
wenzelm@41983: (*  Title:      HOL/Probability/Sigma_Algebra.thy
hoelzl@42067:     Author:     Stefan Richter, Markus Wenzel, TU München
hoelzl@42067:     Author:     Johannes Hölzl, TU München
hoelzl@41981:     Plus material from the Hurd/Coble measure theory development,
hoelzl@41981:     translated by Lawrence Paulson.
paulson@33271: *)
paulson@33271: 
hoelzl@56994: header {* Describing measurable sets *}
paulson@33271: 
wenzelm@41413: theory Sigma_Algebra
wenzelm@41413: imports
hoelzl@42145:   Complex_Main
immler@50245:   "~~/src/HOL/Library/Countable_Set"
wenzelm@41413:   "~~/src/HOL/Library/FuncSet"
wenzelm@41413:   "~~/src/HOL/Library/Indicator_Function"
hoelzl@47694:   "~~/src/HOL/Library/Extended_Real"
wenzelm@41413: begin
paulson@33271: 
paulson@33271: text {* Sigma algebras are an elementary concept in measure
paulson@33271:   theory. To measure --- that is to integrate --- functions, we first have
paulson@33271:   to measure sets. Unfortunately, when dealing with a large universe,
paulson@33271:   it is often not possible to consistently assign a measure to every
paulson@33271:   subset. Therefore it is necessary to define the set of measurable
paulson@33271:   subsets of the universe. A sigma algebra is such a set that has
paulson@33271:   three very natural and desirable properties. *}
paulson@33271: 
hoelzl@47762: subsection {* Families of sets *}
paulson@33271: 
hoelzl@47694: locale subset_class =
hoelzl@47694:   fixes \<Omega> :: "'a set" and M :: "'a set set"
hoelzl@47694:   assumes space_closed: "M \<subseteq> Pow \<Omega>"
paulson@33271: 
hoelzl@47694: lemma (in subset_class) sets_into_space: "x \<in> M \<Longrightarrow> x \<subseteq> \<Omega>"
paulson@33271:   by (metis PowD contra_subsetD space_closed)
paulson@33271: 
hoelzl@56994: subsubsection {* Semiring of sets *}
hoelzl@47762: 
hoelzl@47762: definition "disjoint A \<longleftrightarrow> (\<forall>a\<in>A. \<forall>b\<in>A. a \<noteq> b \<longrightarrow> a \<inter> b = {})"
hoelzl@47762: 
hoelzl@47762: lemma disjointI:
hoelzl@47762:   "(\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> a \<inter> b = {}) \<Longrightarrow> disjoint A"
hoelzl@47762:   unfolding disjoint_def by auto
hoelzl@47762: 
hoelzl@47762: lemma disjointD:
hoelzl@47762:   "disjoint A \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> a \<inter> b = {}"
hoelzl@47762:   unfolding disjoint_def by auto
hoelzl@47762: 
hoelzl@47762: lemma disjoint_empty[iff]: "disjoint {}"
hoelzl@47762:   by (auto simp: disjoint_def)
hoelzl@42065: 
hoelzl@47762: lemma disjoint_union: 
hoelzl@47762:   assumes C: "disjoint C" and B: "disjoint B" and disj: "\<Union>C \<inter> \<Union>B = {}"
hoelzl@47762:   shows "disjoint (C \<union> B)"
hoelzl@47762: proof (rule disjointI)
hoelzl@47762:   fix c d assume sets: "c \<in> C \<union> B" "d \<in> C \<union> B" and "c \<noteq> d"
hoelzl@47762:   show "c \<inter> d = {}"
hoelzl@47762:   proof cases
hoelzl@47762:     assume "(c \<in> C \<and> d \<in> C) \<or> (c \<in> B \<and> d \<in> B)"
hoelzl@47762:     then show ?thesis
hoelzl@47762:     proof 
hoelzl@47762:       assume "c \<in> C \<and> d \<in> C" with `c \<noteq> d` C show "c \<inter> d = {}"
hoelzl@47762:         by (auto simp: disjoint_def)
hoelzl@47762:     next
hoelzl@47762:       assume "c \<in> B \<and> d \<in> B" with `c \<noteq> d` B show "c \<inter> d = {}"
hoelzl@47762:         by (auto simp: disjoint_def)
hoelzl@47762:     qed
hoelzl@47762:   next
hoelzl@47762:     assume "\<not> ((c \<in> C \<and> d \<in> C) \<or> (c \<in> B \<and> d \<in> B))"
hoelzl@47762:     with sets have "(c \<subseteq> \<Union>C \<and> d \<subseteq> \<Union>B) \<or> (c \<subseteq> \<Union>B \<and> d \<subseteq> \<Union>C)"
hoelzl@47762:       by auto
hoelzl@47762:     with disj show "c \<inter> d = {}" by auto
hoelzl@47762:   qed
hoelzl@47762: qed
hoelzl@47762: 
Andreas@53816: lemma disjoint_singleton [simp]: "disjoint {A}"
Andreas@53816: by(simp add: disjoint_def)
Andreas@53816: 
hoelzl@47762: locale semiring_of_sets = subset_class +
hoelzl@47762:   assumes empty_sets[iff]: "{} \<in> M"
hoelzl@47762:   assumes Int[intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<inter> b \<in> M"
hoelzl@47762:   assumes Diff_cover:
hoelzl@47762:     "\<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"
hoelzl@47762: 
hoelzl@47762: lemma (in semiring_of_sets) finite_INT[intro]:
hoelzl@47762:   assumes "finite I" "I \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M"
hoelzl@47762:   shows "(\<Inter>i\<in>I. A i) \<in> M"
hoelzl@47762:   using assms by (induct rule: finite_ne_induct) auto
hoelzl@47762: 
hoelzl@47762: lemma (in semiring_of_sets) Int_space_eq1 [simp]: "x \<in> M \<Longrightarrow> \<Omega> \<inter> x = x"
hoelzl@47762:   by (metis Int_absorb1 sets_into_space)
hoelzl@47762: 
hoelzl@47762: lemma (in semiring_of_sets) Int_space_eq2 [simp]: "x \<in> M \<Longrightarrow> x \<inter> \<Omega> = x"
hoelzl@47762:   by (metis Int_absorb2 sets_into_space)
hoelzl@47762: 
hoelzl@47762: lemma (in semiring_of_sets) sets_Collect_conj:
hoelzl@47762:   assumes "{x\<in>\<Omega>. P x} \<in> M" "{x\<in>\<Omega>. Q x} \<in> M"
hoelzl@47762:   shows "{x\<in>\<Omega>. Q x \<and> P x} \<in> M"
paulson@33271: proof -
hoelzl@47762:   have "{x\<in>\<Omega>. Q x \<and> P x} = {x\<in>\<Omega>. Q x} \<inter> {x\<in>\<Omega>. P x}"
hoelzl@42065:     by auto
hoelzl@47762:   with assms show ?thesis by auto
paulson@33271: qed
paulson@33271: 
hoelzl@50002: lemma (in semiring_of_sets) sets_Collect_finite_All':
hoelzl@47762:   assumes "\<And>i. i \<in> S \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M" "finite S" "S \<noteq> {}"
hoelzl@47762:   shows "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} \<in> M"
hoelzl@47762: proof -
hoelzl@47762:   have "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} = (\<Inter>i\<in>S. {x\<in>\<Omega>. P i x})"
hoelzl@47762:     using `S \<noteq> {}` by auto
hoelzl@47762:   with assms show ?thesis by auto
hoelzl@47762: qed
hoelzl@47762: 
hoelzl@47762: locale ring_of_sets = semiring_of_sets +
hoelzl@47762:   assumes Un [intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<union> b \<in> M"
hoelzl@47762: 
hoelzl@42065: lemma (in ring_of_sets) finite_Union [intro]:
hoelzl@47694:   "finite X \<Longrightarrow> X \<subseteq> M \<Longrightarrow> Union X \<in> M"
hoelzl@38656:   by (induct set: finite) (auto simp add: Un)
paulson@33271: 
hoelzl@42065: lemma (in ring_of_sets) finite_UN[intro]:
hoelzl@47694:   assumes "finite I" and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M"
hoelzl@47694:   shows "(\<Union>i\<in>I. A i) \<in> M"
hoelzl@41981:   using assms by induct auto
hoelzl@41981: 
hoelzl@47762: lemma (in ring_of_sets) Diff [intro]:
hoelzl@47762:   assumes "a \<in> M" "b \<in> M" shows "a - b \<in> M"
hoelzl@47762:   using Diff_cover[OF assms] by auto
hoelzl@47762: 
hoelzl@47762: lemma ring_of_setsI:
hoelzl@47762:   assumes space_closed: "M \<subseteq> Pow \<Omega>"
hoelzl@47762:   assumes empty_sets[iff]: "{} \<in> M"
hoelzl@47762:   assumes Un[intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<union> b \<in> M"
hoelzl@47762:   assumes Diff[intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a - b \<in> M"
hoelzl@47762:   shows "ring_of_sets \<Omega> M"
hoelzl@47762: proof
hoelzl@47762:   fix a b assume ab: "a \<in> M" "b \<in> M"
hoelzl@47762:   from ab show "\<exists>C\<subseteq>M. finite C \<and> disjoint C \<and> a - b = \<Union>C"
hoelzl@47762:     by (intro exI[of _ "{a - b}"]) (auto simp: disjoint_def)
hoelzl@47762:   have "a \<inter> b = a - (a - b)" by auto
hoelzl@47762:   also have "\<dots> \<in> M" using ab by auto
hoelzl@47762:   finally show "a \<inter> b \<in> M" .
hoelzl@47762: qed fact+
hoelzl@47762: 
hoelzl@47762: 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)"
hoelzl@47762: proof
hoelzl@47762:   assume "ring_of_sets \<Omega> M"
hoelzl@47762:   then interpret ring_of_sets \<Omega> M .
hoelzl@47762:   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)"
hoelzl@47762:     using space_closed by auto
hoelzl@47762: qed (auto intro!: ring_of_setsI)
hoelzl@41981: 
hoelzl@42065: lemma (in ring_of_sets) insert_in_sets:
hoelzl@47694:   assumes "{x} \<in> M" "A \<in> M" shows "insert x A \<in> M"
hoelzl@38656: proof -
hoelzl@47694:   have "{x} \<union> A \<in> M" using assms by (rule Un)
hoelzl@38656:   thus ?thesis by auto
hoelzl@38656: qed
hoelzl@38656: 
hoelzl@42867: lemma (in ring_of_sets) sets_Collect_disj:
hoelzl@47694:   assumes "{x\<in>\<Omega>. P x} \<in> M" "{x\<in>\<Omega>. Q x} \<in> M"
hoelzl@47694:   shows "{x\<in>\<Omega>. Q x \<or> P x} \<in> M"
hoelzl@42867: proof -
hoelzl@47694:   have "{x\<in>\<Omega>. Q x \<or> P x} = {x\<in>\<Omega>. Q x} \<union> {x\<in>\<Omega>. P x}"
hoelzl@42867:     by auto
hoelzl@42867:   with assms show ?thesis by auto
hoelzl@42867: qed
hoelzl@42867: 
hoelzl@42867: lemma (in ring_of_sets) sets_Collect_finite_Ex:
hoelzl@47694:   assumes "\<And>i. i \<in> S \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M" "finite S"
hoelzl@47694:   shows "{x\<in>\<Omega>. \<exists>i\<in>S. P i x} \<in> M"
hoelzl@42867: proof -
hoelzl@47694:   have "{x\<in>\<Omega>. \<exists>i\<in>S. P i x} = (\<Union>i\<in>S. {x\<in>\<Omega>. P i x})"
hoelzl@42867:     by auto
hoelzl@42867:   with assms show ?thesis by auto
hoelzl@42867: qed
hoelzl@42867: 
hoelzl@42065: locale algebra = ring_of_sets +
hoelzl@47694:   assumes top [iff]: "\<Omega> \<in> M"
hoelzl@42065: 
hoelzl@42065: lemma (in algebra) compl_sets [intro]:
hoelzl@47694:   "a \<in> M \<Longrightarrow> \<Omega> - a \<in> M"
hoelzl@42065:   by auto
hoelzl@42065: 
hoelzl@42065: lemma algebra_iff_Un:
hoelzl@47694:   "algebra \<Omega> M \<longleftrightarrow>
hoelzl@47694:     M \<subseteq> Pow \<Omega> \<and>
hoelzl@47694:     {} \<in> M \<and>
hoelzl@47694:     (\<forall>a \<in> M. \<Omega> - a \<in> M) \<and>
hoelzl@47694:     (\<forall>a \<in> M. \<forall> b \<in> M. a \<union> b \<in> M)" (is "_ \<longleftrightarrow> ?Un")
hoelzl@42065: proof
hoelzl@47694:   assume "algebra \<Omega> M"
hoelzl@47694:   then interpret algebra \<Omega> M .
hoelzl@42065:   show ?Un using sets_into_space by auto
hoelzl@42065: next
hoelzl@42065:   assume ?Un
hoelzl@47762:   then have "\<Omega> \<in> M" by auto
hoelzl@47762:   interpret ring_of_sets \<Omega> M
hoelzl@47762:   proof (rule ring_of_setsI)
hoelzl@47762:     show \<Omega>: "M \<subseteq> Pow \<Omega>" "{} \<in> M"
hoelzl@42065:       using `?Un` by auto
hoelzl@47694:     fix a b assume a: "a \<in> M" and b: "b \<in> M"
hoelzl@47694:     then show "a \<union> b \<in> M" using `?Un` by auto
hoelzl@47694:     have "a - b = \<Omega> - ((\<Omega> - a) \<union> b)"
hoelzl@47694:       using \<Omega> a b by auto
hoelzl@47694:     then show "a - b \<in> M"
hoelzl@42065:       using a b  `?Un` by auto
hoelzl@42065:   qed
hoelzl@47762:   show "algebra \<Omega> M" proof qed fact
hoelzl@42065: qed
hoelzl@42065: 
hoelzl@42065: lemma algebra_iff_Int:
hoelzl@47694:      "algebra \<Omega> M \<longleftrightarrow>
hoelzl@47694:        M \<subseteq> Pow \<Omega> & {} \<in> M &
hoelzl@47694:        (\<forall>a \<in> M. \<Omega> - a \<in> M) &
hoelzl@47694:        (\<forall>a \<in> M. \<forall> b \<in> M. a \<inter> b \<in> M)" (is "_ \<longleftrightarrow> ?Int")
hoelzl@42065: proof
hoelzl@47694:   assume "algebra \<Omega> M"
hoelzl@47694:   then interpret algebra \<Omega> M .
hoelzl@42065:   show ?Int using sets_into_space by auto
hoelzl@42065: next
hoelzl@42065:   assume ?Int
hoelzl@47694:   show "algebra \<Omega> M"
hoelzl@42065:   proof (unfold algebra_iff_Un, intro conjI ballI)
hoelzl@47694:     show \<Omega>: "M \<subseteq> Pow \<Omega>" "{} \<in> M"
hoelzl@42065:       using `?Int` by auto
hoelzl@47694:     from `?Int` show "\<And>a. a \<in> M \<Longrightarrow> \<Omega> - a \<in> M" by auto
hoelzl@47694:     fix a b assume M: "a \<in> M" "b \<in> M"
hoelzl@47694:     hence "a \<union> b = \<Omega> - ((\<Omega> - a) \<inter> (\<Omega> - b))"
hoelzl@47694:       using \<Omega> by blast
hoelzl@47694:     also have "... \<in> M"
hoelzl@47694:       using M `?Int` by auto
hoelzl@47694:     finally show "a \<union> b \<in> M" .
hoelzl@42065:   qed
hoelzl@42065: qed
hoelzl@42065: 
hoelzl@42867: lemma (in algebra) sets_Collect_neg:
hoelzl@47694:   assumes "{x\<in>\<Omega>. P x} \<in> M"
hoelzl@47694:   shows "{x\<in>\<Omega>. \<not> P x} \<in> M"
hoelzl@42867: proof -
hoelzl@47694:   have "{x\<in>\<Omega>. \<not> P x} = \<Omega> - {x\<in>\<Omega>. P x}" by auto
hoelzl@42867:   with assms show ?thesis by auto
hoelzl@42867: qed
hoelzl@42867: 
hoelzl@42867: lemma (in algebra) sets_Collect_imp:
hoelzl@47694:   "{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"
hoelzl@42867:   unfolding imp_conv_disj by (intro sets_Collect_disj sets_Collect_neg)
hoelzl@42867: 
hoelzl@42867: lemma (in algebra) sets_Collect_const:
hoelzl@47694:   "{x\<in>\<Omega>. P} \<in> M"
hoelzl@42867:   by (cases P) auto
hoelzl@42867: 
hoelzl@42984: lemma algebra_single_set:
hoelzl@47762:   "X \<subseteq> S \<Longrightarrow> algebra S { {}, X, S - X, S }"
hoelzl@47762:   by (auto simp: algebra_iff_Int)
hoelzl@42984: 
hoelzl@56994: subsubsection {* Restricted algebras *}
hoelzl@39092: 
hoelzl@39092: abbreviation (in algebra)
hoelzl@47694:   "restricted_space A \<equiv> (op \<inter> A) ` M"
hoelzl@39092: 
hoelzl@38656: lemma (in algebra) restricted_algebra:
hoelzl@47694:   assumes "A \<in> M" shows "algebra A (restricted_space A)"
hoelzl@47762:   using assms by (auto simp: algebra_iff_Int)
paulson@33271: 
hoelzl@56994: subsubsection {* Sigma Algebras *}
paulson@33271: 
paulson@33271: locale sigma_algebra = algebra +
hoelzl@47694:   assumes countable_nat_UN [intro]: "\<And>A. range A \<subseteq> M \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
paulson@33271: 
hoelzl@42984: lemma (in algebra) is_sigma_algebra:
hoelzl@47694:   assumes "finite M"
hoelzl@47694:   shows "sigma_algebra \<Omega> M"
hoelzl@42984: proof
hoelzl@47694:   fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> M"
hoelzl@47694:   then have "(\<Union>i. A i) = (\<Union>s\<in>M \<inter> range A. s)"
hoelzl@42984:     by auto
hoelzl@47694:   also have "(\<Union>s\<in>M \<inter> range A. s) \<in> M"
hoelzl@47694:     using `finite M` by auto
hoelzl@47694:   finally show "(\<Union>i. A i) \<in> M" .
