(*  Title:      HOL/Probability/Sigma_Algebra.thy

     Author:     Stefan Richter, Markus Wenzel, TU München

     Author:     Johannes Hölzl, TU München

     Plus material from the Hurd/Coble measure theory development,

     translated by Lawrence Paulson.

 *)

 header {* Sigma Algebras *}

 theory Sigma_Algebra

 imports

   Complex_Main

   "~~/src/HOL/Library/Countable"

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

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

 begin

 text {* Sigma algebras are an elementary concept in measure

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

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

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

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

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

   three very natural and desirable properties. *}

 subsection {* Algebras *}

 record 'a algebra =

   space :: "'a set"

   sets :: "'a set set"

 locale subset_class =

   fixes M :: "('a, 'b) algebra_scheme"

   assumes space_closed: "sets M \<subseteq> Pow (space M)"

 lemma (in subset_class) sets_into_space: "x \<in> sets M \<Longrightarrow> x \<subseteq> space M"

   by (metis PowD contra_subsetD space_closed)

 locale ring_of_sets = subset_class +

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

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

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

 lemma (in ring_of_sets) Int [intro]:

   assumes a: "a \<in> sets M" and b: "b \<in> sets M" shows "a \<inter> b \<in> sets M"

 proof -

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

     by auto

   then show "a \<inter> b \<in> sets M"

     using a b by auto

 qed

 lemma (in ring_of_sets) finite_Union [intro]:

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

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

 lemma (in ring_of_sets) finite_UN[intro]:

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

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

   using assms by induct auto

 lemma (in ring_of_sets) finite_INT[intro]:

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

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

   using assms by (induct rule: finite_ne_induct) auto

 lemma (in ring_of_sets) insert_in_sets:

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

 proof -

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

   thus ?thesis by auto

 qed

 lemma (in ring_of_sets) Int_space_eq1 [simp]: "x \<in> sets M \<Longrightarrow> space M \<inter> x = x"

   by (metis Int_absorb1 sets_into_space)

 lemma (in ring_of_sets) Int_space_eq2 [simp]: "x \<in> sets M \<Longrightarrow> x \<inter> space M = x"

   by (metis Int_absorb2 sets_into_space)

 lemma (in ring_of_sets) sets_Collect_conj:

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

   shows "{x\<in>space M. Q x \<and> P x} \<in> sets M"

 proof -

   have "{x\<in>space M. Q x \<and> P x} = {x\<in>space M. Q x} \<inter> {x\<in>space M. P x}"

     by auto

   with assms show ?thesis by auto

 qed

 lemma (in ring_of_sets) sets_Collect_disj:

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

   shows "{x\<in>space M. Q x \<or> P x} \<in> sets M"

 proof -

   have "{x\<in>space M. Q x \<or> P x} = {x\<in>space M. Q x} \<union> {x\<in>space M. P x}"

     by auto

   with assms show ?thesis by auto

 qed

 lemma (in ring_of_sets) sets_Collect_finite_All:

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

   shows "{x\<in>space M. \<forall>i\<in>S. P i x} \<in> sets M"

 proof -

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

     using `S \<noteq> {}` by auto

   with assms show ?thesis by auto

 qed

 lemma (in ring_of_sets) sets_Collect_finite_Ex:

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

   shows "{x\<in>space M. \<exists>i\<in>S. P i x} \<in> sets M"

 proof -

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

     by auto

   with assms show ?thesis by auto

 qed

 locale algebra = ring_of_sets +

   assumes top [iff]: "space M \<in> sets M"

 lemma (in algebra) compl_sets [intro]:

   "a \<in> sets M \<Longrightarrow> space M - a \<in> sets M"

   by auto

 lemma algebra_iff_Un:

   "algebra M \<longleftrightarrow>

     sets M \<subseteq> Pow (space M) &

     {} \<in> sets M &

     (\<forall>a \<in> sets M. space M - a \<in> sets M) &

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

 proof

   assume "algebra M"

   then interpret algebra M .

   show ?Un using sets_into_space by auto

 next

   assume ?Un

   show "algebra M"

   proof

     show space: "sets M \<subseteq> Pow (space M)" "{} \<in> sets M" "space M \<in> sets M"

       using `?Un` by auto

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

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

     have "a - b = space M - ((space M - a) \<union> b)"

       using space a b by auto

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

       using a b  `?Un` by auto

   qed

 qed

 lemma algebra_iff_Int:

      "algebra M \<longleftrightarrow>

        sets M \<subseteq> Pow (space M) & {} \<in> sets M &

        (\<forall>a \<in> sets M. space M - a \<in> sets M) &

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

 proof

   assume "algebra M"

   then interpret algebra M .

   show ?Int using sets_into_space by auto

 next

   assume ?Int

   show "algebra M"

   proof (unfold algebra_iff_Un, intro conjI ballI)

     show space: "sets M \<subseteq> Pow (space M)" "{} \<in> sets M"

       using `?Int` by auto

     from `?Int` show "\<And>a. a \<in> sets M \<Longrightarrow> space M - a \<in> sets M" by auto

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

     hence "a \<union> b = space M - ((space M - a) \<inter> (space M - b))"

       using space by blast

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

       using sets `?Int` by auto

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

   qed

 qed

 lemma (in algebra) sets_Collect_neg:

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

   shows "{x\<in>space M. \<not> P x} \<in> sets M"

 proof -

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

   with assms show ?thesis by auto

 qed

 lemma (in algebra) sets_Collect_imp:

   "{x\<in>space M. P x} \<in> sets M \<Longrightarrow> {x\<in>space M. Q x} \<in> sets M \<Longrightarrow> {x\<in>space M. Q x \<longrightarrow> P x} \<in> sets M"

   unfolding imp_conv_disj by (intro sets_Collect_disj sets_Collect_neg)

 lemma (in algebra) sets_Collect_const:

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

   by (cases P) auto

 lemma algebra_single_set:

   assumes "X \<subseteq> S"

   shows "algebra \<lparr> space = S, sets = { {}, X, S - X, S }\<rparr>"

   by default (insert `X \<subseteq> S`, auto)

 section {* Restricted algebras *}

 abbreviation (in algebra)

   "restricted_space A \<equiv> \<lparr> space = A, sets = (\<lambda>S. (A \<inter> S)) ` sets M, \<dots> = more M \<rparr>"

 lemma (in algebra) restricted_algebra:

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

   using assms by unfold_locales auto

 subsection {* Sigma Algebras *}

 locale sigma_algebra = algebra +

   assumes countable_nat_UN [intro]:

          "!!A. range A \<subseteq> sets M \<Longrightarrow> (\<Union>i::nat. A i) \<in> sets M"

 lemma (in algebra) is_sigma_algebra:

   assumes "finite (sets M)"

   shows "sigma_algebra M"

 proof

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

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

     by auto

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

     using `finite (sets M)` by (auto intro: finite_UN)

