \<Omega> A \<mu>) =

  1703   (\<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)

  1704 proof -

  1705   have "?space \<and> ?sets \<and> ?emeasure"

  1706   proof(cases "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>")

  1707     case True

  1708     from measure_space_closed[OF this] sigma_sets_superset_generator[of A \<Omega>]

  1709     have "A \<subseteq> Pow \<Omega>" by simp

  1710     hence "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A)

  1711       (\<lambda>a. if a \<in> sigma_sets \<Omega> A then \<mu> a else 0)"

  1712       by(rule measure_space_eq) auto

  1713     with True A \<subseteq> Pow \<Omega> show ?thesis

  1714       by(simp add: measure_of_def space_def sets_def emeasure_def Abs_measure_inverse)

  1715   next

  1716     case False thus ?thesis

  1717       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')

  1718   qed

  1719   thus ?space ?sets ?emeasure by simp_all

  1720 qed

  1721

  1722 lemma [simp]:

  1723   assumes A: "A \<subseteq> Pow \<Omega>"

  1724   shows sets_measure_of: "sets (measure_of \<Omega> A \<mu>) = sigma_sets \<Omega> A"

  1725     and space_measure_of: "space (measure_of \<Omega> A \<mu>) = \<Omega>"

  1726 using assms

  1727 by(simp_all add: sets_measure_of_conv space_measure_of_conv)

  1728

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

  1730   using space_closed by (auto intro!: sigma_sets_eq)

  1731

  1732 lemma (in sigma_algebra) space_measure_of_eq[simp]: "space (measure_of \<Omega> M \<mu>) = \<Omega>"

  1733   by (rule space_measure_of_conv)

  1734

  1735 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>')"

  1736   by (auto intro!: sigma_sets_subseteq)

  1737

  1738 lemma emeasure_sigma: "emeasure (sigma \<Omega> A) = (\<lambda>x. 0)"

  1739   unfolding measure_of_def emeasure_def

  1740   by (subst Abs_measure_inverse)

  1741      (auto simp: measure_space_def positive_def countably_additive_def

  1742            intro!: sigma_algebra_sigma_sets sigma_algebra_trivial)

  1743

  1744 lemma sigma_sets_mono'':

  1745   assumes "A \<in> sigma_sets C D"

  1746   assumes "B \<subseteq> D"

  1747   assumes "D \<subseteq> Pow C"

  1748   shows "sigma_sets A B \<subseteq> sigma_sets C D"

  1749 proof

  1750   fix x assume "x \<in> sigma_sets A B"

  1751   thus "x \<in> sigma_sets C D"

  1752   proof induct

  1753     case (Basic a) with assms have "a \<in> D" by auto

  1754     thus ?case ..

  1755   next

  1756     case Empty show ?case by (rule sigma_sets.Empty)

  1757   next

  1758     from assms have "A \<in> sets (sigma C D)" by (subst sets_measure_of[OF D \<subseteq> Pow C])

  1759     moreover case (Compl a) hence "a \<in> sets (sigma C D)" by (subst sets_measure_of[OF D \<subseteq> Pow C])

  1760     ultimately have "A - a \<in> sets (sigma C D)" ..

  1761     thus ?case by (subst (asm) sets_measure_of[OF D \<subseteq> Pow C])

  1762   next

  1763     case (Union a)

  1764     thus ?case by (intro sigma_sets.Union)

  1765   qed

  1766 qed

  1767

  1768 lemma in_measure_of[intro, simp]: "M \<subseteq> Pow \<Omega> \<Longrightarrow> A \<in> M \<Longrightarrow> A \<in> sets (measure_of \<Omega> M \<mu>)"

  1769   by auto

  1770

  1771 lemma space_empty_iff: "space N = {} \<longleftrightarrow> sets N = {{}}"

  1772   by (metis Pow_empty Sup_bot_conv(1) cSup_singleton empty_iff

  1773             sets.sigma_sets_eq sets.space_closed sigma_sets_top subset_singletonD)

  1774

  1775 subsubsection {* Constructing simple @{typ "'a measure"} *}

  1776

  1777 lemma emeasure_measure_of:

  1778   assumes M: "M = measure_of \<Omega> A \<mu>"

  1779   assumes ms: "A \<subseteq> Pow \<Omega>" "positive (sets M) \<mu>" "countably_additive (sets M) \<mu>"

  1780   assumes X: "X \<in> sets M"

  1781   shows "emeasure M X = \<mu> X"

  1782 proof -

  1783   interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" by (rule sigma_algebra_sigma_sets) fact

  1784   have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>"

  1785     using ms M by (simp add: measure_space_def sigma_algebra_sigma_sets)

  1786   thus ?thesis using X ms

  1787     by(simp add: M emeasure_measure_of_conv sets_measure_of_conv)

  1788 qed

  1789

  1790 lemma emeasure_measure_of_sigma:

  1791   assumes ms: "sigma_algebra \<Omega> M" "positive M \<mu>" "countably_additive M \<mu>"

  1792   assumes A: "A \<in> M"

  1793   shows "emeasure (measure_of \<Omega> M \<mu>) A = \<mu> A"

  1794 proof -

  1795   interpret sigma_algebra \<Omega> M by fact

  1796   have "measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"

  1797     using ms sigma_sets_eq by (simp add: measure_space_def)

  1798   thus ?thesis by(simp add: emeasure_measure_of_conv A)

  1799 qed

  1800

  1801 lemma measure_cases[cases type: measure]:

  1802   obtains (measure) \<Omega> A \<mu> where "x = Abs_measure (\<Omega>, A, \<mu>)" "\<forall>a\<in>-A. \<mu> a = 0" "measure_space \<Omega> A \<mu>"

  1803   by atomize_elim (cases x, auto)

  1804

  1805 lemma sets_eq_imp_space_eq:

  1806   "sets M = sets M' \<Longrightarrow> space M = space M'"

  1807   using sets.top[of M] sets.top[of M'] sets.space_closed[of M] sets.space_closed[of M']

  1808   by blast

  1809

  1810 lemma emeasure_notin_sets: "A \<notin> sets M \<Longrightarrow> emeasure M A = 0"

  1811   by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)

  1812

  1813 lemma emeasure_neq_0_sets: "emeasure M A \<noteq> 0 \<Longrightarrow> A \<in> sets M"

  1814   using emeasure_notin_sets[of A M] by blast

  1815

  1816 lemma measure_notin_sets: "A \<notin> sets M \<Longrightarrow> measure M A = 0"

  1817   by (simp add: measure_def emeasure_notin_sets)

  1818

  1819 lemma measure_eqI:

  1820   fixes M N :: "'a measure"