hoelzl@42984: qed
hoelzl@42984: 
hoelzl@38656: lemma countable_UN_eq:
hoelzl@38656:   fixes A :: "'i::countable \<Rightarrow> 'a set"
hoelzl@47694:   shows "(range A \<subseteq> M \<longrightarrow> (\<Union>i. A i) \<in> M) \<longleftrightarrow>
hoelzl@47694:     (range (A \<circ> from_nat) \<subseteq> M \<longrightarrow> (\<Union>i. (A \<circ> from_nat) i) \<in> M)"
hoelzl@38656: proof -
hoelzl@38656:   let ?A' = "A \<circ> from_nat"
hoelzl@38656:   have *: "(\<Union>i. ?A' i) = (\<Union>i. A i)" (is "?l = ?r")
hoelzl@38656:   proof safe
hoelzl@38656:     fix x i assume "x \<in> A i" thus "x \<in> ?l"
hoelzl@38656:       by (auto intro!: exI[of _ "to_nat i"])
hoelzl@38656:   next
hoelzl@38656:     fix x i assume "x \<in> ?A' i" thus "x \<in> ?r"
hoelzl@38656:       by (auto intro!: exI[of _ "from_nat i"])
hoelzl@38656:   qed
hoelzl@38656:   have **: "range ?A' = range A"
hoelzl@40702:     using surj_from_nat
haftmann@56154:     by (auto simp: image_comp [symmetric] intro!: imageI)
hoelzl@38656:   show ?thesis unfolding * ** ..
hoelzl@38656: qed
hoelzl@38656: 
immler@50245: lemma (in sigma_algebra) countable_Union [intro]:
immler@50245:   assumes "countable X" "X \<subseteq> M" shows "Union X \<in> M"
immler@50245: proof cases
immler@50245:   assume "X \<noteq> {}"
immler@50245:   hence "\<Union>X = (\<Union>n. from_nat_into X n)"
immler@50245:     using assms by (auto intro: from_nat_into) (metis from_nat_into_surj)
immler@50245:   also have "\<dots> \<in> M" using assms
immler@50245:     by (auto intro!: countable_nat_UN) (metis `X \<noteq> {}` from_nat_into set_mp)
immler@50245:   finally show ?thesis .
immler@50245: qed simp
immler@50245: 
hoelzl@38656: lemma (in sigma_algebra) countable_UN[intro]:
hoelzl@38656:   fixes A :: "'i::countable \<Rightarrow> 'a set"
hoelzl@47694:   assumes "A`X \<subseteq> M"
hoelzl@47694:   shows  "(\<Union>x\<in>X. A x) \<in> M"
hoelzl@38656: proof -
wenzelm@46731:   let ?A = "\<lambda>i. if i \<in> X then A i else {}"
hoelzl@47694:   from assms have "range ?A \<subseteq> M" by auto
hoelzl@38656:   with countable_nat_UN[of "?A \<circ> from_nat"] countable_UN_eq[of ?A M]
hoelzl@47694:   have "(\<Union>x. ?A x) \<in> M" by auto
hoelzl@38656:   moreover have "(\<Union>x. ?A x) = (\<Union>x\<in>X. A x)" by (auto split: split_if_asm)
hoelzl@38656:   ultimately show ?thesis by simp
hoelzl@38656: qed
hoelzl@38656: 
hoelzl@50526: lemma (in sigma_algebra) countable_UN':
hoelzl@50526:   fixes A :: "'i \<Rightarrow> 'a set"
hoelzl@50526:   assumes X: "countable X"
hoelzl@50526:   assumes A: "A`X \<subseteq> M"
hoelzl@50526:   shows  "(\<Union>x\<in>X. A x) \<in> M"
hoelzl@50526: proof -
hoelzl@50526:   have "(\<Union>x\<in>X. A x) = (\<Union>i\<in>to_nat_on X ` X. A (from_nat_into X i))"
hoelzl@50526:     using X by auto
hoelzl@50526:   also have "\<dots> \<in> M"
hoelzl@50526:     using A X
hoelzl@50526:     by (intro countable_UN) auto
hoelzl@50526:   finally show ?thesis .
hoelzl@50526: qed
hoelzl@50526: 
paulson@33533: lemma (in sigma_algebra) countable_INT [intro]:
hoelzl@38656:   fixes A :: "'i::countable \<Rightarrow> 'a set"
hoelzl@47694:   assumes A: "A`X \<subseteq> M" "X \<noteq> {}"
hoelzl@47694:   shows "(\<Inter>i\<in>X. A i) \<in> M"
paulson@33271: proof -
hoelzl@47694:   from A have "\<forall>i\<in>X. A i \<in> M" by fast
hoelzl@47694:   hence "\<Omega> - (\<Union>i\<in>X. \<Omega> - A i) \<in> M" by blast
paulson@33271:   moreover
hoelzl@47694:   have "(\<Inter>i\<in>X. A i) = \<Omega> - (\<Union>i\<in>X. \<Omega> - A i)" using space_closed A
paulson@33271:     by blast
paulson@33271:   ultimately show ?thesis by metis
paulson@33271: qed
paulson@33271: 
hoelzl@50526: lemma (in sigma_algebra) countable_INT':
hoelzl@50526:   fixes A :: "'i \<Rightarrow> 'a set"
hoelzl@50526:   assumes X: "countable X" "X \<noteq> {}"
hoelzl@50526:   assumes A: "A`X \<subseteq> M"
hoelzl@50526:   shows  "(\<Inter>x\<in>X. A x) \<in> M"
hoelzl@50526: proof -
hoelzl@50526:   have "(\<Inter>x\<in>X. A x) = (\<Inter>i\<in>to_nat_on X ` X. A (from_nat_into X i))"
hoelzl@50526:     using X by auto
hoelzl@50526:   also have "\<dots> \<in> M"
hoelzl@50526:     using A X
hoelzl@50526:     by (intro countable_INT) auto
hoelzl@50526:   finally show ?thesis .
hoelzl@50526: qed
hoelzl@50526: 
hoelzl@57275: 
hoelzl@57275: lemma (in sigma_algebra) countable:
hoelzl@57275:   assumes "\<And>a. a \<in> A \<Longrightarrow> {a} \<in> M" "countable A"
hoelzl@57275:   shows "A \<in> M"
hoelzl@57275: proof -
hoelzl@57275:   have "(\<Union>a\<in>A. {a}) \<in> M"
hoelzl@57275:     using assms by (intro countable_UN') auto
hoelzl@57275:   also have "(\<Union>a\<in>A. {a}) = A" by auto
hoelzl@57275:   finally show ?thesis by auto
hoelzl@57275: qed
hoelzl@57275: 
hoelzl@47694: lemma ring_of_sets_Pow: "ring_of_sets sp (Pow sp)"
hoelzl@47762:   by (auto simp: ring_of_sets_iff)
hoelzl@42145: 
hoelzl@47694: lemma algebra_Pow: "algebra sp (Pow sp)"
hoelzl@47762:   by (auto simp: algebra_iff_Un)
hoelzl@38656: 
hoelzl@38656: lemma sigma_algebra_iff:
hoelzl@47694:   "sigma_algebra \<Omega> M \<longleftrightarrow>
hoelzl@47694:     algebra \<Omega> M \<and> (\<forall>A. range A \<subseteq> M \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"
hoelzl@38656:   by (simp add: sigma_algebra_def sigma_algebra_axioms_def)
paulson@33271: 
hoelzl@47762: lemma sigma_algebra_Pow: "sigma_algebra sp (Pow sp)"
hoelzl@47762:   by (auto simp: sigma_algebra_iff algebra_iff_Int)
hoelzl@47762: 
hoelzl@42867: lemma (in sigma_algebra) sets_Collect_countable_All:
hoelzl@47694:   assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
hoelzl@47694:   shows "{x\<in>\<Omega>. \<forall>i::'i::countable. P i x} \<in> M"
hoelzl@42867: proof -
hoelzl@47694:   have "{x\<in>\<Omega>. \<forall>i::'i::countable. P i x} = (\<Inter>i. {x\<in>\<Omega>. P i x})" by auto
hoelzl@42867:   with assms show ?thesis by auto
hoelzl@42867: qed
hoelzl@42867: 
hoelzl@42867: lemma (in sigma_algebra) sets_Collect_countable_Ex:
hoelzl@47694:   assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
hoelzl@47694:   shows "{x\<in>\<Omega>. \<exists>i::'i::countable. P i x} \<in> M"
hoelzl@42867: proof -
hoelzl@47694:   have "{x\<in>\<Omega>. \<exists>i::'i::countable. P i x} = (\<Union>i. {x\<in>\<Omega>. P i x})" by auto
hoelzl@42867:   with assms show ?thesis by auto
hoelzl@42867: qed
hoelzl@42867: 
hoelzl@50526: lemma (in sigma_algebra) sets_Collect_countable_Ex':
hoelzl@54418:   assumes "\<And>i. i \<in> I \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M"
hoelzl@50526:   assumes "countable I"
hoelzl@50526:   shows "{x\<in>\<Omega>. \<exists>i\<in>I. P i x} \<in> M"
hoelzl@50526: proof -
hoelzl@50526:   have "{x\<in>\<Omega>. \<exists>i\<in>I. P i x} = (\<Union>i\<in>I. {x\<in>\<Omega>. P i x})" by auto
hoelzl@50526:   with assms show ?thesis 
hoelzl@50526:     by (auto intro!: countable_UN')
hoelzl@50526: qed
hoelzl@50526: 
hoelzl@54418: lemma (in sigma_algebra) sets_Collect_countable_All':
hoelzl@54418:   assumes "\<And>i. i \<in> I \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M"
hoelzl@54418:   assumes "countable I"
hoelzl@54418:   shows "{x\<in>\<Omega>. \<forall>i\<in>I. P i x} \<in> M"
hoelzl@54418: proof -
hoelzl@54418:   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
hoelzl@54418:   with assms show ?thesis 
hoelzl@54418:     by (cases "I = {}") (auto intro!: countable_INT')
hoelzl@54418: qed
hoelzl@54418: 
hoelzl@54418: lemma (in sigma_algebra) sets_Collect_countable_Ex1':
hoelzl@54418:   assumes "\<And>i. i \<in> I \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M"
hoelzl@54418:   assumes "countable I"
hoelzl@54418:   shows "{x\<in>\<Omega>. \<exists>!i\<in>I. P i x} \<in> M"
hoelzl@54418: proof -
hoelzl@54418:   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)}"
hoelzl@54418:     by auto
hoelzl@54418:   with assms show ?thesis 
hoelzl@54418:     by (auto intro!: sets_Collect_countable_All' sets_Collect_countable_Ex' sets_Collect_conj sets_Collect_imp sets_Collect_const)
hoelzl@54418: qed
hoelzl@54418: 
hoelzl@42867: lemmas (in sigma_algebra) sets_Collect =
hoelzl@42867:   sets_Collect_imp sets_Collect_disj sets_Collect_conj sets_Collect_neg sets_Collect_const
hoelzl@42867:   sets_Collect_countable_All sets_Collect_countable_Ex sets_Collect_countable_All
hoelzl@42867: 
hoelzl@47694: lemma (in sigma_algebra) sets_Collect_countable_Ball:
hoelzl@47694:   assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
hoelzl@47694:   shows "{x\<in>\<Omega>. \<forall>i::'i::countable\<in>X. P i x} \<in> M"
hoelzl@47694:   unfolding Ball_def by (intro sets_Collect assms)
hoelzl@47694: 
hoelzl@47694: lemma (in sigma_algebra) sets_Collect_countable_Bex:
hoelzl@47694:   assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
hoelzl@47694:   shows "{x\<in>\<Omega>. \<exists>i::'i::countable\<in>X. P i x} \<in> M"
hoelzl@47694:   unfolding Bex_def by (intro sets_Collect assms)
hoelzl@47694: 
hoelzl@42984: lemma sigma_algebra_single_set:
hoelzl@42984:   assumes "X \<subseteq> S"
hoelzl@47694:   shows "sigma_algebra S { {}, X, S - X, S }"
hoelzl@42984:   using algebra.is_sigma_algebra[OF algebra_single_set[OF `X \<subseteq> S`]] by simp
hoelzl@42984: 
hoelzl@56994: subsubsection {* Binary Unions *}
paulson@33271: 
paulson@33271: definition binary :: "'a \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"
wenzelm@50252:   where "binary a b =  (\<lambda>x. b)(0 := a)"
paulson@33271: 
hoelzl@38656: lemma range_binary_eq: "range(binary a b) = {a,b}"
hoelzl@38656:   by (auto simp add: binary_def)
paulson@33271: 
hoelzl@38656: lemma Un_range_binary: "a \<union> b = (\<Union>i::nat. binary a b i)"
haftmann@44106:   by (simp add: SUP_def range_binary_eq)
paulson@33271: 
hoelzl@38656: lemma Int_range_binary: "a \<inter> b = (\<Inter>i::nat. binary a b i)"
haftmann@44106:   by (simp add: INF_def range_binary_eq)
paulson@33271: 
paulson@33271: lemma sigma_algebra_iff2:
hoelzl@47694:      "sigma_algebra \<Omega> M \<longleftrightarrow>
hoelzl@47694:        M \<subseteq> Pow \<Omega> \<and>
hoelzl@47694:        {} \<in> M \<and> (\<forall>s \<in> M. \<Omega> - s \<in> M) \<and>
hoelzl@47694:        (\<forall>A. range A \<subseteq> M \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"
hoelzl@38656:   by (auto simp add: range_binary_eq sigma_algebra_def sigma_algebra_axioms_def
hoelzl@42065:          algebra_iff_Un Un_range_binary)
paulson@33271: 
hoelzl@56994: subsubsection {* Initial Sigma Algebra *}
paulson@33271: 
paulson@33271: text {*Sigma algebras can naturally be created as the closure of any set of
hoelzl@47694:   M with regard to the properties just postulated.  *}
paulson@33271: 
hoelzl@51683: inductive_set sigma_sets :: "'a set \<Rightarrow> 'a set set \<Rightarrow> 'a set set"
paulson@33271:   for sp :: "'a set" and A :: "'a set set"
paulson@33271:   where
hoelzl@47694:     Basic[intro, simp]: "a \<in> A \<Longrightarrow> a \<in> sigma_sets sp A"
paulson@33271:   | Empty: "{} \<in> sigma_sets sp A"
paulson@33271:   | Compl: "a \<in> sigma_sets sp A \<Longrightarrow> sp - a \<in> sigma_sets sp A"
paulson@33271:   | Union: "(\<And>i::nat. a i \<in> sigma_sets sp A) \<Longrightarrow> (\<Union>i. a i) \<in> sigma_sets sp A"
paulson@33271: 
hoelzl@41543: lemma (in sigma_algebra) sigma_sets_subset:
hoelzl@47694:   assumes a: "a \<subseteq> M"
hoelzl@47694:   shows "sigma_sets \<Omega> a \<subseteq> M"
hoelzl@41543: proof
hoelzl@41543:   fix x
hoelzl@47694:   assume "x \<in> sigma_sets \<Omega> a"
hoelzl@47694:   from this show "x \<in> M"
hoelzl@41543:     by (induct rule: sigma_sets.induct, auto) (metis a subsetD)
hoelzl@41543: qed
hoelzl@41543: 
hoelzl@41543: lemma sigma_sets_into_sp: "A \<subseteq> Pow sp \<Longrightarrow> x \<in> sigma_sets sp A \<Longrightarrow> x \<subseteq> sp"
hoelzl@41543:   by (erule sigma_sets.induct, auto)
hoelzl@41543: 
hoelzl@41543: lemma sigma_algebra_sigma_sets:
hoelzl@47694:      "a \<subseteq> Pow \<Omega> \<Longrightarrow> sigma_algebra \<Omega> (sigma_sets \<Omega> a)"
hoelzl@41543:   by (auto simp add: sigma_algebra_iff2 dest: sigma_sets_into_sp
hoelzl@41543:            intro!: sigma_sets.Union sigma_sets.Empty sigma_sets.Compl)
hoelzl@41543: 
hoelzl@41543: lemma sigma_sets_least_sigma_algebra:
hoelzl@41543:   assumes "A \<subseteq> Pow S"
hoelzl@47694:   shows "sigma_sets S A = \<Inter>{B. A \<subseteq> B \<and> sigma_algebra S B}"
hoelzl@41543: proof safe
hoelzl@47694:   fix B X assume "A \<subseteq> B" and sa: "sigma_algebra S B"
hoelzl@41543:     and X: "X \<in> sigma_sets S A"
hoelzl@41543:   from sigma_algebra.sigma_sets_subset[OF sa, simplified, OF `A \<subseteq> B`] X
hoelzl@41543:   show "X \<in> B" by auto
hoelzl@41543: next
hoelzl@47694:   fix X assume "X \<in> \<Inter>{B. A \<subseteq> B \<and> sigma_algebra S B}"
hoelzl@47694:   then have [intro!]: "\<And>B. A \<subseteq> B \<Longrightarrow> sigma_algebra S B \<Longrightarrow> X \<in> B"
hoelzl@41543:      by simp
hoelzl@47694:   have "A \<subseteq> sigma_sets S A" using assms by auto
hoelzl@47694:   moreover have "sigma_algebra S (sigma_sets S A)"
hoelzl@41543:     using assms by (intro sigma_algebra_sigma_sets[of A]) auto
hoelzl@41543:   ultimately show "X \<in> sigma_sets S A" by auto
hoelzl@41543: qed
paulson@33271: 
paulson@33271: lemma sigma_sets_top: "sp \<in> sigma_sets sp A"
paulson@33271:   by (metis Diff_empty sigma_sets.Compl sigma_sets.Empty)
paulson@33271: 
hoelzl@38656: lemma sigma_sets_Un:
paulson@33271:   "a \<in> sigma_sets sp A \<Longrightarrow> b \<in> sigma_sets sp A \<Longrightarrow> a \<union> b \<in> sigma_sets sp A"
hoelzl@38656: apply (simp add: Un_range_binary range_binary_eq)
hoelzl@40859: apply (rule Union, simp add: binary_def)
paulson@33271: done
paulson@33271: 
paulson@33271: lemma sigma_sets_Inter:
paulson@33271:   assumes Asb: "A \<subseteq> Pow sp"
paulson@33271:   shows "(\<And>i::nat. a i \<in> sigma_sets sp A) \<Longrightarrow> (\<Inter>i. a i) \<in> sigma_sets sp A"
paulson@33271: proof -
paulson@33271:   assume ai: "\<And>i::nat. a i \<in> sigma_sets sp A"
hoelzl@38656:   hence "\<And>i::nat. sp-(a i) \<in> sigma_sets sp A"
paulson@33271:     by (rule sigma_sets.Compl)
hoelzl@38656:   hence "(\<Union>i. sp-(a i)) \<in> sigma_sets sp A"
paulson@33271:     by (rule sigma_sets.Union)
hoelzl@38656:   hence "sp-(\<Union>i. sp-(a i)) \<in> sigma_sets sp A"
paulson@33271:     by (rule sigma_sets.Compl)
hoelzl@38656:   also have "sp-(\<Union>i. sp-(a i)) = sp Int (\<Inter>i. a i)"
paulson@33271:     by auto
paulson@33271:   also have "... = (\<Inter>i. a i)" using ai
hoelzl@38656:     by (blast dest: sigma_sets_into_sp [OF Asb])
hoelzl@38656:   finally show ?thesis .