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

 qed

 lemma countable_UN_eq:

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

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

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

 proof -

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

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

   proof safe

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

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

   next

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

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

   qed

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

     using surj_from_nat

     by (auto simp: image_compose intro!: imageI)

   show ?thesis unfolding * ** ..

 qed

 lemma (in sigma_algebra) countable_UN[intro]:

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

   assumes "A`X \<subseteq> sets M"

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

 proof -

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

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

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

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

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

   ultimately show ?thesis by simp

 qed

 lemma (in sigma_algebra) countable_INT [intro]:

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

   assumes A: "A`X \<subseteq> sets M" "X \<noteq> {}"

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

 proof -

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

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

   moreover

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

     by blast

   ultimately show ?thesis by metis

 qed

 lemma ring_of_sets_Pow:

  "ring_of_sets \<lparr> space = sp, sets = Pow sp, \<dots> = X \<rparr>"

   by default auto

 lemma algebra_Pow:

   "algebra \<lparr> space = sp, sets = Pow sp, \<dots> = X \<rparr>"

   by default auto

 lemma sigma_algebra_Pow:

      "sigma_algebra \<lparr> space = sp, sets = Pow sp, \<dots> = X \<rparr>"

   by default auto

 lemma sigma_algebra_iff:

      "sigma_algebra M \<longleftrightarrow>

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

   by (simp add: sigma_algebra_def sigma_algebra_axioms_def)

 lemma (in sigma_algebra) sets_Collect_countable_All:

   assumes "\<And>i. {x\<in>space M. P i x} \<in> sets M"

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

 proof -

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

   with assms show ?thesis by auto

 qed

 lemma (in sigma_algebra) sets_Collect_countable_Ex:

   assumes "\<And>i. {x\<in>space M. P i x} \<in> sets M"

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

 proof -

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

   with assms show ?thesis by auto

 qed

 lemmas (in sigma_algebra) sets_Collect =

   sets_Collect_imp sets_Collect_disj sets_Collect_conj sets_Collect_neg sets_Collect_const

   sets_Collect_countable_All sets_Collect_countable_Ex sets_Collect_countable_All

 lemma sigma_algebra_single_set:

   assumes "X \<subseteq> S"

   shows "sigma_algebra \<lparr> space = S, sets = { {}, X, S - X, S }\<rparr>"

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

 subsection {* Binary Unions *}

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

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

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

   by (auto simp add: binary_def)

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

   by (simp add: UNION_eq_Union_image range_binary_eq)

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

   by (simp add: INTER_eq_Inter_image range_binary_eq)

 lemma sigma_algebra_iff2:

      "sigma_algebra M \<longleftrightarrow>

        sets M \<subseteq> Pow (space M) \<and>

        {} \<in> sets M \<and> (\<forall>s \<in> sets M. space M - s \<in> sets M) \<and>

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

   by (auto simp add: range_binary_eq sigma_algebra_def sigma_algebra_axioms_def

          algebra_iff_Un Un_range_binary)

 subsection {* Initial Sigma Algebra *}

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

   sets with regard to the properties just postulated.  *}

 inductive_set

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

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

   where

     Basic: "a \<in> A \<Longrightarrow> a \<in> sigma_sets sp A"

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

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

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

 definition

   "sigma M = \<lparr> space = space M, sets = sigma_sets (space M) (sets M), \<dots> = more M \<rparr>"

 lemma (in sigma_algebra) sigma_sets_subset:

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

   shows "sigma_sets (space M) a \<subseteq> sets M"

 proof

   fix x

   assume "x \<in> sigma_sets (space M) a"

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

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

 qed

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

   by (erule sigma_sets.induct, auto)

 lemma sigma_algebra_sigma_sets:

      "a \<subseteq> Pow (space M) \<Longrightarrow> sets M = sigma_sets (space M) a \<Longrightarrow> sigma_algebra M"

   by (auto simp add: sigma_algebra_iff2 dest: sigma_sets_into_sp

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

 lemma sigma_sets_least_sigma_algebra:

   assumes "A \<subseteq> Pow S"

   shows "sigma_sets S A = \<Inter>{B. A \<subseteq> B \<and> sigma_algebra \<lparr>space = S, sets = B\<rparr>}"

 proof safe

   fix B X assume "A \<subseteq> B" and sa: "sigma_algebra \<lparr> space = S, sets = B \<rparr>"

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

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

   show "X \<in> B" by auto

 next

   fix X assume "X \<in> \<Inter>{B. A \<subseteq> B \<and> sigma_algebra \<lparr>space = S, sets = B\<rparr>}"

   then have [intro!]: "\<And>B. A \<subseteq> B \<Longrightarrow> sigma_algebra \<lparr>space = S, sets = B\<rparr> \<Longrightarrow> X \<in> B"

      by simp

   have "A \<subseteq> sigma_sets S A" using assms

     by (auto intro!: sigma_sets.Basic)

   moreover have "sigma_algebra \<lparr>space = S, sets = sigma_sets S A\<rparr>"

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

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

 qed

 lemma sets_sigma: "sets (sigma M) = sigma_sets (space M) (sets M)"

   unfolding sigma_def by simp

 lemma space_sigma [simp]: "space (sigma M) = space M"

   by (simp add: sigma_def)

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

   by (metis Diff_empty sigma_sets.Compl sigma_sets.Empty)

 lemma sigma_sets_Un:

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

 apply (simp add: Un_range_binary range_binary_eq)

 apply (rule Union, simp add: binary_def)

 done

 lemma sigma_sets_Inter:

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

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

 proof -

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

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

     by (rule sigma_sets.Compl)

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

     by (rule sigma_sets.Union)

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

     by (rule sigma_sets.Compl)

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

     by auto

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

     by (blast dest: sigma_sets_into_sp [OF Asb])

   finally show ?thesis .

 qed

 lemma sigma_sets_INTER:

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

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

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

 proof -

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

     by (simp add: sigma_sets.intros sigma_sets_top)

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

     by (rule sigma_sets_Inter [OF Asb])

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

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

   finally show ?thesis .

 qed

 lemma (in sigma_algebra) sigma_sets_eq:

      "sigma_sets (space M) (sets M) = sets M"

 proof

   show "sets M \<subseteq> sigma_sets (space M) (sets M)"

     by (metis Set.subsetI sigma_sets.Basic)

   next

   show "sigma_sets (space M) (sets M) \<subseteq> sets M"

     by (metis sigma_sets_subset subset_refl)

 qed

 lemma sigma_sets_eqI:

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

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

   shows "sigma_sets M A = sigma_sets M B"

 proof (intro set_eqI iffI)

