author hoelzl
Fri Feb 19 13:40:50 2016 +0100 (2016-02-19)
changeset 62378 85ed00c1fe7c
parent 62343 24106dc44def
child 62390 842917225d56
permissions -rw-r--r--
generalize more theorems to support enat and ennreal
     1 (*  Title:      HOL/Probability/Sigma_Algebra.thy
     2     Author:     Stefan Richter, Markus Wenzel, TU München
     3     Author:     Johannes Hölzl, TU München
     4     Plus material from the Hurd/Coble measure theory development,
     5     translated by Lawrence Paulson.
     6 *)
     8 section \<open>Describing measurable sets\<close>
    10 theory Sigma_Algebra
    11 imports
    12   Complex_Main
    13   "~~/src/HOL/Library/Countable_Set"
    14   "~~/src/HOL/Library/FuncSet"
    15   "~~/src/HOL/Library/Indicator_Function"
    16   "~~/src/HOL/Library/Extended_Real"
    17   "~~/src/HOL/Library/Disjoint_Sets"
    18 begin
    20 text \<open>Sigma algebras are an elementary concept in measure
    21   theory. To measure --- that is to integrate --- functions, we first have
    22   to measure sets. Unfortunately, when dealing with a large universe,
    23   it is often not possible to consistently assign a measure to every
    24   subset. Therefore it is necessary to define the set of measurable
    25   subsets of the universe. A sigma algebra is such a set that has
    26   three very natural and desirable properties.\<close>
    28 subsection \<open>Families of sets\<close>
    30 locale subset_class =
    31   fixes \<Omega> :: "'a set" and M :: "'a set set"
    32   assumes space_closed: "M \<subseteq> Pow \<Omega>"
    34 lemma (in subset_class) sets_into_space: "x \<in> M \<Longrightarrow> x \<subseteq> \<Omega>"
    35   by (metis PowD contra_subsetD space_closed)
    37 subsubsection \<open>Semiring of sets\<close>
    39 locale semiring_of_sets = subset_class +
    40   assumes empty_sets[iff]: "{} \<in> M"
    41   assumes Int[intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<inter> b \<in> M"
    42   assumes Diff_cover:
    43     "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> \<exists>C\<subseteq>M. finite C \<and> disjoint C \<and> a - b = \<Union>C"
    45 lemma (in semiring_of_sets) finite_INT[intro]:
    46   assumes "finite I" "I \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M"
    47   shows "(\<Inter>i\<in>I. A i) \<in> M"
    48   using assms by (induct rule: finite_ne_induct) auto
    50 lemma (in semiring_of_sets) Int_space_eq1 [simp]: "x \<in> M \<Longrightarrow> \<Omega> \<inter> x = x"
    51   by (metis Int_absorb1 sets_into_space)
    53 lemma (in semiring_of_sets) Int_space_eq2 [simp]: "x \<in> M \<Longrightarrow> x \<inter> \<Omega> = x"
    54   by (metis Int_absorb2 sets_into_space)
    56 lemma (in semiring_of_sets) sets_Collect_conj:
    57   assumes "{x\<in>\<Omega>. P x} \<in> M" "{x\<in>\<Omega>. Q x} \<in> M"
    58   shows "{x\<in>\<Omega>. Q x \<and> P x} \<in> M"
    59 proof -
    60   have "{x\<in>\<Omega>. Q x \<and> P x} = {x\<in>\<Omega>. Q x} \<inter> {x\<in>\<Omega>. P x}"
    61     by auto
    62   with assms show ?thesis by auto
    63 qed
    65 lemma (in semiring_of_sets) sets_Collect_finite_All':
    66   assumes "\<And>i. i \<in> S \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M" "finite S" "S \<noteq> {}"
    67   shows "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} \<in> M"
    68 proof -
    69   have "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} = (\<Inter>i\<in>S. {x\<in>\<Omega>. P i x})"
    70     using \<open>S \<noteq> {}\<close> by auto
    71   with assms show ?thesis by auto
    72 qed
    74 locale ring_of_sets = semiring_of_sets +
    75   assumes Un [intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<union> b \<in> M"
    77 lemma (in ring_of_sets) finite_Union [intro]:
    78   "finite X \<Longrightarrow> X \<subseteq> M \<Longrightarrow> \<Union>X \<in> M"
    79   by (induct set: finite) (auto simp add: Un)
    81 lemma (in ring_of_sets) finite_UN[intro]:
    82   assumes "finite I" and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M"
    83   shows "(\<Union>i\<in>I. A i) \<in> M"
    84   using assms by induct auto
    86 lemma (in ring_of_sets) Diff [intro]:
    87   assumes "a \<in> M" "b \<in> M" shows "a - b \<in> M"
    88   using Diff_cover[OF assms] by auto
    90 lemma ring_of_setsI:
    91   assumes space_closed: "M \<subseteq> Pow \<Omega>"
    92   assumes empty_sets[iff]: "{} \<in> M"
    93   assumes Un[intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<union> b \<in> M"
    94   assumes Diff[intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a - b \<in> M"
    95   shows "ring_of_sets \<Omega> M"
    96 proof
    97   fix a b assume ab: "a \<in> M" "b \<in> M"
    98   from ab show "\<exists>C\<subseteq>M. finite C \<and> disjoint C \<and> a - b = \<Union>C"
    99     by (intro exI[of _ "{a - b}"]) (auto simp: disjoint_def)
   100   have "a \<inter> b = a - (a - b)" by auto
   101   also have "\<dots> \<in> M" using ab by auto
   102   finally show "a \<inter> b \<in> M" .
   103 qed fact+
   105 lemma ring_of_sets_iff: "ring_of_sets \<Omega> M \<longleftrightarrow> M \<subseteq> Pow \<Omega> \<and> {} \<in> M \<and> (\<forall>a\<in>M. \<forall>b\<in>M. a \<union> b \<in> M) \<and> (\<forall>a\<in>M. \<forall>b\<in>M. a - b \<in> M)"
   106 proof
   107   assume "ring_of_sets \<Omega> M"
   108   then interpret ring_of_sets \<Omega> M .
   109   show "M \<subseteq> Pow \<Omega> \<and> {} \<in> M \<and> (\<forall>a\<in>M. \<forall>b\<in>M. a \<union> b \<in> M) \<and> (\<forall>a\<in>M. \<forall>b\<in>M. a - b \<in> M)"
   110     using space_closed by auto
   111 qed (auto intro!: ring_of_setsI)
   113 lemma (in ring_of_sets) insert_in_sets:
   114   assumes "{x} \<in> M" "A \<in> M" shows "insert x A \<in> M"
   115 proof -
   116   have "{x} \<union> A \<in> M" using assms by (rule Un)
   117   thus ?thesis by auto
   118 qed
   120 lemma (in ring_of_sets) sets_Collect_disj:
   121   assumes "{x\<in>\<Omega>. P x} \<in> M" "{x\<in>\<Omega>. Q x} \<in> M"
   122   shows "{x\<in>\<Omega>. Q x \<or> P x} \<in> M"
   123 proof -
   124   have "{x\<in>\<Omega>. Q x \<or> P x} = {x\<in>\<Omega>. Q x} \<union> {x\<in>\<Omega>. P x}"
   125     by auto
   126   with assms show ?thesis by auto
   127 qed
   129 lemma (in ring_of_sets) sets_Collect_finite_Ex:
   130   assumes "\<And>i. i \<in> S \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M" "finite S"
   131   shows "{x\<in>\<Omega>. \<exists>i\<in>S. P i x} \<in> M"
   132 proof -
   133   have "{x\<in>\<Omega>. \<exists>i\<in>S. P i x} = (\<Union>i\<in>S. {x\<in>\<Omega>. P i x})"
   134     by auto
   135   with assms show ?thesis by auto
   136 qed
   138 locale algebra = ring_of_sets +
   139   assumes top [iff]: "\<Omega> \<in> M"
   141 lemma (in algebra) compl_sets [intro]:
   142   "a \<in> M \<Longrightarrow> \<Omega> - a \<in> M"
   143   by auto
   145 lemma algebra_iff_Un:
   146   "algebra \<Omega> M \<longleftrightarrow>
   147     M \<subseteq> Pow \<Omega> \<and>
   148     {} \<in> M \<and>
   149     (\<forall>a \<in> M. \<Omega> - a \<in> M) \<and>
   150     (\<forall>a \<in> M. \<forall> b \<in> M. a \<union> b \<in> M)" (is "_ \<longleftrightarrow> ?Un")
   151 proof
   152   assume "algebra \<Omega> M"
   153   then interpret algebra \<Omega> M .
   154   show ?Un using sets_into_space by auto
   155 next
   156   assume ?Un
   157   then have "\<Omega> \<in> M" by auto
   158   interpret ring_of_sets \<Omega> M
   159   proof (rule ring_of_setsI)
   160     show \<Omega>: "M \<subseteq> Pow \<Omega>" "{} \<in> M"
   161       using \<open>?Un\<close> by auto
   162     fix a b assume a: "a \<in> M" and b: "b \<in> M"
   163     then show "a \<union> b \<in> M" using \<open>?Un\<close> by auto
   164     have "a - b = \<Omega> - ((\<Omega> - a) \<union> b)"
   165       using \<Omega> a b by auto
   166     then show "a - b \<in> M"
   167       using a b  \<open>?Un\<close> by auto
   168   qed
   169   show "algebra \<Omega> M" proof qed fact
   170 qed
   172 lemma algebra_iff_Int:
   173      "algebra \<Omega> M \<longleftrightarrow>
   174        M \<subseteq> Pow \<Omega> & {} \<in> M &
   175        (\<forall>a \<in> M. \<Omega> - a \<in> M) &
   176        (\<forall>a \<in> M. \<forall> b \<in> M. a \<inter> b \<in> M)" (is "_ \<longleftrightarrow> ?Int")
   177 proof
   178   assume "algebra \<Omega> M"
   179   then interpret algebra \<Omega> M .
   180   show ?Int using sets_into_space by auto
   181 next
   182   assume ?Int
   183   show "algebra \<Omega> M"
   184   proof (unfold algebra_iff_Un, intro conjI ballI)
   185     show \<Omega>: "M \<subseteq> Pow \<Omega>" "{} \<in> M"
   186       using \<open>?Int\<close> by auto
   187     from \<open>?Int\<close> show "\<And>a. a \<in> M \<Longrightarrow> \<Omega> - a \<in> M" by auto
   188     fix a b assume M: "a \<in> M" "b \<in> M"
   189     hence "a \<union> b = \<Omega> - ((\<Omega> - a) \<inter> (\<Omega> - b))"
   190       using \<Omega> by blast
   191     also have "... \<in> M"
   192       using M \<open>?Int\<close> by auto
   193     finally show "a \<union> b \<in> M" .
   194   qed
   195 qed
   197 lemma (in algebra) sets_Collect_neg:
   198   assumes "{x\<in>\<Omega>. P x} \<in> M"
   199   shows "{x\<in>\<Omega>. \<not> P x} \<in> M"
   200 proof -
   201   have "{x\<in>\<Omega>. \<not> P x} = \<Omega> - {x\<in>\<Omega>. P x}" by auto
   202   with assms show ?thesis by auto
   203 qed
   205 lemma (in algebra) sets_Collect_imp:
   206   "{x\<in>\<Omega>. P x} \<in> M \<Longrightarrow> {x\<in>\<Omega>. Q x} \<in> M \<Longrightarrow> {x\<in>\<Omega>. Q x \<longrightarrow> P x} \<in> M"
   207   unfolding imp_conv_disj by (intro sets_Collect_disj sets_Collect_neg)
   209 lemma (in algebra) sets_Collect_const:
   210   "{x\<in>\<Omega>. P} \<in> M"
   211   by (cases P) auto
   213 lemma algebra_single_set:
   214   "X \<subseteq> S \<Longrightarrow> algebra S { {}, X, S - X, S }"
   215   by (auto simp: algebra_iff_Int)
   217 subsubsection \<open>Restricted algebras\<close>
   219 abbreviation (in algebra)
   220   "restricted_space A \<equiv> (op \<inter> A) ` M"
   222 lemma (in algebra) restricted_algebra:
   223   assumes "A \<in> M" shows "algebra A (restricted_space A)"
   224   using assms by (auto simp: algebra_iff_Int)
   226 subsubsection \<open>Sigma Algebras\<close>
   228 locale sigma_algebra = algebra +
   229   assumes countable_nat_UN [intro]: "\<And>A. range A \<subseteq> M \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
   231 lemma (in algebra) is_sigma_algebra:
   232   assumes "finite M"
   233   shows "sigma_algebra \<Omega> M"
   234 proof
   235   fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> M"
   236   then have "(\<Union>i. A i) = (\<Union>s\<in>M \<inter> range A. s)"
   237     by auto
   238   also have "(\<Union>s\<in>M \<inter> range A. s) \<in> M"
   239     using \<open>finite M\<close> by auto
   240   finally show "(\<Union>i. A i) \<in> M" .
   241 qed
   243 lemma countable_UN_eq:
   244   fixes A :: "'i::countable \<Rightarrow> 'a set"
   245   shows "(range A \<subseteq> M \<longrightarrow> (\<Union>i. A i) \<in> M) \<longleftrightarrow>
   246     (range (A \<circ> from_nat) \<subseteq> M \<longrightarrow> (\<Union>i. (A \<circ> from_nat) i) \<in> M)"
   247 proof -
   248   let ?A' = "A \<circ> from_nat"
   249   have *: "(\<Union>i. ?A' i) = (\<Union>i. A i)" (is "?l = ?r")
   250   proof safe
   251     fix x i assume "x \<in> A i" thus "x \<in> ?l"
   252       by (auto intro!: exI[of _ "to_nat i"])
   253   next
   254     fix x i assume "x \<in> ?A' i" thus "x \<in> ?r"
   255       by (auto intro!: exI[of _ "from_nat i"])
   256   qed
   257   have **: "range ?A' = range A"
   258     using surj_from_nat
   259     by (auto simp: image_comp [symmetric] intro!: imageI)
   260   show ?thesis unfolding * ** ..
   261 qed
   263 lemma (in sigma_algebra) countable_Union [intro]:
   264   assumes "countable X" "X \<subseteq> M" shows "\<Union>X \<in> M"
   265 proof cases
   266   assume "X \<noteq> {}"
   267   hence "\<Union>X = (\<Union>n. from_nat_into X n)"
   268     using assms by (auto intro: from_nat_into) (metis from_nat_into_surj)
   269   also have "\<dots> \<in> M" using assms
   270     by (auto intro!: countable_nat_UN) (metis \<open>X \<noteq> {}\<close> from_nat_into set_mp)
   271   finally show ?thesis .
   272 qed simp
   274 lemma (in sigma_algebra) countable_UN[intro]:
   275   fixes A :: "'i::countable \<Rightarrow> 'a set"
   276   assumes "A`X \<subseteq> M"
   277   shows  "(\<Union>x\<in>X. A x) \<in> M"
   278 proof -
   279   let ?A = "\<lambda>i. if i \<in> X then A i else {}"
   280   from assms have "range ?A \<subseteq> M" by auto
   281   with countable_nat_UN[of "?A \<circ> from_nat"] countable_UN_eq[of ?A M]
   282   have "(\<Union>x. ?A x) \<in> M" by auto
   283   moreover have "(\<Union>x. ?A x) = (\<Union>x\<in>X. A x)" by (auto split: split_if_asm)
   284   ultimately show ?thesis by simp
   285 qed
   287 lemma (in sigma_algebra) countable_UN':
   288   fixes A :: "'i \<Rightarrow> 'a set"
   289   assumes X: "countable X"
   290   assumes A: "A`X \<subseteq> M"
   291   shows  "(\<Union>x\<in>X. A x) \<in> M"
   292 proof -
   293   have "(\<Union>x\<in>X. A x) = (\<Union>i\<in>to_nat_on X ` X. A (from_nat_into X i))"
   294     using X by auto
   295   also have "\<dots> \<in> M"
   296     using A X
   297     by (intro countable_UN) auto
   298   finally show ?thesis .
   299 qed
   301 lemma (in sigma_algebra) countable_UN'':
   302   "\<lbrakk> countable X; \<And>x y. x \<in> X \<Longrightarrow> A x \<in> M \<rbrakk> \<Longrightarrow> (\<Union>x\<in>X. A x) \<in> M"
   303 by(erule countable_UN')(auto)
   305 lemma (in sigma_algebra) countable_INT [intro]:
   306   fixes A :: "'i::countable \<Rightarrow> 'a set"
   307   assumes A: "A`X \<subseteq> M" "X \<noteq> {}"
   308   shows "(\<Inter>i\<in>X. A i) \<in> M"
   309 proof -
   310   from A have "\<forall>i\<in>X. A i \<in> M" by fast
   311   hence "\<Omega> - (\<Union>i\<in>X. \<Omega> - A i) \<in> M" by blast
   312   moreover
   313   have "(\<Inter>i\<in>X. A i) = \<Omega> - (\<Union>i\<in>X. \<Omega> - A i)" using space_closed A
   314     by blast
   315   ultimately show ?thesis by metis
   316 qed
   318 lemma (in sigma_algebra) countable_INT':
   319   fixes A :: "'i \<Rightarrow> 'a set"
   320   assumes X: "countable X" "X \<noteq> {}"
   321   assumes A: "A`X \<subseteq> M"
   322   shows  "(\<Inter>x\<in>X. A x) \<in> M"
   323 proof -
   324   have "(\<Inter>x\<in>X. A x) = (\<Inter>i\<in>to_nat_on X ` X. A (from_nat_into X i))"
   325     using X by auto
   326   also have "\<dots> \<in> M"
   327     using A X
   328     by (intro countable_INT) auto
   329   finally show ?thesis .