  1821   assumes "sets M = sets N" and eq: "\<And>A. A \<in> sets M \<Longrightarrow> emeasure M A = emeasure N A"

  1822   shows "M = N"

  1823 proof (cases M N rule: measure_cases[case_product measure_cases])

  1824   case (measure_measure \<Omega> A \<mu> \<Omega>' A' \<mu>')

  1825   interpret M: sigma_algebra \<Omega> A using measure_measure by (auto simp: measure_space_def)

  1826   interpret N: sigma_algebra \<Omega>' A' using measure_measure by (auto simp: measure_space_def)

  1827   have "A = sets M" "A' = sets N"

  1828     using measure_measure by (simp_all add: sets_def Abs_measure_inverse)

  1829   with sets M = sets N have AA': "A = A'" by simp

  1830   moreover from M.top N.top M.space_closed N.space_closed AA' have "\<Omega> = \<Omega>'" by auto

  1831   moreover { fix B have "\<mu> B = \<mu>' B"

  1832     proof cases

  1833       assume "B \<in> A"

  1834       with eq A = sets M have "emeasure M B = emeasure N B" by simp

  1835       with measure_measure show "\<mu> B = \<mu>' B"

  1836         by (simp add: emeasure_def Abs_measure_inverse)

  1837     next

  1838       assume "B \<notin> A"

  1839       with A = sets M A' = sets N A = A' have "B \<notin> sets M" "B \<notin> sets N"

  1840         by auto

  1841       then have "emeasure M B = 0" "emeasure N B = 0"

  1842         by (simp_all add: emeasure_notin_sets)

  1843       with measure_measure show "\<mu> B = \<mu>' B"

  1844         by (simp add: emeasure_def Abs_measure_inverse)

  1845     qed }

  1846   then have "\<mu> = \<mu>'" by auto

  1847   ultimately show "M = N"

  1848     by (simp add: measure_measure)

  1849 qed

  1850

  1851 lemma sigma_eqI:

  1852   assumes [simp]: "M \<subseteq> Pow \<Omega>" "N \<subseteq> Pow \<Omega>" "sigma_sets \<Omega> M = sigma_sets \<Omega> N"

  1853   shows "sigma \<Omega> M = sigma \<Omega> N"

  1854   by (rule measure_eqI) (simp_all add: emeasure_sigma)

  1855

  1856 subsubsection {* Measurable functions *}

  1857

  1858 definition measurable :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) set" where

  1859   "measurable A B = {f \<in> space A -> space B. \<forall>y \<in> sets B. f - y \<inter> space A \<in> sets A}"

  1860

  1861 lemma measurableI:

  1862   "(\<And>x. x \<in> space M \<Longrightarrow> f x \<in> space N) \<Longrightarrow> (\<And>A. A \<in> sets N \<Longrightarrow> f - A \<inter> space M \<in> sets M) \<Longrightarrow>

  1863     f \<in> measurable M N"

  1864   by (auto simp: measurable_def)

  1865

  1866 lemma measurable_space:

  1867   "f \<in> measurable M A \<Longrightarrow> x \<in> space M \<Longrightarrow> f x \<in> space A"

  1868    unfolding measurable_def by auto

  1869

  1870 lemma measurable_sets:

  1871   "f \<in> measurable M A \<Longrightarrow> S \<in> sets A \<Longrightarrow> f - S \<inter> space M \<in> sets M"

  1872    unfolding measurable_def by auto

  1873

  1874 lemma measurable_sets_Collect:

  1875   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"

  1876 proof -

  1877   have "f - {x \<in> space N. P x} \<inter> space M = {x\<in>space M. P (f x)}"

  1878     using measurable_space[OF f] by auto

  1879   with measurable_sets[OF f P] show ?thesis

  1880     by simp

  1881 qed

  1882

  1883 lemma measurable_sigma_sets:

  1884   assumes B: "sets N = sigma_sets \<Omega> A" "A \<subseteq> Pow \<Omega>"

  1885       and f: "f \<in> space M \<rightarrow> \<Omega>"

  1886       and ba: "\<And>y. y \<in> A \<Longrightarrow> (f - y) \<inter> space M \<in> sets M"

  1887   shows "f \<in> measurable M N"

  1888 proof -

  1889   interpret A: sigma_algebra \<Omega> "sigma_sets \<Omega> A" using B(2) by (rule sigma_algebra_sigma_sets)

  1890   from B sets.top[of N] A.top sets.space_closed[of N] A.space_closed have \<Omega>: "\<Omega> = space N" by force

  1891

  1892   { fix X assume "X \<in> sigma_sets \<Omega> A"

  1893     then have "f - X \<inter> space M \<in> sets M \<and> X \<subseteq> \<Omega>"

  1894       proof induct

  1895         case (Basic a) then show ?case

  1896           by (auto simp add: ba) (metis B(2) subsetD PowD)

  1897       next

  1898         case (Compl a)

  1899         have [simp]: "f - \<Omega> \<inter> space M = space M"

  1900           by (auto simp add: funcset_mem [OF f])

  1901         then show ?case

  1902           by (auto simp add: vimage_Diff Diff_Int_distrib2 sets.compl_sets Compl)

  1903       next

  1904         case (Union a)

  1905         then show ?case

  1906           by (simp add: vimage_UN, simp only: UN_extend_simps(4)) blast

  1907       qed auto }

  1908   with f show ?thesis

  1909     by (auto simp add: measurable_def B \<Omega>)

  1910 qed

  1911

  1912 lemma measurable_measure_of:

  1913   assumes B: "N \<subseteq> Pow \<Omega>"

  1914       and f: "f \<in> space M \<rightarrow> \<Omega>"

  1915       and ba: "\<And>y. y \<in> N \<Longrightarrow> (f - y) \<inter> space M \<in> sets M"

  1916   shows "f \<in> measurable M (measure_of \<Omega> N \<mu>)"

  1917 proof -

  1918   have "sets (measure_of \<Omega> N \<mu>) = sigma_sets \<Omega> N"

  1919     using B by (rule sets_measure_of)

  1920   from this assms show ?thesis by (rule measurable_sigma_sets)

  1921 qed

  1922

  1923 lemma measurable_iff_measure_of:

  1924   assumes "N \<subseteq> Pow \<Omega>" "f \<in> space M \<rightarrow> \<Omega>"

  1925   shows "f \<in> measurable M (measure_of \<Omega> N \<mu>) \<longleftrightarrow> (\<forall>A\<in>N. f - A \<inter> space M \<in> sets M)"

  1926   by (metis assms in_measure_of measurable_measure_of assms measurable_sets)

  1927

  1928 lemma measurable_cong_sets:

  1929   assumes sets: "sets M = sets M'" "sets N = sets N'"