paulson@33271: qed
paulson@33271: 
paulson@33271: lemma sigma_sets_INTER:
hoelzl@38656:   assumes Asb: "A \<subseteq> Pow sp"
paulson@33271:       and ai: "\<And>i::nat. i \<in> S \<Longrightarrow> a i \<in> sigma_sets sp A" and non: "S \<noteq> {}"
paulson@33271:   shows "(\<Inter>i\<in>S. a i) \<in> sigma_sets sp A"
paulson@33271: proof -
paulson@33271:   from ai have "\<And>i. (if i\<in>S then a i else sp) \<in> sigma_sets sp A"
hoelzl@47694:     by (simp add: sigma_sets.intros(2-) sigma_sets_top)
paulson@33271:   hence "(\<Inter>i. (if i\<in>S then a i else sp)) \<in> sigma_sets sp A"
paulson@33271:     by (rule sigma_sets_Inter [OF Asb])
paulson@33271:   also have "(\<Inter>i. (if i\<in>S then a i else sp)) = (\<Inter>i\<in>S. a i)"
paulson@33271:     by auto (metis ai non sigma_sets_into_sp subset_empty subset_iff Asb)+
paulson@33271:   finally show ?thesis .
paulson@33271: qed
paulson@33271: 
hoelzl@51683: lemma sigma_sets_UNION: "countable B \<Longrightarrow> (\<And>b. b \<in> B \<Longrightarrow> b \<in> sigma_sets X A) \<Longrightarrow> (\<Union>B) \<in> sigma_sets X A"
hoelzl@51683:   using from_nat_into[of B] range_from_nat_into[of B] sigma_sets.Union[of "from_nat_into B" X A]
hoelzl@51683:   apply (cases "B = {}")
hoelzl@51683:   apply (simp add: sigma_sets.Empty)
hoelzl@51683:   apply (simp del: Union_image_eq add: Union_image_eq[symmetric])
hoelzl@51683:   done
hoelzl@51683: 
paulson@33271: lemma (in sigma_algebra) sigma_sets_eq:
hoelzl@47694:      "sigma_sets \<Omega> M = M"
paulson@33271: proof
hoelzl@47694:   show "M \<subseteq> sigma_sets \<Omega> M"
huffman@37032:     by (metis Set.subsetI sigma_sets.Basic)
paulson@33271:   next
hoelzl@47694:   show "sigma_sets \<Omega> M \<subseteq> M"
paulson@33271:     by (metis sigma_sets_subset subset_refl)
paulson@33271: qed
paulson@33271: 
hoelzl@42981: lemma sigma_sets_eqI:
hoelzl@42981:   assumes A: "\<And>a. a \<in> A \<Longrightarrow> a \<in> sigma_sets M B"
hoelzl@42981:   assumes B: "\<And>b. b \<in> B \<Longrightarrow> b \<in> sigma_sets M A"
hoelzl@42981:   shows "sigma_sets M A = sigma_sets M B"
hoelzl@42981: proof (intro set_eqI iffI)
hoelzl@42981:   fix a assume "a \<in> sigma_sets M A"
hoelzl@42981:   from this A show "a \<in> sigma_sets M B"
hoelzl@47694:     by induct (auto intro!: sigma_sets.intros(2-) del: sigma_sets.Basic)
hoelzl@42981: next
hoelzl@42981:   fix b assume "b \<in> sigma_sets M B"
hoelzl@42981:   from this B show "b \<in> sigma_sets M A"
hoelzl@47694:     by induct (auto intro!: sigma_sets.intros(2-) del: sigma_sets.Basic)
hoelzl@42981: qed
hoelzl@42981: 
hoelzl@42984: lemma sigma_sets_subseteq: assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
hoelzl@42984: proof
hoelzl@42984:   fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
hoelzl@47694:     by induct (insert `A \<subseteq> B`, auto intro: sigma_sets.intros(2-))
hoelzl@42984: qed
hoelzl@42984: 
hoelzl@47762: lemma sigma_sets_mono: assumes "A \<subseteq> sigma_sets X B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
hoelzl@47762: proof
hoelzl@47762:   fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
hoelzl@47762:     by induct (insert `A \<subseteq> sigma_sets X B`, auto intro: sigma_sets.intros(2-))
hoelzl@47762: qed
hoelzl@47762: 
hoelzl@47762: lemma sigma_sets_mono': assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
hoelzl@47762: proof
hoelzl@47762:   fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
hoelzl@47762:     by induct (insert `A \<subseteq> B`, auto intro: sigma_sets.intros(2-))
hoelzl@47762: qed
hoelzl@47762: 
hoelzl@47762: lemma sigma_sets_superset_generator: "A \<subseteq> sigma_sets X A"
hoelzl@47762:   by (auto intro: sigma_sets.Basic)
hoelzl@47762: 
hoelzl@38656: lemma (in sigma_algebra) restriction_in_sets:
hoelzl@38656:   fixes A :: "nat \<Rightarrow> 'a set"
hoelzl@47694:   assumes "S \<in> M"
hoelzl@47694:   and *: "range A \<subseteq> (\<lambda>A. S \<inter> A) ` M" (is "_ \<subseteq> ?r")
hoelzl@47694:   shows "range A \<subseteq> M" "(\<Union>i. A i) \<in> (\<lambda>A. S \<inter> A) ` M"
hoelzl@38656: proof -
hoelzl@38656:   { fix i have "A i \<in> ?r" using * by auto
hoelzl@47694:     hence "\<exists>B. A i = B \<inter> S \<and> B \<in> M" by auto
hoelzl@47694:     hence "A i \<subseteq> S" "A i \<in> M" using `S \<in> M` by auto }
hoelzl@47694:   thus "range A \<subseteq> M" "(\<Union>i. A i) \<in> (\<lambda>A. S \<inter> A) ` M"
hoelzl@38656:     by (auto intro!: image_eqI[of _ _ "(\<Union>i. A i)"])
hoelzl@38656: qed
hoelzl@38656: 
hoelzl@38656: lemma (in sigma_algebra) restricted_sigma_algebra:
hoelzl@47694:   assumes "S \<in> M"
hoelzl@47694:   shows "sigma_algebra S (restricted_space S)"
hoelzl@38656:   unfolding sigma_algebra_def sigma_algebra_axioms_def
hoelzl@38656: proof safe
hoelzl@47694:   show "algebra S (restricted_space S)" using restricted_algebra[OF assms] .
hoelzl@38656: next
hoelzl@47694:   fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> restricted_space S"
hoelzl@38656:   from restriction_in_sets[OF assms this[simplified]]
hoelzl@47694:   show "(\<Union>i. A i) \<in> restricted_space S" by simp
hoelzl@38656: qed
hoelzl@38656: 
hoelzl@40859: lemma sigma_sets_Int:
hoelzl@41689:   assumes "A \<in> sigma_sets sp st" "A \<subseteq> sp"
hoelzl@41689:   shows "op \<inter> A ` sigma_sets sp st = sigma_sets A (op \<inter> A ` st)"
hoelzl@40859: proof (intro equalityI subsetI)
hoelzl@40859:   fix x assume "x \<in> op \<inter> A ` sigma_sets sp st"
hoelzl@40859:   then obtain y where "y \<in> sigma_sets sp st" "x = y \<inter> A" by auto
hoelzl@41689:   then have "x \<in> sigma_sets (A \<inter> sp) (op \<inter> A ` st)"
hoelzl@40859:   proof (induct arbitrary: x)
hoelzl@40859:     case (Compl a)
hoelzl@40859:     then show ?case
hoelzl@40859:       by (force intro!: sigma_sets.Compl simp: Diff_Int_distrib ac_simps)
hoelzl@40859:   next
hoelzl@40859:     case (Union a)
hoelzl@40859:     then show ?case
hoelzl@40859:       by (auto intro!: sigma_sets.Union
hoelzl@40859:                simp add: UN_extend_simps simp del: UN_simps)
hoelzl@47694:   qed (auto intro!: sigma_sets.intros(2-))
hoelzl@41689:   then show "x \<in> sigma_sets A (op \<inter> A ` st)"
hoelzl@41689:     using `A \<subseteq> sp` by (simp add: Int_absorb2)
hoelzl@40859: next
hoelzl@41689:   fix x assume "x \<in> sigma_sets A (op \<inter> A ` st)"
hoelzl@40859:   then show "x \<in> op \<inter> A ` sigma_sets sp st"
hoelzl@40859:   proof induct
hoelzl@40859:     case (Compl a)
hoelzl@40859:     then obtain x where "a = A \<inter> x" "x \<in> sigma_sets sp st" by auto
hoelzl@41689:     then show ?case using `A \<subseteq> sp`
hoelzl@40859:       by (force simp add: image_iff intro!: bexI[of _ "sp - x"] sigma_sets.Compl)
hoelzl@40859:   next
hoelzl@40859:     case (Union a)
hoelzl@40859:     then have "\<forall>i. \<exists>x. x \<in> sigma_sets sp st \<and> a i = A \<inter> x"
hoelzl@40859:       by (auto simp: image_iff Bex_def)
hoelzl@40859:     from choice[OF this] guess f ..
hoelzl@40859:     then show ?case
hoelzl@40859:       by (auto intro!: bexI[of _ "(\<Union>x. f x)"] sigma_sets.Union
hoelzl@40859:                simp add: image_iff)
hoelzl@47694:   qed (auto intro!: sigma_sets.intros(2-))
hoelzl@40859: qed
hoelzl@40859: 
hoelzl@47694: lemma sigma_sets_empty_eq: "sigma_sets A {} = {{}, A}"
hoelzl@40859: proof (intro set_eqI iffI)
hoelzl@47694:   fix a assume "a \<in> sigma_sets A {}" then show "a \<in> {{}, A}"
hoelzl@47694:     by induct blast+
hoelzl@47694: qed (auto intro: sigma_sets.Empty sigma_sets_top)
hoelzl@47694: 
hoelzl@47694: lemma sigma_sets_single[simp]: "sigma_sets A {A} = {{}, A}"
hoelzl@47694: proof (intro set_eqI iffI)
hoelzl@47694:   fix x assume "x \<in> sigma_sets A {A}"
hoelzl@47694:   then show "x \<in> {{}, A}"
hoelzl@47694:     by induct blast+
hoelzl@40859: next
hoelzl@47694:   fix x assume "x \<in> {{}, A}"
hoelzl@47694:   then show "x \<in> sigma_sets A {A}"
hoelzl@40859:     by (auto intro: sigma_sets.Empty sigma_sets_top)
hoelzl@40859: qed
hoelzl@40859: 
hoelzl@42987: lemma sigma_sets_sigma_sets_eq:
hoelzl@42987:   "M \<subseteq> Pow S \<Longrightarrow> sigma_sets S (sigma_sets S M) = sigma_sets S M"
hoelzl@47694:   by (rule sigma_algebra.sigma_sets_eq[OF sigma_algebra_sigma_sets, of M S]) auto
hoelzl@42987: 
hoelzl@42984: lemma sigma_sets_singleton:
hoelzl@42984:   assumes "X \<subseteq> S"
hoelzl@42984:   shows "sigma_sets S { X } = { {}, X, S - X, S }"
hoelzl@42984: proof -
hoelzl@47694:   interpret sigma_algebra S "{ {}, X, S - X, S }"
hoelzl@42984:     by (rule sigma_algebra_single_set) fact
hoelzl@42984:   have "sigma_sets S { X } \<subseteq> sigma_sets S { {}, X, S - X, S }"
hoelzl@42984:     by (rule sigma_sets_subseteq) simp
hoelzl@42984:   moreover have "\<dots> = { {}, X, S - X, S }"
hoelzl@47694:     using sigma_sets_eq by simp
hoelzl@42984:   moreover
hoelzl@42984:   { fix A assume "A \<in> { {}, X, S - X, S }"
hoelzl@42984:     then have "A \<in> sigma_sets S { X }"
hoelzl@47694:       by (auto intro: sigma_sets.intros(2-) sigma_sets_top) }
hoelzl@42984:   ultimately have "sigma_sets S { X } = sigma_sets S { {}, X, S - X, S }"
hoelzl@42984:     by (intro antisym) auto
hoelzl@47694:   with sigma_sets_eq show ?thesis by simp
hoelzl@42984: qed
hoelzl@42984: 
hoelzl@42863: lemma restricted_sigma:
hoelzl@47694:   assumes S: "S \<in> sigma_sets \<Omega> M" and M: "M \<subseteq> Pow \<Omega>"
hoelzl@47694:   shows "algebra.restricted_space (sigma_sets \<Omega> M) S =
hoelzl@47694:     sigma_sets S (algebra.restricted_space M S)"
hoelzl@42863: proof -
hoelzl@42863:   from S sigma_sets_into_sp[OF M]
hoelzl@47694:   have "S \<in> sigma_sets \<Omega> M" "S \<subseteq> \<Omega>" by auto
hoelzl@42863:   from sigma_sets_Int[OF this]
hoelzl@47694:   show ?thesis by simp
hoelzl@42863: qed
hoelzl@42863: 
hoelzl@42987: lemma sigma_sets_vimage_commute:
hoelzl@47694:   assumes X: "X \<in> \<Omega> \<rightarrow> \<Omega>'"
hoelzl@47694:   shows "{X -` A \<inter> \<Omega> |A. A \<in> sigma_sets \<Omega>' M'}
hoelzl@47694:        = sigma_sets \<Omega> {X -` A \<inter> \<Omega> |A. A \<in> M'}" (is "?L = ?R")
hoelzl@42987: proof
hoelzl@42987:   show "?L \<subseteq> ?R"
hoelzl@42987:   proof clarify
hoelzl@47694:     fix A assume "A \<in> sigma_sets \<Omega>' M'"
hoelzl@47694:     then show "X -` A \<inter> \<Omega> \<in> ?R"
hoelzl@42987:     proof induct
hoelzl@42987:       case Empty then show ?case
hoelzl@42987:         by (auto intro!: sigma_sets.Empty)
hoelzl@42987:     next
hoelzl@42987:       case (Compl B)
hoelzl@47694:       have [simp]: "X -` (\<Omega>' - B) \<inter> \<Omega> = \<Omega> - (X -` B \<inter> \<Omega>)"
hoelzl@42987:         by (auto simp add: funcset_mem [OF X])
hoelzl@42987:       with Compl show ?case
hoelzl@42987:         by (auto intro!: sigma_sets.Compl)
hoelzl@42987:     next
hoelzl@42987:       case (Union F)
hoelzl@42987:       then show ?case
hoelzl@42987:         by (auto simp add: vimage_UN UN_extend_simps(4) simp del: UN_simps
hoelzl@42987:                  intro!: sigma_sets.Union)
hoelzl@47694:     qed auto
hoelzl@42987:   qed
hoelzl@42987:   show "?R \<subseteq> ?L"
hoelzl@42987:   proof clarify
hoelzl@42987:     fix A assume "A \<in> ?R"
hoelzl@47694:     then show "\<exists>B. A = X -` B \<inter> \<Omega> \<and> B \<in> sigma_sets \<Omega>' M'"
hoelzl@42987:     proof induct
hoelzl@42987:       case (Basic B) then show ?case by auto
hoelzl@42987:     next
hoelzl@42987:       case Empty then show ?case
hoelzl@47694:         by (auto intro!: sigma_sets.Empty exI[of _ "{}"])
hoelzl@42987:     next
hoelzl@42987:       case (Compl B)
hoelzl@47694:       then obtain A where A: "B = X -` A \<inter> \<Omega>" "A \<in> sigma_sets \<Omega>' M'" by auto
hoelzl@47694:       then have [simp]: "\<Omega> - B = X -` (\<Omega>' - A) \<inter> \<Omega>"
hoelzl@42987:         by (auto simp add: funcset_mem [OF X])
hoelzl@42987:       with A(2) show ?case
hoelzl@47694:         by (auto intro: sigma_sets.Compl)
hoelzl@42987:     next
hoelzl@42987:       case (Union F)
hoelzl@47694:       then have "\<forall>i. \<exists>B. F i = X -` B \<inter> \<Omega> \<and> B \<in> sigma_sets \<Omega>' M'" by auto
hoelzl@42987:       from choice[OF this] guess A .. note A = this
hoelzl@42987:       with A show ?case
hoelzl@47694:         by (auto simp: vimage_UN[symmetric] intro: sigma_sets.Union)
hoelzl@42987:     qed
hoelzl@42987:   qed
hoelzl@42987: qed
hoelzl@42987: 
hoelzl@56994: subsubsection "Disjoint families"
hoelzl@38656: 
hoelzl@38656: definition
hoelzl@38656:   disjoint_family_on  where
hoelzl@38656:   "disjoint_family_on A S \<longleftrightarrow> (\<forall>m\<in>S. \<forall>n\<in>S. m \<noteq> n \<longrightarrow> A m \<inter> A n = {})"
hoelzl@38656: 
hoelzl@38656: abbreviation
hoelzl@38656:   "disjoint_family A \<equiv> disjoint_family_on A UNIV"
hoelzl@38656: 
hoelzl@38656: lemma range_subsetD: "range f \<subseteq> B \<Longrightarrow> f i \<in> B"
hoelzl@38656:   by blast
hoelzl@38656: 
hoelzl@57447: lemma disjoint_family_onD: "disjoint_family_on A I \<Longrightarrow> i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> i \<noteq> j \<Longrightarrow> A i \<inter> A j = {}"
hoelzl@57447:   by (auto simp: disjoint_family_on_def)
hoelzl@57447: 
hoelzl@38656: lemma Int_Diff_disjoint: "A \<inter> B \<inter> (A - B) = {}"
hoelzl@38656:   by blast
hoelzl@38656: 
hoelzl@38656: lemma Int_Diff_Un: "A \<inter> B \<union> (A - B) = A"