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

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

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

 next

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

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

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

 qed

 lemma sigma_algebra_sigma:

     "sets M \<subseteq> Pow (space M) \<Longrightarrow> sigma_algebra (sigma M)"

   apply (rule sigma_algebra_sigma_sets)

   apply (auto simp add: sigma_def)

   done

 lemma (in sigma_algebra) sigma_subset:

     "sets N \<subseteq> sets M \<Longrightarrow> space N = space M \<Longrightarrow> sets (sigma N) \<subseteq> (sets M)"

   by (simp add: sigma_def sigma_sets_subset)

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

 proof

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

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

 qed

 lemma (in sigma_algebra) restriction_in_sets:

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

   assumes "S \<in> sets M"

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

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

 proof -

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

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

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

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

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

 qed

 lemma (in sigma_algebra) restricted_sigma_algebra:

   assumes "S \<in> sets M"

   shows "sigma_algebra (restricted_space S)"

   unfolding sigma_algebra_def sigma_algebra_axioms_def

 proof safe

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

 next

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

   from restriction_in_sets[OF assms this[simplified]]

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

 qed

 lemma sigma_sets_Int:

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

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

 proof (intro equalityI subsetI)

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

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

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

   proof (induct arbitrary: x)

     case (Compl a)

     then show ?case

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

   next

     case (Union a)

     then show ?case

       by (auto intro!: sigma_sets.Union

                simp add: UN_extend_simps simp del: UN_simps)

   qed (auto intro!: sigma_sets.intros)

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

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

 next

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

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

   proof induct

     case (Compl a)

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

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

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

   next

     case (Union a)

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

       by (auto simp: image_iff Bex_def)

     from choice[OF this] guess f ..

     then show ?case

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

                simp add: image_iff)

   qed (auto intro!: sigma_sets.intros)

 qed

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

 proof (intro set_eqI iffI)

   fix x assume "x \<in> sigma_sets {X} {{X}}"

   from sigma_sets_into_sp[OF _ this]

   show "x \<in> {{}, {X}}" by auto

 next

   fix x assume "x \<in> {{}, {X}}"

   then show "x \<in> sigma_sets {X} {{X}}"

     by (auto intro: sigma_sets.Empty sigma_sets_top)

 qed

 lemma (in sigma_algebra) sets_sigma_subset:

   assumes "space N = space M"

   assumes "sets N \<subseteq> sets M"

   shows "sets (sigma N) \<subseteq> sets M"

   by (unfold assms sets_sigma, rule sigma_sets_subset, rule assms)

 lemma in_sigma[intro, simp]: "A \<in> sets M \<Longrightarrow> A \<in> sets (sigma M)"

   unfolding sigma_def by (auto intro!: sigma_sets.Basic)

 lemma (in sigma_algebra) sigma_eq[simp]: "sigma M = M"

   unfolding sigma_def sigma_sets_eq by simp

 lemma sigma_sets_singleton:

   assumes "X \<subseteq> S"

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

 proof -

   interpret sigma_algebra "\<lparr> space = S, sets = { {}, X, S - X, S }\<rparr>"

     by (rule sigma_algebra_single_set) fact

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

     by (rule sigma_sets_subseteq) simp

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

     using sigma_eq unfolding sigma_def by simp

   moreover

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

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

       by (auto intro: sigma_sets.intros sigma_sets_top) }

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

     by (intro antisym) auto

   with sigma_eq show ?thesis

     unfolding sigma_def by simp

 qed

 lemma restricted_sigma:

   assumes S: "S \<in> sets (sigma M)" and M: "sets M \<subseteq> Pow (space M)"

   shows "algebra.restricted_space (sigma M) S = sigma (algebra.restricted_space M S)"

 proof -

   from S sigma_sets_into_sp[OF M]

   have "S \<in> sigma_sets (space M) (sets M)" "S \<subseteq> space M"

     by (auto simp: sigma_def)

   from sigma_sets_Int[OF this]

   show ?thesis

     by (simp add: sigma_def)

 qed

 section {* Measurable functions *}

 definition

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

 lemma (in sigma_algebra) measurable_sigma:

   assumes B: "sets N \<subseteq> Pow (space N)"

       and f: "f \<in> space M -> space N"

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

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

 proof -

   have "sigma_sets (space N) (sets N) \<subseteq> {y . (f -` y) \<inter> space M \<in> sets M & y \<subseteq> space N}"

     proof clarify

       fix x

       assume "x \<in> sigma_sets (space N) (sets N)"

       thus "f -` x \<inter> space M \<in> sets M \<and> x \<subseteq> space N"

         proof induct

           case (Basic a)

           thus ?case

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

         next

           case Empty

           thus ?case

             by auto

         next

           case (Compl a)

           have [simp]: "f -` space N \<inter> space M = space M"

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

           thus ?case

             by (auto simp add: vimage_Diff Diff_Int_distrib2 compl_sets Compl)

         next

           case (Union a)

           thus ?case

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

         qed

     qed

   thus ?thesis

     by (simp add: measurable_def sigma_algebra_axioms sigma_algebra_sigma B f)

        (auto simp add: sigma_def)

 qed

 lemma measurable_cong:

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

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

   unfolding measurable_def using assms

   by (simp cong: vimage_inter_cong Pi_cong)

 lemma measurable_space:

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

    unfolding measurable_def by auto

 lemma measurable_sets:

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

    unfolding measurable_def by auto

 lemma (in sigma_algebra) measurable_subset:

      "(\<And>S. S \<in> sets A \<Longrightarrow> S \<subseteq> space A) \<Longrightarrow> measurable M A \<subseteq> measurable M (sigma A)"

   by (auto intro: measurable_sigma measurable_sets measurable_space)

 lemma measurable_eqI:

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

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

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

   by (simp add: measurable_def sigma_algebra_iff2)

 lemma (in sigma_algebra) measurable_const[intro, simp]:

   assumes "c \<in> space M'"

   shows "(\<lambda>x. c) \<in> measurable M M'"

   using assms by (auto simp add: measurable_def)

 lemma (in sigma_algebra) measurable_If:

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

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

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

   unfolding measurable_def

 proof safe

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

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

     using measure unfolding measurable_def by auto

 next

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

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

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

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

     using measure unfolding measurable_def by (auto split: split_if_asm)

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

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

     by (auto intro!: Un)

 qed

 lemma (in sigma_algebra) measurable_If_set:

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

   assumes P: "A \<in> sets M"

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

 proof (rule measurable_If[OF measure])

   have "{x \<in> space M. x \<in> A} = A" using `A \<in> sets M` sets_into_space by auto