  1930   shows "measurable M N = measurable M' N'"

  1931   using sets[THEN sets_eq_imp_space_eq] sets by (simp add: measurable_def)

  1932

  1933 lemma measurable_cong:

  1934   assumes "\<And>w. w \<in> space M \<Longrightarrow> f w = g w"

  1935   shows "f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable M M'"

  1936   unfolding measurable_def using assms

  1937   by (simp cong: vimage_inter_cong Pi_cong)

  1938

  1939 lemma measurable_cong':

  1940   assumes "\<And>w. w \<in> space M =simp=> f w = g w"

  1941   shows "f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable M M'"

  1942   unfolding measurable_def using assms

  1943   by (simp cong: vimage_inter_cong Pi_cong add: simp_implies_def)

  1944

  1945 lemma measurable_cong_strong:

  1946   "M = N \<Longrightarrow> M' = N' \<Longrightarrow> (\<And>w. w \<in> space M \<Longrightarrow> f w = g w) \<Longrightarrow>

  1947     f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable N N'"

  1948   by (metis measurable_cong)

  1949

  1950 lemma measurable_compose:

  1951   assumes f: "f \<in> measurable M N" and g: "g \<in> measurable N L"

  1952   shows "(\<lambda>x. g (f x)) \<in> measurable M L"

  1953 proof -

  1954   have "\<And>A. (\<lambda>x. g (f x)) - A \<inter> space M = f - (g - A \<inter> space N) \<inter> space M"

  1955     using measurable_space[OF f] by auto

  1956   with measurable_space[OF f] measurable_space[OF g] show ?thesis

  1957     by (auto intro: measurable_sets[OF f] measurable_sets[OF g]

  1958              simp del: vimage_Int simp add: measurable_def)

  1959 qed

  1960

  1961 lemma measurable_comp:

  1962   "f \<in> measurable M N \<Longrightarrow> g \<in> measurable N L \<Longrightarrow> g \<circ> f \<in> measurable M L"

  1963   using measurable_compose[of f M N g L] by (simp add: comp_def)

  1964

  1965 lemma measurable_const:

  1966   "c \<in> space M' \<Longrightarrow> (\<lambda>x. c) \<in> measurable M M'"

  1967   by (auto simp add: measurable_def)

  1968

  1969 lemma measurable_ident: "id \<in> measurable M M"

  1970   by (auto simp add: measurable_def)

  1971

  1972 lemma measurable_id: "(\<lambda>x. x) \<in> measurable M M"

  1973   by (simp add: measurable_def)

  1974

  1975 lemma measurable_ident_sets:

  1976   assumes eq: "sets M = sets M'" shows "(\<lambda>x. x) \<in> measurable M M'"

  1977   using measurable_ident[of M]

  1978   unfolding id_def measurable_def eq sets_eq_imp_space_eq[OF eq] .

  1979

  1980 lemma sets_Least:

  1981   assumes meas: "\<And>i::nat. {x\<in>space M. P i x} \<in> M"

  1982   shows "(\<lambda>x. LEAST j. P j x) - A \<inter> space M \<in> sets M"

  1983 proof -

  1984   { fix i have "(\<lambda>x. LEAST j. P j x) - {i} \<inter> space M \<in> sets M"

  1985     proof cases

  1986       assume i: "(LEAST j. False) = i"

  1987       have "(\<lambda>x. LEAST j. P j x) - {i} \<inter> space M =

  1988         {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}))"

  1989         by (simp add: set_eq_iff, safe)

  1990            (insert i, auto dest: Least_le intro: LeastI intro!: Least_equality)

  1991       with meas show ?thesis

  1992         by (auto intro!: sets.Int)

  1993     next

  1994       assume i: "(LEAST j. False) \<noteq> i"

  1995       then have "(\<lambda>x. LEAST j. P j x) - {i} \<inter> space M =

  1996         {x\<in>space M. P i x} \<inter> (space M - (\<Union>j<i. {x\<in>space M. P j x}))"

  1997       proof (simp add: set_eq_iff, safe)

  1998         fix x assume neq: "(LEAST j. False) \<noteq> (LEAST j. P j x)"

  1999         have "\<exists>j. P j x"

  2000           by (rule ccontr) (insert neq, auto)

  2001         then show "P (LEAST j. P j x) x" by (rule LeastI_ex)

  2002       qed (auto dest: Least_le intro!: Least_equality)

  2003       with meas show ?thesis

  2004         by auto

  2005     qed }

  2006   then have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) - {i} \<inter> space M) \<in> sets M"

  2007     by (intro sets.countable_UN) auto

  2008   moreover have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) - {i} \<inter> space M) =

  2009     (\<lambda>x. LEAST j. P j x) - A \<inter> space M" by auto

  2010   ultimately show ?thesis by auto

  2011 qed

  2012

  2013 lemma measurable_mono1:

  2014   "M' \<subseteq> Pow \<Omega> \<Longrightarrow> M \<subseteq> M' \<Longrightarrow>

  2015     measurable (measure_of \<Omega> M \<mu>) N \<subseteq> measurable (measure_of \<Omega> M' \<mu>') N"

  2016   using measure_of_subset[of M' \<Omega> M] by (auto simp add: measurable_def)

  2017

  2018 subsubsection {* Counting space *}

  2019

  2020 definition count_space :: "'a set \<Rightarrow> 'a measure" where

  2021   "count_space \<Omega> = measure_of \<Omega> (Pow \<Omega>) (\<lambda>A. if finite A then ereal (card A) else \<infinity>)"

  2022

  2023 lemma

  2024   shows space_count_space[simp]: "space (count_space \<Omega>) = \<Omega>"

  2025     and sets_count_space[simp]: "sets (count_space \<Omega>) = Pow \<Omega>"

  2026   using sigma_sets_into_sp[of "Pow \<Omega>" \<Omega>]

  2027   by (auto simp: count_space_def)

  2028

  2029 lemma measurable_count_space_eq1[simp]:

  2030   "f \<in> measurable (count_space A) M \<longleftrightarrow> f \<in> A \<rightarrow> space M"

  2031  unfolding measurable_def by simp

  2032

  2033 lemma measurable_compose_countable':

  2034   assumes f: "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. f i x) \<in> measurable M N"

  2035   and g: "g \<in> measurable M (count_space I)" and I: "countable I"

  2036   shows "(\<lambda>x. f (g x) x) \<in> measurable M N"

  2037   unfolding measurable_def

  2038 proof safe

  2039   fix x assume "x \<in> space M" then show "f (g x) x \<in> space N"

  2040     using measurable_space[OF f] g[THEN measurable_space] by auto

  2041 next

  2042   fix A assume A: "A \<in> sets N"

  2043   have "(\<lambda>x. f (g x) x) - A \<inter> space M = (\<Union>i\<in>I. (g - {i} \<inter> space M) \<inter> (f i - A \<inter> space M))"

  2044     using measurable_space[OF g] by auto

  2045   also have "\<dots> \<in> sets M"

  2046     using f[THEN measurable_sets, OF _ A] g[THEN measurable_sets]

  2047     by (auto intro!: sets.countable_UN' I intro: sets.Int[OF measurable_sets measurable_sets])

  2048   finally show "(\<lambda>x. f (g x) x) - A \<inter> space M \<in> sets M" .