hoelzl@38656:   by blast
hoelzl@38656: 
hoelzl@38656: lemma disjoint_family_subset:
hoelzl@38656:      "disjoint_family A \<Longrightarrow> (!!x. B x \<subseteq> A x) \<Longrightarrow> disjoint_family B"
hoelzl@38656:   by (force simp add: disjoint_family_on_def)
hoelzl@38656: 
hoelzl@40859: lemma disjoint_family_on_bisimulation:
hoelzl@40859:   assumes "disjoint_family_on f S"
hoelzl@40859:   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 = {}"
hoelzl@40859:   shows "disjoint_family_on g S"
hoelzl@40859:   using assms unfolding disjoint_family_on_def by auto
hoelzl@40859: 
hoelzl@38656: lemma disjoint_family_on_mono:
hoelzl@38656:   "A \<subseteq> B \<Longrightarrow> disjoint_family_on f B \<Longrightarrow> disjoint_family_on f A"
hoelzl@38656:   unfolding disjoint_family_on_def by auto
hoelzl@38656: 
hoelzl@38656: lemma disjoint_family_Suc:
hoelzl@38656:   assumes Suc: "!!n. A n \<subseteq> A (Suc n)"
hoelzl@38656:   shows "disjoint_family (\<lambda>i. A (Suc i) - A i)"
hoelzl@38656: proof -
hoelzl@38656:   {
hoelzl@38656:     fix m
hoelzl@38656:     have "!!n. A n \<subseteq> A (m+n)"
hoelzl@38656:     proof (induct m)
hoelzl@38656:       case 0 show ?case by simp
hoelzl@38656:     next
hoelzl@38656:       case (Suc m) thus ?case
haftmann@57512:         by (metis Suc_eq_plus1 assms add.commute add.left_commute subset_trans)
hoelzl@38656:     qed
hoelzl@38656:   }
hoelzl@38656:   hence "!!m n. m < n \<Longrightarrow> A m \<subseteq> A n"
haftmann@57512:     by (metis add.commute le_add_diff_inverse nat_less_le)
hoelzl@38656:   thus ?thesis
hoelzl@38656:     by (auto simp add: disjoint_family_on_def)
hoelzl@38656:       (metis insert_absorb insert_subset le_SucE le_antisym not_leE)
hoelzl@38656: qed
hoelzl@38656: 
hoelzl@39092: lemma setsum_indicator_disjoint_family:
hoelzl@39092:   fixes f :: "'d \<Rightarrow> 'e::semiring_1"
hoelzl@39092:   assumes d: "disjoint_family_on A P" and "x \<in> A j" and "finite P" and "j \<in> P"
hoelzl@39092:   shows "(\<Sum>i\<in>P. f i * indicator (A i) x) = f j"
hoelzl@39092: proof -
hoelzl@39092:   have "P \<inter> {i. x \<in> A i} = {j}"
hoelzl@39092:     using d `x \<in> A j` `j \<in> P` unfolding disjoint_family_on_def
hoelzl@39092:     by auto
hoelzl@39092:   thus ?thesis
hoelzl@39092:     unfolding indicator_def
haftmann@57418:     by (simp add: if_distrib setsum.If_cases[OF `finite P`])
hoelzl@39092: qed
hoelzl@39092: 
hoelzl@38656: definition disjointed :: "(nat \<Rightarrow> 'a set) \<Rightarrow> nat \<Rightarrow> 'a set "
hoelzl@38656:   where "disjointed A n = A n - (\<Union>i\<in>{0..<n}. A i)"
hoelzl@38656: 
hoelzl@38656: lemma finite_UN_disjointed_eq: "(\<Union>i\<in>{0..<n}. disjointed A i) = (\<Union>i\<in>{0..<n}. A i)"
hoelzl@38656: proof (induct n)
hoelzl@38656:   case 0 show ?case by simp
hoelzl@38656: next
hoelzl@38656:   case (Suc n)
hoelzl@38656:   thus ?case by (simp add: atLeastLessThanSuc disjointed_def)
hoelzl@38656: qed
hoelzl@38656: 
hoelzl@38656: lemma UN_disjointed_eq: "(\<Union>i. disjointed A i) = (\<Union>i. A i)"
hoelzl@38656:   apply (rule UN_finite2_eq [where k=0])
hoelzl@38656:   apply (simp add: finite_UN_disjointed_eq)
hoelzl@38656:   done
hoelzl@38656: 
hoelzl@38656: lemma less_disjoint_disjointed: "m<n \<Longrightarrow> disjointed A m \<inter> disjointed A n = {}"
hoelzl@38656:   by (auto simp add: disjointed_def)
hoelzl@38656: 
hoelzl@38656: lemma disjoint_family_disjointed: "disjoint_family (disjointed A)"
hoelzl@38656:   by (simp add: disjoint_family_on_def)
hoelzl@38656:      (metis neq_iff Int_commute less_disjoint_disjointed)
hoelzl@38656: 
hoelzl@38656: lemma disjointed_subset: "disjointed A n \<subseteq> A n"
hoelzl@38656:   by (auto simp add: disjointed_def)
hoelzl@38656: 
hoelzl@42065: lemma (in ring_of_sets) UNION_in_sets:
hoelzl@38656:   fixes A:: "nat \<Rightarrow> 'a set"
hoelzl@47694:   assumes A: "range A \<subseteq> M"
hoelzl@47694:   shows  "(\<Union>i\<in>{0..<n}. A i) \<in> M"
hoelzl@38656: proof (induct n)
hoelzl@38656:   case 0 show ?case by simp
hoelzl@38656: next
hoelzl@38656:   case (Suc n)
hoelzl@38656:   thus ?case
hoelzl@38656:     by (simp add: atLeastLessThanSuc) (metis A Un UNIV_I image_subset_iff)
hoelzl@38656: qed
hoelzl@38656: 
hoelzl@42065: lemma (in ring_of_sets) range_disjointed_sets:
hoelzl@47694:   assumes A: "range A \<subseteq> M"
hoelzl@47694:   shows  "range (disjointed A) \<subseteq> M"
hoelzl@38656: proof (auto simp add: disjointed_def)
hoelzl@38656:   fix n
hoelzl@47694:   show "A n - (\<Union>i\<in>{0..<n}. A i) \<in> M" using UNION_in_sets
hoelzl@38656:     by (metis A Diff UNIV_I image_subset_iff)
hoelzl@38656: qed
hoelzl@38656: 
hoelzl@42065: lemma (in algebra) range_disjointed_sets':
hoelzl@47694:   "range A \<subseteq> M \<Longrightarrow> range (disjointed A) \<subseteq> M"
hoelzl@42065:   using range_disjointed_sets .
hoelzl@42065: 
hoelzl@42145: lemma disjointed_0[simp]: "disjointed A 0 = A 0"
hoelzl@42145:   by (simp add: disjointed_def)
hoelzl@42145: 
hoelzl@42145: lemma incseq_Un:
hoelzl@42145:   "incseq A \<Longrightarrow> (\<Union>i\<le>n. A i) = A n"
hoelzl@42145:   unfolding incseq_def by auto
hoelzl@42145: 
hoelzl@42145: lemma disjointed_incseq:
hoelzl@42145:   "incseq A \<Longrightarrow> disjointed A (Suc n) = A (Suc n) - A n"
hoelzl@42145:   using incseq_Un[of A]
hoelzl@42145:   by (simp add: disjointed_def atLeastLessThanSuc_atLeastAtMost atLeast0AtMost)
hoelzl@42145: 
hoelzl@38656: lemma sigma_algebra_disjoint_iff:
hoelzl@47694:   "sigma_algebra \<Omega> M \<longleftrightarrow> algebra \<Omega> M \<and>
hoelzl@47694:     (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"
hoelzl@38656: proof (auto simp add: sigma_algebra_iff)
hoelzl@38656:   fix A :: "nat \<Rightarrow> 'a set"
hoelzl@47694:   assume M: "algebra \<Omega> M"
hoelzl@47694:      and A: "range A \<subseteq> M"
hoelzl@47694:      and UnA: "\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M"
hoelzl@47694:   hence "range (disjointed A) \<subseteq> M \<longrightarrow>
hoelzl@38656:          disjoint_family (disjointed A) \<longrightarrow>
hoelzl@47694:          (\<Union>i. disjointed A i) \<in> M" by blast
hoelzl@47694:   hence "(\<Union>i. disjointed A i) \<in> M"
hoelzl@47694:     by (simp add: algebra.range_disjointed_sets'[of \<Omega>] M A disjoint_family_disjointed)
hoelzl@47694:   thus "(\<Union>i::nat. A i) \<in> M" by (simp add: UN_disjointed_eq)
hoelzl@47694: qed
hoelzl@47694: 
hoelzl@47762: lemma disjoint_family_on_disjoint_image:
hoelzl@47762:   "disjoint_family_on A I \<Longrightarrow> disjoint (A ` I)"
hoelzl@47762:   unfolding disjoint_family_on_def disjoint_def by force
hoelzl@47762: 
hoelzl@47762: lemma disjoint_image_disjoint_family_on:
hoelzl@47762:   assumes d: "disjoint (A ` I)" and i: "inj_on A I"
hoelzl@47762:   shows "disjoint_family_on A I"
hoelzl@47762:   unfolding disjoint_family_on_def
hoelzl@47762: proof (intro ballI impI)
hoelzl@47762:   fix n m assume nm: "m \<in> I" "n \<in> I" and "n \<noteq> m"
hoelzl@47762:   with i[THEN inj_onD, of n m] show "A n \<inter> A m = {}"
hoelzl@47762:     by (intro disjointD[OF d]) auto
hoelzl@47762: qed
hoelzl@47762: 
hoelzl@56994: subsubsection {* Ring generated by a semiring *}
hoelzl@47762: 
hoelzl@47762: definition (in semiring_of_sets)
hoelzl@47762:   "generated_ring = { \<Union>C | C. C \<subseteq> M \<and> finite C \<and> disjoint C }"
hoelzl@47762: 
hoelzl@47762: lemma (in semiring_of_sets) generated_ringE[elim?]:
hoelzl@47762:   assumes "a \<in> generated_ring"
hoelzl@47762:   obtains C where "finite C" "disjoint C" "C \<subseteq> M" "a = \<Union>C"
hoelzl@47762:   using assms unfolding generated_ring_def by auto
hoelzl@47762: 
hoelzl@47762: lemma (in semiring_of_sets) generated_ringI[intro?]:
hoelzl@47762:   assumes "finite C" "disjoint C" "C \<subseteq> M" "a = \<Union>C"
hoelzl@47762:   shows "a \<in> generated_ring"
hoelzl@47762:   using assms unfolding generated_ring_def by auto
hoelzl@47762: 
hoelzl@47762: lemma (in semiring_of_sets) generated_ringI_Basic:
hoelzl@47762:   "A \<in> M \<Longrightarrow> A \<in> generated_ring"
hoelzl@47762:   by (rule generated_ringI[of "{A}"]) (auto simp: disjoint_def)
hoelzl@47762: 
hoelzl@47762: lemma (in semiring_of_sets) generated_ring_disjoint_Un[intro]:
hoelzl@47762:   assumes a: "a \<in> generated_ring" and b: "b \<in> generated_ring"
hoelzl@47762:   and "a \<inter> b = {}"
hoelzl@47762:   shows "a \<union> b \<in> generated_ring"
hoelzl@47762: proof -
hoelzl@47762:   from a guess Ca .. note Ca = this
hoelzl@47762:   from b guess Cb .. note Cb = this
hoelzl@47762:   show ?thesis
hoelzl@47762:   proof
hoelzl@47762:     show "disjoint (Ca \<union> Cb)"
hoelzl@47762:       using `a \<inter> b = {}` Ca Cb by (auto intro!: disjoint_union)
hoelzl@47762:   qed (insert Ca Cb, auto)
hoelzl@47762: qed
hoelzl@47762: 
hoelzl@47762: lemma (in semiring_of_sets) generated_ring_empty: "{} \<in> generated_ring"
hoelzl@47762:   by (auto simp: generated_ring_def disjoint_def)
hoelzl@47762: 
hoelzl@47762: lemma (in semiring_of_sets) generated_ring_disjoint_Union:
hoelzl@47762:   assumes "finite A" shows "A \<subseteq> generated_ring \<Longrightarrow> disjoint A \<Longrightarrow> \<Union>A \<in> generated_ring"
hoelzl@47762:   using assms by (induct A) (auto simp: disjoint_def intro!: generated_ring_disjoint_Un generated_ring_empty)
hoelzl@47762: 
hoelzl@47762: lemma (in semiring_of_sets) generated_ring_disjoint_UNION:
hoelzl@47762:   "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"
hoelzl@47762:   unfolding SUP_def by (intro generated_ring_disjoint_Union) auto
hoelzl@47762: 
hoelzl@47762: lemma (in semiring_of_sets) generated_ring_Int:
hoelzl@47762:   assumes a: "a \<in> generated_ring" and b: "b \<in> generated_ring"
hoelzl@47762:   shows "a \<inter> b \<in> generated_ring"
hoelzl@47762: proof -
hoelzl@47762:   from a guess Ca .. note Ca = this
hoelzl@47762:   from b guess Cb .. note Cb = this
hoelzl@47762:   def C \<equiv> "(\<lambda>(a,b). a \<inter> b)` (Ca\<times>Cb)"
hoelzl@47762:   show ?thesis
hoelzl@47762:   proof
hoelzl@47762:     show "disjoint C"
hoelzl@47762:     proof (simp add: disjoint_def C_def, intro ballI impI)
hoelzl@47762:       fix a1 b1 a2 b2 assume sets: "a1 \<in> Ca" "b1 \<in> Cb" "a2 \<in> Ca" "b2 \<in> Cb"
hoelzl@47762:       assume "a1 \<inter> b1 \<noteq> a2 \<inter> b2"
hoelzl@47762:       then have "a1 \<noteq> a2 \<or> b1 \<noteq> b2" by auto
hoelzl@47762:       then show "(a1 \<inter> b1) \<inter> (a2 \<inter> b2) = {}"
hoelzl@47762:       proof
hoelzl@47762:         assume "a1 \<noteq> a2"
hoelzl@47762:         with sets Ca have "a1 \<inter> a2 = {}"
hoelzl@47762:           by (auto simp: disjoint_def)
hoelzl@47762:         then show ?thesis by auto
hoelzl@47762:       next
hoelzl@47762:         assume "b1 \<noteq> b2"
hoelzl@47762:         with sets Cb have "b1 \<inter> b2 = {}"
hoelzl@47762:           by (auto simp: disjoint_def)
hoelzl@47762:         then show ?thesis by auto
hoelzl@47762:       qed
hoelzl@47762:     qed
hoelzl@47762:   qed (insert Ca Cb, auto simp: C_def)
hoelzl@47762: qed
hoelzl@47762: 
hoelzl@47762: lemma (in semiring_of_sets) generated_ring_Inter:
hoelzl@47762:   assumes "finite A" "A \<noteq> {}" shows "A \<subseteq> generated_ring \<Longrightarrow> \<Inter>A \<in> generated_ring"
hoelzl@47762:   using assms by (induct A rule: finite_ne_induct) (auto intro: generated_ring_Int)
hoelzl@47762: 
hoelzl@47762: lemma (in semiring_of_sets) generated_ring_INTER:
hoelzl@47762:   "finite I \<Longrightarrow> I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> A i \<in> generated_ring) \<Longrightarrow> INTER I A \<in> generated_ring"
hoelzl@47762:   unfolding INF_def by (intro generated_ring_Inter) auto
hoelzl@47762: 
hoelzl@47762: lemma (in semiring_of_sets) generating_ring:
hoelzl@47762:   "ring_of_sets \<Omega> generated_ring"
hoelzl@47762: proof (rule ring_of_setsI)
hoelzl@47762:   let ?R = generated_ring
hoelzl@47762:   show "?R \<subseteq> Pow \<Omega>"
hoelzl@47762:     using sets_into_space by (auto simp: generated_ring_def generated_ring_empty)
hoelzl@47762:   show "{} \<in> ?R" by (rule generated_ring_empty)
hoelzl@47762: 
hoelzl@47762:   { fix a assume a: "a \<in> ?R" then guess Ca .. note Ca = this
hoelzl@47762:     fix b assume b: "b \<in> ?R" then guess Cb .. note Cb = this
hoelzl@47762:   
hoelzl@47762:     show "a - b \<in> ?R"
hoelzl@47762:     proof cases
hoelzl@47762:       assume "Cb = {}" with Cb `a \<in> ?R` show ?thesis
hoelzl@47762:         by simp
hoelzl@47762:     next
hoelzl@47762:       assume "Cb \<noteq> {}"
hoelzl@47762:       with Ca Cb have "a - b = (\<Union>a'\<in>Ca. \<Inter>b'\<in>Cb. a' - b')" by auto
hoelzl@47762:       also have "\<dots> \<in> ?R"
hoelzl@47762:       proof (intro generated_ring_INTER generated_ring_disjoint_UNION)
hoelzl@47762:         fix a b assume "a \<in> Ca" "b \<in> Cb"
hoelzl@47762:         with Ca Cb Diff_cover[of a b] show "a - b \<in> ?R"
hoelzl@47762:           by (auto simp add: generated_ring_def)
hoelzl@47762:       next
hoelzl@47762:         show "disjoint ((\<lambda>a'. \<Inter>b'\<in>Cb. a' - b')`Ca)"
hoelzl@47762:           using Ca by (auto simp add: disjoint_def `Cb \<noteq> {}`)
hoelzl@47762:       next
hoelzl@47762:         show "finite Ca" "finite Cb" "Cb \<noteq> {}" by fact+
hoelzl@47762:       qed
hoelzl@47762:       finally show "a - b \<in> ?R" .
