src/HOL/Probability/Sigma_Algebra.thy
author haftmann
Fri Jun 19 07:53:35 2015 +0200 (2015-06-19)
changeset 60517 f16e4fb20652
parent 60063 81835db730e8
child 60727 53697011b03a
permissions -rw-r--r--
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
     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 "A`X \<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: "A`X \<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: "A`X \<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: "A`X \<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> (G`I)) \<mu>') \<and> \<not> (\<forall>i\<in>I. \<mu> i = 0)
  2100       then measure_of \<Omega> (G`I) (SOME \<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G`I)) \<mu>')
  2101       else measure_of \<Omega> (G`I) (\<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> (G`I)"
  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> (G`I) (\<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> (G`I)) \<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> (G`I) (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> (G`I)"
  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