  2049 qed

  2050

  2051 lemma measurable_count_space_eq_countable:

  2052   assumes "countable A"

  2053   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))"

  2054 proof -

  2055   { fix X assume "X \<subseteq> A" "f \<in> space M \<rightarrow> A"

  2056     with countable A have "f - X \<inter> space M = (\<Union>a\<in>X. f - {a} \<inter> space M)" "countable X"

  2057       by (auto dest: countable_subset)

  2058     moreover assume "\<forall>a\<in>A. f - {a} \<inter> space M \<in> sets M"

  2059     ultimately have "f - X \<inter> space M \<in> sets M"

  2060       using X \<subseteq> A by (auto intro!: sets.countable_UN' simp del: UN_simps) }

  2061   then show ?thesis

  2062     unfolding measurable_def by auto

  2063 qed

  2064

  2065 lemma measurable_count_space_eq2:

  2066   "finite A \<Longrightarrow> f \<in> measurable M (count_space A) \<longleftrightarrow> (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f - {a} \<inter> space M \<in> sets M))"

  2067   by (intro measurable_count_space_eq_countable countable_finite)

  2068

  2069 lemma measurable_count_space_eq2_countable:

  2070   fixes f :: "'a => 'c::countable"

  2071   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))"

  2072   by (intro measurable_count_space_eq_countable countableI_type)

  2073

  2074 lemma measurable_compose_countable:

  2075   assumes f: "\<And>i::'i::countable. (\<lambda>x. f i x) \<in> measurable M N" and g: "g \<in> measurable M (count_space UNIV)"

  2076   shows "(\<lambda>x. f (g x) x) \<in> measurable M N"

  2077   by (rule measurable_compose_countable'[OF assms]) auto

  2078

  2079 lemma measurable_count_space_const:

  2080   "(\<lambda>x. c) \<in> measurable M (count_space UNIV)"

  2081   by (simp add: measurable_const)

  2082

  2083 lemma measurable_count_space:

  2084   "f \<in> measurable (count_space A) (count_space UNIV)"

  2085   by simp

  2086

  2087 lemma measurable_compose_rev:

  2088   assumes f: "f \<in> measurable L N" and g: "g \<in> measurable M L"

  2089   shows "(\<lambda>x. f (g x)) \<in> measurable M N"

  2090   using measurable_compose[OF g f] .

  2091

  2092 lemma measurable_empty_iff:

  2093   "space N = {} \<Longrightarrow> f \<in> measurable M N \<longleftrightarrow> space M = {}"

  2094   by (auto simp add: measurable_def Pi_iff)

  2095

  2096 subsubsection {* Extend measure *}

  2097

  2098 definition "extend_measure \<Omega> I G \<mu> =

  2099   (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)

  2100       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>')

  2101       else measure_of \<Omega> (GI) (\<lambda>_. 0))"

  2102

  2103 lemma space_extend_measure: "G  I \<subseteq> Pow \<Omega> \<Longrightarrow> space (extend_measure \<Omega> I G \<mu>) = \<Omega>"

  2104   unfolding extend_measure_def by simp

  2105

  2106 lemma sets_extend_measure: "G  I \<subseteq> Pow \<Omega> \<Longrightarrow> sets (extend_measure \<Omega> I G \<mu>) = sigma_sets \<Omega> (GI)"

  2107   unfolding extend_measure_def by simp

  2108

  2109 lemma emeasure_extend_measure:

  2110   assumes M: "M = extend_measure \<Omega> I G \<mu>"

  2111     and eq: "\<And>i. i \<in> I \<Longrightarrow> \<mu>' (G i) = \<mu> i"

  2112     and ms: "G  I \<subseteq> Pow \<Omega>" "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'"

  2113     and "i \<in> I"

  2114   shows "emeasure M (G i) = \<mu> i"

  2115 proof cases

  2116   assume *: "(\<forall>i\<in>I. \<mu> i = 0)"

  2117   with M have M_eq: "M = measure_of \<Omega> (GI) (\<lambda>_. 0)"

  2118    by (simp add: extend_measure_def)

  2119   from measure_space_0[OF ms(1)] ms i\<in>I

  2120   have "emeasure M (G i) = 0"

  2121     by (intro emeasure_measure_of[OF M_eq]) (auto simp add: M measure_space_def sets_extend_measure)

  2122   with i\<in>I * show ?thesis

  2123     by simp

  2124 next

  2125   def P \<equiv> "\<lambda>\<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (GI)) \<mu>'"

  2126   assume "\<not> (\<forall>i\<in>I. \<mu> i = 0)"

  2127   moreover

  2128   have "measure_space (space M) (sets M) \<mu>'"

  2129     using ms unfolding measure_space_def by auto default

  2130   with ms eq have "\<exists>\<mu>'. P \<mu>'"

  2131     unfolding P_def

  2132     by (intro exI[of _ \<mu>']) (auto simp add: M space_extend_measure sets_extend_measure)

  2133   ultimately have M_eq: "M = measure_of \<Omega> (GI) (Eps P)"

  2134     by (simp add: M extend_measure_def P_def[symmetric])

  2135

  2136   from \<exists>\<mu>'. P \<mu>' have P: "P (Eps P)" by (rule someI_ex)

  2137   show "emeasure M (G i) = \<mu> i"

  2138   proof (subst emeasure_measure_of[OF M_eq])

  2139     have sets_M: "sets M = sigma_sets \<Omega> (GI)"

  2140       using M_eq ms by (auto simp: sets_extend_measure)

  2141     then show "G i \<in> sets M" using i \<in> I by auto

  2142     show "positive (sets M) (Eps P)" "countably_additive (sets M) (Eps P)" "Eps P (G i) = \<mu> i"

  2143       using P i\<in>I by (auto simp add: sets_M measure_space_def P_def)

  2144   qed fact

  2145 qed

  2146

  2147 lemma emeasure_extend_measure_Pair:

  2148   assumes M: "M = extend_measure \<Omega> {(i, j). I i j} (\<lambda>(i, j). G i j) (\<lambda>(i, j). \<mu> i j)"