hoelzl@47762:     qed }
hoelzl@47762:   note Diff = this
hoelzl@47762: 
hoelzl@47762:   fix a b assume sets: "a \<in> ?R" "b \<in> ?R"
hoelzl@47762:   have "a \<union> b = (a - b) \<union> (a \<inter> b) \<union> (b - a)" by auto
hoelzl@47762:   also have "\<dots> \<in> ?R"
hoelzl@47762:     by (intro sets generated_ring_disjoint_Un generated_ring_Int Diff) auto
hoelzl@47762:   finally show "a \<union> b \<in> ?R" .
hoelzl@47762: qed
hoelzl@47762: 
hoelzl@47762: lemma (in semiring_of_sets) sigma_sets_generated_ring_eq: "sigma_sets \<Omega> generated_ring = sigma_sets \<Omega> M"
hoelzl@47762: proof
hoelzl@47762:   interpret M: sigma_algebra \<Omega> "sigma_sets \<Omega> M"
hoelzl@47762:     using space_closed by (rule sigma_algebra_sigma_sets)
hoelzl@47762:   show "sigma_sets \<Omega> generated_ring \<subseteq> sigma_sets \<Omega> M"
hoelzl@47762:     by (blast intro!: sigma_sets_mono elim: generated_ringE)
hoelzl@47762: qed (auto intro!: generated_ringI_Basic sigma_sets_mono)
hoelzl@47762: 
hoelzl@56994: subsubsection {* A Two-Element Series *}
hoelzl@38656: 
hoelzl@38656: definition binaryset :: "'a set \<Rightarrow> 'a set \<Rightarrow> nat \<Rightarrow> 'a set "
wenzelm@50252:   where "binaryset A B = (\<lambda>x. {})(0 := A, Suc 0 := B)"
hoelzl@38656: 
hoelzl@38656: lemma range_binaryset_eq: "range(binaryset A B) = {A,B,{}}"
hoelzl@38656:   apply (simp add: binaryset_def)
nipkow@39302:   apply (rule set_eqI)
hoelzl@38656:   apply (auto simp add: image_iff)
hoelzl@38656:   done
hoelzl@38656: 
hoelzl@38656: lemma UN_binaryset_eq: "(\<Union>i. binaryset A B i) = A \<union> B"
haftmann@44106:   by (simp add: SUP_def range_binaryset_eq)
hoelzl@38656: 
hoelzl@56994: subsubsection {* Closed CDI *}
hoelzl@38656: 
hoelzl@47694: definition closed_cdi where
hoelzl@47694:   "closed_cdi \<Omega> M \<longleftrightarrow>
hoelzl@47694:    M \<subseteq> Pow \<Omega> &
hoelzl@47694:    (\<forall>s \<in> M. \<Omega> - s \<in> M) &
hoelzl@47694:    (\<forall>A. (range A \<subseteq> M) & (A 0 = {}) & (\<forall>n. A n \<subseteq> A (Suc n)) \<longrightarrow>
hoelzl@47694:         (\<Union>i. A i) \<in> M) &
hoelzl@47694:    (\<forall>A. (range A \<subseteq> M) & disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"
hoelzl@38656: 
hoelzl@38656: inductive_set
hoelzl@47694:   smallest_ccdi_sets :: "'a set \<Rightarrow> 'a set set \<Rightarrow> 'a set set"
hoelzl@47694:   for \<Omega> M
hoelzl@38656:   where
hoelzl@38656:     Basic [intro]:
hoelzl@47694:       "a \<in> M \<Longrightarrow> a \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656:   | Compl [intro]:
hoelzl@47694:       "a \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> \<Omega> - a \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656:   | Inc:
hoelzl@47694:       "range A \<in> Pow(smallest_ccdi_sets \<Omega> M) \<Longrightarrow> A 0 = {} \<Longrightarrow> (\<And>n. A n \<subseteq> A (Suc n))
hoelzl@47694:        \<Longrightarrow> (\<Union>i. A i) \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656:   | Disj:
hoelzl@47694:       "range A \<in> Pow(smallest_ccdi_sets \<Omega> M) \<Longrightarrow> disjoint_family A
hoelzl@47694:        \<Longrightarrow> (\<Union>i::nat. A i) \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656: 
hoelzl@47694: lemma (in subset_class) smallest_closed_cdi1: "M \<subseteq> smallest_ccdi_sets \<Omega> M"
hoelzl@47694:   by auto
hoelzl@38656: 
hoelzl@47694: lemma (in subset_class) smallest_ccdi_sets: "smallest_ccdi_sets \<Omega> M \<subseteq> Pow \<Omega>"
hoelzl@38656:   apply (rule subsetI)
hoelzl@38656:   apply (erule smallest_ccdi_sets.induct)
hoelzl@38656:   apply (auto intro: range_subsetD dest: sets_into_space)
hoelzl@38656:   done
hoelzl@38656: 
hoelzl@47694: lemma (in subset_class) smallest_closed_cdi2: "closed_cdi \<Omega> (smallest_ccdi_sets \<Omega> M)"
hoelzl@47694:   apply (auto simp add: closed_cdi_def smallest_ccdi_sets)
hoelzl@38656:   apply (blast intro: smallest_ccdi_sets.Inc smallest_ccdi_sets.Disj) +
hoelzl@38656:   done
hoelzl@38656: 
hoelzl@47694: lemma closed_cdi_subset: "closed_cdi \<Omega> M \<Longrightarrow> M \<subseteq> Pow \<Omega>"
hoelzl@38656:   by (simp add: closed_cdi_def)
hoelzl@38656: 
hoelzl@47694: lemma closed_cdi_Compl: "closed_cdi \<Omega> M \<Longrightarrow> s \<in> M \<Longrightarrow> \<Omega> - s \<in> M"
hoelzl@38656:   by (simp add: closed_cdi_def)
hoelzl@38656: 
hoelzl@38656: lemma closed_cdi_Inc:
hoelzl@47694:   "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"
hoelzl@38656:   by (simp add: closed_cdi_def)
hoelzl@38656: 
hoelzl@38656: lemma closed_cdi_Disj:
hoelzl@47694:   "closed_cdi \<Omega> M \<Longrightarrow> range A \<subseteq> M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
hoelzl@38656:   by (simp add: closed_cdi_def)
hoelzl@38656: 
hoelzl@38656: lemma closed_cdi_Un:
hoelzl@47694:   assumes cdi: "closed_cdi \<Omega> M" and empty: "{} \<in> M"
hoelzl@47694:       and A: "A \<in> M" and B: "B \<in> M"
hoelzl@38656:       and disj: "A \<inter> B = {}"
hoelzl@47694:     shows "A \<union> B \<in> M"
hoelzl@38656: proof -
hoelzl@47694:   have ra: "range (binaryset A B) \<subseteq> M"
hoelzl@38656:    by (simp add: range_binaryset_eq empty A B)
hoelzl@38656:  have di:  "disjoint_family (binaryset A B)" using disj
hoelzl@38656:    by (simp add: disjoint_family_on_def binaryset_def Int_commute)
hoelzl@38656:  from closed_cdi_Disj [OF cdi ra di]
hoelzl@38656:  show ?thesis
hoelzl@38656:    by (simp add: UN_binaryset_eq)
hoelzl@38656: qed
hoelzl@38656: 
hoelzl@38656: lemma (in algebra) smallest_ccdi_sets_Un:
hoelzl@47694:   assumes A: "A \<in> smallest_ccdi_sets \<Omega> M" and B: "B \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656:       and disj: "A \<inter> B = {}"
hoelzl@47694:     shows "A \<union> B \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656: proof -
hoelzl@47694:   have ra: "range (binaryset A B) \<in> Pow (smallest_ccdi_sets \<Omega> M)"
hoelzl@38656:     by (simp add: range_binaryset_eq  A B smallest_ccdi_sets.Basic)
hoelzl@38656:   have di:  "disjoint_family (binaryset A B)" using disj
hoelzl@38656:     by (simp add: disjoint_family_on_def binaryset_def Int_commute)
hoelzl@38656:   from Disj [OF ra di]
hoelzl@38656:   show ?thesis
hoelzl@38656:     by (simp add: UN_binaryset_eq)
hoelzl@38656: qed
hoelzl@38656: 
hoelzl@38656: lemma (in algebra) smallest_ccdi_sets_Int1:
hoelzl@47694:   assumes a: "a \<in> M"
hoelzl@47694:   shows "b \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656: proof (induct rule: smallest_ccdi_sets.induct)
hoelzl@38656:   case (Basic x)
hoelzl@38656:   thus ?case
hoelzl@38656:     by (metis a Int smallest_ccdi_sets.Basic)
hoelzl@38656: next
hoelzl@38656:   case (Compl x)
hoelzl@47694:   have "a \<inter> (\<Omega> - x) = \<Omega> - ((\<Omega> - a) \<union> (a \<inter> x))"
hoelzl@38656:     by blast
hoelzl@47694:   also have "... \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656:     by (metis smallest_ccdi_sets.Compl a Compl(2) Diff_Int2 Diff_Int_distrib2
hoelzl@47694:            Diff_disjoint Int_Diff Int_empty_right smallest_ccdi_sets_Un
hoelzl@47694:            smallest_ccdi_sets.Basic smallest_ccdi_sets.Compl)
hoelzl@38656:   finally show ?case .
hoelzl@38656: next
hoelzl@38656:   case (Inc A)
hoelzl@38656:   have 1: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) = a \<inter> (\<Union>i. A i)"
hoelzl@38656:     by blast
hoelzl@47694:   have "range (\<lambda>i. a \<inter> A i) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Inc
hoelzl@38656:     by blast
hoelzl@38656:   moreover have "(\<lambda>i. a \<inter> A i) 0 = {}"
hoelzl@38656:     by (simp add: Inc)
hoelzl@38656:   moreover have "!!n. (\<lambda>i. a \<inter> A i) n \<subseteq> (\<lambda>i. a \<inter> A i) (Suc n)" using Inc
hoelzl@38656:     by blast
hoelzl@47694:   ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656:     by (rule smallest_ccdi_sets.Inc)
hoelzl@38656:   show ?case
hoelzl@38656:     by (metis 1 2)
hoelzl@38656: next
hoelzl@38656:   case (Disj A)
hoelzl@38656:   have 1: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) = a \<inter> (\<Union>i. A i)"
hoelzl@38656:     by blast
hoelzl@47694:   have "range (\<lambda>i. a \<inter> A i) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Disj
hoelzl@38656:     by blast
hoelzl@38656:   moreover have "disjoint_family (\<lambda>i. a \<inter> A i)" using Disj
hoelzl@38656:     by (auto simp add: disjoint_family_on_def)
hoelzl@47694:   ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656:     by (rule smallest_ccdi_sets.Disj)
hoelzl@38656:   show ?case
hoelzl@38656:     by (metis 1 2)
hoelzl@38656: qed
hoelzl@38656: 
hoelzl@38656: 
hoelzl@38656: lemma (in algebra) smallest_ccdi_sets_Int:
hoelzl@47694:   assumes b: "b \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@47694:   shows "a \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656: proof (induct rule: smallest_ccdi_sets.induct)
hoelzl@38656:   case (Basic x)
hoelzl@38656:   thus ?case
hoelzl@38656:     by (metis b smallest_ccdi_sets_Int1)
hoelzl@38656: next
hoelzl@38656:   case (Compl x)
hoelzl@47694:   have "(\<Omega> - x) \<inter> b = \<Omega> - (x \<inter> b \<union> (\<Omega> - b))"
hoelzl@38656:     by blast
hoelzl@47694:   also have "... \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656:     by (metis Compl(2) Diff_disjoint Int_Diff Int_commute Int_empty_right b
hoelzl@38656:            smallest_ccdi_sets.Compl smallest_ccdi_sets_Un)
hoelzl@38656:   finally show ?case .
hoelzl@38656: next
hoelzl@38656:   case (Inc A)
hoelzl@38656:   have 1: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) = (\<Union>i. A i) \<inter> b"
hoelzl@38656:     by blast
hoelzl@47694:   have "range (\<lambda>i. A i \<inter> b) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Inc
hoelzl@38656:     by blast
hoelzl@38656:   moreover have "(\<lambda>i. A i \<inter> b) 0 = {}"
hoelzl@38656:     by (simp add: Inc)
hoelzl@38656:   moreover have "!!n. (\<lambda>i. A i \<inter> b) n \<subseteq> (\<lambda>i. A i \<inter> b) (Suc n)" using Inc
hoelzl@38656:     by blast
hoelzl@47694:   ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656:     by (rule smallest_ccdi_sets.Inc)
hoelzl@38656:   show ?case
hoelzl@38656:     by (metis 1 2)
hoelzl@38656: next
hoelzl@38656:   case (Disj A)
hoelzl@38656:   have 1: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) = (\<Union>i. A i) \<inter> b"
hoelzl@38656:     by blast
hoelzl@47694:   have "range (\<lambda>i. A i \<inter> b) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Disj
hoelzl@38656:     by blast
hoelzl@38656:   moreover have "disjoint_family (\<lambda>i. A i \<inter> b)" using Disj
hoelzl@38656:     by (auto simp add: disjoint_family_on_def)
hoelzl@47694:   ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656:     by (rule smallest_ccdi_sets.Disj)
hoelzl@38656:   show ?case
hoelzl@38656:     by (metis 1 2)
hoelzl@38656: qed
hoelzl@38656: 
hoelzl@38656: lemma (in algebra) sigma_property_disjoint_lemma:
hoelzl@47694:   assumes sbC: "M \<subseteq> C"
hoelzl@47694:       and ccdi: "closed_cdi \<Omega> C"
hoelzl@47694:   shows "sigma_sets \<Omega> M \<subseteq> C"
hoelzl@38656: proof -
hoelzl@47694:   have "smallest_ccdi_sets \<Omega> M \<in> {B . M \<subseteq> B \<and> sigma_algebra \<Omega> B}"
hoelzl@38656:     apply (auto simp add: sigma_algebra_disjoint_iff algebra_iff_Int
hoelzl@38656:             smallest_ccdi_sets_Int)
hoelzl@38656:     apply (metis Union_Pow_eq Union_upper subsetD smallest_ccdi_sets)
hoelzl@38656:     apply (blast intro: smallest_ccdi_sets.Disj)
hoelzl@38656:     done
hoelzl@47694:   hence "sigma_sets (\<Omega>) (M) \<subseteq> smallest_ccdi_sets \<Omega> M"
hoelzl@38656:     by clarsimp
hoelzl@47694:        (drule sigma_algebra.sigma_sets_subset [where a="M"], auto)
hoelzl@38656:   also have "...  \<subseteq> C"
hoelzl@38656:     proof
hoelzl@38656:       fix x
hoelzl@47694:       assume x: "x \<in> smallest_ccdi_sets \<Omega> M"
hoelzl@38656:       thus "x \<in> C"
hoelzl@38656:         proof (induct rule: smallest_ccdi_sets.induct)
hoelzl@38656:           case (Basic x)
hoelzl@38656:           thus ?case
hoelzl@38656:             by (metis Basic subsetD sbC)
hoelzl@38656:         next
hoelzl@38656:           case (Compl x)
hoelzl@38656:           thus ?case
hoelzl@38656:             by (blast intro: closed_cdi_Compl [OF ccdi, simplified])
hoelzl@38656:         next
hoelzl@38656:           case (Inc A)
hoelzl@38656:           thus ?case
hoelzl@38656:                by (auto intro: closed_cdi_Inc [OF ccdi, simplified])
hoelzl@38656:         next
hoelzl@38656:           case (Disj A)
hoelzl@38656:           thus ?case
hoelzl@38656:                by (auto intro: closed_cdi_Disj [OF ccdi, simplified])
hoelzl@38656:         qed
hoelzl@38656:     qed
hoelzl@38656:   finally show ?thesis .