   330 qed
   332 lemma (in sigma_algebra) countable_INT'':
   333   "UNIV \<in> M \<Longrightarrow> countable I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> F i \<in> M) \<Longrightarrow> (\<Inter>i\<in>I. F i) \<in> M"
   334   by (cases "I = {}") (auto intro: countable_INT')
   336 lemma (in sigma_algebra) countable:
   337   assumes "\<And>a. a \<in> A \<Longrightarrow> {a} \<in> M" "countable A"
   338   shows "A \<in> M"
   339 proof -
   340   have "(\<Union>a\<in>A. {a}) \<in> M"
   341     using assms by (intro countable_UN') auto
   342   also have "(\<Union>a\<in>A. {a}) = A" by auto
   343   finally show ?thesis by auto
   344 qed
   346 lemma ring_of_sets_Pow: "ring_of_sets sp (Pow sp)"
   347   by (auto simp: ring_of_sets_iff)
   349 lemma algebra_Pow: "algebra sp (Pow sp)"
   350   by (auto simp: algebra_iff_Un)
   352 lemma sigma_algebra_iff:
   353   "sigma_algebra \<Omega> M \<longleftrightarrow>
   354     algebra \<Omega> M \<and> (\<forall>A. range A \<subseteq> M \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"
   355   by (simp add: sigma_algebra_def sigma_algebra_axioms_def)
   357 lemma sigma_algebra_Pow: "sigma_algebra sp (Pow sp)"
   358   by (auto simp: sigma_algebra_iff algebra_iff_Int)
   360 lemma (in sigma_algebra) sets_Collect_countable_All:
   361   assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
   362   shows "{x\<in>\<Omega>. \<forall>i::'i::countable. P i x} \<in> M"
   363 proof -
   364   have "{x\<in>\<Omega>. \<forall>i::'i::countable. P i x} = (\<Inter>i. {x\<in>\<Omega>. P i x})" by auto
   365   with assms show ?thesis by auto
   366 qed
   368 lemma (in sigma_algebra) sets_Collect_countable_Ex:
   369   assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
   370   shows "{x\<in>\<Omega>. \<exists>i::'i::countable. P i x} \<in> M"
   371 proof -
   372   have "{x\<in>\<Omega>. \<exists>i::'i::countable. P i x} = (\<Union>i. {x\<in>\<Omega>. P i x})" by auto
   373   with assms show ?thesis by auto
   374 qed
   376 lemma (in sigma_algebra) sets_Collect_countable_Ex':
   377   assumes "\<And>i. i \<in> I \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M"
   378   assumes "countable I"
   379   shows "{x\<in>\<Omega>. \<exists>i\<in>I. P i x} \<in> M"
   380 proof -
   381   have "{x\<in>\<Omega>. \<exists>i\<in>I. P i x} = (\<Union>i\<in>I. {x\<in>\<Omega>. P i x})" by auto
   382   with assms show ?thesis 
   383     by (auto intro!: countable_UN')
   384 qed
   386 lemma (in sigma_algebra) sets_Collect_countable_All':
   387   assumes "\<And>i. i \<in> I \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M"
   388   assumes "countable I"
   389   shows "{x\<in>\<Omega>. \<forall>i\<in>I. P i x} \<in> M"
   390 proof -
   391   have "{x\<in>\<Omega>. \<forall>i\<in>I. P i x} = (\<Inter>i\<in>I. {x\<in>\<Omega>. P i x}) \<inter> \<Omega>" by auto
   392   with assms show ?thesis 
   393     by (cases "I = {}") (auto intro!: countable_INT')
   394 qed
   396 lemma (in sigma_algebra) sets_Collect_countable_Ex1':
   397   assumes "\<And>i. i \<in> I \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M"
   398   assumes "countable I"
   399   shows "{x\<in>\<Omega>. \<exists>!i\<in>I. P i x} \<in> M"
   400 proof -
   401   have "{x\<in>\<Omega>. \<exists>!i\<in>I. P i x} = {x\<in>\<Omega>. \<exists>i\<in>I. P i x \<and> (\<forall>j\<in>I. P j x \<longrightarrow> i = j)}"
   402     by auto
   403   with assms show ?thesis 
   404     by (auto intro!: sets_Collect_countable_All' sets_Collect_countable_Ex' sets_Collect_conj sets_Collect_imp sets_Collect_const)
   405 qed
   407 lemmas (in sigma_algebra) sets_Collect =
   408   sets_Collect_imp sets_Collect_disj sets_Collect_conj sets_Collect_neg sets_Collect_const
   409   sets_Collect_countable_All sets_Collect_countable_Ex sets_Collect_countable_All
   411 lemma (in sigma_algebra) sets_Collect_countable_Ball:
   412   assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
   413   shows "{x\<in>\<Omega>. \<forall>i::'i::countable\<in>X. P i x} \<in> M"
   414   unfolding Ball_def by (intro sets_Collect assms)
   416 lemma (in sigma_algebra) sets_Collect_countable_Bex:
   417   assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
   418   shows "{x\<in>\<Omega>. \<exists>i::'i::countable\<in>X. P i x} \<in> M"
   419   unfolding Bex_def by (intro sets_Collect assms)
   421 lemma sigma_algebra_single_set:
   422   assumes "X \<subseteq> S"
   423   shows "sigma_algebra S { {}, X, S - X, S }"
   424   using algebra.is_sigma_algebra[OF algebra_single_set[OF \<open>X \<subseteq> S\<close>]] by simp
   426 subsubsection \<open>Binary Unions\<close>
   428 definition binary :: "'a \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"
   429   where "binary a b =  (\<lambda>x. b)(0 := a)"
   431 lemma range_binary_eq: "range(binary a b) = {a,b}"
   432   by (auto simp add: binary_def)
   434 lemma Un_range_binary: "a \<union> b = (\<Union>i::nat. binary a b i)"
   435   by (simp add: range_binary_eq cong del: strong_SUP_cong)
   437 lemma Int_range_binary: "a \<inter> b = (\<Inter>i::nat. binary a b i)"
   438   by (simp add: range_binary_eq cong del: strong_INF_cong)
   440 lemma sigma_algebra_iff2:
   441      "sigma_algebra \<Omega> M \<longleftrightarrow>
   442        M \<subseteq> Pow \<Omega> \<and>
   443        {} \<in> M \<and> (\<forall>s \<in> M. \<Omega> - s \<in> M) \<and>
   444        (\<forall>A. range A \<subseteq> M \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"
   445   by (auto simp add: range_binary_eq sigma_algebra_def sigma_algebra_axioms_def
   446          algebra_iff_Un Un_range_binary)
   448 subsubsection \<open>Initial Sigma Algebra\<close>
   450 text \<open>Sigma algebras can naturally be created as the closure of any set of
   451   M with regard to the properties just postulated.\<close>
   453 inductive_set sigma_sets :: "'a set \<Rightarrow> 'a set set \<Rightarrow> 'a set set"
   454   for sp :: "'a set" and A :: "'a set set"
   455   where
   456     Basic[intro, simp]: "a \<in> A \<Longrightarrow> a \<in> sigma_sets sp A"
   457   | Empty: "{} \<in> sigma_sets sp A"
   458   | Compl: "a \<in> sigma_sets sp A \<Longrightarrow> sp - a \<in> sigma_sets sp A"
   459   | Union: "(\<And>i::nat. a i \<in> sigma_sets sp A) \<Longrightarrow> (\<Union>i. a i) \<in> sigma_sets sp A"
   461 lemma (in sigma_algebra) sigma_sets_subset:
   462   assumes a: "a \<subseteq> M"
   463   shows "sigma_sets \<Omega> a \<subseteq> M"
   464 proof
   465   fix x
   466   assume "x \<in> sigma_sets \<Omega> a"
   467   from this show "x \<in> M"
   468     by (induct rule: sigma_sets.induct, auto) (metis a subsetD)
   469 qed
   471 lemma sigma_sets_into_sp: "A \<subseteq> Pow sp \<Longrightarrow> x \<in> sigma_sets sp A \<Longrightarrow> x \<subseteq> sp"
   472   by (erule sigma_sets.induct, auto)
   474 lemma sigma_algebra_sigma_sets:
   475      "a \<subseteq> Pow \<Omega> \<Longrightarrow> sigma_algebra \<Omega> (sigma_sets \<Omega> a)"
   476   by (auto simp add: sigma_algebra_iff2 dest: sigma_sets_into_sp
   477            intro!: sigma_sets.Union sigma_sets.Empty sigma_sets.Compl)
   479 lemma sigma_sets_least_sigma_algebra:
   480   assumes "A \<subseteq> Pow S"
   481   shows "sigma_sets S A = \<Inter>{B. A \<subseteq> B \<and> sigma_algebra S B}"
   482 proof safe
   483   fix B X assume "A \<subseteq> B" and sa: "sigma_algebra S B"
   484     and X: "X \<in> sigma_sets S A"
   485   from sigma_algebra.sigma_sets_subset[OF sa, simplified, OF \<open>A \<subseteq> B\<close>] X
   486   show "X \<in> B" by auto
   487 next
   488   fix X assume "X \<in> \<Inter>{B. A \<subseteq> B \<and> sigma_algebra S B}"
   489   then have [intro!]: "\<And>B. A \<subseteq> B \<Longrightarrow> sigma_algebra S B \<Longrightarrow> X \<in> B"
   490      by simp
   491   have "A \<subseteq> sigma_sets S A" using assms by auto
   492   moreover have "sigma_algebra S (sigma_sets S A)"
   493     using assms by (intro sigma_algebra_sigma_sets[of A]) auto
   494   ultimately show "X \<in> sigma_sets S A" by auto
   495 qed
   497 lemma sigma_sets_top: "sp \<in> sigma_sets sp A"
   498   by (metis Diff_empty sigma_sets.Compl sigma_sets.Empty)
   500 lemma sigma_sets_Un:
   501   "a \<in> sigma_sets sp A \<Longrightarrow> b \<in> sigma_sets sp A \<Longrightarrow> a \<union> b \<in> sigma_sets sp A"
   502 apply (simp add: Un_range_binary range_binary_eq)
   503 apply (rule Union, simp add: binary_def)
   504 done
   506 lemma sigma_sets_Inter:
   507   assumes Asb: "A \<subseteq> Pow sp"
   508   shows "(\<And>i::nat. a i \<in> sigma_sets sp A) \<Longrightarrow> (\<Inter>i. a i) \<in> sigma_sets sp A"
   509 proof -
   510   assume ai: "\<And>i::nat. a i \<in> sigma_sets sp A"
   511   hence "\<And>i::nat. sp-(a i) \<in> sigma_sets sp A"
   512     by (rule sigma_sets.Compl)
   513   hence "(\<Union>i. sp-(a i)) \<in> sigma_sets sp A"
   514     by (rule sigma_sets.Union)
   515   hence "sp-(\<Union>i. sp-(a i)) \<in> sigma_sets sp A"
   516     by (rule sigma_sets.Compl)
   517   also have "sp-(\<Union>i. sp-(a i)) = sp Int (\<Inter>i. a i)"
   518     by auto
   519   also have "... = (\<Inter>i. a i)" using ai
   520     by (blast dest: sigma_sets_into_sp [OF Asb])
   521   finally show ?thesis .
   522 qed
   524 lemma sigma_sets_INTER:
   525   assumes Asb: "A \<subseteq> Pow sp"
   526       and ai: "\<And>i::nat. i \<in> S \<Longrightarrow> a i \<in> sigma_sets sp A" and non: "S \<noteq> {}"
   527   shows "(\<Inter>i\<in>S. a i) \<in> sigma_sets sp A"
   528 proof -
   529   from ai have "\<And>i. (if i\<in>S then a i else sp) \<in> sigma_sets sp A"
   530     by (simp add: sigma_sets.intros(2-) sigma_sets_top)
   531   hence "(\<Inter>i. (if i\<in>S then a i else sp)) \<in> sigma_sets sp A"
   532     by (rule sigma_sets_Inter [OF Asb])
   533   also have "(\<Inter>i. (if i\<in>S then a i else sp)) = (\<Inter>i\<in>S. a i)"
   534     by auto (metis ai non sigma_sets_into_sp subset_empty subset_iff Asb)+
   535   finally show ?thesis .
   536 qed
   538 lemma sigma_sets_UNION:
   539   "countable B \<Longrightarrow> (\<And>b. b \<in> B \<Longrightarrow> b \<in> sigma_sets X A) \<Longrightarrow> (\<Union>B) \<in> sigma_sets X A"
   540   apply (cases "B = {}")
   541   apply (simp add: sigma_sets.Empty)
   542   using from_nat_into [of B] range_from_nat_into [of B] sigma_sets.Union [of "from_nat_into B" X A]
   543   apply simp
   544   apply auto
   545   apply (metis Sup_bot_conv(1) Union_empty `\<lbrakk>B \<noteq> {}; countable B\<rbrakk> \<Longrightarrow> range (from_nat_into B) = B`)
   546   done
   548 lemma (in sigma_algebra) sigma_sets_eq:
   549      "sigma_sets \<Omega> M = M"
   550 proof
   551   show "M \<subseteq> sigma_sets \<Omega> M"
   552     by (metis Set.subsetI sigma_sets.Basic)
   553   next
   554   show "sigma_sets \<Omega> M \<subseteq> M"
   555     by (metis sigma_sets_subset subset_refl)
   556 qed
   558 lemma sigma_sets_eqI:
   559   assumes A: "\<And>a. a \<in> A \<Longrightarrow> a \<in> sigma_sets M B"
   560   assumes B: "\<And>b. b \<in> B \<Longrightarrow> b \<in> sigma_sets M A"
   561   shows "sigma_sets M A = sigma_sets M B"
   562 proof (intro set_eqI iffI)
   563   fix a assume "a \<in> sigma_sets M A"
   564   from this A show "a \<in> sigma_sets M B"
   565     by induct (auto intro!: sigma_sets.intros(2-) del: sigma_sets.Basic)
   566 next
   567   fix b assume "b \<in> sigma_sets M B"
   568   from this B show "b \<in> sigma_sets M A"
   569     by induct (auto intro!: sigma_sets.intros(2-) del: sigma_sets.Basic)
   570 qed
   572 lemma sigma_sets_subseteq: assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
   573 proof
   574   fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
   575     by induct (insert \<open>A \<subseteq> B\<close>, auto intro: sigma_sets.intros(2-))
   576 qed
   578 lemma sigma_sets_mono: assumes "A \<subseteq> sigma_sets X B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
   579 proof
   580   fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
   581     by induct (insert \<open>A \<subseteq> sigma_sets X B\<close>, auto intro: sigma_sets.intros(2-))
   582 qed
   584 lemma sigma_sets_mono': assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
   585 proof
   586   fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
   587     by induct (insert \<open>A \<subseteq> B\<close>, auto intro: sigma_sets.intros(2-))
   588 qed
   590 lemma sigma_sets_superset_generator: "A \<subseteq> sigma_sets X A"
   591   by (auto intro: sigma_sets.Basic)
   593 lemma (in sigma_algebra) restriction_in_sets:
   594   fixes A :: "nat \<Rightarrow> 'a set"
   595   assumes "S \<in> M"
   596   and *: "range A \<subseteq> (\<lambda>A. S \<inter> A) ` M" (is "_ \<subseteq> ?r")
   597   shows "range A \<subseteq> M" "(\<Union>i. A i) \<in> (\<lambda>A. S \<inter> A) ` M"
   598 proof -
   599   { fix i have "A i \<in> ?r" using * by auto
   600     hence "\<exists>B. A i = B \<inter> S \<and> B \<in> M" by auto
   601     hence "A i \<subseteq> S" "A i \<in> M" using \<open>S \<in> M\<close> by auto }
   602   thus "range A \<subseteq> M" "(\<Union>i. A i) \<in> (\<lambda>A. S \<inter> A) ` M"
   603     by (auto intro!: image_eqI[of _ _ "(\<Union>i. A i)"])
   604 qed
   606 lemma (in sigma_algebra) restricted_sigma_algebra:
   607   assumes "S \<in> M"
   608   shows "sigma_algebra S (restricted_space S)"
   609   unfolding sigma_algebra_def sigma_algebra_axioms_def
   610 proof safe
   611   show "algebra S (restricted_space S)" using restricted_algebra[OF assms] .