  2149     and eq: "\<And>i j. I i j \<Longrightarrow> \<mu>' (G i j) = \<mu> i j"

  2150     and ms: "\<And>i j. I i j \<Longrightarrow> G i j \<in> Pow \<Omega>" "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'"

  2151     and "I i j"

  2152   shows "emeasure M (G i j) = \<mu> i j"

  2153   using emeasure_extend_measure[OF M _ _ ms(2,3), of "(i,j)"] eq ms(1) I i j

  2154   by (auto simp: subset_eq)

  2155

  2156 subsubsection {* Supremum of a set of $\sigma$-algebras *}

  2157

  2158 definition "Sup_sigma M = sigma (\<Union>x\<in>M. space x) (\<Union>x\<in>M. sets x)"

  2159

  2160 syntax

  2161   "_SUP_sigma"   :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>\<^sub>\<sigma> _\<in>_./ _)" [0, 0, 10] 10)

  2162

  2163 translations

  2164   "\<Squnion>\<^sub>\<sigma> x\<in>A. B"   == "CONST Sup_sigma ((\<lambda>x. B)  A)"

  2165

  2166 lemma space_Sup_sigma: "space (Sup_sigma M) = (\<Union>x\<in>M. space x)"

  2167   unfolding Sup_sigma_def by (rule space_measure_of) (auto dest: sets.sets_into_space)

  2168

  2169 lemma sets_Sup_sigma: "sets (Sup_sigma M) = sigma_sets (\<Union>x\<in>M. space x) (\<Union>x\<in>M. sets x)"

  2170   unfolding Sup_sigma_def by (rule sets_measure_of) (auto dest: sets.sets_into_space)

  2171

  2172 lemma in_Sup_sigma: "m \<in> M \<Longrightarrow> A \<in> sets m \<Longrightarrow> A \<in> sets (Sup_sigma M)"

  2173   unfolding sets_Sup_sigma by auto

  2174

  2175 lemma SUP_sigma_cong:

  2176   assumes *: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets (N i)" shows "sets (\<Squnion>\<^sub>\<sigma> i\<in>I. M i) = sets (\<Squnion>\<^sub>\<sigma> i\<in>I. N i)"

  2177   using * sets_eq_imp_space_eq[OF *] by (simp add: Sup_sigma_def)

  2178

  2179 lemma sets_Sup_in_sets:

  2180   assumes "M \<noteq> {}"

  2181   assumes "\<And>m. m \<in> M \<Longrightarrow> space m = space N"

  2182   assumes "\<And>m. m \<in> M \<Longrightarrow> sets m \<subseteq> sets N"

  2183   shows "sets (Sup_sigma M) \<subseteq> sets N"

  2184 proof -

  2185   have *: "UNION M space = space N"

  2186     using assms by auto

  2187   show ?thesis

  2188     unfolding sets_Sup_sigma * using assms by (auto intro!: sets.sigma_sets_subset)

  2189 qed

  2190

  2191 lemma measurable_Sup_sigma1:

  2192   assumes m: "m \<in> M" and f: "f \<in> measurable m N"

  2193     and const_space: "\<And>m n. m \<in> M \<Longrightarrow> n \<in> M \<Longrightarrow> space m = space n"

  2194   shows "f \<in> measurable (Sup_sigma M) N"

  2195 proof -

  2196   have "space (Sup_sigma M) = space m"

  2197     using m by (auto simp add: space_Sup_sigma dest: const_space)

  2198   then show ?thesis

  2199     using m f unfolding measurable_def by (auto intro: in_Sup_sigma)

  2200 qed

  2201

  2202 lemma measurable_Sup_sigma2:

  2203   assumes M: "M \<noteq> {}"

  2204   assumes f: "\<And>m. m \<in> M \<Longrightarrow> f \<in> measurable N m"

  2205   shows "f \<in> measurable N (Sup_sigma M)"

  2206   unfolding Sup_sigma_def

  2207 proof (rule measurable_measure_of)

  2208   show "f \<in> space N \<rightarrow> UNION M space"

  2209     using measurable_space[OF f] M by auto

  2210 qed (auto intro: measurable_sets f dest: sets.sets_into_space)

  2211

  2212 lemma Sup_sigma_sigma:

  2213   assumes [simp]: "M \<noteq> {}" and M: "\<And>m. m \<in> M \<Longrightarrow> m \<subseteq> Pow \<Omega>"

  2214   shows "(\<Squnion>\<^sub>\<sigma> m\<in>M. sigma \<Omega> m) = sigma \<Omega> (\<Union>M)"

  2215 proof (rule measure_eqI)

  2216   { fix a m assume "a \<in> sigma_sets \<Omega> m" "m \<in> M"

  2217     then have "a \<in> sigma_sets \<Omega> (\<Union>M)"

  2218      by induction (auto intro: sigma_sets.intros) }

  2219   then show "sets (\<Squnion>\<^sub>\<sigma> m\<in>M. sigma \<Omega> m) = sets (sigma \<Omega> (\<Union>M))"

  2220     apply (simp add: sets_Sup_sigma space_measure_of_conv M Union_least)

  2221     apply (rule sigma_sets_eqI)

  2222     apply auto

  2223     done

  2224 qed (simp add: Sup_sigma_def emeasure_sigma)

  2225

  2226 lemma SUP_sigma_sigma:

  2227   assumes M: "M \<noteq> {}" "\<And>m. m \<in> M \<Longrightarrow> f m \<subseteq> Pow \<Omega>"

  2228   shows "(\<Squnion>\<^sub>\<sigma> m\<in>M. sigma \<Omega> (f m)) = sigma \<Omega> (\<Union>m\<in>M. f m)"

  2229 proof -

  2230   have "Sup_sigma (sigma \<Omega>  f  M) = sigma \<Omega> (\<Union>(f  M))"

  2231     using M by (intro Sup_sigma_sigma) auto

  2232   then show ?thesis

  2233     by (simp add: image_image)

  2234 qed

  2235

  2236 subsection {* The smallest $\sigma$-algebra regarding a function *}

  2237

  2238 definition

  2239   "vimage_algebra X f M = sigma X {f - A \<inter> X | A. A \<in> sets M}"

  2240

  2241 lemma space_vimage_algebra[simp]: "space (vimage_algebra X f M) = X"

  2242   unfolding vimage_algebra_def by (rule space_measure_of) auto

  2243

  2244 lemma sets_vimage_algebra: "sets (vimage_algebra X f M) = sigma_sets X {f - A \<inter> X | A. A \<in> sets M}"

  2245   unfolding vimage_algebra_def by (rule sets_measure_of) auto

  2246

  2247 lemma sets_vimage_algebra2:

  2248   "f \<in> X \<rightarrow> space M \<Longrightarrow> sets (vimage_algebra X f M) = {f - A \<inter> X | A. A \<in> sets M}"

  2249   using sigma_sets_vimage_commute[of f X "space M" "sets M"]

  2250   unfolding sets_vimage_algebra sets.sigma_sets_eq by simp

  2251

  2252 lemma sets_vimage_algebra_cong: "sets M = sets N \<Longrightarrow> sets (vimage_algebra X f M) = sets (vimage_algebra X f N)"

  2253   by (simp add: sets_vimage_algebra)

  2254

  2255 lemma vimage_algebra_cong:

  2256   assumes "X = Y"

  2257   assumes "\<And>x. x \<in> Y \<Longrightarrow> f x = g x"

  2258   assumes "sets M = sets N"

  2259   shows "vimage_algebra X f M = vimage_algebra Y g N"

  2260   by (auto simp: vimage_algebra_def assms intro!: arg_cong2[where f=sigma])

  2261

  2262 lemma in_vimage_algebra: "A \<in> sets M \<Longrightarrow> f - A \<inter> X \<in> sets (vimage_algebra X f M)"

  2263   by (auto simp: vimage_algebra_def)

  2264

  2265 lemma sets_image_in_sets:

  2266   assumes N: "space N = X"

  2267   assumes f: "f \<in> measurable N M"

  2268   shows "sets (vimage_algebra X f M) \<subseteq> sets N"

  2269   unfolding sets_vimage_algebra N[symmetric]

  2270   by (rule sets.sigma_sets_subset) (auto intro!: measurable_sets f)

  2271

  2272 lemma measurable_vimage_algebra1: "f \<in> X \<rightarrow> space M \<Longrightarrow> f \<in> measurable (vimage_algebra X f M) M"

  2273   unfolding measurable_def by (auto intro: in_vimage_algebra)

  2274

  2275 lemma measurable_vimage_algebra2:

  2276   assumes g: "g \<in> space N \<rightarrow> X" and f: "(\<lambda>x. f (g x)) \<in> measurable N M"

  2277   shows "g \<in> measurable N (vimage_algebra X f M)"

  2278   unfolding vimage_algebra_def

  2279 proof (rule measurable_measure_of)

  2280   fix A assume "A \<in> {f - A \<inter> X | A. A \<in> sets M}"

  2281   then obtain Y where Y: "Y \<in> sets M" and A: "A = f - Y \<inter> X"

  2282     by auto

  2283   then have "g - A \<inter> space N = (\<lambda>x. f (g x)) - Y \<inter> space N"

  2284     using g by auto

  2285   also have "\<dots> \<in> sets N"

  2286     using f Y by (rule measurable_sets)

  2287   finally show "g - A \<inter> space N \<in> sets N" .

  2288 qed (insert g, auto)

  2289

  2290 lemma vimage_algebra_sigma:

  2291   assumes X: "X \<subseteq> Pow \<Omega>'" and f: "f \<in> \<Omega> \<rightarrow> \<Omega>'"

  2292   shows "vimage_algebra \<Omega> f (sigma \<Omega>' X) = sigma \<Omega> {f - A \<inter> \<Omega> | A. A \<in> X }" (is "?V = ?S")

  2293 proof (rule measure_eqI)

  2294   have \<Omega>: "{f - A \<inter> \<Omega> |A. A \<in> X} \<subseteq> Pow \<Omega>" by auto

  2295   show "sets ?V = sets ?S"

  2296     using sigma_sets_vimage_commute[OF f, of X]

  2297     by (simp add: space_measure_of_conv f sets_vimage_algebra2 \<Omega> X)

  2298 qed (simp add: vimage_algebra_def emeasure_sigma)

  2299

  2300 lemma vimage_algebra_vimage_algebra_eq:

  2301   assumes *: "f \<in> X \<rightarrow> Y" "g \<in> Y \<rightarrow> space M"

  2302   shows "vimage_algebra X f (vimage_algebra Y g M) = vimage_algebra X (\<lambda>x. g (f x)) M"

  2303     (is "?VV = ?V")

  2304 proof (rule measure_eqI)

  2305   have "(\<lambda>x. g (f x)) \<in> X \<rightarrow> space M" "\<And>A. A \<inter> f - Y \<inter> X = A \<inter> X"

  2306     using * by auto

  2307   with * show "sets ?VV = sets ?V"

  2308     by (simp add: sets_vimage_algebra2 ex_simps[symmetric] vimage_comp comp_def del: ex_simps)

  2309 qed (simp add: vimage_algebra_def emeasure_sigma)

  2310

  2311 lemma sets_vimage_Sup_eq:

  2312   assumes *: "M \<noteq> {}" "\<And>m. m \<in> M \<Longrightarrow> f \<in> X \<rightarrow> space m"

  2313   shows "sets (vimage_algebra X f (Sup_sigma M)) = sets (\<Squnion>\<^sub>\<sigma> m \<in> M. vimage_algebra X f m)"

  2314   (is "?IS = ?SI")

  2315 proof

  2316   show "?IS \<subseteq> ?SI"

  2317     by (intro sets_image_in_sets measurable_Sup_sigma2 measurable_Sup_sigma1)

  2318        (auto simp: space_Sup_sigma measurable_vimage_algebra1 *)

  2319   { fix m assume "m \<in> M"

  2320     moreover then have "f \<in> X \<rightarrow> space (Sup_sigma M)" "f \<in> X \<rightarrow> space m"

  2321       using * by (auto simp: space_Sup_sigma)

  2322     ultimately have "f \<in> measurable (vimage_algebra X f (Sup_sigma M)) m"

  2323       by (auto simp add: measurable_def sets_vimage_algebra2 intro: in_Sup_sigma) }

  2324   then show "?SI \<subseteq> ?IS"

  2325     by (auto intro!: sets_image_in_sets sets_Sup_in_sets del: subsetI simp: *)

  2326 qed

  2327

  2328 lemma vimage_algebra_Sup_sigma:

  2329   assumes [simp]: "MM \<noteq> {}" and "\<And>M. M \<in> MM \<Longrightarrow> f \<in> X \<rightarrow> space M"

  2330   shows "vimage_algebra X f (Sup_sigma MM) = Sup_sigma (vimage_algebra X f  MM)"

  2331 proof (rule measure_eqI)

  2332   show "sets (vimage_algebra X f (Sup_sigma MM)) = sets (Sup_sigma (vimage_algebra X f  MM))"

  2333     using assms by (rule sets_vimage_Sup_eq)

  2334 qed (simp add: vimage_algebra_def Sup_sigma_def emeasure_sigma)