hoelzl@38656: qed
hoelzl@38656: 
hoelzl@38656: lemma (in algebra) sigma_property_disjoint:
hoelzl@47694:   assumes sbC: "M \<subseteq> C"
hoelzl@47694:       and compl: "!!s. s \<in> C \<inter> sigma_sets (\<Omega>) (M) \<Longrightarrow> \<Omega> - s \<in> C"
hoelzl@47694:       and inc: "!!A. range A \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)
hoelzl@38656:                      \<Longrightarrow> A 0 = {} \<Longrightarrow> (!!n. A n \<subseteq> A (Suc n))
hoelzl@38656:                      \<Longrightarrow> (\<Union>i. A i) \<in> C"
hoelzl@47694:       and disj: "!!A. range A \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)
hoelzl@38656:                       \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> C"
hoelzl@47694:   shows "sigma_sets (\<Omega>) (M) \<subseteq> C"
hoelzl@38656: proof -
hoelzl@47694:   have "sigma_sets (\<Omega>) (M) \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)"
hoelzl@38656:     proof (rule sigma_property_disjoint_lemma)
hoelzl@47694:       show "M \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)"
hoelzl@38656:         by (metis Int_greatest Set.subsetI sbC sigma_sets.Basic)
hoelzl@38656:     next
hoelzl@47694:       show "closed_cdi \<Omega> (C \<inter> sigma_sets (\<Omega>) (M))"
hoelzl@38656:         by (simp add: closed_cdi_def compl inc disj)
hoelzl@38656:            (metis PowI Set.subsetI le_infI2 sigma_sets_into_sp space_closed
hoelzl@38656:              IntE sigma_sets.Compl range_subsetD sigma_sets.Union)
hoelzl@38656:     qed
hoelzl@38656:   thus ?thesis
hoelzl@38656:     by blast
hoelzl@38656: qed
hoelzl@38656: 
hoelzl@56994: subsubsection {* Dynkin systems *}
hoelzl@40859: 
hoelzl@42065: locale dynkin_system = subset_class +
hoelzl@47694:   assumes space: "\<Omega> \<in> M"
hoelzl@47694:     and   compl[intro!]: "\<And>A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"
hoelzl@47694:     and   UN[intro!]: "\<And>A. disjoint_family A \<Longrightarrow> range A \<subseteq> M
hoelzl@47694:                            \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
hoelzl@40859: 
hoelzl@47694: lemma (in dynkin_system) empty[intro, simp]: "{} \<in> M"
hoelzl@47694:   using space compl[of "\<Omega>"] by simp
hoelzl@40859: 
hoelzl@40859: lemma (in dynkin_system) diff:
hoelzl@47694:   assumes sets: "D \<in> M" "E \<in> M" and "D \<subseteq> E"
hoelzl@47694:   shows "E - D \<in> M"
hoelzl@40859: proof -
hoelzl@47694:   let ?f = "\<lambda>x. if x = 0 then D else if x = Suc 0 then \<Omega> - E else {}"
hoelzl@47694:   have "range ?f = {D, \<Omega> - E, {}}"
hoelzl@40859:     by (auto simp: image_iff)
hoelzl@47694:   moreover have "D \<union> (\<Omega> - E) = (\<Union>i. ?f i)"
hoelzl@40859:     by (auto simp: image_iff split: split_if_asm)
hoelzl@40859:   moreover
wenzelm@53374:   have "disjoint_family ?f" unfolding disjoint_family_on_def
hoelzl@47694:     using `D \<in> M`[THEN sets_into_space] `D \<subseteq> E` by auto
hoelzl@47694:   ultimately have "\<Omega> - (D \<union> (\<Omega> - E)) \<in> M"
hoelzl@40859:     using sets by auto
hoelzl@47694:   also have "\<Omega> - (D \<union> (\<Omega> - E)) = E - D"
hoelzl@40859:     using assms sets_into_space by auto
hoelzl@40859:   finally show ?thesis .
hoelzl@40859: qed
hoelzl@40859: 
hoelzl@40859: lemma dynkin_systemI:
hoelzl@47694:   assumes "\<And> A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>" "\<Omega> \<in> M"
hoelzl@47694:   assumes "\<And> A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"
hoelzl@47694:   assumes "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> M
hoelzl@47694:           \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
hoelzl@47694:   shows "dynkin_system \<Omega> M"
hoelzl@42065:   using assms by (auto simp: dynkin_system_def dynkin_system_axioms_def subset_class_def)
hoelzl@40859: 
hoelzl@42988: lemma dynkin_systemI':
hoelzl@47694:   assumes 1: "\<And> A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>"
hoelzl@47694:   assumes empty: "{} \<in> M"
hoelzl@47694:   assumes Diff: "\<And> A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"
hoelzl@47694:   assumes 2: "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> M
hoelzl@47694:           \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
hoelzl@47694:   shows "dynkin_system \<Omega> M"
hoelzl@42988: proof -
hoelzl@47694:   from Diff[OF empty] have "\<Omega> \<in> M" by auto
hoelzl@42988:   from 1 this Diff 2 show ?thesis
hoelzl@42988:     by (intro dynkin_systemI) auto
hoelzl@42988: qed
hoelzl@42988: 
hoelzl@40859: lemma dynkin_system_trivial:
hoelzl@47694:   shows "dynkin_system A (Pow A)"
hoelzl@40859:   by (rule dynkin_systemI) auto
hoelzl@40859: 
hoelzl@40859: lemma sigma_algebra_imp_dynkin_system:
hoelzl@47694:   assumes "sigma_algebra \<Omega> M" shows "dynkin_system \<Omega> M"
hoelzl@40859: proof -
hoelzl@47694:   interpret sigma_algebra \<Omega> M by fact
nipkow@44890:   show ?thesis using sets_into_space by (fastforce intro!: dynkin_systemI)
hoelzl@40859: qed
hoelzl@40859: 
hoelzl@56994: subsubsection "Intersection sets systems"
hoelzl@40859: 
hoelzl@47694: definition "Int_stable M \<longleftrightarrow> (\<forall> a \<in> M. \<forall> b \<in> M. a \<inter> b \<in> M)"
hoelzl@40859: 
hoelzl@40859: lemma (in algebra) Int_stable: "Int_stable M"
hoelzl@40859:   unfolding Int_stable_def by auto
hoelzl@40859: 
hoelzl@42981: lemma Int_stableI:
hoelzl@47694:   "(\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<inter> b \<in> A) \<Longrightarrow> Int_stable A"
hoelzl@42981:   unfolding Int_stable_def by auto
hoelzl@42981: 
hoelzl@42981: lemma Int_stableD:
hoelzl@47694:   "Int_stable M \<Longrightarrow> a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<inter> b \<in> M"
hoelzl@42981:   unfolding Int_stable_def by auto
hoelzl@42981: 
hoelzl@40859: lemma (in dynkin_system) sigma_algebra_eq_Int_stable:
hoelzl@47694:   "sigma_algebra \<Omega> M \<longleftrightarrow> Int_stable M"
hoelzl@40859: proof
hoelzl@47694:   assume "sigma_algebra \<Omega> M" then show "Int_stable M"
hoelzl@40859:     unfolding sigma_algebra_def using algebra.Int_stable by auto
hoelzl@40859: next
hoelzl@40859:   assume "Int_stable M"
hoelzl@47694:   show "sigma_algebra \<Omega> M"
hoelzl@42065:     unfolding sigma_algebra_disjoint_iff algebra_iff_Un
hoelzl@40859:   proof (intro conjI ballI allI impI)
hoelzl@47694:     show "M \<subseteq> Pow (\<Omega>)" using sets_into_space by auto
hoelzl@40859:   next
hoelzl@47694:     fix A B assume "A \<in> M" "B \<in> M"
hoelzl@47694:     then have "A \<union> B = \<Omega> - ((\<Omega> - A) \<inter> (\<Omega> - B))"
hoelzl@47694:               "\<Omega> - A \<in> M" "\<Omega> - B \<in> M"
hoelzl@40859:       using sets_into_space by auto
hoelzl@47694:     then show "A \<union> B \<in> M"
hoelzl@40859:       using `Int_stable M` unfolding Int_stable_def by auto
hoelzl@40859:   qed auto
hoelzl@40859: qed
hoelzl@40859: 
hoelzl@56994: subsubsection "Smallest Dynkin systems"
hoelzl@40859: 
hoelzl@41689: definition dynkin where
hoelzl@47694:   "dynkin \<Omega> M =  (\<Inter>{D. dynkin_system \<Omega> D \<and> M \<subseteq> D})"
hoelzl@40859: 
hoelzl@40859: lemma dynkin_system_dynkin:
hoelzl@47694:   assumes "M \<subseteq> Pow (\<Omega>)"
hoelzl@47694:   shows "dynkin_system \<Omega> (dynkin \<Omega> M)"
hoelzl@40859: proof (rule dynkin_systemI)
hoelzl@47694:   fix A assume "A \<in> dynkin \<Omega> M"
hoelzl@40859:   moreover
hoelzl@47694:   { fix D assume "A \<in> D" and d: "dynkin_system \<Omega> D"
hoelzl@47694:     then have "A \<subseteq> \<Omega>" by (auto simp: dynkin_system_def subset_class_def) }
hoelzl@47694:   moreover have "{D. dynkin_system \<Omega> D \<and> M \<subseteq> D} \<noteq> {}"
nipkow@44890:     using assms dynkin_system_trivial by fastforce
hoelzl@47694:   ultimately show "A \<subseteq> \<Omega>"
hoelzl@40859:     unfolding dynkin_def using assms
hoelzl@47694:     by auto
hoelzl@40859: next
hoelzl@47694:   show "\<Omega> \<in> dynkin \<Omega> M"
nipkow@44890:     unfolding dynkin_def using dynkin_system.space by fastforce
hoelzl@40859: next
hoelzl@47694:   fix A assume "A \<in> dynkin \<Omega> M"
hoelzl@47694:   then show "\<Omega> - A \<in> dynkin \<Omega> M"
hoelzl@40859:     unfolding dynkin_def using dynkin_system.compl by force
hoelzl@40859: next
hoelzl@40859:   fix A :: "nat \<Rightarrow> 'a set"
hoelzl@47694:   assume A: "disjoint_family A" "range A \<subseteq> dynkin \<Omega> M"
hoelzl@47694:   show "(\<Union>i. A i) \<in> dynkin \<Omega> M" unfolding dynkin_def
hoelzl@40859:   proof (simp, safe)
hoelzl@47694:     fix D assume "dynkin_system \<Omega> D" "M \<subseteq> D"
hoelzl@47694:     with A have "(\<Union>i. A i) \<in> D"
hoelzl@40859:       by (intro dynkin_system.UN) (auto simp: dynkin_def)
hoelzl@40859:     then show "(\<Union>i. A i) \<in> D" by auto
hoelzl@40859:   qed
hoelzl@40859: qed
hoelzl@40859: 
hoelzl@47694: lemma dynkin_Basic[intro]: "A \<in> M \<Longrightarrow> A \<in> dynkin \<Omega> M"
hoelzl@40859:   unfolding dynkin_def by auto
hoelzl@40859: 
hoelzl@40859: lemma (in dynkin_system) restricted_dynkin_system:
hoelzl@47694:   assumes "D \<in> M"
hoelzl@47694:   shows "dynkin_system \<Omega> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}"
hoelzl@40859: proof (rule dynkin_systemI, simp_all)
hoelzl@47694:   have "\<Omega> \<inter> D = D"
hoelzl@47694:     using `D \<in> M` sets_into_space by auto
hoelzl@47694:   then show "\<Omega> \<inter> D \<in> M"
hoelzl@47694:     using `D \<in> M` by auto
hoelzl@40859: next
hoelzl@47694:   fix A assume "A \<subseteq> \<Omega> \<and> A \<inter> D \<in> M"
hoelzl@47694:   moreover have "(\<Omega> - A) \<inter> D = (\<Omega> - (A \<inter> D)) - (\<Omega> - D)"
hoelzl@40859:     by auto
hoelzl@47694:   ultimately show "\<Omega> - A \<subseteq> \<Omega> \<and> (\<Omega> - A) \<inter> D \<in> M"
hoelzl@47694:     using  `D \<in> M` by (auto intro: diff)
hoelzl@40859: next
hoelzl@40859:   fix A :: "nat \<Rightarrow> 'a set"
hoelzl@47694:   assume "disjoint_family A" "range A \<subseteq> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}"
hoelzl@47694:   then have "\<And>i. A i \<subseteq> \<Omega>" "disjoint_family (\<lambda>i. A i \<inter> D)"
hoelzl@47694:     "range (\<lambda>i. A i \<inter> D) \<subseteq> M" "(\<Union>x. A x) \<inter> D = (\<Union>x. A x \<inter> D)"
nipkow@44890:     by ((fastforce simp: disjoint_family_on_def)+)
hoelzl@47694:   then show "(\<Union>x. A x) \<subseteq> \<Omega> \<and> (\<Union>x. A x) \<inter> D \<in> M"
hoelzl@40859:     by (auto simp del: UN_simps)
hoelzl@40859: qed
hoelzl@40859: 
hoelzl@40859: lemma (in dynkin_system) dynkin_subset:
hoelzl@47694:   assumes "N \<subseteq> M"
hoelzl@47694:   shows "dynkin \<Omega> N \<subseteq> M"
hoelzl@40859: proof -
hoelzl@47694:   have "dynkin_system \<Omega> M" by default
hoelzl@47694:   then have "dynkin_system \<Omega> M"
hoelzl@42065:     using assms unfolding dynkin_system_def dynkin_system_axioms_def subset_class_def by simp
hoelzl@47694:   with `N \<subseteq> M` show ?thesis by (auto simp add: dynkin_def)
hoelzl@40859: qed
hoelzl@40859: 
hoelzl@40859: lemma sigma_eq_dynkin:
hoelzl@47694:   assumes sets: "M \<subseteq> Pow \<Omega>"
hoelzl@40859:   assumes "Int_stable M"
hoelzl@47694:   shows "sigma_sets \<Omega> M = dynkin \<Omega> M"
hoelzl@40859: proof -
hoelzl@47694:   have "dynkin \<Omega> M \<subseteq> sigma_sets (\<Omega>) (M)"
hoelzl@40859:     using sigma_algebra_imp_dynkin_system
hoelzl@47694:     unfolding dynkin_def sigma_sets_least_sigma_algebra[OF sets] by auto
hoelzl@40859:   moreover
hoelzl@47694:   interpret dynkin_system \<Omega> "dynkin \<Omega> M"
hoelzl@40859:     using dynkin_system_dynkin[OF sets] .