   612 next
   613   fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> restricted_space S"
   614   from restriction_in_sets[OF assms this[simplified]]
   615   show "(\<Union>i. A i) \<in> restricted_space S" by simp
   616 qed
   618 lemma sigma_sets_Int:
   619   assumes "A \<in> sigma_sets sp st" "A \<subseteq> sp"
   620   shows "op \<inter> A ` sigma_sets sp st = sigma_sets A (op \<inter> A ` st)"
   621 proof (intro equalityI subsetI)
   622   fix x assume "x \<in> op \<inter> A ` sigma_sets sp st"
   623   then obtain y where "y \<in> sigma_sets sp st" "x = y \<inter> A" by auto
   624   then have "x \<in> sigma_sets (A \<inter> sp) (op \<inter> A ` st)"
   625   proof (induct arbitrary: x)
   626     case (Compl a)
   627     then show ?case
   628       by (force intro!: sigma_sets.Compl simp: Diff_Int_distrib ac_simps)
   629   next
   630     case (Union a)
   631     then show ?case
   632       by (auto intro!: sigma_sets.Union
   633                simp add: UN_extend_simps simp del: UN_simps)
   634   qed (auto intro!: sigma_sets.intros(2-))
   635   then show "x \<in> sigma_sets A (op \<inter> A ` st)"
   636     using \<open>A \<subseteq> sp\<close> by (simp add: Int_absorb2)
   637 next
   638   fix x assume "x \<in> sigma_sets A (op \<inter> A ` st)"
   639   then show "x \<in> op \<inter> A ` sigma_sets sp st"
   640   proof induct
   641     case (Compl a)
   642     then obtain x where "a = A \<inter> x" "x \<in> sigma_sets sp st" by auto
   643     then show ?case using \<open>A \<subseteq> sp\<close>
   644       by (force simp add: image_iff intro!: bexI[of _ "sp - x"] sigma_sets.Compl)
   645   next
   646     case (Union a)
   647     then have "\<forall>i. \<exists>x. x \<in> sigma_sets sp st \<and> a i = A \<inter> x"
   648       by (auto simp: image_iff Bex_def)
   649     from choice[OF this] guess f ..
   650     then show ?case
   651       by (auto intro!: bexI[of _ "(\<Union>x. f x)"] sigma_sets.Union
   652                simp add: image_iff)
   653   qed (auto intro!: sigma_sets.intros(2-))
   654 qed
   656 lemma sigma_sets_empty_eq: "sigma_sets A {} = {{}, A}"
   657 proof (intro set_eqI iffI)
   658   fix a assume "a \<in> sigma_sets A {}" then show "a \<in> {{}, A}"
   659     by induct blast+
   660 qed (auto intro: sigma_sets.Empty sigma_sets_top)
   662 lemma sigma_sets_single[simp]: "sigma_sets A {A} = {{}, A}"
   663 proof (intro set_eqI iffI)
   664   fix x assume "x \<in> sigma_sets A {A}"
   665   then show "x \<in> {{}, A}"
   666     by induct blast+
   667 next
   668   fix x assume "x \<in> {{}, A}"
   669   then show "x \<in> sigma_sets A {A}"
   670     by (auto intro: sigma_sets.Empty sigma_sets_top)
   671 qed
   673 lemma sigma_sets_sigma_sets_eq:
   674   "M \<subseteq> Pow S \<Longrightarrow> sigma_sets S (sigma_sets S M) = sigma_sets S M"
   675   by (rule sigma_algebra.sigma_sets_eq[OF sigma_algebra_sigma_sets, of M S]) auto
   677 lemma sigma_sets_singleton:
   678   assumes "X \<subseteq> S"
   679   shows "sigma_sets S { X } = { {}, X, S - X, S }"
   680 proof -
   681   interpret sigma_algebra S "{ {}, X, S - X, S }"
   682     by (rule sigma_algebra_single_set) fact
   683   have "sigma_sets S { X } \<subseteq> sigma_sets S { {}, X, S - X, S }"
   684     by (rule sigma_sets_subseteq) simp
   685   moreover have "\<dots> = { {}, X, S - X, S }"
   686     using sigma_sets_eq by simp
   687   moreover
   688   { fix A assume "A \<in> { {}, X, S - X, S }"
   689     then have "A \<in> sigma_sets S { X }"
   690       by (auto intro: sigma_sets.intros(2-) sigma_sets_top) }
   691   ultimately have "sigma_sets S { X } = sigma_sets S { {}, X, S - X, S }"
   692     by (intro antisym) auto
   693   with sigma_sets_eq show ?thesis by simp
   694 qed
   696 lemma restricted_sigma:
   697   assumes S: "S \<in> sigma_sets \<Omega> M" and M: "M \<subseteq> Pow \<Omega>"
   698   shows "algebra.restricted_space (sigma_sets \<Omega> M) S =
   699     sigma_sets S (algebra.restricted_space M S)"
   700 proof -
   701   from S sigma_sets_into_sp[OF M]
   702   have "S \<in> sigma_sets \<Omega> M" "S \<subseteq> \<Omega>" by auto
   703   from sigma_sets_Int[OF this]
   704   show ?thesis by simp
   705 qed
   707 lemma sigma_sets_vimage_commute:
   708   assumes X: "X \<in> \<Omega> \<rightarrow> \<Omega>'"
   709   shows "{X -` A \<inter> \<Omega> |A. A \<in> sigma_sets \<Omega>' M'}
   710        = sigma_sets \<Omega> {X -` A \<inter> \<Omega> |A. A \<in> M'}" (is "?L = ?R")
   711 proof
   712   show "?L \<subseteq> ?R"
   713   proof clarify
   714     fix A assume "A \<in> sigma_sets \<Omega>' M'"
   715     then show "X -` A \<inter> \<Omega> \<in> ?R"
   716     proof induct
   717       case Empty then show ?case
   718         by (auto intro!: sigma_sets.Empty)
   719     next
   720       case (Compl B)
   721       have [simp]: "X -` (\<Omega>' - B) \<inter> \<Omega> = \<Omega> - (X -` B \<inter> \<Omega>)"
   722         by (auto simp add: funcset_mem [OF X])
   723       with Compl show ?case
   724         by (auto intro!: sigma_sets.Compl)
   725     next
   726       case (Union F)
   727       then show ?case
   728         by (auto simp add: vimage_UN UN_extend_simps(4) simp del: UN_simps
   729                  intro!: sigma_sets.Union)
   730     qed auto
   731   qed
   732   show "?R \<subseteq> ?L"
   733   proof clarify
   734     fix A assume "A \<in> ?R"
   735     then show "\<exists>B. A = X -` B \<inter> \<Omega> \<and> B \<in> sigma_sets \<Omega>' M'"
   736     proof induct
   737       case (Basic B) then show ?case by auto
   738     next
   739       case Empty then show ?case
   740         by (auto intro!: sigma_sets.Empty exI[of _ "{}"])
   741     next
   742       case (Compl B)
   743       then obtain A where A: "B = X -` A \<inter> \<Omega>" "A \<in> sigma_sets \<Omega>' M'" by auto
   744       then have [simp]: "\<Omega> - B = X -` (\<Omega>' - A) \<inter> \<Omega>"
   745         by (auto simp add: funcset_mem [OF X])
   746       with A(2) show ?case
   747         by (auto intro: sigma_sets.Compl)
   748     next
   749       case (Union F)
   750       then have "\<forall>i. \<exists>B. F i = X -` B \<inter> \<Omega> \<and> B \<in> sigma_sets \<Omega>' M'" by auto
   751       from choice[OF this] guess A .. note A = this
   752       with A show ?case
   753         by (auto simp: vimage_UN[symmetric] intro: sigma_sets.Union)
   754     qed
   755   qed
   756 qed
   758 lemma (in ring_of_sets) UNION_in_sets:
   759   fixes A:: "nat \<Rightarrow> 'a set"
   760   assumes A: "range A \<subseteq> M"
   761   shows  "(\<Union>i\<in>{0..<n}. A i) \<in> M"
   762 proof (induct n)
   763   case 0 show ?case by simp
   764 next
   765   case (Suc n)
   766   thus ?case
   767     by (simp add: atLeastLessThanSuc) (metis A Un UNIV_I image_subset_iff)
   768 qed
   770 lemma (in ring_of_sets) range_disjointed_sets:
   771   assumes A: "range A \<subseteq> M"
   772   shows  "range (disjointed A) \<subseteq> M"
   773 proof (auto simp add: disjointed_def)
   774   fix n
   775   show "A n - (\<Union>i\<in>{0..<n}. A i) \<in> M" using UNION_in_sets
   776     by (metis A Diff UNIV_I image_subset_iff)
   777 qed
   779 lemma (in algebra) range_disjointed_sets':
   780   "range A \<subseteq> M \<Longrightarrow> range (disjointed A) \<subseteq> M"
   781   using range_disjointed_sets .
   783 lemma sigma_algebra_disjoint_iff:
   784   "sigma_algebra \<Omega> M \<longleftrightarrow> algebra \<Omega> M \<and>
   785     (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"
   786 proof (auto simp add: sigma_algebra_iff)
   787   fix A :: "nat \<Rightarrow> 'a set"
   788   assume M: "algebra \<Omega> M"
   789      and A: "range A \<subseteq> M"
   790      and UnA: "\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M"
   791   hence "range (disjointed A) \<subseteq> M \<longrightarrow>
   792          disjoint_family (disjointed A) \<longrightarrow>
   793          (\<Union>i. disjointed A i) \<in> M" by blast
   794   hence "(\<Union>i. disjointed A i) \<in> M"
   795     by (simp add: algebra.range_disjointed_sets'[of \<Omega>] M A disjoint_family_disjointed)
   796   thus "(\<Union>i::nat. A i) \<in> M" by (simp add: UN_disjointed_eq)
   797 qed
   799 subsubsection \<open>Ring generated by a semiring\<close>
   801 definition (in semiring_of_sets)
   802   "generated_ring = { \<Union>C | C. C \<subseteq> M \<and> finite C \<and> disjoint C }"
   804 lemma (in semiring_of_sets) generated_ringE[elim?]:
   805   assumes "a \<in> generated_ring"
   806   obtains C where "finite C" "disjoint C" "C \<subseteq> M" "a = \<Union>C"
   807   using assms unfolding generated_ring_def by auto
   809 lemma (in semiring_of_sets) generated_ringI[intro?]:
   810   assumes "finite C" "disjoint C" "C \<subseteq> M" "a = \<Union>C"
   811   shows "a \<in> generated_ring"
   812   using assms unfolding generated_ring_def by auto
   814 lemma (in semiring_of_sets) generated_ringI_Basic:
   815   "A \<in> M \<Longrightarrow> A \<in> generated_ring"
   816   by (rule generated_ringI[of "{A}"]) (auto simp: disjoint_def)
   818 lemma (in semiring_of_sets) generated_ring_disjoint_Un[intro]:
   819   assumes a: "a \<in> generated_ring" and b: "b \<in> generated_ring"
   820   and "a \<inter> b = {}"
   821   shows "a \<union> b \<in> generated_ring"
   822 proof -
   823   from a guess Ca .. note Ca = this
   824   from b guess Cb .. note Cb = this
   825   show ?thesis
   826   proof
   827     show "disjoint (Ca \<union> Cb)"
   828       using \<open>a \<inter> b = {}\<close> Ca Cb by (auto intro!: disjoint_union)
   829   qed (insert Ca Cb, auto)
   830 qed
   832 lemma (in semiring_of_sets) generated_ring_empty: "{} \<in> generated_ring"
   833   by (auto simp: generated_ring_def disjoint_def)
   835 lemma (in semiring_of_sets) generated_ring_disjoint_Union:
   836   assumes "finite A" shows "A \<subseteq> generated_ring \<Longrightarrow> disjoint A \<Longrightarrow> \<Union>A \<in> generated_ring"
   837   using assms by (induct A) (auto simp: disjoint_def intro!: generated_ring_disjoint_Un generated_ring_empty)
   839 lemma (in semiring_of_sets) generated_ring_disjoint_UNION:
   840   "finite I \<Longrightarrow> disjoint (A ` I) \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> A i \<in> generated_ring) \<Longrightarrow> UNION I A \<in> generated_ring"
   841   by (intro generated_ring_disjoint_Union) auto
   843 lemma (in semiring_of_sets) generated_ring_Int:
   844   assumes a: "a \<in> generated_ring" and b: "b \<in> generated_ring"
   845   shows "a \<inter> b \<in> generated_ring"
   846 proof -
   847   from a guess Ca .. note Ca = this
   848   from b guess Cb .. note Cb = this
   849   def C \<equiv> "(\<lambda>(a,b). a \<inter> b)` (Ca\<times>Cb)"
   850   show ?thesis
   851   proof
   852     show "disjoint C"
   853     proof (simp add: disjoint_def C_def, intro ballI impI)
   854       fix a1 b1 a2 b2 assume sets: "a1 \<in> Ca" "b1 \<in> Cb" "a2 \<in> Ca" "b2 \<in> Cb"
   855       assume "a1 \<inter> b1 \<noteq> a2 \<inter> b2"
   856       then have "a1 \<noteq> a2 \<or> b1 \<noteq> b2" by auto
   857       then show "(a1 \<inter> b1) \<inter> (a2 \<inter> b2) = {}"
   858       proof
   859         assume "a1 \<noteq> a2"
   860         with sets Ca have "a1 \<inter> a2 = {}"
   861           by (auto simp: disjoint_def)
   862         then show ?thesis by auto
   863       next
   864         assume "b1 \<noteq> b2"
   865         with sets Cb have "b1 \<inter> b2 = {}"
   866           by (auto simp: disjoint_def)
   867         then show ?thesis by auto
   868       qed
   869     qed
   870   qed (insert Ca Cb, auto simp: C_def)
   871 qed
   873 lemma (in semiring_of_sets) generated_ring_Inter:
   874   assumes "finite A" "A \<noteq> {}" shows "A \<subseteq> generated_ring \<Longrightarrow> \<Inter>A \<in> generated_ring"
   875   using assms by (induct A rule: finite_ne_induct) (auto intro: generated_ring_Int)
   877 lemma (in semiring_of_sets) generated_ring_INTER:
   878   "finite I \<Longrightarrow> I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> A i \<in> generated_ring) \<Longrightarrow> INTER I A \<in> generated_ring"
   879   by (intro generated_ring_Inter) auto
   881 lemma (in semiring_of_sets) generating_ring:
   882   "ring_of_sets \<Omega> generated_ring"
   883 proof (rule ring_of_setsI)
   884   let ?R = generated_ring
   885   show "?R \<subseteq> Pow \<Omega>"
   886     using sets_into_space by (auto simp: generated_ring_def generated_ring_empty)
   887   show "{} \<in> ?R" by (rule generated_ring_empty)
   889   { fix a assume a: "a \<in> ?R" then guess Ca .. note Ca = this
   890     fix b assume b: "b \<in> ?R" then guess Cb .. note Cb = this
   892     show "a - b \<in> ?R"
   893     proof cases
   894       assume "Cb = {}" with Cb \<open>a \<in> ?R\<close> show ?thesis
   895         by simp
   896     next
   897       assume "Cb \<noteq> {}"
   898       with Ca Cb have "a - b = (\<Union>a'\<in>Ca. \<Inter>b'\<in>Cb. a' - b')" by auto
   899       also have "\<dots> \<in> ?R"
   900       proof (intro generated_ring_INTER generated_ring_disjoint_UNION)
   901         fix a b assume "a \<in> Ca" "b \<in> Cb"
   902         with Ca Cb Diff_cover[of a b] show "a - b \<in> ?R"
   903           by (auto simp add: generated_ring_def)
   904             (metis DiffI Diff_eq_empty_iff empty_iff)
   905       next
   906         show "disjoint ((\<lambda>a'. \<Inter>b'\<in>Cb. a' - b')`Ca)"
   907           using Ca by (auto simp add: disjoint_def \<open>Cb \<noteq> {}\<close>)
   908       next
   909         show "finite Ca" "finite Cb" "Cb \<noteq> {}" by fact+
   910       qed
   911       finally show "a - b \<in> ?R" .
   912     qed }
   913   note Diff = this
   915   fix a b assume sets: "a \<in> ?R" "b \<in> ?R"
   916   have "a \<union> b = (a - b) \<union> (a \<inter> b) \<union> (b - a)" by auto
   917   also have "\<dots> \<in> ?R"
   918     by (intro sets generated_ring_disjoint_Un generated_ring_Int Diff) auto
   919   finally show "a \<union> b \<in> ?R" .