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

 qed

 lemma (in ring_of_sets) measurable_ident[intro, simp]: "id \<in> measurable M M"

   by (auto simp add: measurable_def)

 lemma measurable_comp[intro]:

   fixes f :: "'a \<Rightarrow> 'b" and g :: "'b \<Rightarrow> 'c"

   shows "f \<in> measurable a b \<Longrightarrow> g \<in> measurable b c \<Longrightarrow> (g o f) \<in> measurable a c"

   apply (auto simp add: measurable_def vimage_compose)

   apply (subgoal_tac "f -` g -` y \<inter> space a = f -` (g -` y \<inter> space b) \<inter> space a")

   apply force+

   done

 lemma measurable_strong:

   fixes f :: "'a \<Rightarrow> 'b" and g :: "'b \<Rightarrow> 'c"

   assumes f: "f \<in> measurable a b" and g: "g \<in> (space b -> space c)"

       and a: "sigma_algebra a" and b: "sigma_algebra b" and c: "sigma_algebra c"

       and t: "f ` (space a) \<subseteq> t"

       and cb: "\<And>s. s \<in> sets c \<Longrightarrow> (g -` s) \<inter> t \<in> sets b"

   shows "(g o f) \<in> measurable a c"

 proof -

   have fab: "f \<in> (space a -> space b)"

    and ba: "\<And>y. y \<in> sets b \<Longrightarrow> (f -` y) \<inter> (space a) \<in> sets a" using f

      by (auto simp add: measurable_def)

   have eq: "f -` g -` y \<inter> space a = f -` (g -` y \<inter> t) \<inter> space a" using t

     by force

   show ?thesis

     apply (auto simp add: measurable_def vimage_compose a c)

     apply (metis funcset_mem fab g)

     apply (subst eq, metis ba cb)

     done

 qed

 lemma measurable_mono1:

      "a \<subseteq> b \<Longrightarrow> sigma_algebra \<lparr>space = X, sets = b\<rparr>

       \<Longrightarrow> measurable \<lparr>space = X, sets = a\<rparr> c \<subseteq> measurable \<lparr>space = X, sets = b\<rparr> c"

   by (auto simp add: measurable_def)

 lemma measurable_up_sigma:

   "measurable A M \<subseteq> measurable (sigma A) M"

   unfolding measurable_def

   by (auto simp: sigma_def intro: sigma_sets.Basic)

 lemma (in sigma_algebra) measurable_range_reduce:

    "\<lbrakk> f \<in> measurable M \<lparr>space = s, sets = Pow s\<rparr> ; s \<noteq> {} \<rbrakk>

     \<Longrightarrow> f \<in> measurable M \<lparr>space = s \<inter> (f ` space M), sets = Pow (s \<inter> (f ` space M))\<rparr>"

   by (simp add: measurable_def sigma_algebra_Pow del: Pow_Int_eq) blast

 lemma (in sigma_algebra) measurable_Pow_to_Pow:

    "(sets M = Pow (space M)) \<Longrightarrow> f \<in> measurable M \<lparr>space = UNIV, sets = Pow UNIV\<rparr>"

   by (auto simp add: measurable_def sigma_algebra_def sigma_algebra_axioms_def algebra_def)

 lemma (in sigma_algebra) measurable_Pow_to_Pow_image:

    "sets M = Pow (space M)

     \<Longrightarrow> f \<in> measurable M \<lparr>space = f ` space M, sets = Pow (f ` space M)\<rparr>"

   by (simp add: measurable_def sigma_algebra_Pow) intro_locales

 lemma (in sigma_algebra) measurable_iff_sigma:

   assumes "sets E \<subseteq> Pow (space E)" and "f \<in> space M \<rightarrow> space E"

   shows "f \<in> measurable M (sigma E) \<longleftrightarrow> (\<forall>A\<in>sets E. f -` A \<inter> space M \<in> sets M)"

   using measurable_sigma[OF assms]

   by (fastsimp simp: measurable_def sets_sigma intro: sigma_sets.intros)

 section "Disjoint families"

 definition

   disjoint_family_on  where

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

 abbreviation

   "disjoint_family A \<equiv> disjoint_family_on A UNIV"

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

   by blast

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

   by blast

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

   by blast

 lemma disjoint_family_subset:

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

   by (force simp add: disjoint_family_on_def)

 lemma disjoint_family_on_bisimulation:

   assumes "disjoint_family_on f S"

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

   shows "disjoint_family_on g S"

   using assms unfolding disjoint_family_on_def by auto

 lemma disjoint_family_on_mono:

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

   unfolding disjoint_family_on_def by auto

 lemma disjoint_family_Suc:

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

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

 proof -

   {

     fix m

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

     proof (induct m)

       case 0 show ?case by simp

     next

       case (Suc m) thus ?case

         by (metis Suc_eq_plus1 assms nat_add_commute nat_add_left_commute subset_trans)

     qed

   }

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

     by (metis add_commute le_add_diff_inverse nat_less_le)

   thus ?thesis

     by (auto simp add: disjoint_family_on_def)

       (metis insert_absorb insert_subset le_SucE le_antisym not_leE)

 qed

 lemma setsum_indicator_disjoint_family:

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

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

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

 proof -

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

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

     by auto

   thus ?thesis

     unfolding indicator_def

     by (simp add: if_distrib setsum_cases[OF `finite P`])

 qed

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

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

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

 proof (induct n)

   case 0 show ?case by simp

 next

   case (Suc n)

   thus ?case by (simp add: atLeastLessThanSuc disjointed_def)

 qed

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

   apply (rule UN_finite2_eq [where k=0])

   apply (simp add: finite_UN_disjointed_eq)

   done

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

   by (auto simp add: disjointed_def)

 lemma disjoint_family_disjointed: "disjoint_family (disjointed A)"

   by (simp add: disjoint_family_on_def)

      (metis neq_iff Int_commute less_disjoint_disjointed)

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

   by (auto simp add: disjointed_def)

 lemma (in ring_of_sets) UNION_in_sets:

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

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

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

 proof (induct n)

   case 0 show ?case by simp

 next

   case (Suc n)

   thus ?case

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

 qed

 lemma (in ring_of_sets) range_disjointed_sets:

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

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

 proof (auto simp add: disjointed_def)

   fix n

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

     by (metis A Diff UNIV_I image_subset_iff)

 qed

 lemma (in algebra) range_disjointed_sets':

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

   using range_disjointed_sets .