hoelzl@47694:   have "sigma_algebra \<Omega> (dynkin \<Omega> M)"
hoelzl@40859:     unfolding sigma_algebra_eq_Int_stable Int_stable_def
hoelzl@40859:   proof (intro ballI)
hoelzl@47694:     fix A B assume "A \<in> dynkin \<Omega> M" "B \<in> dynkin \<Omega> M"
hoelzl@47694:     let ?D = "\<lambda>E. {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> E \<in> dynkin \<Omega> M}"
hoelzl@47694:     have "M \<subseteq> ?D B"
hoelzl@40859:     proof
hoelzl@47694:       fix E assume "E \<in> M"
hoelzl@47694:       then have "M \<subseteq> ?D E" "E \<in> dynkin \<Omega> M"
hoelzl@40859:         using sets_into_space `Int_stable M` by (auto simp: Int_stable_def)
hoelzl@47694:       then have "dynkin \<Omega> M \<subseteq> ?D E"
hoelzl@47694:         using restricted_dynkin_system `E \<in> dynkin \<Omega> M`
hoelzl@40859:         by (intro dynkin_system.dynkin_subset) simp_all
hoelzl@47694:       then have "B \<in> ?D E"
hoelzl@47694:         using `B \<in> dynkin \<Omega> M` by auto
hoelzl@47694:       then have "E \<inter> B \<in> dynkin \<Omega> M"
hoelzl@40859:         by (subst Int_commute) simp
hoelzl@47694:       then show "E \<in> ?D B"
hoelzl@47694:         using sets `E \<in> M` by auto
hoelzl@40859:     qed
hoelzl@47694:     then have "dynkin \<Omega> M \<subseteq> ?D B"
hoelzl@47694:       using restricted_dynkin_system `B \<in> dynkin \<Omega> M`
hoelzl@40859:       by (intro dynkin_system.dynkin_subset) simp_all
hoelzl@47694:     then show "A \<inter> B \<in> dynkin \<Omega> M"
hoelzl@47694:       using `A \<in> dynkin \<Omega> M` sets_into_space by auto
hoelzl@40859:   qed
hoelzl@47694:   from sigma_algebra.sigma_sets_subset[OF this, of "M"]
hoelzl@47694:   have "sigma_sets (\<Omega>) (M) \<subseteq> dynkin \<Omega> M" by auto
hoelzl@47694:   ultimately have "sigma_sets (\<Omega>) (M) = dynkin \<Omega> M" by auto
hoelzl@40859:   then show ?thesis
hoelzl@47694:     by (auto simp: dynkin_def)
hoelzl@40859: qed
hoelzl@40859: 
hoelzl@40859: lemma (in dynkin_system) dynkin_idem:
hoelzl@47694:   "dynkin \<Omega> M = M"
hoelzl@40859: proof -
hoelzl@47694:   have "dynkin \<Omega> M = M"
hoelzl@40859:   proof
hoelzl@47694:     show "M \<subseteq> dynkin \<Omega> M"
hoelzl@40859:       using dynkin_Basic by auto
hoelzl@47694:     show "dynkin \<Omega> M \<subseteq> M"
hoelzl@40859:       by (intro dynkin_subset) auto
hoelzl@40859:   qed
hoelzl@40859:   then show ?thesis
hoelzl@47694:     by (auto simp: dynkin_def)
hoelzl@40859: qed
hoelzl@40859: 
hoelzl@40859: lemma (in dynkin_system) dynkin_lemma:
hoelzl@41689:   assumes "Int_stable E"
hoelzl@47694:   and E: "E \<subseteq> M" "M \<subseteq> sigma_sets \<Omega> E"
hoelzl@47694:   shows "sigma_sets \<Omega> E = M"
hoelzl@40859: proof -
hoelzl@47694:   have "E \<subseteq> Pow \<Omega>"
hoelzl@41689:     using E sets_into_space by force
wenzelm@53374:   then have *: "sigma_sets \<Omega> E = dynkin \<Omega> E"
hoelzl@40859:     using `Int_stable E` by (rule sigma_eq_dynkin)
wenzelm@53374:   then have "dynkin \<Omega> E = M"
hoelzl@47694:     using assms dynkin_subset[OF E(1)] by simp
wenzelm@53374:   with * show ?thesis
hoelzl@47694:     using assms by (auto simp: dynkin_def)
hoelzl@42864: qed
hoelzl@42864: 
hoelzl@56994: subsubsection {* Induction rule for intersection-stable generators *}
hoelzl@56994: 
hoelzl@56994: text {* The reason to introduce Dynkin-systems is the following induction rules for $\sigma$-algebras
hoelzl@56994: generated by a generator closed under intersection. *}
hoelzl@56994: 
hoelzl@49789: lemma sigma_sets_induct_disjoint[consumes 3, case_names basic empty compl union]:
hoelzl@49789:   assumes "Int_stable G"
hoelzl@49789:     and closed: "G \<subseteq> Pow \<Omega>"
hoelzl@49789:     and A: "A \<in> sigma_sets \<Omega> G"
hoelzl@49789:   assumes basic: "\<And>A. A \<in> G \<Longrightarrow> P A"
hoelzl@49789:     and empty: "P {}"
hoelzl@49789:     and compl: "\<And>A. A \<in> sigma_sets \<Omega> G \<Longrightarrow> P A \<Longrightarrow> P (\<Omega> - A)"
hoelzl@49789:     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)"
hoelzl@49789:   shows "P A"
hoelzl@49789: proof -
hoelzl@49789:   let ?D = "{ A \<in> sigma_sets \<Omega> G. P A }"
hoelzl@49789:   interpret sigma_algebra \<Omega> "sigma_sets \<Omega> G"
hoelzl@49789:     using closed by (rule sigma_algebra_sigma_sets)
hoelzl@49789:   from compl[OF _ empty] closed have space: "P \<Omega>" by simp
hoelzl@49789:   interpret dynkin_system \<Omega> ?D
hoelzl@49789:     by default (auto dest: sets_into_space intro!: space compl union)
hoelzl@49789:   have "sigma_sets \<Omega> G = ?D"
hoelzl@49789:     by (rule dynkin_lemma) (auto simp: basic `Int_stable G`)
hoelzl@49789:   with A show ?thesis by auto
hoelzl@49789: qed
hoelzl@49789: 
hoelzl@56994: subsection {* Measure type *}
hoelzl@56994: 
hoelzl@56994: definition positive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
hoelzl@56994:   "positive M \<mu> \<longleftrightarrow> \<mu> {} = 0 \<and> (\<forall>A\<in>M. 0 \<le> \<mu> A)"
hoelzl@56994: 
hoelzl@56994: definition countably_additive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
hoelzl@56994:   "countably_additive M f \<longleftrightarrow> (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow>
hoelzl@56994:     (\<Sum>i. f (A i)) = f (\<Union>i. A i))"
hoelzl@56994: 
hoelzl@56994: definition measure_space :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
hoelzl@56994:   "measure_space \<Omega> A \<mu> \<longleftrightarrow> sigma_algebra \<Omega> A \<and> positive A \<mu> \<and> countably_additive A \<mu>"
hoelzl@56994: 
hoelzl@56994: typedef 'a measure = "{(\<Omega>::'a set, A, \<mu>). (\<forall>a\<in>-A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu> }"
hoelzl@56994: proof
hoelzl@56994:   have "sigma_algebra UNIV {{}, UNIV}"
hoelzl@56994:     by (auto simp: sigma_algebra_iff2)
hoelzl@56994:   then show "(UNIV, {{}, UNIV}, \<lambda>A. 0) \<in> {(\<Omega>, A, \<mu>). (\<forall>a\<in>-A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu>} "
hoelzl@56994:     by (auto simp: measure_space_def positive_def countably_additive_def)
hoelzl@56994: qed
hoelzl@56994: 
hoelzl@56994: definition space :: "'a measure \<Rightarrow> 'a set" where
hoelzl@56994:   "space M = fst (Rep_measure M)"
hoelzl@56994: 
hoelzl@56994: definition sets :: "'a measure \<Rightarrow> 'a set set" where
hoelzl@56994:   "sets M = fst (snd (Rep_measure M))"
hoelzl@56994: 
hoelzl@56994: definition emeasure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> ereal" where
hoelzl@56994:   "emeasure M = snd (snd (Rep_measure M))"
hoelzl@56994: 
hoelzl@56994: definition measure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> real" where
hoelzl@56994:   "measure M A = real (emeasure M A)"
hoelzl@56994: 
hoelzl@56994: declare [[coercion sets]]
hoelzl@56994: 
hoelzl@56994: declare [[coercion measure]]
hoelzl@56994: 
hoelzl@56994: declare [[coercion emeasure]]
hoelzl@56994: 
hoelzl@56994: lemma measure_space: "measure_space (space M) (sets M) (emeasure M)"
hoelzl@56994:   by (cases M) (auto simp: space_def sets_def emeasure_def Abs_measure_inverse)
hoelzl@56994: 
hoelzl@56994: interpretation sets!: sigma_algebra "space M" "sets M" for M :: "'a measure"
hoelzl@56994:   using measure_space[of M] by (auto simp: measure_space_def)
hoelzl@56994: 
hoelzl@56994: definition measure_of :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> 'a measure" where
hoelzl@56994:   "measure_of \<Omega> A \<mu> = Abs_measure (\<Omega>, if A \<subseteq> Pow \<Omega> then sigma_sets \<Omega> A else {{}, \<Omega>},
hoelzl@56994:     \<lambda>a. if a \<in> sigma_sets \<Omega> A \<and> measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> then \<mu> a else 0)"
hoelzl@56994: 
hoelzl@56994: abbreviation "sigma \<Omega> A \<equiv> measure_of \<Omega> A (\<lambda>x. 0)"
hoelzl@56994: 
hoelzl@56994: lemma measure_space_0: "A \<subseteq> Pow \<Omega> \<Longrightarrow> measure_space \<Omega> (sigma_sets \<Omega> A) (\<lambda>x. 0)"
hoelzl@56994:   unfolding measure_space_def
hoelzl@56994:   by (auto intro!: sigma_algebra_sigma_sets simp: positive_def countably_additive_def)
hoelzl@56994: 
hoelzl@56994: lemma sigma_algebra_trivial: "sigma_algebra \<Omega> {{}, \<Omega>}"
hoelzl@56994: by unfold_locales(fastforce intro: exI[where x="{{}}"] exI[where x="{\<Omega>}"])+
hoelzl@56994: 
hoelzl@56994: lemma measure_space_0': "measure_space \<Omega> {{}, \<Omega>} (\<lambda>x. 0)"
hoelzl@56994: by(simp add: measure_space_def positive_def countably_additive_def sigma_algebra_trivial)
hoelzl@56994: 
hoelzl@56994: lemma measure_space_closed:
hoelzl@56994:   assumes "measure_space \<Omega> M \<mu>"
hoelzl@56994:   shows "M \<subseteq> Pow \<Omega>"
hoelzl@56994: proof -
hoelzl@56994:   interpret sigma_algebra \<Omega> M using assms by(simp add: measure_space_def)
hoelzl@56994:   show ?thesis by(rule space_closed)
hoelzl@56994: qed
hoelzl@56994: 
hoelzl@56994: lemma (in ring_of_sets) positive_cong_eq:
hoelzl@56994:   "(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> positive M \<mu>' = positive M \<mu>"
hoelzl@56994:   by (auto simp add: positive_def)
hoelzl@56994: 
hoelzl@56994: lemma (in sigma_algebra) countably_additive_eq:
hoelzl@56994:   "(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> countably_additive M \<mu>' = countably_additive M \<mu>"
hoelzl@56994:   unfolding countably_additive_def
hoelzl@56994:   by (intro arg_cong[where f=All] ext) (auto simp add: countably_additive_def subset_eq)
hoelzl@56994: 
hoelzl@56994: lemma measure_space_eq:
hoelzl@56994:   assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a"
hoelzl@56994:   shows "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'"
hoelzl@56994: proof -
hoelzl@56994:   interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" using closed by (rule sigma_algebra_sigma_sets)
hoelzl@56994:   from positive_cong_eq[OF eq, of "\<lambda>i. i"] countably_additive_eq[OF eq, of "\<lambda>i. i"] show ?thesis
hoelzl@56994:     by (auto simp: measure_space_def)
hoelzl@56994: qed
hoelzl@56994: 
hoelzl@56994: lemma measure_of_eq:
hoelzl@56994:   assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "(\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a)"
hoelzl@56994:   shows "measure_of \<Omega> A \<mu> = measure_of \<Omega> A \<mu>'"
hoelzl@56994: proof -
hoelzl@56994:   have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'"
hoelzl@56994:     using assms by (rule measure_space_eq)
hoelzl@56994:   with eq show ?thesis
hoelzl@56994:     by (auto simp add: measure_of_def intro!: arg_cong[where f=Abs_measure])
hoelzl@56994: qed
hoelzl@56994: 
hoelzl@56994: lemma
hoelzl@56994:   shows space_measure_of_conv: "space (measure_of \<Omega> A \<mu>) = \<Omega>" (is ?space)
hoelzl@56994:   and sets_measure_of_conv:
hoelzl@56994:   "sets (measure_of \<Omega> A \<mu>) = (if A \<subseteq> Pow \<Omega> then sigma_sets \<Omega> A else {{}, \<Omega>})" (is ?sets)
hoelzl@56994:   and emeasure_measure_of_conv: 
hoelzl@56994:   "emeasure (measure_of \<Omega> A \<mu>) = 
hoelzl@56994:   (\<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)
hoelzl@56994: proof -
hoelzl@56994:   have "?space \<and> ?sets \<and> ?emeasure"
hoelzl@56994:   proof(cases "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>")
hoelzl@56994:     case True
hoelzl@56994:     from measure_space_closed[OF this] sigma_sets_superset_generator[of A \<Omega>]
hoelzl@56994:     have "A \<subseteq> Pow \<Omega>" by simp
hoelzl@56994:     hence "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A)
hoelzl@56994:       (\<lambda>a. if a \<in> sigma_sets \<Omega> A then \<mu> a else 0)"
hoelzl@56994:       by(rule measure_space_eq) auto
hoelzl@56994:     with True `A \<subseteq> Pow \<Omega>` show ?thesis
hoelzl@56994:       by(simp add: measure_of_def space_def sets_def emeasure_def Abs_measure_inverse)
hoelzl@56994:   next
hoelzl@56994:     case False thus ?thesis
hoelzl@56994:       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')
hoelzl@56994:   qed
hoelzl@56994:   thus ?space ?sets ?emeasure by simp_all
hoelzl@56994: qed
hoelzl@56994: 
hoelzl@56994: lemma [simp]:
hoelzl@56994:   assumes A: "A \<subseteq> Pow \<Omega>"
hoelzl@56994:   shows sets_measure_of: "sets (measure_of \<Omega> A \<mu>) = sigma_sets \<Omega> A"
hoelzl@56994:     and space_measure_of: "space (measure_of \<Omega> A \<mu>) = \<Omega>"
hoelzl@56994: using assms
hoelzl@56994: by(simp_all add: sets_measure_of_conv space_measure_of_conv)
hoelzl@56994: 
hoelzl@56994: lemma (in sigma_algebra) sets_measure_of_eq[simp]: "sets (measure_of \<Omega> M \<mu>) = M"
hoelzl@56994:   using space_closed by (auto intro!: sigma_sets_eq)
hoelzl@56994: 
hoelzl@56994: lemma (in sigma_algebra) space_measure_of_eq[simp]: "space (measure_of \<Omega> M \<mu>) = \<Omega>"
hoelzl@56994:   by (rule space_measure_of_conv)
hoelzl@56994: 
hoelzl@56994: 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>')"
hoelzl@56994:   by (auto intro!: sigma_sets_subseteq)
hoelzl@56994: 
hoelzl@56994: lemma sigma_sets_mono'':
hoelzl@56994:   assumes "A \<in> sigma_sets C D"
hoelzl@56994:   assumes "B \<subseteq> D"
hoelzl@56994:   assumes "D \<subseteq> Pow C"
hoelzl@56994:   shows "sigma_sets A B \<subseteq> sigma_sets C D"
hoelzl@56994: proof
hoelzl@56994:   fix x assume "x \<in> sigma_sets A B"
hoelzl@56994:   thus "x \<in> sigma_sets C D"
hoelzl@56994:   proof induct
hoelzl@56994:     case (Basic a) with assms have "a \<in> D" by auto
hoelzl@56994:     thus ?case ..
hoelzl@56994:   next
hoelzl@56994:     case Empty show ?case by (rule sigma_sets.Empty)
hoelzl@56994:   next
hoelzl@56994:     from assms have "A \<in> sets (sigma C D)" by (subst sets_measure_of[OF `D \<subseteq> Pow C`])
hoelzl@56994:     moreover case (Compl a) hence "a \<in> sets (sigma C D)" by (subst sets_measure_of[OF `D \<subseteq> Pow C`])
hoelzl@56994:     ultimately have "A - a \<in> sets (sigma C D)" ..