   920 qed
   922 lemma (in semiring_of_sets) sigma_sets_generated_ring_eq: "sigma_sets \<Omega> generated_ring = sigma_sets \<Omega> M"
   923 proof
   924   interpret M: sigma_algebra \<Omega> "sigma_sets \<Omega> M"
   925     using space_closed by (rule sigma_algebra_sigma_sets)
   926   show "sigma_sets \<Omega> generated_ring \<subseteq> sigma_sets \<Omega> M"
   927     by (blast intro!: sigma_sets_mono elim: generated_ringE)
   928 qed (auto intro!: generated_ringI_Basic sigma_sets_mono)
   930 subsubsection \<open>A Two-Element Series\<close>
   932 definition binaryset :: "'a set \<Rightarrow> 'a set \<Rightarrow> nat \<Rightarrow> 'a set "
   933   where "binaryset A B = (\<lambda>x. {})(0 := A, Suc 0 := B)"
   935 lemma range_binaryset_eq: "range(binaryset A B) = {A,B,{}}"
   936   apply (simp add: binaryset_def)
   937   apply (rule set_eqI)
   938   apply (auto simp add: image_iff)
   939   done
   941 lemma UN_binaryset_eq: "(\<Union>i. binaryset A B i) = A \<union> B"
   942   by (simp add: range_binaryset_eq cong del: strong_SUP_cong)
   944 subsubsection \<open>Closed CDI\<close>
   946 definition closed_cdi where
   947   "closed_cdi \<Omega> M \<longleftrightarrow>
   948    M \<subseteq> Pow \<Omega> &
   949    (\<forall>s \<in> M. \<Omega> - s \<in> M) &
   950    (\<forall>A. (range A \<subseteq> M) & (A 0 = {}) & (\<forall>n. A n \<subseteq> A (Suc n)) \<longrightarrow>
   951         (\<Union>i. A i) \<in> M) &
   952    (\<forall>A. (range A \<subseteq> M) & disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"
   954 inductive_set
   955   smallest_ccdi_sets :: "'a set \<Rightarrow> 'a set set \<Rightarrow> 'a set set"
   956   for \<Omega> M
   957   where
   958     Basic [intro]:
   959       "a \<in> M \<Longrightarrow> a \<in> smallest_ccdi_sets \<Omega> M"
   960   | Compl [intro]:
   961       "a \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> \<Omega> - a \<in> smallest_ccdi_sets \<Omega> M"
   962   | Inc:
   963       "range A \<in> Pow(smallest_ccdi_sets \<Omega> M) \<Longrightarrow> A 0 = {} \<Longrightarrow> (\<And>n. A n \<subseteq> A (Suc n))
   964        \<Longrightarrow> (\<Union>i. A i) \<in> smallest_ccdi_sets \<Omega> M"
   965   | Disj:
   966       "range A \<in> Pow(smallest_ccdi_sets \<Omega> M) \<Longrightarrow> disjoint_family A
   967        \<Longrightarrow> (\<Union>i::nat. A i) \<in> smallest_ccdi_sets \<Omega> M"
   969 lemma (in subset_class) smallest_closed_cdi1: "M \<subseteq> smallest_ccdi_sets \<Omega> M"
   970   by auto
   972 lemma (in subset_class) smallest_ccdi_sets: "smallest_ccdi_sets \<Omega> M \<subseteq> Pow \<Omega>"
   973   apply (rule subsetI)
   974   apply (erule smallest_ccdi_sets.induct)
   975   apply (auto intro: range_subsetD dest: sets_into_space)
   976   done
   978 lemma (in subset_class) smallest_closed_cdi2: "closed_cdi \<Omega> (smallest_ccdi_sets \<Omega> M)"
   979   apply (auto simp add: closed_cdi_def smallest_ccdi_sets)
   980   apply (blast intro: smallest_ccdi_sets.Inc smallest_ccdi_sets.Disj) +
   981   done
   983 lemma closed_cdi_subset: "closed_cdi \<Omega> M \<Longrightarrow> M \<subseteq> Pow \<Omega>"
   984   by (simp add: closed_cdi_def)
   986 lemma closed_cdi_Compl: "closed_cdi \<Omega> M \<Longrightarrow> s \<in> M \<Longrightarrow> \<Omega> - s \<in> M"
   987   by (simp add: closed_cdi_def)
   989 lemma closed_cdi_Inc:
   990   "closed_cdi \<Omega> M \<Longrightarrow> range A \<subseteq> M \<Longrightarrow> A 0 = {} \<Longrightarrow> (!!n. A n \<subseteq> A (Suc n)) \<Longrightarrow> (\<Union>i. A i) \<in> M"
   991   by (simp add: closed_cdi_def)
   993 lemma closed_cdi_Disj:
   994   "closed_cdi \<Omega> M \<Longrightarrow> range A \<subseteq> M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
   995   by (simp add: closed_cdi_def)
   997 lemma closed_cdi_Un:
   998   assumes cdi: "closed_cdi \<Omega> M" and empty: "{} \<in> M"
   999       and A: "A \<in> M" and B: "B \<in> M"
  1000       and disj: "A \<inter> B = {}"
  1001     shows "A \<union> B \<in> M"
  1002 proof -
  1003   have ra: "range (binaryset A B) \<subseteq> M"
  1004    by (simp add: range_binaryset_eq empty A B)
  1005  have di:  "disjoint_family (binaryset A B)" using disj
  1006    by (simp add: disjoint_family_on_def binaryset_def Int_commute)
  1007  from closed_cdi_Disj [OF cdi ra di]
  1008  show ?thesis
  1009    by (simp add: UN_binaryset_eq)
  1010 qed
  1012 lemma (in algebra) smallest_ccdi_sets_Un:
  1013   assumes A: "A \<in> smallest_ccdi_sets \<Omega> M" and B: "B \<in> smallest_ccdi_sets \<Omega> M"
  1014       and disj: "A \<inter> B = {}"
  1015     shows "A \<union> B \<in> smallest_ccdi_sets \<Omega> M"
  1016 proof -
  1017   have ra: "range (binaryset A B) \<in> Pow (smallest_ccdi_sets \<Omega> M)"
  1018     by (simp add: range_binaryset_eq  A B smallest_ccdi_sets.Basic)
  1019   have di:  "disjoint_family (binaryset A B)" using disj
  1020     by (simp add: disjoint_family_on_def binaryset_def Int_commute)
  1021   from Disj [OF ra di]
  1022   show ?thesis
  1023     by (simp add: UN_binaryset_eq)
  1024 qed
  1026 lemma (in algebra) smallest_ccdi_sets_Int1:
  1027   assumes a: "a \<in> M"
  1028   shows "b \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets \<Omega> M"
  1029 proof (induct rule: smallest_ccdi_sets.induct)
  1030   case (Basic x)
  1031   thus ?case
  1032     by (metis a Int smallest_ccdi_sets.Basic)
  1033 next
  1034   case (Compl x)
  1035   have "a \<inter> (\<Omega> - x) = \<Omega> - ((\<Omega> - a) \<union> (a \<inter> x))"
  1036     by blast
  1037   also have "... \<in> smallest_ccdi_sets \<Omega> M"
  1038     by (metis smallest_ccdi_sets.Compl a Compl(2) Diff_Int2 Diff_Int_distrib2
  1039            Diff_disjoint Int_Diff Int_empty_right smallest_ccdi_sets_Un
  1040            smallest_ccdi_sets.Basic smallest_ccdi_sets.Compl)
  1041   finally show ?case .
  1042 next
  1043   case (Inc A)
  1044   have 1: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) = a \<inter> (\<Union>i. A i)"
  1045     by blast
  1046   have "range (\<lambda>i. a \<inter> A i) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Inc
  1047     by blast
  1048   moreover have "(\<lambda>i. a \<inter> A i) 0 = {}"
  1049     by (simp add: Inc)
  1050   moreover have "!!n. (\<lambda>i. a \<inter> A i) n \<subseteq> (\<lambda>i. a \<inter> A i) (Suc n)" using Inc
  1051     by blast
  1052   ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets \<Omega> M"
  1053     by (rule smallest_ccdi_sets.Inc)
  1054   show ?case
  1055     by (metis 1 2)
  1056 next
  1057   case (Disj A)
  1058   have 1: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) = a \<inter> (\<Union>i. A i)"
  1059     by blast
  1060   have "range (\<lambda>i. a \<inter> A i) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Disj
  1061     by blast
  1062   moreover have "disjoint_family (\<lambda>i. a \<inter> A i)" using Disj
  1063     by (auto simp add: disjoint_family_on_def)
  1064   ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets \<Omega> M"
  1065     by (rule smallest_ccdi_sets.Disj)
  1066   show ?case
  1067     by (metis 1 2)
  1068 qed
  1071 lemma (in algebra) smallest_ccdi_sets_Int:
  1072   assumes b: "b \<in> smallest_ccdi_sets \<Omega> M"
  1073   shows "a \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets \<Omega> M"
  1074 proof (induct rule: smallest_ccdi_sets.induct)
  1075   case (Basic x)
  1076   thus ?case
  1077     by (metis b smallest_ccdi_sets_Int1)
  1078 next
  1079   case (Compl x)
  1080   have "(\<Omega> - x) \<inter> b = \<Omega> - (x \<inter> b \<union> (\<Omega> - b))"
  1081     by blast
  1082   also have "... \<in> smallest_ccdi_sets \<Omega> M"
  1083     by (metis Compl(2) Diff_disjoint Int_Diff Int_commute Int_empty_right b
  1084            smallest_ccdi_sets.Compl smallest_ccdi_sets_Un)
  1085   finally show ?case .
  1086 next
  1087   case (Inc A)
  1088   have 1: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) = (\<Union>i. A i) \<inter> b"
  1089     by blast
  1090   have "range (\<lambda>i. A i \<inter> b) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Inc
  1091     by blast
  1092   moreover have "(\<lambda>i. A i \<inter> b) 0 = {}"
  1093     by (simp add: Inc)
  1094   moreover have "!!n. (\<lambda>i. A i \<inter> b) n \<subseteq> (\<lambda>i. A i \<inter> b) (Suc n)" using Inc
  1095     by blast
  1096   ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets \<Omega> M"
  1097     by (rule smallest_ccdi_sets.Inc)
  1098   show ?case
  1099     by (metis 1 2)
  1100 next
  1101   case (Disj A)
  1102   have 1: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) = (\<Union>i. A i) \<inter> b"
  1103     by blast
  1104   have "range (\<lambda>i. A i \<inter> b) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Disj
  1105     by blast
  1106   moreover have "disjoint_family (\<lambda>i. A i \<inter> b)" using Disj
  1107     by (auto simp add: disjoint_family_on_def)
  1108   ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets \<Omega> M"
  1109     by (rule smallest_ccdi_sets.Disj)
  1110   show ?case
  1111     by (metis 1 2)
  1112 qed
  1114 lemma (in algebra) sigma_property_disjoint_lemma:
  1115   assumes sbC: "M \<subseteq> C"
  1116       and ccdi: "closed_cdi \<Omega> C"
  1117   shows "sigma_sets \<Omega> M \<subseteq> C"
  1118 proof -
  1119   have "smallest_ccdi_sets \<Omega> M \<in> {B . M \<subseteq> B \<and> sigma_algebra \<Omega> B}"
  1120     apply (auto simp add: sigma_algebra_disjoint_iff algebra_iff_Int
  1121             smallest_ccdi_sets_Int)
  1122     apply (metis Union_Pow_eq Union_upper subsetD smallest_ccdi_sets)
  1123     apply (blast intro: smallest_ccdi_sets.Disj)
  1124     done
  1125   hence "sigma_sets (\<Omega>) (M) \<subseteq> smallest_ccdi_sets \<Omega> M"
  1126     by clarsimp
  1127        (drule sigma_algebra.sigma_sets_subset [where a="M"], auto)
  1128   also have "...  \<subseteq> C"
  1129     proof
  1130       fix x
  1131       assume x: "x \<in> smallest_ccdi_sets \<Omega> M"
  1132       thus "x \<in> C"
  1133         proof (induct rule: smallest_ccdi_sets.induct)
  1134           case (Basic x)
  1135           thus ?case
  1136             by (metis Basic subsetD sbC)
  1137         next
  1138           case (Compl x)
  1139           thus ?case
  1140             by (blast intro: closed_cdi_Compl [OF ccdi, simplified])
  1141         next
  1142           case (Inc A)
  1143           thus ?case
  1144                by (auto intro: closed_cdi_Inc [OF ccdi, simplified])
  1145         next
  1146           case (Disj A)
  1147           thus ?case
  1148                by (auto intro: closed_cdi_Disj [OF ccdi, simplified])
  1149         qed
  1150     qed
  1151   finally show ?thesis .
  1152 qed
  1154 lemma (in algebra) sigma_property_disjoint:
  1155   assumes sbC: "M \<subseteq> C"
  1156       and compl: "!!s. s \<in> C \<inter> sigma_sets (\<Omega>) (M) \<Longrightarrow> \<Omega> - s \<in> C"
  1157       and inc: "!!A. range A \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)
  1158                      \<Longrightarrow> A 0 = {} \<Longrightarrow> (!!n. A n \<subseteq> A (Suc n))
  1159                      \<Longrightarrow> (\<Union>i. A i) \<in> C"
  1160       and disj: "!!A. range A \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)
  1161                       \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> C"
  1162   shows "sigma_sets (\<Omega>) (M) \<subseteq> C"
  1163 proof -
  1164   have "sigma_sets (\<Omega>) (M) \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)"
  1165     proof (rule sigma_property_disjoint_lemma)
  1166       show "M \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)"
  1167         by (metis Int_greatest Set.subsetI sbC sigma_sets.Basic)
  1168     next
  1169       show "closed_cdi \<Omega> (C \<inter> sigma_sets (\<Omega>) (M))"
  1170         by (simp add: closed_cdi_def compl inc disj)
  1171            (metis PowI Set.subsetI le_infI2 sigma_sets_into_sp space_closed
  1172              IntE sigma_sets.Compl range_subsetD sigma_sets.Union)
  1173     qed
  1174   thus ?thesis
  1175     by blast
  1176 qed
  1178 subsubsection \<open>Dynkin systems\<close>
  1180 locale dynkin_system = subset_class +
  1181   assumes space: "\<Omega> \<in> M"
  1182     and   compl[intro!]: "\<And>A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"
  1183     and   UN[intro!]: "\<And>A. disjoint_family A \<Longrightarrow> range A \<subseteq> M
  1184                            \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
  1186 lemma (in dynkin_system) empty[intro, simp]: "{} \<in> M"
  1187   using space compl[of "\<Omega>"] by simp
  1189 lemma (in dynkin_system) diff:
  1190   assumes sets: "D \<in> M" "E \<in> M" and "D \<subseteq> E"
  1191   shows "E - D \<in> M"
  1192 proof -
  1193   let ?f = "\<lambda>x. if x = 0 then D else if x = Suc 0 then \<Omega> - E else {}"
  1194   have "range ?f = {D, \<Omega> - E, {}}"
  1195     by (auto simp: image_iff)
  1196   moreover have "D \<union> (\<Omega> - E) = (\<Union>i. ?f i)"
  1197     by (auto simp: image_iff split: split_if_asm)
  1198   moreover
  1199   have "disjoint_family ?f" unfolding disjoint_family_on_def
  1200     using \<open>D \<in> M\<close>[THEN sets_into_space] \<open>D \<subseteq> E\<close> by auto
  1201   ultimately have "\<Omega> - (D \<union> (\<Omega> - E)) \<in> M"
  1202     using sets by auto
  1203   also have "\<Omega> - (D \<union> (\<Omega> - E)) = E - D"
  1204     using assms sets_into_space by auto
  1205   finally show ?thesis .