  2335

  2336 subsubsection {* Restricted Space Sigma Algebra *}

  2337

  2338 definition restrict_space where

  2339   "restrict_space M \<Omega> = measure_of (\<Omega> \<inter> space M) ((op \<inter> \<Omega>)  sets M) (emeasure M)"

  2340

  2341 lemma space_restrict_space: "space (restrict_space M \<Omega>) = \<Omega> \<inter> space M"

  2342   using sets.sets_into_space unfolding restrict_space_def by (subst space_measure_of) auto

  2343

  2344 lemma space_restrict_space2: "\<Omega> \<in> sets M \<Longrightarrow> space (restrict_space M \<Omega>) = \<Omega>"

  2345   by (simp add: space_restrict_space sets.sets_into_space)

  2346

  2347 lemma sets_restrict_space: "sets (restrict_space M \<Omega>) = (op \<inter> \<Omega>)  sets M"

  2348   unfolding restrict_space_def

  2349 proof (subst sets_measure_of)

  2350   show "op \<inter> \<Omega>  sets M \<subseteq> Pow (\<Omega> \<inter> space M)"

  2351     by (auto dest: sets.sets_into_space)

  2352   have "sigma_sets (\<Omega> \<inter> space M) {((\<lambda>x. x) - X) \<inter> (\<Omega> \<inter> space M) | X. X \<in> sets M} =

  2353     (\<lambda>X. X \<inter> (\<Omega> \<inter> space M))  sets M"

  2354     by (subst sigma_sets_vimage_commute[symmetric, where \<Omega>' = "space M"])

  2355        (auto simp add: sets.sigma_sets_eq)

  2356   moreover have "{((\<lambda>x. x) - X) \<inter> (\<Omega> \<inter> space M) | X. X \<in> sets M} = (\<lambda>X. X \<inter> (\<Omega> \<inter> space M))   sets M"

  2357     by auto

  2358   moreover have "(\<lambda>X. X \<inter> (\<Omega> \<inter> space M))   sets M = (op \<inter> \<Omega>)  sets M"

  2359     by (intro image_cong) (auto dest: sets.sets_into_space)

  2360   ultimately show "sigma_sets (\<Omega> \<inter> space M) (op \<inter> \<Omega>  sets M) = op \<inter> \<Omega>  sets M"

  2361     by simp

  2362 qed

  2363

  2364 lemma sets_restrict_space_count_space :

  2365   "sets (restrict_space (count_space A) B) = sets (count_space (A \<inter> B))"

  2366 by(auto simp add: sets_restrict_space)

  2367

  2368 lemma sets_restrict_UNIV[simp]: "sets (restrict_space M UNIV) = sets M"

  2369   by (auto simp add: sets_restrict_space)

  2370

  2371 lemma sets_restrict_restrict_space:

  2372   "sets (restrict_space (restrict_space M A) B) = sets (restrict_space M (A \<inter> B))"

  2373   unfolding sets_restrict_space image_comp by (intro image_cong) auto

  2374

  2375 lemma sets_restrict_space_iff:

  2376   "\<Omega> \<inter> space M \<in> sets M \<Longrightarrow> A \<in> sets (restrict_space M \<Omega>) \<longleftrightarrow> (A \<subseteq> \<Omega> \<and> A \<in> sets M)"

  2377 proof (subst sets_restrict_space, safe)

  2378   fix A assume "\<Omega> \<inter> space M \<in> sets M" and A: "A \<in> sets M"

  2379   then have "(\<Omega> \<inter> space M) \<inter> A \<in> sets M"

  2380     by rule

  2381   also have "(\<Omega> \<inter> space M) \<inter> A = \<Omega> \<inter> A"

  2382     using sets.sets_into_space[OF A] by auto

  2383   finally show "\<Omega> \<inter> A \<in> sets M"

  2384     by auto

  2385 qed auto

  2386

  2387 lemma sets_restrict_space_cong: "sets M = sets N \<Longrightarrow> sets (restrict_space M \<Omega>) = sets (restrict_space N \<Omega>)"

  2388   by (simp add: sets_restrict_space)

  2389

  2390 lemma restrict_space_eq_vimage_algebra:

  2391   "\<Omega> \<subseteq> space M \<Longrightarrow> sets (restrict_space M \<Omega>) = sets (vimage_algebra \<Omega> (\<lambda>x. x) M)"

  2392   unfolding restrict_space_def

  2393   apply (subst sets_measure_of)

  2394   apply (auto simp add: image_subset_iff dest: sets.sets_into_space) []

  2395   apply (auto simp add: sets_vimage_algebra intro!: arg_cong2[where f=sigma_sets])

  2396   done

  2397

  2398 lemma sets_Collect_restrict_space_iff:

  2399   assumes "S \<in> sets M"

  2400   shows "{x\<in>space (restrict_space M S). P x} \<in> sets (restrict_space M S) \<longleftrightarrow> {x\<in>space M. x \<in> S \<and> P x} \<in> sets M"

  2401 proof -

  2402   have "{x\<in>S. P x} = {x\<in>space M. x \<in> S \<and> P x}"

  2403     using sets.sets_into_space[OF assms] by auto

  2404   then show ?thesis

  2405     by (subst sets_restrict_space_iff) (auto simp add: space_restrict_space assms)

  2406 qed

  2407

  2408 lemma measurable_restrict_space1:

  2409   assumes f: "f \<in> measurable M N"

  2410   shows "f \<in> measurable (restrict_space M \<Omega>) N"

  2411   unfolding measurable_def

  2412 proof (intro CollectI conjI ballI)

  2413   show sp: "f \<in> space (restrict_space M \<Omega>) \<rightarrow> space N"

  2414     using measurable_space[OF f] by (auto simp: space_restrict_space)

  2415

  2416   fix A assume "A \<in> sets N"

  2417   have "f - A \<inter> space (restrict_space M \<Omega>) = (f - A \<inter> space M) \<inter> (\<Omega> \<inter> space M)"

  2418     by (auto simp: space_restrict_space)

  2419   also have "\<dots> \<in> sets (restrict_space M \<Omega>)"

  2420     unfolding sets_restrict_space

  2421     using measurable_sets[OF f A \<in> sets N] by blast

  2422   finally show "f - A \<inter> space (restrict_space M \<Omega>) \<in> sets (restrict_space M \<Omega>)" .