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

   by (simp add: disjointed_def)

 lemma incseq_Un:

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

   unfolding incseq_def by auto

 lemma disjointed_incseq:

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

   using incseq_Un[of A]

   by (simp add: disjointed_def atLeastLessThanSuc_atLeastAtMost atLeast0AtMost)

 lemma sigma_algebra_disjoint_iff:

      "sigma_algebra M \<longleftrightarrow>

       algebra M &

       (\<forall>A. range A \<subseteq> sets M \<longrightarrow> disjoint_family A \<longrightarrow>

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

 proof (auto simp add: sigma_algebra_iff)

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

   assume M: "algebra M"

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

      and UnA: "\<forall>A. range A \<subseteq> sets M \<longrightarrow>

                disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> sets M"

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

          disjoint_family (disjointed A) \<longrightarrow>

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

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

     by (simp add: algebra.range_disjointed_sets' M A disjoint_family_disjointed)

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

 qed

 subsection {* Sigma algebra generated by function preimages *}

 definition (in sigma_algebra)

   "vimage_algebra S f = \<lparr> space = S, sets = (\<lambda>A. f -` A \<inter> S) ` sets M, \<dots> = more M \<rparr>"

 lemma (in sigma_algebra) in_vimage_algebra[simp]:

   "A \<in> sets (vimage_algebra S f) \<longleftrightarrow> (\<exists>B\<in>sets M. A = f -` B \<inter> S)"

   by (simp add: vimage_algebra_def image_iff)

 lemma (in sigma_algebra) space_vimage_algebra[simp]:

   "space (vimage_algebra S f) = S"

   by (simp add: vimage_algebra_def)

 lemma (in sigma_algebra) sigma_algebra_preimages:

   fixes f :: "'x \<Rightarrow> 'a"

   assumes "f \<in> A \<rightarrow> space M"

   shows "sigma_algebra \<lparr> space = A, sets = (\<lambda>M. f -` M \<inter> A) ` sets M \<rparr>"

     (is "sigma_algebra \<lparr> space = _, sets = ?F ` sets M \<rparr>")

 proof (simp add: sigma_algebra_iff2, safe)

   show "{} \<in> ?F ` sets M" by blast

 next

   fix S assume "S \<in> sets M"

   moreover have "A - ?F S = ?F (space M - S)"

     using assms by auto

   ultimately show "A - ?F S \<in> ?F ` sets M"

     by blast

 next

   fix S :: "nat \<Rightarrow> 'x set" assume *: "range S \<subseteq> ?F ` sets M"

   have "\<forall>i. \<exists>b. b \<in> sets M \<and> S i = ?F b"

   proof safe

     fix i

     have "S i \<in> ?F ` sets M" using * by auto

     then show "\<exists>b. b \<in> sets M \<and> S i = ?F b" by auto

   qed

   from choice[OF this] obtain b where b: "range b \<subseteq> sets M" "\<And>i. S i = ?F (b i)"

     by auto

   then have "(\<Union>i. S i) = ?F (\<Union>i. b i)" by auto

   then show "(\<Union>i. S i) \<in> ?F ` sets M" using b(1) by blast

 qed

 lemma (in sigma_algebra) sigma_algebra_vimage:

   fixes S :: "'c set" assumes "f \<in> S \<rightarrow> space M"

   shows "sigma_algebra (vimage_algebra S f)"

 proof -

   from sigma_algebra_preimages[OF assms]

   show ?thesis unfolding vimage_algebra_def by (auto simp: sigma_algebra_iff2)

 qed

 lemma (in sigma_algebra) measurable_vimage_algebra:

   fixes S :: "'c set" assumes "f \<in> S \<rightarrow> space M"

   shows "f \<in> measurable (vimage_algebra S f) M"

     unfolding measurable_def using assms by force

 lemma (in sigma_algebra) measurable_vimage:

   fixes g :: "'a \<Rightarrow> 'c" and f :: "'d \<Rightarrow> 'a"

   assumes "g \<in> measurable M M2" "f \<in> S \<rightarrow> space M"

   shows "(\<lambda>x. g (f x)) \<in> measurable (vimage_algebra S f) M2"

 proof -

   note measurable_vimage_algebra[OF assms(2)]

   from measurable_comp[OF this assms(1)]

   show ?thesis by (simp add: comp_def)

 qed

 lemma sigma_sets_vimage:

   assumes "f \<in> S' \<rightarrow> S" and "A \<subseteq> Pow S"

   shows "sigma_sets S' ((\<lambda>X. f -` X \<inter> S') ` A) = (\<lambda>X. f -` X \<inter> S') ` sigma_sets S A"

 proof (intro set_eqI iffI)

   let ?F = "\<lambda>X. f -` X \<inter> S'"

   fix X assume "X \<in> sigma_sets S' (?F ` A)"

   then show "X \<in> ?F ` sigma_sets S A"

   proof induct

     case (Basic X) then obtain X' where "X = ?F X'" "X' \<in> A"

       by auto

     then show ?case by (auto intro!: sigma_sets.Basic)

   next

     case Empty then show ?case

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

   next

     case (Compl X) then obtain X' where X: "X = ?F X'" and "X' \<in> sigma_sets S A"

       by auto

     then have "S - X' \<in> sigma_sets S A"

       by (auto intro!: sigma_sets.Compl)

     then show ?case

       using X assms by (auto intro!: image_eqI[where x="S - X'"])

   next

     case (Union F)

     then have "\<forall>i. \<exists>F'.  F' \<in> sigma_sets S A \<and> F i = f -` F' \<inter> S'"

       by (auto simp: image_iff Bex_def)

     from choice[OF this] obtain F' where

       "\<And>i. F' i \<in> sigma_sets S A" and "\<And>i. F i = f -` F' i \<inter> S'"

       by auto

     then show ?case

       by (auto intro!: sigma_sets.Union image_eqI[where x="\<Union>i. F' i"])

   qed

 next

   let ?F = "\<lambda>X. f -` X \<inter> S'"

   fix X assume "X \<in> ?F ` sigma_sets S A"

   then obtain X' where "X' \<in> sigma_sets S A" "X = ?F X'" by auto

   then show "X \<in> sigma_sets S' (?F ` A)"

   proof (induct arbitrary: X)

     case (Basic X') then show ?case by (auto intro: sigma_sets.Basic)

   next

     case Empty then show ?case by (auto intro: sigma_sets.Empty)

   next

     case (Compl X')

     have "S' - (S' - X) \<in> sigma_sets S' (?F ` A)"

       apply (rule sigma_sets.Compl)

       using assms by (auto intro!: Compl.hyps simp: Compl.prems)

     also have "S' - (S' - X) = X"

       using assms Compl by auto

     finally show ?case .

   next

     case (Union F)

     have "(\<Union>i. f -` F i \<inter> S') \<in> sigma_sets S' (?F ` A)"

       by (intro sigma_sets.Union Union.hyps) simp

     also have "(\<Union>i. f -` F i \<inter> S') = X"

       using assms Union by auto

     finally show ?case .

   qed

 qed

 section {* Conditional space *}

 definition (in algebra)