  1206 qed
  1208 lemma dynkin_systemI:
  1209   assumes "\<And> A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>" "\<Omega> \<in> M"
  1210   assumes "\<And> A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"
  1211   assumes "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> M
  1212           \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
  1213   shows "dynkin_system \<Omega> M"
  1214   using assms by (auto simp: dynkin_system_def dynkin_system_axioms_def subset_class_def)
  1216 lemma dynkin_systemI':
  1217   assumes 1: "\<And> A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>"
  1218   assumes empty: "{} \<in> M"
  1219   assumes Diff: "\<And> A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"
  1220   assumes 2: "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> M
  1221           \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
  1222   shows "dynkin_system \<Omega> M"
  1223 proof -
  1224   from Diff[OF empty] have "\<Omega> \<in> M" by auto
  1225   from 1 this Diff 2 show ?thesis
  1226     by (intro dynkin_systemI) auto
  1227 qed
  1229 lemma dynkin_system_trivial:
  1230   shows "dynkin_system A (Pow A)"
  1231   by (rule dynkin_systemI) auto
  1233 lemma sigma_algebra_imp_dynkin_system:
  1234   assumes "sigma_algebra \<Omega> M" shows "dynkin_system \<Omega> M"
  1235 proof -
  1236   interpret sigma_algebra \<Omega> M by fact
  1237   show ?thesis using sets_into_space by (fastforce intro!: dynkin_systemI)
  1238 qed
  1240 subsubsection "Intersection sets systems"
  1242 definition "Int_stable M \<longleftrightarrow> (\<forall> a \<in> M. \<forall> b \<in> M. a \<inter> b \<in> M)"
  1244 lemma (in algebra) Int_stable: "Int_stable M"
  1245   unfolding Int_stable_def by auto
  1247 lemma Int_stableI:
  1248   "(\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<inter> b \<in> A) \<Longrightarrow> Int_stable A"
  1249   unfolding Int_stable_def by auto
  1251 lemma Int_stableD:
  1252   "Int_stable M \<Longrightarrow> a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<inter> b \<in> M"
  1253   unfolding Int_stable_def by auto
  1255 lemma (in dynkin_system) sigma_algebra_eq_Int_stable:
  1256   "sigma_algebra \<Omega> M \<longleftrightarrow> Int_stable M"
  1257 proof
  1258   assume "sigma_algebra \<Omega> M" then show "Int_stable M"
  1259     unfolding sigma_algebra_def using algebra.Int_stable by auto
  1260 next
  1261   assume "Int_stable M"
  1262   show "sigma_algebra \<Omega> M"
  1263     unfolding sigma_algebra_disjoint_iff algebra_iff_Un
  1264   proof (intro conjI ballI allI impI)
  1265     show "M \<subseteq> Pow (\<Omega>)" using sets_into_space by auto
  1266   next
  1267     fix A B assume "A \<in> M" "B \<in> M"
  1268     then have "A \<union> B = \<Omega> - ((\<Omega> - A) \<inter> (\<Omega> - B))"
  1269               "\<Omega> - A \<in> M" "\<Omega> - B \<in> M"
  1270       using sets_into_space by auto
  1271     then show "A \<union> B \<in> M"
  1272       using \<open>Int_stable M\<close> unfolding Int_stable_def by auto
  1273   qed auto
  1274 qed
  1276 subsubsection "Smallest Dynkin systems"
  1278 definition dynkin where
  1279   "dynkin \<Omega> M =  (\<Inter>{D. dynkin_system \<Omega> D \<and> M \<subseteq> D})"
  1281 lemma dynkin_system_dynkin:
  1282   assumes "M \<subseteq> Pow (\<Omega>)"
  1283   shows "dynkin_system \<Omega> (dynkin \<Omega> M)"
  1284 proof (rule dynkin_systemI)
  1285   fix A assume "A \<in> dynkin \<Omega> M"
  1286   moreover
  1287   { fix D assume "A \<in> D" and d: "dynkin_system \<Omega> D"
  1288     then have "A \<subseteq> \<Omega>" by (auto simp: dynkin_system_def subset_class_def) }
  1289   moreover have "{D. dynkin_system \<Omega> D \<and> M \<subseteq> D} \<noteq> {}"
  1290     using assms dynkin_system_trivial by fastforce
  1291   ultimately show "A \<subseteq> \<Omega>"
  1292     unfolding dynkin_def using assms
  1293     by auto
  1294 next
  1295   show "\<Omega> \<in> dynkin \<Omega> M"
  1296     unfolding dynkin_def using by fastforce
  1297 next
  1298   fix A assume "A \<in> dynkin \<Omega> M"
  1299   then show "\<Omega> - A \<in> dynkin \<Omega> M"
  1300     unfolding dynkin_def using dynkin_system.compl by force
  1301 next
  1302   fix A :: "nat \<Rightarrow> 'a set"
  1303   assume A: "disjoint_family A" "range A \<subseteq> dynkin \<Omega> M"
  1304   show "(\<Union>i. A i) \<in> dynkin \<Omega> M" unfolding dynkin_def
  1305   proof (simp, safe)
  1306     fix D assume "dynkin_system \<Omega> D" "M \<subseteq> D"
  1307     with A have "(\<Union>i. A i) \<in> D"
  1308       by (intro dynkin_system.UN) (auto simp: dynkin_def)
  1309     then show "(\<Union>i. A i) \<in> D" by auto
  1310   qed
  1311 qed
  1313 lemma dynkin_Basic[intro]: "A \<in> M \<Longrightarrow> A \<in> dynkin \<Omega> M"
  1314   unfolding dynkin_def by auto
  1316 lemma (in dynkin_system) restricted_dynkin_system:
  1317   assumes "D \<in> M"
  1318   shows "dynkin_system \<Omega> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}"
  1319 proof (rule dynkin_systemI, simp_all)
  1320   have "\<Omega> \<inter> D = D"
  1321     using \<open>D \<in> M\<close> sets_into_space by auto
  1322   then show "\<Omega> \<inter> D \<in> M"
  1323     using \<open>D \<in> M\<close> by auto
  1324 next
  1325   fix A assume "A \<subseteq> \<Omega> \<and> A \<inter> D \<in> M"
  1326   moreover have "(\<Omega> - A) \<inter> D = (\<Omega> - (A \<inter> D)) - (\<Omega> - D)"
  1327     by auto
  1328   ultimately show "\<Omega> - A \<subseteq> \<Omega> \<and> (\<Omega> - A) \<inter> D \<in> M"
  1329     using  \<open>D \<in> M\<close> by (auto intro: diff)
  1330 next
  1331   fix A :: "nat \<Rightarrow> 'a set"
  1332   assume "disjoint_family A" "range A \<subseteq> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}"
  1333   then have "\<And>i. A i \<subseteq> \<Omega>" "disjoint_family (\<lambda>i. A i \<inter> D)"
  1334     "range (\<lambda>i. A i \<inter> D) \<subseteq> M" "(\<Union>x. A x) \<inter> D = (\<Union>x. A x \<inter> D)"
  1335     by ((fastforce simp: disjoint_family_on_def)+)
  1336   then show "(\<Union>x. A x) \<subseteq> \<Omega> \<and> (\<Union>x. A x) \<inter> D \<in> M"
  1337     by (auto simp del: UN_simps)
  1338 qed
  1340 lemma (in dynkin_system) dynkin_subset:
  1341   assumes "N \<subseteq> M"
  1342   shows "dynkin \<Omega> N \<subseteq> M"
  1343 proof -
  1344   have "dynkin_system \<Omega> M" ..
  1345   then have "dynkin_system \<Omega> M"
  1346     using assms unfolding dynkin_system_def dynkin_system_axioms_def subset_class_def by simp
  1347   with \<open>N \<subseteq> M\<close> show ?thesis by (auto simp add: dynkin_def)
  1348 qed
  1350 lemma sigma_eq_dynkin:
  1351   assumes sets: "M \<subseteq> Pow \<Omega>"
  1352   assumes "Int_stable M"
  1353   shows "sigma_sets \<Omega> M = dynkin \<Omega> M"
  1354 proof -
  1355   have "dynkin \<Omega> M \<subseteq> sigma_sets (\<Omega>) (M)"
  1356     using sigma_algebra_imp_dynkin_system
  1357     unfolding dynkin_def sigma_sets_least_sigma_algebra[OF sets] by auto
  1358   moreover
  1359   interpret dynkin_system \<Omega> "dynkin \<Omega> M"
  1360     using dynkin_system_dynkin[OF sets] .
  1361   have "sigma_algebra \<Omega> (dynkin \<Omega> M)"
  1362     unfolding sigma_algebra_eq_Int_stable Int_stable_def
  1363   proof (intro ballI)
  1364     fix A B assume "A \<in> dynkin \<Omega> M" "B \<in> dynkin \<Omega> M"
  1365     let ?D = "\<lambda>E. {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> E \<in> dynkin \<Omega> M}"
  1366     have "M \<subseteq> ?D B"
  1367     proof
  1368       fix E assume "E \<in> M"
  1369       then have "M \<subseteq> ?D E" "E \<in> dynkin \<Omega> M"
  1370         using sets_into_space \<open>Int_stable M\<close> by (auto simp: Int_stable_def)
  1371       then have "dynkin \<Omega> M \<subseteq> ?D E"
  1372         using restricted_dynkin_system \<open>E \<in> dynkin \<Omega> M\<close>
  1373         by (intro dynkin_system.dynkin_subset) simp_all
  1374       then have "B \<in> ?D E"
  1375         using \<open>B \<in> dynkin \<Omega> M\<close> by auto
  1376       then have "E \<inter> B \<in> dynkin \<Omega> M"
  1377         by (subst Int_commute) simp
  1378       then show "E \<in> ?D B"
  1379         using sets \<open>E \<in> M\<close> by auto
  1380     qed
  1381     then have "dynkin \<Omega> M \<subseteq> ?D B"
  1382       using restricted_dynkin_system \<open>B \<in> dynkin \<Omega> M\<close>
  1383       by (intro dynkin_system.dynkin_subset) simp_all
  1384     then show "A \<inter> B \<in> dynkin \<Omega> M"
  1385       using \<open>A \<in> dynkin \<Omega> M\<close> sets_into_space by auto
  1386   qed
  1387   from sigma_algebra.sigma_sets_subset[OF this, of "M"]
  1388   have "sigma_sets (\<Omega>) (M) \<subseteq> dynkin \<Omega> M" by auto
  1389   ultimately have "sigma_sets (\<Omega>) (M) = dynkin \<Omega> M" by auto
  1390   then show ?thesis
  1391     by (auto simp: dynkin_def)
  1392 qed
  1394 lemma (in dynkin_system) dynkin_idem:
  1395   "dynkin \<Omega> M = M"
  1396 proof -
  1397   have "dynkin \<Omega> M = M"
  1398   proof
  1399     show "M \<subseteq> dynkin \<Omega> M"
  1400       using dynkin_Basic by auto
  1401     show "dynkin \<Omega> M \<subseteq> M"
  1402       by (intro dynkin_subset) auto
  1403   qed
  1404   then show ?thesis
  1405     by (auto simp: dynkin_def)
  1406 qed
  1408 lemma (in dynkin_system) dynkin_lemma:
  1409   assumes "Int_stable E"
  1410   and E: "E \<subseteq> M" "M \<subseteq> sigma_sets \<Omega> E"
  1411   shows "sigma_sets \<Omega> E = M"
  1412 proof -
  1413   have "E \<subseteq> Pow \<Omega>"
  1414     using E sets_into_space by force
  1415   then have *: "sigma_sets \<Omega> E = dynkin \<Omega> E"
  1416     using \<open>Int_stable E\<close> by (rule sigma_eq_dynkin)
  1417   then have "dynkin \<Omega> E = M"
  1418     using assms dynkin_subset[OF E(1)] by simp
  1419   with * show ?thesis
  1420     using assms by (auto simp: dynkin_def)
  1421 qed
  1423 subsubsection \<open>Induction rule for intersection-stable generators\<close>
  1425 text \<open>The reason to introduce Dynkin-systems is the following induction rules for $\sigma$-algebras
  1426 generated by a generator closed under intersection.\<close>
  1428 lemma sigma_sets_induct_disjoint[consumes 3, case_names basic empty compl union]:
  1429   assumes "Int_stable G"
  1430     and closed: "G \<subseteq> Pow \<Omega>"
  1431     and A: "A \<in> sigma_sets \<Omega> G"
  1432   assumes basic: "\<And>A. A \<in> G \<Longrightarrow> P A"
  1433     and empty: "P {}"
  1434     and compl: "\<And>A. A \<in> sigma_sets \<Omega> G \<Longrightarrow> P A \<Longrightarrow> P (\<Omega> - A)"
  1435     and union: "\<And>A. disjoint_family A \<Longrightarrow> range A \<subseteq> sigma_sets \<Omega> G \<Longrightarrow> (\<And>i. P (A i)) \<Longrightarrow> P (\<Union>i::nat. A i)"
  1436   shows "P A"
  1437 proof -
  1438   let ?D = "{ A \<in> sigma_sets \<Omega> G. P A }"
  1439   interpret sigma_algebra \<Omega> "sigma_sets \<Omega> G"
  1440     using closed by (rule sigma_algebra_sigma_sets)
  1441   from compl[OF _ empty] closed have space: "P \<Omega>" by simp
  1442   interpret dynkin_system \<Omega> ?D
  1443     by standard (auto dest: sets_into_space intro!: space compl union)
  1444   have "sigma_sets \<Omega> G = ?D"
  1445     by (rule dynkin_lemma) (auto simp: basic \<open>Int_stable G\<close>)
  1446   with A show ?thesis by auto
  1447 qed
  1449 subsection \<open>Measure type\<close>
  1451 definition positive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
  1452   "positive M \<mu> \<longleftrightarrow> \<mu> {} = 0 \<and> (\<forall>A\<in>M. 0 \<le> \<mu> A)"
  1454 definition countably_additive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
  1455   "countably_additive M f \<longleftrightarrow> (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow>
  1456     (\<Sum>i. f (A i)) = f (\<Union>i. A i))"
  1458 definition measure_space :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
  1459   "measure_space \<Omega> A \<mu> \<longleftrightarrow> sigma_algebra \<Omega> A \<and> positive A \<mu> \<and> countably_additive A \<mu>"
  1461 typedef 'a measure = "{(\<Omega>::'a set, A, \<mu>). (\<forall>a\<in>-A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu> }"
  1462 proof
  1463   have "sigma_algebra UNIV {{}, UNIV}"
  1464     by (auto simp: sigma_algebra_iff2)
  1465   then show "(UNIV, {{}, UNIV}, \<lambda>A. 0) \<in> {(\<Omega>, A, \<mu>). (\<forall>a\<in>-A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu>} "
  1466     by (auto simp: measure_space_def positive_def countably_additive_def)
  1467 qed
  1469 definition space :: "'a measure \<Rightarrow> 'a set" where
  1470   "space M = fst (Rep_measure M)"
  1472 definition sets :: "'a measure \<Rightarrow> 'a set set" where
  1473   "sets M = fst (snd (Rep_measure M))"
  1475 definition emeasure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> ereal" where
  1476   "emeasure M = snd (snd (Rep_measure M))"
  1478 definition measure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> real" where
  1479   "measure M A = real_of_ereal (emeasure M A)"
  1481 declare [[coercion sets]]
  1483 declare [[coercion measure]]
  1485 declare [[coercion emeasure]]
  1487 lemma measure_space: "measure_space (space M) (sets M) (emeasure M)"
  1488   by (cases M) (auto simp: space_def sets_def emeasure_def Abs_measure_inverse)
  1490 interpretation sets: sigma_algebra "space M" "sets M" for M :: "'a measure"
  1491   using measure_space[of M] by (auto simp: measure_space_def)
  1493 definition measure_of :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> 'a measure" where
  1494   "measure_of \<Omega> A \<mu> = Abs_measure (\<Omega>, if A \<subseteq> Pow \<Omega> then sigma_sets \<Omega> A else {{}, \<Omega>},
  1495     \<lambda>a. if a \<in> sigma_sets \<Omega> A \<and> measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> then \<mu> a else 0)"
  1497 abbreviation "sigma \<Omega> A \<equiv> measure_of \<Omega> A (\<lambda>x. 0)"
  1499 lemma measure_space_0: "A \<subseteq> Pow \<Omega> \<Longrightarrow> measure_space \<Omega> (sigma_sets \<Omega> A) (\<lambda>x. 0)"
  1500   unfolding measure_space_def
  1501   by (auto intro!: sigma_algebra_sigma_sets simp: positive_def countably_additive_def)
  1503 lemma sigma_algebra_trivial: "sigma_algebra \<Omega> {{}, \<Omega>}"
  1504 by unfold_locales(fastforce intro: exI[where x="{{}}"] exI[where x="{\<Omega>}"])+
  1506 lemma measure_space_0': "measure_space \<Omega> {{}, \<Omega>} (\<lambda>x. 0)"
  1507 by(simp add: measure_space_def positive_def countably_additive_def sigma_algebra_trivial)
  1509 lemma measure_space_closed:
  1510   assumes "measure_space \<Omega> M \<mu>"
  1511   shows "M \<subseteq> Pow \<Omega>"
  1512 proof -
  1513   interpret sigma_algebra \<Omega> M using assms by(simp add: measure_space_def)
  1514   show ?thesis by(rule space_closed)
  1515 qed
  1517 lemma (in ring_of_sets) positive_cong_eq:
  1518   "(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> positive M \<mu>' = positive M \<mu>"
  1519   by (auto simp add: positive_def)
  1521 lemma (in sigma_algebra) countably_additive_eq:
  1522   "(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> countably_additive M \<mu>' = countably_additive M \<mu>"
  1523   unfolding countably_additive_def
  1524   by (intro arg_cong[where f=All] ext) (auto simp add: countably_additive_def subset_eq)
  1526 lemma measure_space_eq:
  1527   assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a"
  1528   shows "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'"
  1529 proof -
  1530   interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" using closed by (rule sigma_algebra_sigma_sets)
  1531   from positive_cong_eq[OF eq, of "\<lambda>i. i"] countably_additive_eq[OF eq, of "\<lambda>i. i"] show ?thesis
  1532     by (auto simp: measure_space_def)
  1533 qed
  1535 lemma measure_of_eq:
  1536   assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "(\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a)"
  1537   shows "measure_of \<Omega> A \<mu> = measure_of \<Omega> A \<mu>'"
  1538 proof -
  1539   have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'"
  1540     using assms by (rule measure_space_eq)
  1541   with eq show ?thesis
  1542     by (auto simp add: measure_of_def intro!: arg_cong[where f=Abs_measure])
  1543 qed
  1545 lemma
  1546   shows space_measure_of_conv: "space (measure_of \<Omega> A \<mu>) = \<Omega>" (is ?space)
  1547   and sets_measure_of_conv:
  1548   "sets (measure_of \<Omega> A \<mu>) = (if A \<subseteq> Pow \<Omega> then sigma_sets \<Omega> A else {{}, \<Omega>})" (is ?sets)
  1549   and emeasure_measure_of_conv: 
  1550   "emeasure (measure_of \<Omega> A \<mu>) = 
  1551   (\<lambda>B. if B \<in> sigma_sets \<Omega> A \<and> measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> then \<mu> B else 0)" (is ?emeasure)
  1552 proof -
  1553   have "?space \<and> ?sets \<and> ?emeasure"
  1554   proof(cases "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>")
  1555     case True
  1556     from measure_space_closed[OF this] sigma_sets_superset_generator[of A \<Omega>]
  1557     have "A \<subseteq> Pow \<Omega>" by simp
  1558     hence "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A)
  1559       (\<lambda>a. if a \<in> sigma_sets \<Omega> A then \<mu> a else 0)"
  1560       by(rule measure_space_eq) auto
  1561     with True \<open>A \<subseteq> Pow \<Omega>\<close> show ?thesis
  1562       by(simp add: measure_of_def space_def sets_def emeasure_def Abs_measure_inverse)
  1563   next
  1564     case False thus ?thesis
  1565       by(cases "A \<subseteq> Pow \<Omega>")(simp_all add: Abs_measure_inverse measure_of_def sets_def space_def emeasure_def measure_space_0 measure_space_0')
  1566   qed
  1567   thus ?space ?sets ?emeasure by simp_all
  1568 qed
  1570 lemma [simp]:
  1571   assumes A: "A \<subseteq> Pow \<Omega>"
  1572   shows sets_measure_of: "sets (measure_of \<Omega> A \<mu>) = sigma_sets \<Omega> A"
  1573     and space_measure_of: "space (measure_of \<Omega> A \<mu>) = \<Omega>"
  1574 using assms
  1575 by(simp_all add: sets_measure_of_conv space_measure_of_conv)
  1577 lemma (in sigma_algebra) sets_measure_of_eq[simp]: "sets (measure_of \<Omega> M \<mu>) = M"
  1578   using space_closed by (auto intro!: sigma_sets_eq)
  1580 lemma (in sigma_algebra) space_measure_of_eq[simp]: "space (measure_of \<Omega> M \<mu>) = \<Omega>"
  1581   by (rule space_measure_of_conv)
  1583 lemma measure_of_subset: "M \<subseteq> Pow \<Omega> \<Longrightarrow> M' \<subseteq> M \<Longrightarrow> sets (measure_of \<Omega> M' \<mu>) \<subseteq> sets (measure_of \<Omega> M \<mu>')"
  1584   by (auto intro!: sigma_sets_subseteq)
  1586 lemma emeasure_sigma: "emeasure (sigma \<Omega> A) = (\<lambda>x. 0)"
  1587   unfolding measure_of_def emeasure_def
  1588   by (subst Abs_measure_inverse)
  1589      (auto simp: measure_space_def positive_def countably_additive_def
  1590            intro!: sigma_algebra_sigma_sets sigma_algebra_trivial)
  1592 lemma sigma_sets_mono'':
  1593   assumes "A \<in> sigma_sets C D"
  1594   assumes "B \<subseteq> D"
  1595   assumes "D \<subseteq> Pow C"
  1596   shows "sigma_sets A B \<subseteq> sigma_sets C D"
  1597 proof
  1598   fix x assume "x \<in> sigma_sets A B"
  1599   thus "x \<in> sigma_sets C D"
  1600   proof induct
  1601     case (Basic a) with assms have "a \<in> D" by auto
  1602     thus ?case ..