  2423 qed

  2424

  2425 lemma measurable_restrict_space2_iff:

  2426   "f \<in> measurable M (restrict_space N \<Omega>) \<longleftrightarrow> (f \<in> measurable M N \<and> f \<in> space M \<rightarrow> \<Omega>)"

  2427 proof -

  2428   have "\<And>A. f \<in> space M \<rightarrow> \<Omega> \<Longrightarrow> f - \<Omega> \<inter> f - A \<inter> space M = f - A \<inter> space M"

  2429     by auto

  2430   then show ?thesis

  2431     by (auto simp: measurable_def space_restrict_space Pi_Int[symmetric] sets_restrict_space)

  2432 qed

  2433

  2434 lemma measurable_restrict_space2:

  2435   "f \<in> space M \<rightarrow> \<Omega> \<Longrightarrow> f \<in> measurable M N \<Longrightarrow> f \<in> measurable M (restrict_space N \<Omega>)"

  2436   by (simp add: measurable_restrict_space2_iff)

  2437

  2438 lemma measurable_piecewise_restrict:

  2439   assumes I: "countable C"

  2440     and X: "\<And>\<Omega>. \<Omega> \<in> C \<Longrightarrow> \<Omega> \<inter> space M \<in> sets M" "space M \<subseteq> \<Union>C"

  2441     and f: "\<And>\<Omega>. \<Omega> \<in> C \<Longrightarrow> f \<in> measurable (restrict_space M \<Omega>) N"

  2442   shows "f \<in> measurable M N"

  2443 proof (rule measurableI)

  2444   fix x assume "x \<in> space M"

  2445   with X obtain \<Omega> where "\<Omega> \<in> C" "x \<in> \<Omega>" "x \<in> space M" by auto

  2446   then show "f x \<in> space N"

  2447     by (auto simp: space_restrict_space intro: f measurable_space)

  2448 next

  2449   fix A assume A: "A \<in> sets N"

  2450   have "f - A \<inter> space M = (\<Union>\<Omega>\<in>C. (f - A \<inter> (\<Omega> \<inter> space M)))"

  2451     using X by (auto simp: subset_eq)

  2452   also have "\<dots> \<in> sets M"

  2453     using measurable_sets[OF f A] X I

  2454     by (intro sets.countable_UN') (auto simp: sets_restrict_space_iff space_restrict_space)

  2455   finally show "f - A \<inter> space M \<in> sets M" .

  2456 qed

  2457

  2458 lemma measurable_piecewise_restrict_iff:

  2459   "countable C \<Longrightarrow> (\<And>\<Omega>. \<Omega> \<in> C \<Longrightarrow> \<Omega> \<inter> space M \<in> sets M) \<Longrightarrow> space M \<subseteq> (\<Union>C) \<Longrightarrow>

  2460     f \<in> measurable M N \<longleftrightarrow> (\<forall>\<Omega>\<in>C. f \<in> measurable (restrict_space M \<Omega>) N)"

  2461   by (auto intro: measurable_piecewise_restrict measurable_restrict_space1)

  2462

  2463 lemma measurable_If_restrict_space_iff:

  2464   "{x\<in>space M. P x} \<in> sets M \<Longrightarrow>

  2465     (\<lambda>x. if P x then f x else g x) \<in> measurable M N \<longleftrightarrow>

  2466     (f \<in> measurable (restrict_space M {x. P x}) N \<and> g \<in> measurable (restrict_space M {x. \<not> P x}) N)"

  2467   by (subst measurable_piecewise_restrict_iff[where C="{{x. P x}, {x. \<not> P x}}"])

  2468      (auto simp: Int_def sets.sets_Collect_neg space_restrict_space conj_commute[of _ "x \<in> space M" for x]

  2469            cong: measurable_cong')

  2470

  2471 lemma measurable_If:

  2472   "f \<in> measurable M M' \<Longrightarrow> g \<in> measurable M M' \<Longrightarrow> {x\<in>space M. P x} \<in> sets M \<Longrightarrow>

  2473     (\<lambda>x. if P x then f x else g x) \<in> measurable M M'"

  2474   unfolding measurable_If_restrict_space_iff by (auto intro: measurable_restrict_space1)

  2475

  2476 lemma measurable_If_set:

  2477   assumes measure: "f \<in> measurable M M'" "g \<in> measurable M M'"

  2478   assumes P: "A \<inter> space M \<in> sets M"

  2479   shows "(\<lambda>x. if x \<in> A then f x else g x) \<in> measurable M M'"

  2480 proof (rule measurable_If[OF measure])

  2481   have "{x \<in> space M. x \<in> A} = A \<inter> space M" by auto

  2482   thus "{x \<in> space M. x \<in> A} \<in> sets M" using A \<inter> space M \<in> sets M by auto

  2483 qed

  2484

  2485 lemma measurable_restrict_space_iff:

  2486   "\<Omega> \<inter> space M \<in> sets M \<Longrightarrow> c \<in> space N \<Longrightarrow>

  2487     f \<in> measurable (restrict_space M \<Omega>) N \<longleftrightarrow> (\<lambda>x. if x \<in> \<Omega> then f x else c) \<in> measurable M N"

  2488   by (subst measurable_If_restrict_space_iff)

  2489      (simp_all add: Int_def conj_commute measurable_const)

  2490

  2491 lemma restrict_space_singleton: "{x} \<in> sets M \<Longrightarrow> sets (restrict_space M {x}) = sets (count_space {x})"

  2492   using sets_restrict_space_iff[of "{x}" M]

  2493   by (auto simp add: sets_restrict_space_iff dest!: subset_singletonD)

  2494

  2495 lemma measurable_restrict_countable:

  2496   assumes X[intro]: "countable X"

  2497   assumes sets[simp]: "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"

  2498   assumes space[simp]: "\<And>x. x \<in> X \<Longrightarrow> f x \<in> space N"

  2499   assumes f: "f \<in> measurable (restrict_space M (- X)) N"

  2500   shows "f \<in> measurable M N"

  2501   using f sets.countable[OF sets X]

  2502   by (intro measurable_piecewise_restrict[where M=M and C="{- X} \<union> ((\<lambda>x. {x})  X)"])

  2503      (auto simp: Diff_Int_distrib2 Compl_eq_Diff_UNIV Int_insert_left sets.Diff restrict_space_singleton

  2504            simp del: sets_count_space  cong: measurable_cong_sets)

  2505

  2506 lemma measurable_discrete_difference:

  2507   assumes f: "f \<in> measurable M N"

  2508   assumes X: "countable X" "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M" "\<And>x. x \<in> X \<Longrightarrow> g x \<in> space N"

  2509   assumes eq: "\<And>x. x \<in> space M \<Longrightarrow> x \<notin> X \<Longrightarrow> f x = g x"

  2510   shows "g \<in> measurable M N"

  2511   by (rule measurable_restrict_countable[OF X])

  2512      (auto simp: eq[symmetric] space_restrict_space cong: measurable_cong' intro: f measurable_restrict_space1)

  2513

  2514 end

  2515