   "image_space X = \<lparr> space = X`space M, sets = (\<lambda>S. X`S) ` sets M, \<dots> = more M \<rparr>"

 definition (in algebra)

   "conditional_space X A = algebra.image_space (restricted_space A) X"

 lemma (in algebra) space_conditional_space:

   assumes "A \<in> sets M" shows "space (conditional_space X A) = X`A"

 proof -

   interpret r: algebra "restricted_space A" using assms by (rule restricted_algebra)

   show ?thesis unfolding conditional_space_def r.image_space_def

     by simp

 qed

 subsection {* A Two-Element Series *}

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

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

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

   apply (simp add: binaryset_def)

   apply (rule set_eqI)

   apply (auto simp add: image_iff)

   done

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

   by (simp add: UNION_eq_Union_image range_binaryset_eq)

 section {* Closed CDI *}

 definition

   closed_cdi  where

   "closed_cdi M \<longleftrightarrow>

    sets M \<subseteq> Pow (space M) &

    (\<forall>s \<in> sets M. space M - s \<in> sets M) &

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

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

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

 inductive_set

   smallest_ccdi_sets :: "('a, 'b) algebra_scheme \<Rightarrow> 'a set set"

   for M

   where

     Basic [intro]:

       "a \<in> sets M \<Longrightarrow> a \<in> smallest_ccdi_sets M"

   | Compl [intro]:

       "a \<in> smallest_ccdi_sets M \<Longrightarrow> space M - a \<in> smallest_ccdi_sets M"

   | Inc:

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

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

   | Disj:

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

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

   monos Pow_mono

 definition

   smallest_closed_cdi  where

   "smallest_closed_cdi M = (|space = space M, sets = smallest_ccdi_sets M|)"

 lemma space_smallest_closed_cdi [simp]:

      "space (smallest_closed_cdi M) = space M"

   by (simp add: smallest_closed_cdi_def)

 lemma (in algebra) smallest_closed_cdi1: "sets M \<subseteq> sets (smallest_closed_cdi M)"

   by (auto simp add: smallest_closed_cdi_def)

 lemma (in algebra) smallest_ccdi_sets:

      "smallest_ccdi_sets M \<subseteq> Pow (space M)"

   apply (rule subsetI)

   apply (erule smallest_ccdi_sets.induct)

   apply (auto intro: range_subsetD dest: sets_into_space)

   done

 lemma (in algebra) smallest_closed_cdi2: "closed_cdi (smallest_closed_cdi M)"

   apply (auto simp add: closed_cdi_def smallest_closed_cdi_def smallest_ccdi_sets)

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

   done

 lemma (in algebra) smallest_closed_cdi3:

      "sets (smallest_closed_cdi M) \<subseteq> Pow (space M)"

   by (simp add: smallest_closed_cdi_def smallest_ccdi_sets)

 lemma closed_cdi_subset: "closed_cdi M \<Longrightarrow> sets M \<subseteq> Pow (space M)"

   by (simp add: closed_cdi_def)

 lemma closed_cdi_Compl: "closed_cdi M \<Longrightarrow> s \<in> sets M \<Longrightarrow> space M - s \<in> sets M"

   by (simp add: closed_cdi_def)

 lemma closed_cdi_Inc:

      "closed_cdi M \<Longrightarrow> range A \<subseteq> sets M \<Longrightarrow> A 0 = {} \<Longrightarrow> (!!n. A n \<subseteq> A (Suc n)) \<Longrightarrow>

         (\<Union>i. A i) \<in> sets M"

   by (simp add: closed_cdi_def)

 lemma closed_cdi_Disj:

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

   by (simp add: closed_cdi_def)

 lemma closed_cdi_Un:

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

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

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

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

 proof -

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

    by (simp add: range_binaryset_eq empty A B)

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

    by (simp add: disjoint_family_on_def binaryset_def Int_commute)

  from closed_cdi_Disj [OF cdi ra di]

  show ?thesis

    by (simp add: UN_binaryset_eq)

 qed

 lemma (in algebra) smallest_ccdi_sets_Un:

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

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

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

 proof -

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

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

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

     by (simp add: disjoint_family_on_def binaryset_def Int_commute)

   from Disj [OF ra di]

   show ?thesis

     by (simp add: UN_binaryset_eq)

 qed

 lemma (in algebra) smallest_ccdi_sets_Int1:

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

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

 proof (induct rule: smallest_ccdi_sets.induct)

   case (Basic x)

   thus ?case

     by (metis a Int smallest_ccdi_sets.Basic)

 next

   case (Compl x)

   have "a \<inter> (space M - x) = space M - ((space M - a) \<union> (a \<inter> x))"

     by blast

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

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

            Diff_disjoint Int_Diff Int_empty_right Un_commute

            smallest_ccdi_sets.Basic smallest_ccdi_sets.Compl

            smallest_ccdi_sets_Un)

   finally show ?case .

 next

   case (Inc A)

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

     by blast

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

     by blast

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

     by (simp add: Inc)

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

     by blast

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

     by (rule smallest_ccdi_sets.Inc)

   show ?case

     by (metis 1 2)

 next

   case (Disj A)

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

     by blast

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

     by blast

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

     by (auto simp add: disjoint_family_on_def)

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

     by (rule smallest_ccdi_sets.Disj)

   show ?case

     by (metis 1 2)

 qed

 lemma (in algebra) smallest_ccdi_sets_Int:

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

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

 proof (induct rule: smallest_ccdi_sets.induct)

   case (Basic x)

   thus ?case

     by (metis b smallest_ccdi_sets_Int1)

 next

   case (Compl x)

   have "(space M - x) \<inter> b = space M - (x \<inter> b \<union> (space M - b))"

     by blast

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

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

            smallest_ccdi_sets.Compl smallest_ccdi_sets_Un)

   finally show ?case .

 next

   case (Inc A)

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

     by blast

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

     by blast

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

     by (simp add: Inc)

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

     by blast

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

     by (rule smallest_ccdi_sets.Inc)

   show ?case

     by (metis 1 2)

 next

   case (Disj A)

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

     by blast

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

     by blast

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

     by (auto simp add: disjoint_family_on_def)

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

     by (rule smallest_ccdi_sets.Disj)

   show ?case

     by (metis 1 2)

 qed

 lemma (in algebra) sets_smallest_closed_cdi_Int:

    "a \<in> sets (smallest_closed_cdi M) \<Longrightarrow> b \<in> sets (smallest_closed_cdi M)

     \<Longrightarrow> a \<inter> b \<in> sets (smallest_closed_cdi M)"

   by (simp add: smallest_ccdi_sets_Int smallest_closed_cdi_def)

 lemma (in algebra) sigma_property_disjoint_lemma:

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

       and ccdi: "closed_cdi (|space = space M, sets = C|)"

   shows "sigma_sets (space M) (sets M) \<subseteq> C"

 proof -

   have "smallest_ccdi_sets M \<in> {B . sets M \<subseteq> B \<and> sigma_algebra (|space = space M, sets = B|)}"

     apply (auto simp add: sigma_algebra_disjoint_iff algebra_iff_Int

             smallest_ccdi_sets_Int)

     apply (metis Union_Pow_eq Union_upper subsetD smallest_ccdi_sets)

     apply (blast intro: smallest_ccdi_sets.Disj)

     done

   hence "sigma_sets (space M) (sets M) \<subseteq> smallest_ccdi_sets M"

     by clarsimp

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

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

     proof

       fix x

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

       thus "x \<in> C"

         proof (induct rule: smallest_ccdi_sets.induct)

           case (Basic x)

           thus ?case

             by (metis Basic subsetD sbC)

         next

           case (Compl x)

           thus ?case

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

         next

           case (Inc A)

           thus ?case

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

         next

           case (Disj A)

           thus ?case

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

         qed

     qed

   finally show ?thesis .

 qed

 lemma (in algebra) sigma_property_disjoint:

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

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

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

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

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

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

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

   shows "sigma_sets (space M) (sets M) \<subseteq> C"

 proof -

   have "sigma_sets (space M) (sets M) \<subseteq> C \<inter> sigma_sets (space M) (sets M)"

     proof (rule sigma_property_disjoint_lemma)

       show "sets M \<subseteq> C \<inter> sigma_sets (space M) (sets M)"

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

     next

       show "closed_cdi \<lparr>space = space M, sets = C \<inter> sigma_sets (space M) (sets M)\<rparr>"

         by (simp add: closed_cdi_def compl inc disj)

            (metis PowI Set.subsetI le_infI2 sigma_sets_into_sp space_closed

              IntE sigma_sets.Compl range_subsetD sigma_sets.Union)

     qed

   thus ?thesis

     by blast

 qed

 section {* Dynkin systems *}

 locale dynkin_system = subset_class +

   assumes space: "space M \<in> sets M"

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

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

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

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

   using space compl[of "space M"] by simp

 lemma (in dynkin_system) diff:

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

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

 proof -

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

   have "range ?f = {D, space M - E, {}}"

     by (auto simp: image_iff)

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

     by (auto simp: image_iff split: split_if_asm)

   moreover

   then have "disjoint_family ?f" unfolding disjoint_family_on_def

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

   ultimately have "space M - (D \<union> (space M - E)) \<in> sets M"

     using sets by auto

   also have "space M - (D \<union> (space M - E)) = E - D"

     using assms sets_into_space by auto

   finally show ?thesis .

 qed

 lemma dynkin_systemI:

   assumes "\<And> A. A \<in> sets M \<Longrightarrow> A \<subseteq> space M" "space M \<in> sets M"

   assumes "\<And> A. A \<in> sets M \<Longrightarrow> space M - A \<in> sets M"

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

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

   shows "dynkin_system M"

   using assms by (auto simp: dynkin_system_def dynkin_system_axioms_def subset_class_def)

 lemma dynkin_system_trivial:

   shows "dynkin_system \<lparr> space = A, sets = Pow A \<rparr>"

   by (rule dynkin_systemI) auto

 lemma sigma_algebra_imp_dynkin_system:

   assumes "sigma_algebra M" shows "dynkin_system M"

 proof -

   interpret sigma_algebra M by fact

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

 qed

 subsection "Intersection stable algebras"

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

 lemma (in algebra) Int_stable: "Int_stable M"

   unfolding Int_stable_def by auto

 lemma Int_stableI:

   "(\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<inter> b \<in> A) \<Longrightarrow> Int_stable \<lparr> space = \<Omega>, sets = A \<rparr>"

   unfolding Int_stable_def by auto

 lemma Int_stableD:

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

   unfolding Int_stable_def by auto

 lemma (in dynkin_system) sigma_algebra_eq_Int_stable:

   "sigma_algebra M \<longleftrightarrow> Int_stable M"

 proof

   assume "sigma_algebra M" then show "Int_stable M"

     unfolding sigma_algebra_def using algebra.Int_stable by auto

 next

   assume "Int_stable M"

   show "sigma_algebra M"

     unfolding sigma_algebra_disjoint_iff algebra_iff_Un

   proof (intro conjI ballI allI impI)

     show "sets M \<subseteq> Pow (space M)" using sets_into_space by auto

   next

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

     then have "A \<union> B = space M - ((space M - A) \<inter> (space M - B))"

               "space M - A \<in> sets M" "space M - B \<in> sets M"

       using sets_into_space by auto

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

       using `Int_stable M` unfolding Int_stable_def by auto

   qed auto

 qed

 subsection "Smallest Dynkin systems"

 definition dynkin where

   "dynkin M = \<lparr> space = space M,

      sets = \<Inter>{D. dynkin_system \<lparr> space = space M, sets = D \<rparr> \<and> sets M \<subseteq> D},

      \<dots> = more M \<rparr>"

 lemma dynkin_system_dynkin:

   assumes "sets M \<subseteq> Pow (space M)"

   shows "dynkin_system (dynkin M)"

 proof (rule dynkin_systemI)

   fix A assume "A \<in> sets (dynkin M)"

   moreover

   { fix D assume "A \<in> D" and d: "dynkin_system \<lparr> space = space M, sets = D \<rparr>"

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

   moreover have "{D. dynkin_system \<lparr> space = space M, sets = D\<rparr> \<and> sets M \<subseteq> D} \<noteq> {}"

     using assms dynkin_system_trivial by fastsimp

   ultimately show "A \<subseteq> space (dynkin M)"

     unfolding dynkin_def using assms

     by simp (metis dynkin_system_def subset_class_def in_mono mem_def)

 next

   show "space (dynkin M) \<in> sets (dynkin M)"

     unfolding dynkin_def using dynkin_system.space by fastsimp

 next

   fix A assume "A \<in> sets (dynkin M)"

   then show "space (dynkin M) - A \<in> sets (dynkin M)"

     unfolding dynkin_def using dynkin_system.compl by force

 next

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

   assume A: "disjoint_family A" "range A \<subseteq> sets (dynkin M)"

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

   proof (simp, safe)

     fix D assume "dynkin_system \<lparr>space = space M, sets = D\<rparr>" "sets M \<subseteq> D"

     with A have "(\<Union>i. A i) \<in> sets \<lparr>space = space M, sets = D\<rparr>"

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

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

   qed

 qed

 lemma dynkin_Basic[intro]:

   "A \<in> sets M \<Longrightarrow> A \<in> sets (dynkin M)"

   unfolding dynkin_def by auto

 lemma dynkin_space[simp]:

   "space (dynkin M) = space M"

   unfolding dynkin_def by auto

 lemma (in dynkin_system) restricted_dynkin_system:

   assumes "D \<in> sets M"

   shows "dynkin_system \<lparr> space = space M,

                          sets = {Q. Q \<subseteq> space M \<and> Q \<inter> D \<in> sets M} \<rparr>"

 proof (rule dynkin_systemI, simp_all)

   have "space M \<inter> D = D"

     using `D \<in> sets M` sets_into_space by auto

   then show "space M \<inter> D \<in> sets M"

     using `D \<in> sets M` by auto

 next

   fix A assume "A \<subseteq> space M \<and> A \<inter> D \<in> sets M"

   moreover have "(space M - A) \<inter> D = (space M - (A \<inter> D)) - (space M - D)"

     by auto

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

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

 next

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

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

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

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

     by ((fastsimp simp: disjoint_family_on_def)+)

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

     by (auto simp del: UN_simps)

 qed

 lemma (in dynkin_system) dynkin_subset:

   assumes "sets N \<subseteq> sets M"

   assumes "space N = space M"

   shows "sets (dynkin N) \<subseteq> sets M"

 proof -

   have "dynkin_system M" by default

   then have "dynkin_system \<lparr>space = space N, sets = sets M \<rparr>"

     using assms unfolding dynkin_system_def dynkin_system_axioms_def subset_class_def by simp

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

 qed

 lemma sigma_eq_dynkin:

   assumes sets: "sets M \<subseteq> Pow (space M)"

   assumes "Int_stable M"

   shows "sigma M = dynkin M"

 proof -

   have "sets (dynkin M) \<subseteq> sigma_sets (space M) (sets M)"

     using sigma_algebra_imp_dynkin_system

     unfolding dynkin_def sigma_def sigma_sets_least_sigma_algebra[OF sets] by auto

   moreover

   interpret dynkin_system "dynkin M"

     using dynkin_system_dynkin[OF sets] .

   have "sigma_algebra (dynkin M)"

     unfolding sigma_algebra_eq_Int_stable Int_stable_def

   proof (intro ballI)

     fix A B assume "A \<in> sets (dynkin M)" "B \<in> sets (dynkin M)"

     let "?D E" = "\<lparr> space = space M,

                     sets = {Q. Q \<subseteq> space M \<and> Q \<inter> E \<in> sets (dynkin M)} \<rparr>"

     have "sets M \<subseteq> sets (?D B)"

     proof

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

       then have "sets M \<subseteq> sets (?D E)" "E \<in> sets (dynkin M)"

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

       then have "sets (dynkin M) \<subseteq> sets (?D E)"

         using restricted_dynkin_system `E \<in> sets (dynkin M)`

         by (intro dynkin_system.dynkin_subset) simp_all

       then have "B \<in> sets (?D E)"

         using `B \<in> sets (dynkin M)` by auto

       then have "E \<inter> B \<in> sets (dynkin M)"

         by (subst Int_commute) simp

       then show "E \<in> sets (?D B)"

         using sets `E \<in> sets M` by auto

     qed

     then have "sets (dynkin M) \<subseteq> sets (?D B)"

       using restricted_dynkin_system `B \<in> sets (dynkin M)`

       by (intro dynkin_system.dynkin_subset) simp_all

     then show "A \<inter> B \<in> sets (dynkin M)"

       using `A \<in> sets (dynkin M)` sets_into_space by auto

   qed

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

   have "sigma_sets (space M) (sets M) \<subseteq> sets (dynkin M)" by auto

   ultimately have "sigma_sets (space M) (sets M) = sets (dynkin M)" by auto

   then show ?thesis

     by (auto intro!: algebra.equality simp: sigma_def dynkin_def)

 qed

 lemma (in dynkin_system) dynkin_idem:

   "dynkin M = M"

 proof -

   have "sets (dynkin M) = sets M"

   proof

     show "sets M \<subseteq> sets (dynkin M)"

       using dynkin_Basic by auto

     show "sets (dynkin M) \<subseteq> sets M"

       by (intro dynkin_subset) auto

   qed

   then show ?thesis

     by (auto intro!: algebra.equality simp: dynkin_def)

 qed

 lemma (in dynkin_system) dynkin_lemma:

   assumes "Int_stable E"

   and E: "sets E \<subseteq> sets M" "space E = space M" "sets M \<subseteq> sets (sigma E)"

   shows "sets (sigma E) = sets M"

 proof -

   have "sets E \<subseteq> Pow (space E)"

     using E sets_into_space by force

   then have "sigma E = dynkin E"

     using `Int_stable E` by (rule sigma_eq_dynkin)

   moreover then have "sets (dynkin E) = sets M"

     using assms dynkin_subset[OF E(1,2)] by simp

   ultimately show ?thesis

     using assms by (auto intro!: algebra.equality simp: dynkin_def)

 qed

 subsection "Sigma algebras on finite sets"

 locale finite_sigma_algebra = sigma_algebra +

   assumes finite_space: "finite (space M)"

   and sets_eq_Pow[simp]: "sets M = Pow (space M)"

 lemma (in finite_sigma_algebra) sets_image_space_eq_Pow:

   "sets (image_space X) = Pow (space (image_space X))"

 proof safe

   fix x S assume "S \<in> sets (image_space X)" "x \<in> S"

   then show "x \<in> space (image_space X)"

     using sets_into_space by (auto intro!: imageI simp: image_space_def)

 next

   fix S assume "S \<subseteq> space (image_space X)"

   then obtain S' where "S = X`S'" "S'\<in>sets M"

     by (auto simp: subset_image_iff sets_eq_Pow image_space_def)

   then show "S \<in> sets (image_space X)"

     by (auto simp: image_space_def)

 qed

 lemma measurable_sigma_sigma:

   assumes M: "sets M \<subseteq> Pow (space M)" and N: "sets N \<subseteq> Pow (space N)"

   shows "f \<in> measurable M N \<Longrightarrow> f \<in> measurable (sigma M) (sigma N)"

   using sigma_algebra.measurable_subset[OF sigma_algebra_sigma[OF M], of N]

   using measurable_up_sigma[of M N] N by auto

 lemma (in sigma_algebra) measurable_Least:

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

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

 proof -

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

     proof cases

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

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

         {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}))"

         by (simp add: set_eq_iff, safe)

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

       with meas show ?thesis

         by (auto intro!: Int)

     next

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

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

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

       proof (simp add: set_eq_iff, safe)

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

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

           by (rule ccontr) (insert neq, auto)

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

       qed (auto dest: Least_le intro!: Least_equality)

       with meas show ?thesis

         by (auto intro!: Int)

     qed }

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

     by (intro countable_UN) auto

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

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

   ultimately show ?thesis by auto

 qed

 end