  1603   next
  1604     case Empty show ?case by (rule sigma_sets.Empty)
  1605   next
  1606     from assms have "A \<in> sets (sigma C D)" by (subst sets_measure_of[OF \<open>D \<subseteq> Pow C\<close>])
  1607     moreover case (Compl a) hence "a \<in> sets (sigma C D)" by (subst sets_measure_of[OF \<open>D \<subseteq> Pow C\<close>])
  1608     ultimately have "A - a \<in> sets (sigma C D)" ..
  1609     thus ?case by (subst (asm) sets_measure_of[OF \<open>D \<subseteq> Pow C\<close>])
  1610   next
  1611     case (Union a)
  1612     thus ?case by (intro sigma_sets.Union)
  1613   qed
  1614 qed
  1616 lemma in_measure_of[intro, simp]: "M \<subseteq> Pow \<Omega> \<Longrightarrow> A \<in> M \<Longrightarrow> A \<in> sets (measure_of \<Omega> M \<mu>)"
  1617   by auto
  1619 lemma space_empty_iff: "space N = {} \<longleftrightarrow> sets N = {{}}"
  1620   by (metis Pow_empty Sup_bot_conv(1) cSup_singleton empty_iff
  1621             sets.sigma_sets_eq sets.space_closed sigma_sets_top subset_singletonD)
  1623 subsubsection \<open>Constructing simple @{typ "'a measure"}\<close>
  1625 lemma emeasure_measure_of:
  1626   assumes M: "M = measure_of \<Omega> A \<mu>"
  1627   assumes ms: "A \<subseteq> Pow \<Omega>" "positive (sets M) \<mu>" "countably_additive (sets M) \<mu>"
  1628   assumes X: "X \<in> sets M"
  1629   shows "emeasure M X = \<mu> X"
  1630 proof -
  1631   interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" by (rule sigma_algebra_sigma_sets) fact
  1632   have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>"
  1633     using ms M by (simp add: measure_space_def sigma_algebra_sigma_sets)
  1634   thus ?thesis using X ms
  1635     by(simp add: M emeasure_measure_of_conv sets_measure_of_conv)
  1636 qed
  1638 lemma emeasure_measure_of_sigma:
  1639   assumes ms: "sigma_algebra \<Omega> M" "positive M \<mu>" "countably_additive M \<mu>"
  1640   assumes A: "A \<in> M"
  1641   shows "emeasure (measure_of \<Omega> M \<mu>) A = \<mu> A"
  1642 proof -
  1643   interpret sigma_algebra \<Omega> M by fact
  1644   have "measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"
  1645     using ms sigma_sets_eq by (simp add: measure_space_def)
  1646   thus ?thesis by(simp add: emeasure_measure_of_conv A)
  1647 qed
  1649 lemma measure_cases[cases type: measure]:
  1650   obtains (measure) \<Omega> A \<mu> where "x = Abs_measure (\<Omega>, A, \<mu>)" "\<forall>a\<in>-A. \<mu> a = 0" "measure_space \<Omega> A \<mu>"
  1651   by atomize_elim (cases x, auto)
  1653 lemma sets_le_imp_space_le: "sets A \<subseteq> sets B \<Longrightarrow> space A \<subseteq> space B"
  1654   by (auto dest: sets.sets_into_space)
  1656 lemma sets_eq_imp_space_eq: "sets M = sets M' \<Longrightarrow> space M = space M'"
  1657   by (auto intro!: antisym sets_le_imp_space_le)
  1659 lemma emeasure_notin_sets: "A \<notin> sets M \<Longrightarrow> emeasure M A = 0"
  1660   by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)
  1662 lemma emeasure_neq_0_sets: "emeasure M A \<noteq> 0 \<Longrightarrow> A \<in> sets M"
  1663   using emeasure_notin_sets[of A M] by blast
  1665 lemma measure_notin_sets: "A \<notin> sets M \<Longrightarrow> measure M A = 0"
  1666   by (simp add: measure_def emeasure_notin_sets)
  1668 lemma measure_eqI:
  1669   fixes M N :: "'a measure"
  1670   assumes "sets M = sets N" and eq: "\<And>A. A \<in> sets M \<Longrightarrow> emeasure M A = emeasure N A"
  1671   shows "M = N"
  1672 proof (cases M N rule: measure_cases[case_product measure_cases])
  1673   case (measure_measure \<Omega> A \<mu> \<Omega>' A' \<mu>')
  1674   interpret M: sigma_algebra \<Omega> A using measure_measure by (auto simp: measure_space_def)
  1675   interpret N: sigma_algebra \<Omega>' A' using measure_measure by (auto simp: measure_space_def)
  1676   have "A = sets M" "A' = sets N"
  1677     using measure_measure by (simp_all add: sets_def Abs_measure_inverse)
  1678   with \<open>sets M = sets N\<close> have AA': "A = A'" by simp
  1679   moreover from M.space_closed N.space_closed AA' have "\<Omega> = \<Omega>'" by auto
  1680   moreover { fix B have "\<mu> B = \<mu>' B"
  1681     proof cases
  1682       assume "B \<in> A"
  1683       with eq \<open>A = sets M\<close> have "emeasure M B = emeasure N B" by simp
  1684       with measure_measure show "\<mu> B = \<mu>' B"
  1685         by (simp add: emeasure_def Abs_measure_inverse)
  1686     next
  1687       assume "B \<notin> A"
  1688       with \<open>A = sets M\<close> \<open>A' = sets N\<close> \<open>A = A'\<close> have "B \<notin> sets M" "B \<notin> sets N"
  1689         by auto
  1690       then have "emeasure M B = 0" "emeasure N B = 0"
  1691         by (simp_all add: emeasure_notin_sets)
  1692       with measure_measure show "\<mu> B = \<mu>' B"
  1693         by (simp add: emeasure_def Abs_measure_inverse)
  1694     qed }
  1695   then have "\<mu> = \<mu>'" by auto
  1696   ultimately show "M = N"
  1697     by (simp add: measure_measure)
  1698 qed
  1700 lemma sigma_eqI:
  1701   assumes [simp]: "M \<subseteq> Pow \<Omega>" "N \<subseteq> Pow \<Omega>" "sigma_sets \<Omega> M = sigma_sets \<Omega> N"
  1702   shows "sigma \<Omega> M = sigma \<Omega> N"
  1703   by (rule measure_eqI) (simp_all add: emeasure_sigma)
  1705 subsubsection \<open>Measurable functions\<close>
  1707 definition measurable :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) set" (infixr "\<rightarrow>\<^sub>M" 60) where
  1708   "measurable A B = {f \<in> space A \<rightarrow> space B. \<forall>y \<in> sets B. f -` y \<inter> space A \<in> sets A}"
  1710 lemma measurableI:
  1711   "(\<And>x. x \<in> space M \<Longrightarrow> f x \<in> space N) \<Longrightarrow> (\<And>A. A \<in> sets N \<Longrightarrow> f -` A \<inter> space M \<in> sets M) \<Longrightarrow>
  1712     f \<in> measurable M N"
  1713   by (auto simp: measurable_def)
  1715 lemma measurable_space:
  1716   "f \<in> measurable M A \<Longrightarrow> x \<in> space M \<Longrightarrow> f x \<in> space A"
  1717    unfolding measurable_def by auto
  1719 lemma measurable_sets:
  1720   "f \<in> measurable M A \<Longrightarrow> S \<in> sets A \<Longrightarrow> f -` S \<inter> space M \<in> sets M"
  1721    unfolding measurable_def by auto
  1723 lemma measurable_sets_Collect:
  1724   assumes f: "f \<in> measurable M N" and P: "{x\<in>space N. P x} \<in> sets N" shows "{x\<in>space M. P (f x)} \<in> sets M"
  1725 proof -
  1726   have "f -` {x \<in> space N. P x} \<inter> space M = {x\<in>space M. P (f x)}"
  1727     using measurable_space[OF f] by auto
  1728   with measurable_sets[OF f P] show ?thesis
  1729     by simp
  1730 qed
  1732 lemma measurable_sigma_sets:
  1733   assumes B: "sets N = sigma_sets \<Omega> A" "A \<subseteq> Pow \<Omega>"
  1734       and f: "f \<in> space M \<rightarrow> \<Omega>"
  1735       and ba: "\<And>y. y \<in> A \<Longrightarrow> (f -` y) \<inter> space M \<in> sets M"
  1736   shows "f \<in> measurable M N"
  1737 proof -
  1738   interpret A: sigma_algebra \<Omega> "sigma_sets \<Omega> A" using B(2) by (rule sigma_algebra_sigma_sets)
  1739   from B[of N] sets.space_closed[of N] A.space_closed have \<Omega>: "\<Omega> = space N" by force
  1741   { fix X assume "X \<in> sigma_sets \<Omega> A"
  1742     then have "f -` X \<inter> space M \<in> sets M \<and> X \<subseteq> \<Omega>"
  1743       proof induct
  1744         case (Basic a) then show ?case
  1745           by (auto simp add: ba) (metis B(2) subsetD PowD)
  1746       next
  1747         case (Compl a)
  1748         have [simp]: "f -` \<Omega> \<inter> space M = space M"
  1749           by (auto simp add: funcset_mem [OF f])
  1750         then show ?case
  1751           by (auto simp add: vimage_Diff Diff_Int_distrib2 sets.compl_sets Compl)
  1752       next
  1753         case (Union a)
  1754         then show ?case
  1755           by (simp add: vimage_UN, simp only: UN_extend_simps(4)) blast
  1756       qed auto }
  1757   with f show ?thesis
  1758     by (auto simp add: measurable_def B \<Omega>)
  1759 qed
  1761 lemma measurable_measure_of:
  1762   assumes B: "N \<subseteq> Pow \<Omega>"
  1763       and f: "f \<in> space M \<rightarrow> \<Omega>"
  1764       and ba: "\<And>y. y \<in> N \<Longrightarrow> (f -` y) \<inter> space M \<in> sets M"
  1765   shows "f \<in> measurable M (measure_of \<Omega> N \<mu>)"
  1766 proof -
  1767   have "sets (measure_of \<Omega> N \<mu>) = sigma_sets \<Omega> N"
  1768     using B by (rule sets_measure_of)
  1769   from this assms show ?thesis by (rule measurable_sigma_sets)
  1770 qed
  1772 lemma measurable_iff_measure_of:
  1773   assumes "N \<subseteq> Pow \<Omega>" "f \<in> space M \<rightarrow> \<Omega>"
  1774   shows "f \<in> measurable M (measure_of \<Omega> N \<mu>) \<longleftrightarrow> (\<forall>A\<in>N. f -` A \<inter> space M \<in> sets M)"
  1775   by (metis assms in_measure_of measurable_measure_of assms measurable_sets)
  1777 lemma measurable_cong_sets:
  1778   assumes sets: "sets M = sets M'" "sets N = sets N'"
  1779   shows "measurable M N = measurable M' N'"
  1780   using sets[THEN sets_eq_imp_space_eq] sets by (simp add: measurable_def)
  1782 lemma measurable_cong:
  1783   assumes "\<And>w. w \<in> space M \<Longrightarrow> f w = g w"
  1784   shows "f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable M M'"
  1785   unfolding measurable_def using assms
  1786   by (simp cong: vimage_inter_cong Pi_cong)
  1788 lemma measurable_cong':
  1789   assumes "\<And>w. w \<in> space M =simp=> f w = g w"
  1790   shows "f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable M M'"
  1791   unfolding measurable_def using assms
  1792   by (simp cong: vimage_inter_cong Pi_cong add: simp_implies_def)
  1794 lemma measurable_cong_strong:
  1795   "M = N \<Longrightarrow> M' = N' \<Longrightarrow> (\<And>w. w \<in> space M \<Longrightarrow> f w = g w) \<Longrightarrow>
  1796     f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable N N'"
  1797   by (metis measurable_cong)
  1799 lemma measurable_compose:
  1800   assumes f: "f \<in> measurable M N" and g: "g \<in> measurable N L"
  1801   shows "(\<lambda>x. g (f x)) \<in> measurable M L"
  1802 proof -
  1803   have "\<And>A. (\<lambda>x. g (f x)) -` A \<inter> space M = f -` (g -` A \<inter> space N) \<inter> space M"
  1804     using measurable_space[OF f] by auto
  1805   with measurable_space[OF f] measurable_space[OF g] show ?thesis
  1806     by (auto intro: measurable_sets[OF f] measurable_sets[OF g]
  1807              simp del: vimage_Int simp add: measurable_def)
  1808 qed
  1810 lemma measurable_comp:
  1811   "f \<in> measurable M N \<Longrightarrow> g \<in> measurable N L \<Longrightarrow> g \<circ> f \<in> measurable M L"
  1812   using measurable_compose[of f M N g L] by (simp add: comp_def)
  1814 lemma measurable_const:
  1815   "c \<in> space M' \<Longrightarrow> (\<lambda>x. c) \<in> measurable M M'"
  1816   by (auto simp add: measurable_def)
  1818 lemma measurable_ident: "id \<in> measurable M M"
  1819   by (auto simp add: measurable_def)
  1821 lemma measurable_id: "(\<lambda>x. x) \<in> measurable M M"
  1822   by (simp add: measurable_def)
  1824 lemma measurable_ident_sets:
  1825   assumes eq: "sets M = sets M'" shows "(\<lambda>x. x) \<in> measurable M M'"
  1826   using measurable_ident[of M]
  1827   unfolding id_def measurable_def eq sets_eq_imp_space_eq[OF eq] .
  1829 lemma sets_Least:
  1830   assumes meas: "\<And>i::nat. {x\<in>space M. P i x} \<in> M"
  1831   shows "(\<lambda>x. LEAST j. P j x) -` A \<inter> space M \<in> sets M"
  1832 proof -
  1833   { fix i have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M \<in> sets M"
  1834     proof cases
  1835       assume i: "(LEAST j. False) = i"
  1836       have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M =
  1837         {x\<in>space M. P i x} \<inter> (space M - (\<Union>j<i. {x\<in>space M. P j x})) \<union> (space M - (\<Union>i. {x\<in>space M. P i x}))"
  1838         by (simp add: set_eq_iff, safe)
  1839            (insert i, auto dest: Least_le intro: LeastI intro!: Least_equality)
  1840       with meas show ?thesis
  1841         by (auto intro!: sets.Int)
  1842     next
  1843       assume i: "(LEAST j. False) \<noteq> i"
  1844       then have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M =
  1845         {x\<in>space M. P i x} \<inter> (space M - (\<Union>j<i. {x\<in>space M. P j x}))"
  1846       proof (simp add: set_eq_iff, safe)
  1847         fix x assume neq: "(LEAST j. False) \<noteq> (LEAST j. P j x)"
  1848         have "\<exists>j. P j x"
  1849           by (rule ccontr) (insert neq, auto)
  1850         then show "P (LEAST j. P j x) x" by (rule LeastI_ex)
  1851       qed (auto dest: Least_le intro!: Least_equality)
  1852       with meas show ?thesis
  1853         by auto
  1854     qed }
  1855   then have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M) \<in> sets M"
  1856     by (intro sets.countable_UN) auto
  1857   moreover have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M) =
  1858     (\<lambda>x. LEAST j. P j x) -` A \<inter> space M" by auto
  1859   ultimately show ?thesis by auto
  1860 qed
  1862 lemma measurable_mono1:
  1863   "M' \<subseteq> Pow \<Omega> \<Longrightarrow> M \<subseteq> M' \<Longrightarrow>
  1864     measurable (measure_of \<Omega> M \<mu>) N \<subseteq> measurable (measure_of \<Omega> M' \<mu>') N"
  1865   using measure_of_subset[of M' \<Omega> M] by (auto simp add: measurable_def)
  1867 subsubsection \<open>Counting space\<close>
  1869 definition count_space :: "'a set \<Rightarrow> 'a measure" where
  1870   "count_space \<Omega> = measure_of \<Omega> (Pow \<Omega>) (\<lambda>A. if finite A then ereal (card A) else \<infinity>)"
  1872 lemma 
  1873   shows space_count_space[simp]: "space (count_space \<Omega>) = \<Omega>"
  1874     and sets_count_space[simp]: "sets (count_space \<Omega>) = Pow \<Omega>"
  1875   using sigma_sets_into_sp[of "Pow \<Omega>" \<Omega>]
  1876   by (auto simp: count_space_def)
  1878 lemma measurable_count_space_eq1[simp]:
  1879   "f \<in> measurable (count_space A) M \<longleftrightarrow> f \<in> A \<rightarrow> space M"
  1880  unfolding measurable_def by simp
  1882 lemma measurable_compose_countable':
  1883   assumes f: "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. f i x) \<in> measurable M N"
  1884   and g: "g \<in> measurable M (count_space I)" and I: "countable I"
  1885   shows "(\<lambda>x. f (g x) x) \<in> measurable M N"
  1886   unfolding measurable_def
  1887 proof safe
  1888   fix x assume "x \<in> space M" then show "f (g x) x \<in> space N"
  1889     using measurable_space[OF f] g[THEN measurable_space] by auto
  1890 next
  1891   fix A assume A: "A \<in> sets N"
  1892   have "(\<lambda>x. f (g x) x) -` A \<inter> space M = (\<Union>i\<in>I. (g -` {i} \<inter> space M) \<inter> (f i -` A \<inter> space M))"
  1893     using measurable_space[OF g] by auto
  1894   also have "\<dots> \<in> sets M"
  1895     using f[THEN measurable_sets, OF _ A] g[THEN measurable_sets]
  1896     by (auto intro!: sets.countable_UN' I intro: sets.Int[OF measurable_sets measurable_sets])
  1897   finally show "(\<lambda>x. f (g x) x) -` A \<inter> space M \<in> sets M" .
  1898 qed
  1900 lemma measurable_count_space_eq_countable:
  1901   assumes "countable A"
  1902   shows "f \<in> measurable M (count_space A) \<longleftrightarrow> (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))"
  1903 proof -
  1904   { fix X assume "X \<subseteq> A" "f \<in> space M \<rightarrow> A"
  1905     with \<open>countable A\<close> have "f -` X \<inter> space M = (\<Union>a\<in>X. f -` {a} \<inter> space M)" "countable X"
  1906       by (auto dest: countable_subset)
  1907     moreover assume "\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M"
  1908     ultimately have "f -` X \<inter> space M \<in> sets M"
  1909       using \<open>X \<subseteq> A\<close> by (auto intro!: sets.countable_UN' simp del: UN_simps) }
  1910   then show ?thesis
  1911     unfolding measurable_def by auto
  1912 qed
  1914 lemma measurable_count_space_eq2:
  1915   "finite A \<Longrightarrow> f \<in> measurable M (count_space A) \<longleftrightarrow> (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))"
  1916   by (intro measurable_count_space_eq_countable countable_finite)
  1918 lemma measurable_count_space_eq2_countable:
  1919   fixes f :: "'a => 'c::countable"
  1920   shows "f \<in> measurable M (count_space A) \<longleftrightarrow> (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))"
  1921   by (intro measurable_count_space_eq_countable countableI_type)
  1923 lemma measurable_compose_countable:
  1924   assumes f: "\<And>i::'i::countable. (\<lambda>x. f i x) \<in> measurable M N" and g: "g \<in> measurable M (count_space UNIV)"
  1925   shows "(\<lambda>x. f (g x) x) \<in> measurable M N"
  1926   by (rule measurable_compose_countable'[OF assms]) auto
  1928 lemma measurable_count_space_const:
  1929   "(\<lambda>x. c) \<in> measurable M (count_space UNIV)"
  1930   by (simp add: measurable_const)
  1932 lemma measurable_count_space:
  1933   "f \<in> measurable (count_space A) (count_space UNIV)"
  1934   by simp
  1936 lemma measurable_compose_rev:
  1937   assumes f: "f \<in> measurable L N" and g: "g \<in> measurable M L"
  1938   shows "(\<lambda>x. f (g x)) \<in> measurable M N"
  1939   using measurable_compose[OF g f] .
  1941 lemma measurable_empty_iff: 
  1942   "space N = {} \<Longrightarrow> f \<in> measurable M N \<longleftrightarrow> space M = {}"
  1943   by (auto simp add: measurable_def Pi_iff)
  1945 subsubsection \<open>Extend measure\<close>
  1947 definition "extend_measure \<Omega> I G \<mu> =
  1948   (if (\<exists>\<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G`I)) \<mu>') \<and> \<not> (\<forall>i\<in>I. \<mu> i = 0)
  1949       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>')
  1950       else measure_of \<Omega> (G`I) (\<lambda>_. 0))"
  1952 lemma space_extend_measure: "G ` I \<subseteq> Pow \<Omega> \<Longrightarrow> space (extend_measure \<Omega> I G \<mu>) = \<Omega>"
  1953   unfolding extend_measure_def by simp
  1955 lemma sets_extend_measure: "G ` I \<subseteq> Pow \<Omega> \<Longrightarrow> sets (extend_measure \<Omega> I G \<mu>) = sigma_sets \<Omega> (G`I)"
  1956   unfolding extend_measure_def by simp
  1958 lemma emeasure_extend_measure:
  1959   assumes M: "M = extend_measure \<Omega> I G \<mu>"
  1960     and eq: "\<And>i. i \<in> I \<Longrightarrow> \<mu>' (G i) = \<mu> i"
  1961     and ms: "G ` I \<subseteq> Pow \<Omega>" "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'"
  1962     and "i \<in> I"
  1963   shows "emeasure M (G i) = \<mu> i"
  1964 proof cases
  1965   assume *: "(\<forall>i\<in>I. \<mu> i = 0)"
  1966   with M have M_eq: "M = measure_of \<Omega> (G`I) (\<lambda>_. 0)"
  1967    by (simp add: extend_measure_def)
  1968   from measure_space_0[OF ms(1)] ms \<open>i\<in>I\<close>
  1969   have "emeasure M (G i) = 0"
  1970     by (intro emeasure_measure_of[OF M_eq]) (auto simp add: M measure_space_def sets_extend_measure)
  1971   with \<open>i\<in>I\<close> * show ?thesis
  1972     by simp
  1973 next
  1974   def P \<equiv> "\<lambda>\<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G`I)) \<mu>'"
  1975   assume "\<not> (\<forall>i\<in>I. \<mu> i = 0)"
  1976   moreover
  1977   have "measure_space (space M) (sets M) \<mu>'"
  1978     using ms unfolding measure_space_def by auto standard
  1979   with ms eq have "\<exists>\<mu>'. P \<mu>'"
  1980     unfolding P_def
  1981     by (intro exI[of _ \<mu>']) (auto simp add: M space_extend_measure sets_extend_measure)
  1982   ultimately have M_eq: "M = measure_of \<Omega> (G`I) (Eps P)"
  1983     by (simp add: M extend_measure_def P_def[symmetric])
  1985   from \<open>\<exists>\<mu>'. P \<mu>'\<close> have P: "P (Eps P)" by (rule someI_ex)
  1986   show "emeasure M (G i) = \<mu> i"
  1987   proof (subst emeasure_measure_of[OF M_eq])
  1988     have sets_M: "sets M = sigma_sets \<Omega> (G`I)"
  1989       using M_eq ms by (auto simp: sets_extend_measure)
  1990     then show "G i \<in> sets M" using \<open>i \<in> I\<close> by auto
  1991     show "positive (sets M) (Eps P)" "countably_additive (sets M) (Eps P)" "Eps P (G i) = \<mu> i"
  1992       using P \<open>i\<in>I\<close> by (auto simp add: sets_M measure_space_def P_def)
  1993   qed fact
  1994 qed
  1996 lemma emeasure_extend_measure_Pair:
  1997   assumes M: "M = extend_measure \<Omega> {(i, j). I i j} (\<lambda>(i, j). G i j) (\<lambda>(i, j). \<mu> i j)"
  1998     and eq: "\<And>i j. I i j \<Longrightarrow> \<mu>' (G i j) = \<mu> i j"
  1999     and ms: "\<And>i j. I i j \<Longrightarrow> G i j \<in> Pow \<Omega>" "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'"
  2000     and "I i j"
  2001   shows "emeasure M (G i j) = \<mu> i j"
  2002   using emeasure_extend_measure[OF M _ _ ms(2,3), of "(i,j)"] eq ms(1) \<open>I i j\<close>
  2003   by (auto simp: subset_eq)
  2005 subsubsection \<open>Supremum of a set of $\sigma$-algebras\<close>
  2007 definition "Sup_sigma M = sigma (\<Union>x\<in>M. space x) (\<Union>x\<in>M. sets x)"
  2009 syntax
  2010   "_SUP_sigma"   :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>\<^sub>\<sigma> _\<in>_./ _)" [0, 0, 10] 10)
  2012 translations
  2013   "\<Squnion>\<^sub>\<sigma> x\<in>A. B"   == "CONST Sup_sigma ((\<lambda>x. B) ` A)"
  2015 lemma space_Sup_sigma: "space (Sup_sigma M) = (\<Union>x\<in>M. space x)"
  2016   unfolding Sup_sigma_def by (rule space_measure_of) (auto dest: sets.sets_into_space)
  2018 lemma sets_Sup_sigma: "sets (Sup_sigma M) = sigma_sets (\<Union>x\<in>M. space x) (\<Union>x\<in>M. sets x)"
  2019   unfolding Sup_sigma_def by (rule sets_measure_of) (auto dest: sets.sets_into_space)
  2021 lemma in_Sup_sigma: "m \<in> M \<Longrightarrow> A \<in> sets m \<Longrightarrow> A \<in> sets (Sup_sigma M)"
  2022   unfolding sets_Sup_sigma by auto
  2024 lemma SUP_sigma_cong: 
  2025   assumes *: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets (N i)" shows "sets (\<Squnion>\<^sub>\<sigma> i\<in>I. M i) = sets (\<Squnion>\<^sub>\<sigma> i\<in>I. N i)"
  2026   using * sets_eq_imp_space_eq[OF *] by (simp add: Sup_sigma_def)
  2028 lemma sets_Sup_in_sets: 
  2029   assumes "M \<noteq> {}"
  2030   assumes "\<And>m. m \<in> M \<Longrightarrow> space m = space N"
  2031   assumes "\<And>m. m \<in> M \<Longrightarrow> sets m \<subseteq> sets N"
  2032   shows "sets (Sup_sigma M) \<subseteq> sets N"
  2033 proof -
  2034   have *: "UNION M space = space N"
  2035     using assms by auto
  2036   show ?thesis
  2037     unfolding sets_Sup_sigma * using assms by (auto intro!: sets.sigma_sets_subset)
  2038 qed
  2040 lemma measurable_Sup_sigma1:
  2041   assumes m: "m \<in> M" and f: "f \<in> measurable m N"
  2042     and const_space: "\<And>m n. m \<in> M \<Longrightarrow> n \<in> M \<Longrightarrow> space m = space n"
  2043   shows "f \<in> measurable (Sup_sigma M) N"
  2044 proof -
  2045   have "space (Sup_sigma M) = space m"
  2046     using m by (auto simp add: space_Sup_sigma dest: const_space)
  2047   then show ?thesis
  2048     using m f unfolding measurable_def by (auto intro: in_Sup_sigma)
  2049 qed
  2051 lemma measurable_Sup_sigma2:
  2052   assumes M: "M \<noteq> {}"
  2053   assumes f: "\<And>m. m \<in> M \<Longrightarrow> f \<in> measurable N m"
  2054   shows "f \<in> measurable N (Sup_sigma M)"
  2055   unfolding Sup_sigma_def
  2056 proof (rule measurable_measure_of)
  2057   show "f \<in> space N \<rightarrow> UNION M space"
  2058     using measurable_space[OF f] M by auto
  2059 qed (auto intro: measurable_sets f dest: sets.sets_into_space)
  2061 lemma Sup_sigma_sigma:
  2062   assumes [simp]: "M \<noteq> {}" and M: "\<And>m. m \<in> M \<Longrightarrow> m \<subseteq> Pow \<Omega>"
  2063   shows "(\<Squnion>\<^sub>\<sigma> m\<in>M. sigma \<Omega> m) = sigma \<Omega> (\<Union>M)"
  2064 proof (rule measure_eqI)
  2065   { fix a m assume "a \<in> sigma_sets \<Omega> m" "m \<in> M"
  2066     then have "a \<in> sigma_sets \<Omega> (\<Union>M)"
  2067      by induction (auto intro: sigma_sets.intros) }
  2068   then show "sets (\<Squnion>\<^sub>\<sigma> m\<in>M. sigma \<Omega> m) = sets (sigma \<Omega> (\<Union>M))"
  2069     apply (simp add: sets_Sup_sigma space_measure_of_conv M Union_least)
  2070     apply (rule sigma_sets_eqI)
  2071     apply auto
  2072     done
  2073 qed (simp add: Sup_sigma_def emeasure_sigma)
  2075 lemma SUP_sigma_sigma:
  2076   assumes M: "M \<noteq> {}" "\<And>m. m \<in> M \<Longrightarrow> f m \<subseteq> Pow \<Omega>"
  2077   shows "(\<Squnion>\<^sub>\<sigma> m\<in>M. sigma \<Omega> (f m)) = sigma \<Omega> (\<Union>m\<in>M. f m)"
  2078 proof -
  2079   have "Sup_sigma (sigma \<Omega> ` f ` M) = sigma \<Omega> (\<Union>(f ` M))"
  2080     using M by (intro Sup_sigma_sigma) auto
  2081   then show ?thesis
  2082     by (simp add: image_image)
  2083 qed
  2085 subsection \<open>The smallest $\sigma$-algebra regarding a function\<close>
  2087 definition
  2088   "vimage_algebra X f M = sigma X {f -` A \<inter> X | A. A \<in> sets M}"
  2090 lemma space_vimage_algebra[simp]: "space (vimage_algebra X f M) = X"
  2091   unfolding vimage_algebra_def by (rule space_measure_of) auto
  2093 lemma sets_vimage_algebra: "sets (vimage_algebra X f M) = sigma_sets X {f -` A \<inter> X | A. A \<in> sets M}"
  2094   unfolding vimage_algebra_def by (rule sets_measure_of) auto
  2096 lemma sets_vimage_algebra2:
  2097   "f \<in> X \<rightarrow> space M \<Longrightarrow> sets (vimage_algebra X f M) = {f -` A \<inter> X | A. A \<in> sets M}"
  2098   using sigma_sets_vimage_commute[of f X "space M" "sets M"]
  2099   unfolding sets_vimage_algebra sets.sigma_sets_eq by simp
  2101 lemma sets_vimage_algebra_cong: "sets M = sets N \<Longrightarrow> sets (vimage_algebra X f M) = sets (vimage_algebra X f N)"
  2102   by (simp add: sets_vimage_algebra)
  2104 lemma vimage_algebra_cong:
  2105   assumes "X = Y"
  2106   assumes "\<And>x. x \<in> Y \<Longrightarrow> f x = g x"
  2107   assumes "sets M = sets N"
  2108   shows "vimage_algebra X f M = vimage_algebra Y g N"
  2109   by (auto simp: vimage_algebra_def assms intro!: arg_cong2[where f=sigma])
  2111 lemma in_vimage_algebra: "A \<in> sets M \<Longrightarrow> f -` A \<inter> X \<in> sets (vimage_algebra X f M)"
  2112   by (auto simp: vimage_algebra_def)
  2114 lemma sets_image_in_sets:
  2115   assumes N: "space N = X"
  2116   assumes f: "f \<in> measurable N M"
  2117   shows "sets (vimage_algebra X f M) \<subseteq> sets N"
  2118   unfolding sets_vimage_algebra N[symmetric]
  2119   by (rule sets.sigma_sets_subset) (auto intro!: measurable_sets f)
  2121 lemma measurable_vimage_algebra1: "f \<in> X \<rightarrow> space M \<Longrightarrow> f \<in> measurable (vimage_algebra X f M) M"
  2122   unfolding measurable_def by (auto intro: in_vimage_algebra)
  2124 lemma measurable_vimage_algebra2:
  2125   assumes g: "g \<in> space N \<rightarrow> X" and f: "(\<lambda>x. f (g x)) \<in> measurable N M"
  2126   shows "g \<in> measurable N (vimage_algebra X f M)"
  2127   unfolding vimage_algebra_def
  2128 proof (rule measurable_measure_of)
  2129   fix A assume "A \<in> {f -` A \<inter> X | A. A \<in> sets M}"
  2130   then obtain Y where Y: "Y \<in> sets M" and A: "A = f -` Y \<inter> X"
  2131     by auto
  2132   then have "g -` A \<inter> space N = (\<lambda>x. f (g x)) -` Y \<inter> space N"
  2133     using g by auto
  2134   also have "\<dots> \<in> sets N"
  2135     using f Y by (rule measurable_sets)
  2136   finally show "g -` A \<inter> space N \<in> sets N" .
  2137 qed (insert g, auto)
  2139 lemma vimage_algebra_sigma:
  2140   assumes X: "X \<subseteq> Pow \<Omega>'" and f: "f \<in> \<Omega> \<rightarrow> \<Omega>'"
  2141   shows "vimage_algebra \<Omega> f (sigma \<Omega>' X) = sigma \<Omega> {f -` A \<inter> \<Omega> | A. A \<in> X }" (is "?V = ?S")
  2142 proof (rule measure_eqI)
  2143   have \<Omega>: "{f -` A \<inter> \<Omega> |A. A \<in> X} \<subseteq> Pow \<Omega>" by auto
  2144   show "sets ?V = sets ?S"
  2145     using sigma_sets_vimage_commute[OF f, of X]
  2146     by (simp add: space_measure_of_conv f sets_vimage_algebra2 \<Omega> X)
  2147 qed (simp add: vimage_algebra_def emeasure_sigma)
  2149 lemma vimage_algebra_vimage_algebra_eq:
  2150   assumes *: "f \<in> X \<rightarrow> Y" "g \<in> Y \<rightarrow> space M"
  2151   shows "vimage_algebra X f (vimage_algebra Y g M) = vimage_algebra X (\<lambda>x. g (f x)) M"
  2152     (is "?VV = ?V")
  2153 proof (rule measure_eqI)
  2154   have "(\<lambda>x. g (f x)) \<in> X \<rightarrow> space M" "\<And>A. A \<inter> f -` Y \<inter> X = A \<inter> X"
  2155     using * by auto
  2156   with * show "sets ?VV = sets ?V"
  2157     by (simp add: sets_vimage_algebra2 ex_simps[symmetric] vimage_comp comp_def del: ex_simps)
  2158 qed (simp add: vimage_algebra_def emeasure_sigma)
  2160 lemma sets_vimage_Sup_eq:
  2161   assumes *: "M \<noteq> {}" "\<And>m. m \<in> M \<Longrightarrow> f \<in> X \<rightarrow> space m"
  2162   shows "sets (vimage_algebra X f (Sup_sigma M)) = sets (\<Squnion>\<^sub>\<sigma> m \<in> M. vimage_algebra X f m)"
  2163   (is "?IS = ?SI")
  2164 proof
  2165   show "?IS \<subseteq> ?SI"
  2166     by (intro sets_image_in_sets measurable_Sup_sigma2 measurable_Sup_sigma1)
  2167        (auto simp: space_Sup_sigma measurable_vimage_algebra1 *)
  2168   { fix m assume "m \<in> M"
  2169     moreover then have "f \<in> X \<rightarrow> space (Sup_sigma M)" "f \<in> X \<rightarrow> space m"
  2170       using * by (auto simp: space_Sup_sigma)
  2171     ultimately have "f \<in> measurable (vimage_algebra X f (Sup_sigma M)) m"
  2172       by (auto simp add: measurable_def sets_vimage_algebra2 intro: in_Sup_sigma) }
  2173   then show "?SI \<subseteq> ?IS"
  2174     by (auto intro!: sets_image_in_sets sets_Sup_in_sets del: subsetI simp: *)
  2175 qed
  2177 lemma vimage_algebra_Sup_sigma:
  2178   assumes [simp]: "MM \<noteq> {}" and "\<And>M. M \<in> MM \<Longrightarrow> f \<in> X \<rightarrow> space M"
  2179   shows "vimage_algebra X f (Sup_sigma MM) = Sup_sigma (vimage_algebra X f ` MM)"
  2180 proof (rule measure_eqI)
  2181   show "sets (vimage_algebra X f (Sup_sigma MM)) = sets (Sup_sigma (vimage_algebra X f ` MM))"
  2182     using assms by (rule sets_vimage_Sup_eq)
  2183 qed (simp add: vimage_algebra_def Sup_sigma_def emeasure_sigma)
  2185 subsubsection \<open>Restricted Space Sigma Algebra\<close>
  2187 definition restrict_space where
  2188   "restrict_space M \<Omega> = measure_of (\<Omega> \<inter> space M) ((op \<inter> \<Omega>) ` sets M) (emeasure M)"
  2190 lemma space_restrict_space: "space (restrict_space M \<Omega>) = \<Omega> \<inter> space M"
  2191   using sets.sets_into_space unfolding restrict_space_def by (subst space_measure_of) auto
  2193 lemma space_restrict_space2: "\<Omega> \<in> sets M \<Longrightarrow> space (restrict_space M \<Omega>) = \<Omega>"
  2194   by (simp add: space_restrict_space sets.sets_into_space)
  2196 lemma sets_restrict_space: "sets (restrict_space M \<Omega>) = (op \<inter> \<Omega>) ` sets M"
  2197   unfolding restrict_space_def
  2198 proof (subst sets_measure_of)
  2199   show "op \<inter> \<Omega> ` sets M \<subseteq> Pow (\<Omega> \<inter> space M)"
  2200     by (auto dest: sets.sets_into_space)
  2201   have "sigma_sets (\<Omega> \<inter> space M) {((\<lambda>x. x) -` X) \<inter> (\<Omega> \<inter> space M) | X. X \<in> sets M} =
  2202     (\<lambda>X. X \<inter> (\<Omega> \<inter> space M)) ` sets M"
  2203     by (subst sigma_sets_vimage_commute[symmetric, where \<Omega>' = "space M"])
  2204        (auto simp add: sets.sigma_sets_eq)
  2205   moreover have "{((\<lambda>x. x) -` X) \<inter> (\<Omega> \<inter> space M) | X. X \<in> sets M} = (\<lambda>X. X \<inter> (\<Omega> \<inter> space M)) `  sets M"
  2206     by auto
  2207   moreover have "(\<lambda>X. X \<inter> (\<Omega> \<inter> space M)) `  sets M = (op \<inter> \<Omega>) ` sets M"
  2208     by (intro image_cong) (auto dest: sets.sets_into_space)
  2209   ultimately show "sigma_sets (\<Omega> \<inter> space M) (op \<inter> \<Omega> ` sets M) = op \<inter> \<Omega> ` sets M"
  2210     by simp
  2211 qed
  2213 lemma restrict_space_sets_cong:
  2214   "A = B \<Longrightarrow> sets M = sets N \<Longrightarrow> sets (restrict_space M A) = sets (restrict_space N B)"
  2215   by (auto simp: sets_restrict_space)
  2217 lemma sets_restrict_space_count_space :
  2218   "sets (restrict_space (count_space A) B) = sets (count_space (A \<inter> B))"
  2219 by(auto simp add: sets_restrict_space)
  2221 lemma sets_restrict_UNIV[simp]: "sets (restrict_space M UNIV) = sets M"
  2222   by (auto simp add: sets_restrict_space)
  2224 lemma sets_restrict_restrict_space:
  2225   "sets (restrict_space (restrict_space M A) B) = sets (restrict_space M (A \<inter> B))"
  2226   unfolding sets_restrict_space image_comp by (intro image_cong) auto
  2228 lemma sets_restrict_space_iff:
  2229   "\<Omega> \<inter> space M \<in> sets M \<Longrightarrow> A \<in> sets (restrict_space M \<Omega>) \<longleftrightarrow> (A \<subseteq> \<Omega> \<and> A \<in> sets M)"
  2230 proof (subst sets_restrict_space, safe)
  2231   fix A assume "\<Omega> \<inter> space M \<in> sets M" and A: "A \<in> sets M"
  2232   then have "(\<Omega> \<inter> space M) \<inter> A \<in> sets M"
  2233     by rule
  2234   also have "(\<Omega> \<inter> space M) \<inter> A = \<Omega> \<inter> A"
  2235     using sets.sets_into_space[OF A] by auto
  2236   finally show "\<Omega> \<inter> A \<in> sets M"
  2237     by auto
  2238 qed auto
  2240 lemma sets_restrict_space_cong: "sets M = sets N \<Longrightarrow> sets (restrict_space M \<Omega>) = sets (restrict_space N \<Omega>)"
  2241   by (simp add: sets_restrict_space)
  2243 lemma restrict_space_eq_vimage_algebra:
  2244   "\<Omega> \<subseteq> space M \<Longrightarrow> sets (restrict_space M \<Omega>) = sets (vimage_algebra \<Omega> (\<lambda>x. x) M)"
  2245   unfolding restrict_space_def
  2246   apply (subst sets_measure_of)
  2247   apply (auto simp add: image_subset_iff dest: sets.sets_into_space) []
  2248   apply (auto simp add: sets_vimage_algebra intro!: arg_cong2[where f=sigma_sets])
  2249   done
  2251 lemma sets_Collect_restrict_space_iff: 
  2252   assumes "S \<in> sets M"
  2253   shows "{x\<in>space (restrict_space M S). P x} \<in> sets (restrict_space M S) \<longleftrightarrow> {x\<in>space M. x \<in> S \<and> P x} \<in> sets M"
  2254 proof -
  2255   have "{x\<in>S. P x} = {x\<in>space M. x \<in> S \<and> P x}"
  2256     using sets.sets_into_space[OF assms] by auto
  2257   then show ?thesis
  2258     by (subst sets_restrict_space_iff) (auto simp add: space_restrict_space assms)
  2259 qed
  2261 lemma measurable_restrict_space1:
  2262   assumes f: "f \<in> measurable M N"
  2263   shows "f \<in> measurable (restrict_space M \<Omega>) N"
  2264   unfolding measurable_def
  2265 proof (intro CollectI conjI ballI)
  2266   show sp: "f \<in> space (restrict_space M \<Omega>) \<rightarrow> space N"
  2267     using measurable_space[OF f] by (auto simp: space_restrict_space)
  2269   fix A assume "A \<in> sets N"
  2270   have "f -` A \<inter> space (restrict_space M \<Omega>) = (f -` A \<inter> space M) \<inter> (\<Omega> \<inter> space M)"
  2271     by (auto simp: space_restrict_space)
  2272   also have "\<dots> \<in> sets (restrict_space M \<Omega>)"
  2273     unfolding sets_restrict_space
  2274     using measurable_sets[OF f \<open>A \<in> sets N\<close>] by blast
  2275   finally show "f -` A \<inter> space (restrict_space M \<Omega>) \<in> sets (restrict_space M \<Omega>)" .
  2276 qed
  2278 lemma measurable_restrict_space2_iff:
  2279   "f \<in> measurable M (restrict_space N \<Omega>) \<longleftrightarrow> (f \<in> measurable M N \<and> f \<in> space M \<rightarrow> \<Omega>)"
  2280 proof -
  2281   have "\<And>A. f \<in> space M \<rightarrow> \<Omega> \<Longrightarrow> f -` \<Omega> \<inter> f -` A \<inter> space M = f -` A \<inter> space M"
  2282     by auto
  2283   then show ?thesis
  2284     by (auto simp: measurable_def space_restrict_space Pi_Int[symmetric] sets_restrict_space)
  2285 qed
  2287 lemma measurable_restrict_space2:
  2288   "f \<in> space M \<rightarrow> \<Omega> \<Longrightarrow> f \<in> measurable M N \<Longrightarrow> f \<in> measurable M (restrict_space N \<Omega>)"
  2289   by (simp add: measurable_restrict_space2_iff)
  2291 lemma measurable_piecewise_restrict:
  2292   assumes I: "countable C"
  2293     and X: "\<And>\<Omega>. \<Omega> \<in> C \<Longrightarrow> \<Omega> \<inter> space M \<in> sets M" "space M \<subseteq> \<Union>C"
  2294     and f: "\<And>\<Omega>. \<Omega> \<in> C \<Longrightarrow> f \<in> measurable (restrict_space M \<Omega>) N"
  2295   shows "f \<in> measurable M N"
  2296 proof (rule measurableI)
  2297   fix x assume "x \<in> space M"
  2298   with X obtain \<Omega> where "\<Omega> \<in> C" "x \<in> \<Omega>" "x \<in> space M" by auto
  2299   then show "f x \<in> space N"
  2300     by (auto simp: space_restrict_space intro: f measurable_space)
  2301 next
  2302   fix A assume A: "A \<in> sets N"
  2303   have "f -` A \<inter> space M = (\<Union>\<Omega>\<in>C. (f -` A \<inter> (\<Omega> \<inter> space M)))"
  2304     using X by (auto simp: subset_eq)
  2305   also have "\<dots> \<in> sets M"
  2306     using measurable_sets[OF f A] X I
  2307     by (intro sets.countable_UN') (auto simp: sets_restrict_space_iff space_restrict_space)
  2308   finally show "f -` A \<inter> space M \<in> sets M" .
  2309 qed
  2311 lemma measurable_piecewise_restrict_iff:
  2312   "countable C \<Longrightarrow> (\<And>\<Omega>. \<Omega> \<in> C \<Longrightarrow> \<Omega> \<inter> space M \<in> sets M) \<Longrightarrow> space M \<subseteq> (\<Union>C) \<Longrightarrow>
  2313     f \<in> measurable M N \<longleftrightarrow> (\<forall>\<Omega>\<in>C. f \<in> measurable (restrict_space M \<Omega>) N)"
  2314   by (auto intro: measurable_piecewise_restrict measurable_restrict_space1)
  2316 lemma measurable_If_restrict_space_iff:
  2317   "{x\<in>space M. P x} \<in> sets M \<Longrightarrow>
  2318     (\<lambda>x. if P x then f x else g x) \<in> measurable M N \<longleftrightarrow>
  2319     (f \<in> measurable (restrict_space M {x. P x}) N \<and> g \<in> measurable (restrict_space M {x. \<not> P x}) N)"
  2320   by (subst measurable_piecewise_restrict_iff[where C="{{x. P x}, {x. \<not> P x}}"])
  2321      (auto simp: Int_def sets.sets_Collect_neg space_restrict_space conj_commute[of _ "x \<in> space M" for x]
  2322            cong: measurable_cong')
  2324 lemma measurable_If:
  2325   "f \<in> measurable M M' \<Longrightarrow> g \<in> measurable M M' \<Longrightarrow> {x\<in>space M. P x} \<in> sets M \<Longrightarrow>
  2326     (\<lambda>x. if P x then f x else g x) \<in> measurable M M'"
  2327   unfolding measurable_If_restrict_space_iff by (auto intro: measurable_restrict_space1)
  2329 lemma measurable_If_set:
  2330   assumes measure: "f \<in> measurable M M'" "g \<in> measurable M M'"
  2331   assumes P: "A \<inter> space M \<in> sets M"
  2332   shows "(\<lambda>x. if x \<in> A then f x else g x) \<in> measurable M M'"
  2333 proof (rule measurable_If[OF measure])
  2334   have "{x \<in> space M. x \<in> A} = A \<inter> space M" by auto
  2335   thus "{x \<in> space M. x \<in> A} \<in> sets M" using \<open>A \<inter> space M \<in> sets M\<close> by auto
  2336 qed
  2338 lemma measurable_restrict_space_iff:
  2339   "\<Omega> \<inter> space M \<in> sets M \<Longrightarrow> c \<in> space N \<Longrightarrow>
  2340     f \<in> measurable (restrict_space M \<Omega>) N \<longleftrightarrow> (\<lambda>x. if x \<in> \<Omega> then f x else c) \<in> measurable M N"
  2341   by (subst measurable_If_restrict_space_iff)
  2342      (simp_all add: Int_def conj_commute measurable_const)
  2344 lemma restrict_space_singleton: "{x} \<in> sets M \<Longrightarrow> sets (restrict_space M {x}) = sets (count_space {x})"
  2345   using sets_restrict_space_iff[of "{x}" M]
  2346   by (auto simp add: sets_restrict_space_iff dest!: subset_singletonD)
  2348 lemma measurable_restrict_countable:
  2349   assumes X[intro]: "countable X"
  2350   assumes sets[simp]: "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"
  2351   assumes space[simp]: "\<And>x. x \<in> X \<Longrightarrow> f x \<in> space N"
  2352   assumes f: "f \<in> measurable (restrict_space M (- X)) N"
  2353   shows "f \<in> measurable M N"
  2354   using f sets.countable[OF sets X]
  2355   by (intro measurable_piecewise_restrict[where M=M and C="{- X} \<union> ((\<lambda>x. {x}) ` X)"])
  2356      (auto simp: Diff_Int_distrib2 Compl_eq_Diff_UNIV Int_insert_left sets.Diff restrict_space_singleton
  2357            simp del: sets_count_space  cong: measurable_cong_sets)
  2359 lemma measurable_discrete_difference:
  2360   assumes f: "f \<in> measurable M N"
  2361   assumes X: "countable X" "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M" "\<And>x. x \<in> X \<Longrightarrow> g x \<in> space N"
  2362   assumes eq: "\<And>x. x \<in> space M \<Longrightarrow> x \<notin> X \<Longrightarrow> f x = g x"
  2363   shows "g \<in> measurable M N"
  2364   by (rule measurable_restrict_countable[OF X])
  2365      (auto simp: eq[symmetric] space_restrict_space cong: measurable_cong' intro: f measurable_restrict_space1)
  2367 end