hoelzl@56994:     thus ?case by (subst (asm) sets_measure_of[OF `D \<subseteq> Pow C`])
hoelzl@56994:   next
hoelzl@56994:     case (Union a)
hoelzl@56994:     thus ?case by (intro sigma_sets.Union)
hoelzl@56994:   qed
hoelzl@56994: qed
hoelzl@56994: 
hoelzl@56994: lemma in_measure_of[intro, simp]: "M \<subseteq> Pow \<Omega> \<Longrightarrow> A \<in> M \<Longrightarrow> A \<in> sets (measure_of \<Omega> M \<mu>)"
hoelzl@56994:   by auto
hoelzl@56994: 
hoelzl@56994: subsubsection {* Constructing simple @{typ "'a measure"} *}
hoelzl@56994: 
hoelzl@56994: lemma emeasure_measure_of:
hoelzl@56994:   assumes M: "M = measure_of \<Omega> A \<mu>"
hoelzl@56994:   assumes ms: "A \<subseteq> Pow \<Omega>" "positive (sets M) \<mu>" "countably_additive (sets M) \<mu>"
hoelzl@56994:   assumes X: "X \<in> sets M"
hoelzl@56994:   shows "emeasure M X = \<mu> X"
hoelzl@56994: proof -
hoelzl@56994:   interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" by (rule sigma_algebra_sigma_sets) fact
hoelzl@56994:   have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>"
hoelzl@56994:     using ms M by (simp add: measure_space_def sigma_algebra_sigma_sets)
hoelzl@56994:   thus ?thesis using X ms
hoelzl@56994:     by(simp add: M emeasure_measure_of_conv sets_measure_of_conv)
hoelzl@56994: qed
hoelzl@56994: 
hoelzl@56994: lemma emeasure_measure_of_sigma:
hoelzl@56994:   assumes ms: "sigma_algebra \<Omega> M" "positive M \<mu>" "countably_additive M \<mu>"
hoelzl@56994:   assumes A: "A \<in> M"
hoelzl@56994:   shows "emeasure (measure_of \<Omega> M \<mu>) A = \<mu> A"
hoelzl@56994: proof -
hoelzl@56994:   interpret sigma_algebra \<Omega> M by fact
hoelzl@56994:   have "measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"
hoelzl@56994:     using ms sigma_sets_eq by (simp add: measure_space_def)
hoelzl@56994:   thus ?thesis by(simp add: emeasure_measure_of_conv A)
hoelzl@56994: qed
hoelzl@56994: 
hoelzl@56994: lemma measure_cases[cases type: measure]:
hoelzl@56994:   obtains (measure) \<Omega> A \<mu> where "x = Abs_measure (\<Omega>, A, \<mu>)" "\<forall>a\<in>-A. \<mu> a = 0" "measure_space \<Omega> A \<mu>"
hoelzl@56994:   by atomize_elim (cases x, auto)
hoelzl@56994: 
hoelzl@56994: lemma sets_eq_imp_space_eq:
hoelzl@56994:   "sets M = sets M' \<Longrightarrow> space M = space M'"
hoelzl@56994:   using sets.top[of M] sets.top[of M'] sets.space_closed[of M] sets.space_closed[of M']
hoelzl@56994:   by blast
hoelzl@56994: 
hoelzl@56994: lemma emeasure_notin_sets: "A \<notin> sets M \<Longrightarrow> emeasure M A = 0"
hoelzl@56994:   by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)
hoelzl@56994: 
hoelzl@56994: lemma emeasure_neq_0_sets: "emeasure M A \<noteq> 0 \<Longrightarrow> A \<in> sets M"
hoelzl@56994:   using emeasure_notin_sets[of A M] by blast
hoelzl@56994: 
hoelzl@56994: lemma measure_notin_sets: "A \<notin> sets M \<Longrightarrow> measure M A = 0"
hoelzl@56994:   by (simp add: measure_def emeasure_notin_sets)
hoelzl@56994: 
hoelzl@56994: lemma measure_eqI:
hoelzl@56994:   fixes M N :: "'a measure"
hoelzl@56994:   assumes "sets M = sets N" and eq: "\<And>A. A \<in> sets M \<Longrightarrow> emeasure M A = emeasure N A"
hoelzl@56994:   shows "M = N"
hoelzl@56994: proof (cases M N rule: measure_cases[case_product measure_cases])
hoelzl@56994:   case (measure_measure \<Omega> A \<mu> \<Omega>' A' \<mu>')
hoelzl@56994:   interpret M: sigma_algebra \<Omega> A using measure_measure by (auto simp: measure_space_def)
hoelzl@56994:   interpret N: sigma_algebra \<Omega>' A' using measure_measure by (auto simp: measure_space_def)
hoelzl@56994:   have "A = sets M" "A' = sets N"
hoelzl@56994:     using measure_measure by (simp_all add: sets_def Abs_measure_inverse)
hoelzl@56994:   with `sets M = sets N` have AA': "A = A'" by simp
hoelzl@56994:   moreover from M.top N.top M.space_closed N.space_closed AA' have "\<Omega> = \<Omega>'" by auto
hoelzl@56994:   moreover { fix B have "\<mu> B = \<mu>' B"
hoelzl@56994:     proof cases
hoelzl@56994:       assume "B \<in> A"
hoelzl@56994:       with eq `A = sets M` have "emeasure M B = emeasure N B" by simp
hoelzl@56994:       with measure_measure show "\<mu> B = \<mu>' B"
hoelzl@56994:         by (simp add: emeasure_def Abs_measure_inverse)
hoelzl@56994:     next
hoelzl@56994:       assume "B \<notin> A"
hoelzl@56994:       with `A = sets M` `A' = sets N` `A = A'` have "B \<notin> sets M" "B \<notin> sets N"
hoelzl@56994:         by auto
hoelzl@56994:       then have "emeasure M B = 0" "emeasure N B = 0"
hoelzl@56994:         by (simp_all add: emeasure_notin_sets)
hoelzl@56994:       with measure_measure show "\<mu> B = \<mu>' B"
hoelzl@56994:         by (simp add: emeasure_def Abs_measure_inverse)
hoelzl@56994:     qed }
hoelzl@56994:   then have "\<mu> = \<mu>'" by auto
hoelzl@56994:   ultimately show "M = N"
hoelzl@56994:     by (simp add: measure_measure)
hoelzl@56994: qed
hoelzl@56994: 
hoelzl@56994: lemma emeasure_sigma: "A \<subseteq> Pow \<Omega> \<Longrightarrow> emeasure (sigma \<Omega> A) = (\<lambda>_. 0)"
hoelzl@56994:   using measure_space_0[of A \<Omega>]
hoelzl@56994:   by (simp add: measure_of_def emeasure_def Abs_measure_inverse)
hoelzl@56994: 
hoelzl@56994: lemma sigma_eqI:
hoelzl@56994:   assumes [simp]: "M \<subseteq> Pow \<Omega>" "N \<subseteq> Pow \<Omega>" "sigma_sets \<Omega> M = sigma_sets \<Omega> N"
hoelzl@56994:   shows "sigma \<Omega> M = sigma \<Omega> N"
hoelzl@56994:   by (rule measure_eqI) (simp_all add: emeasure_sigma)
hoelzl@56994: 
hoelzl@56994: subsubsection {* Measurable functions *}
hoelzl@56994: 
hoelzl@56994: definition measurable :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) set" where
hoelzl@56994:   "measurable A B = {f \<in> space A -> space B. \<forall>y \<in> sets B. f -` y \<inter> space A \<in> sets A}"
hoelzl@56994: 
hoelzl@56994: lemma measurable_space:
hoelzl@56994:   "f \<in> measurable M A \<Longrightarrow> x \<in> space M \<Longrightarrow> f x \<in> space A"
hoelzl@56994:    unfolding measurable_def by auto
hoelzl@56994: 
hoelzl@56994: lemma measurable_sets:
hoelzl@56994:   "f \<in> measurable M A \<Longrightarrow> S \<in> sets A \<Longrightarrow> f -` S \<inter> space M \<in> sets M"
hoelzl@56994:    unfolding measurable_def by auto
hoelzl@56994: 
hoelzl@56994: lemma measurable_sets_Collect:
hoelzl@56994:   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"
hoelzl@56994: proof -
hoelzl@56994:   have "f -` {x \<in> space N. P x} \<inter> space M = {x\<in>space M. P (f x)}"
hoelzl@56994:     using measurable_space[OF f] by auto
hoelzl@56994:   with measurable_sets[OF f P] show ?thesis
hoelzl@56994:     by simp
hoelzl@56994: qed
hoelzl@56994: 
hoelzl@56994: lemma measurable_sigma_sets:
hoelzl@56994:   assumes B: "sets N = sigma_sets \<Omega> A" "A \<subseteq> Pow \<Omega>"
hoelzl@56994:       and f: "f \<in> space M \<rightarrow> \<Omega>"
hoelzl@56994:       and ba: "\<And>y. y \<in> A \<Longrightarrow> (f -` y) \<inter> space M \<in> sets M"
hoelzl@56994:   shows "f \<in> measurable M N"
hoelzl@56994: proof -
hoelzl@56994:   interpret A: sigma_algebra \<Omega> "sigma_sets \<Omega> A" using B(2) by (rule sigma_algebra_sigma_sets)
hoelzl@56994:   from B sets.top[of N] A.top sets.space_closed[of N] A.space_closed have \<Omega>: "\<Omega> = space N" by force
hoelzl@56994:   
hoelzl@56994:   { fix X assume "X \<in> sigma_sets \<Omega> A"
hoelzl@56994:     then have "f -` X \<inter> space M \<in> sets M \<and> X \<subseteq> \<Omega>"
hoelzl@56994:       proof induct
hoelzl@56994:         case (Basic a) then show ?case
hoelzl@56994:           by (auto simp add: ba) (metis B(2) subsetD PowD)
hoelzl@56994:       next
hoelzl@56994:         case (Compl a)
hoelzl@56994:         have [simp]: "f -` \<Omega> \<inter> space M = space M"
hoelzl@56994:           by (auto simp add: funcset_mem [OF f])
hoelzl@56994:         then show ?case
hoelzl@56994:           by (auto simp add: vimage_Diff Diff_Int_distrib2 sets.compl_sets Compl)
hoelzl@56994:       next
hoelzl@56994:         case (Union a)
hoelzl@56994:         then show ?case
hoelzl@56994:           by (simp add: vimage_UN, simp only: UN_extend_simps(4)) blast
hoelzl@56994:       qed auto }
hoelzl@56994:   with f show ?thesis
hoelzl@56994:     by (auto simp add: measurable_def B \<Omega>)
hoelzl@56994: qed
hoelzl@56994: 
hoelzl@56994: lemma measurable_measure_of:
hoelzl@56994:   assumes B: "N \<subseteq> Pow \<Omega>"
hoelzl@56994:       and f: "f \<in> space M \<rightarrow> \<Omega>"
hoelzl@56994:       and ba: "\<And>y. y \<in> N \<Longrightarrow> (f -` y) \<inter> space M \<in> sets M"
hoelzl@56994:   shows "f \<in> measurable M (measure_of \<Omega> N \<mu>)"
hoelzl@56994: proof -
hoelzl@56994:   have "sets (measure_of \<Omega> N \<mu>) = sigma_sets \<Omega> N"
hoelzl@56994:     using B by (rule sets_measure_of)
hoelzl@56994:   from this assms show ?thesis by (rule measurable_sigma_sets)
hoelzl@56994: qed
hoelzl@56994: 
hoelzl@56994: lemma measurable_iff_measure_of:
hoelzl@56994:   assumes "N \<subseteq> Pow \<Omega>" "f \<in> space M \<rightarrow> \<Omega>"
hoelzl@56994:   shows "f \<in> measurable M (measure_of \<Omega> N \<mu>) \<longleftrightarrow> (\<forall>A\<in>N. f -` A \<inter> space M \<in> sets M)"
hoelzl@56994:   by (metis assms in_measure_of measurable_measure_of assms measurable_sets)
hoelzl@56994: 
hoelzl@56994: lemma measurable_cong_sets:
hoelzl@56994:   assumes sets: "sets M = sets M'" "sets N = sets N'"
hoelzl@56994:   shows "measurable M N = measurable M' N'"
hoelzl@56994:   using sets[THEN sets_eq_imp_space_eq] sets by (simp add: measurable_def)
hoelzl@56994: 
hoelzl@56994: lemma measurable_cong:
hoelzl@56994:   assumes "\<And> w. w \<in> space M \<Longrightarrow> f w = g w"
hoelzl@56994:   shows "f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable M M'"
hoelzl@56994:   unfolding measurable_def using assms
hoelzl@56994:   by (simp cong: vimage_inter_cong Pi_cong)
hoelzl@56994: 
hoelzl@56994: lemma measurable_cong_strong:
hoelzl@56994:   "M = N \<Longrightarrow> M' = N' \<Longrightarrow> (\<And>w. w \<in> space M \<Longrightarrow> f w = g w) \<Longrightarrow>
hoelzl@56994:     f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable N N'"
hoelzl@56994:   by (metis measurable_cong)
hoelzl@56994: 
hoelzl@56994: lemma measurable_eqI:
hoelzl@56994:      "\<lbrakk> space m1 = space m1' ; space m2 = space m2' ;
hoelzl@56994:         sets m1 = sets m1' ; sets m2 = sets m2' \<rbrakk>
hoelzl@56994:       \<Longrightarrow> measurable m1 m2 = measurable m1' m2'"
hoelzl@56994:   by (simp add: measurable_def sigma_algebra_iff2)
hoelzl@56994: 
hoelzl@56994: lemma measurable_compose:
hoelzl@56994:   assumes f: "f \<in> measurable M N" and g: "g \<in> measurable N L"
hoelzl@56994:   shows "(\<lambda>x. g (f x)) \<in> measurable M L"
hoelzl@56994: proof -
hoelzl@56994:   have "\<And>A. (\<lambda>x. g (f x)) -` A \<inter> space M = f -` (g -` A \<inter> space N) \<inter> space M"
hoelzl@56994:     using measurable_space[OF f] by auto
hoelzl@56994:   with measurable_space[OF f] measurable_space[OF g] show ?thesis
hoelzl@56994:     by (auto intro: measurable_sets[OF f] measurable_sets[OF g]
hoelzl@56994:              simp del: vimage_Int simp add: measurable_def)
hoelzl@56994: qed
hoelzl@56994: 
hoelzl@56994: lemma measurable_comp:
hoelzl@56994:   "f \<in> measurable M N \<Longrightarrow> g \<in> measurable N L \<Longrightarrow> g \<circ> f \<in> measurable M L"
hoelzl@56994:   using measurable_compose[of f M N g L] by (simp add: comp_def)
hoelzl@56994: 
hoelzl@56994: lemma measurable_const:
hoelzl@56994:   "c \<in> space M' \<Longrightarrow> (\<lambda>x. c) \<in> measurable M M'"
hoelzl@56994:   by (auto simp add: measurable_def)
hoelzl@56994: 
hoelzl@56994: lemma measurable_If:
hoelzl@56994:   assumes measure: "f \<in> measurable M M'" "g \<in> measurable M M'"
hoelzl@56994:   assumes P: "{x\<in>space M. P x} \<in> sets M"
hoelzl@56994:   shows "(\<lambda>x. if P x then f x else g x) \<in> measurable M M'"
hoelzl@56994:   unfolding measurable_def
hoelzl@56994: proof safe
hoelzl@56994:   fix x assume "x \<in> space M"
hoelzl@56994:   thus "(if P x then f x else g x) \<in> space M'"
hoelzl@56994:     using measure unfolding measurable_def by auto
hoelzl@56994: next
hoelzl@56994:   fix A assume "A \<in> sets M'"
hoelzl@56994:   hence *: "(\<lambda>x. if P x then f x else g x) -` A \<inter> space M =
hoelzl@56994:     ((f -` A \<inter> space M) \<inter> {x\<in>space M. P x}) \<union>
hoelzl@56994:     ((g -` A \<inter> space M) \<inter> (space M - {x\<in>space M. P x}))"
hoelzl@56994:     using measure unfolding measurable_def by (auto split: split_if_asm)
hoelzl@56994:   show "(\<lambda>x. if P x then f x else g x) -` A \<inter> space M \<in> sets M"
hoelzl@56994:     using `A \<in> sets M'` measure P unfolding * measurable_def
hoelzl@56994:     by (auto intro!: sets.Un)
hoelzl@56994: qed
hoelzl@56994: 
hoelzl@56994: lemma measurable_If_set:
hoelzl@56994:   assumes measure: "f \<in> measurable M M'" "g \<in> measurable M M'"
hoelzl@56994:   assumes P: "A \<inter> space M \<in> sets M"
hoelzl@56994:   shows "(\<lambda>x. if x \<in> A then f x else g x) \<in> measurable M M'"
hoelzl@56994: proof (rule measurable_If[OF measure])
hoelzl@56994:   have "{x \<in> space M. x \<in> A} = A \<inter> space M" by auto
hoelzl@56994:   thus "{x \<in> space M. x \<in> A} \<in> sets M" using `A \<inter> space M \<in> sets M` by auto
hoelzl@56994: qed
hoelzl@56994: 
hoelzl@56994: lemma measurable_ident: "id \<in> measurable M M"
hoelzl@56994:   by (auto simp add: measurable_def)
hoelzl@56994: 
hoelzl@56994: lemma measurable_ident_sets:
hoelzl@56994:   assumes eq: "sets M = sets M'" shows "(\<lambda>x. x) \<in> measurable M M'"
hoelzl@56994:   using measurable_ident[of M]
hoelzl@56994:   unfolding id_def measurable_def eq sets_eq_imp_space_eq[OF eq] .
hoelzl@56994: 
hoelzl@56994: lemma sets_Least:
hoelzl@56994:   assumes meas: "\<And>i::nat. {x\<in>space M. P i x} \<in> M"
hoelzl@56994:   shows "(\<lambda>x. LEAST j. P j x) -` A \<inter> space M \<in> sets M"
hoelzl@56994: proof -
hoelzl@56994:   { fix i have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M \<in> sets M"
hoelzl@56994:     proof cases
hoelzl@56994:       assume i: "(LEAST j. False) = i"
hoelzl@56994:       have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M =
hoelzl@56994:         {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}))"
hoelzl@56994:         by (simp add: set_eq_iff, safe)
hoelzl@56994:            (insert i, auto dest: Least_le intro: LeastI intro!: Least_equality)
hoelzl@56994:       with meas show ?thesis
hoelzl@56994:         by (auto intro!: sets.Int)
hoelzl@56994:     next
hoelzl@56994:       assume i: "(LEAST j. False) \<noteq> i"
hoelzl@56994:       then have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M =
hoelzl@56994:         {x\<in>space M. P i x} \<inter> (space M - (\<Union>j<i. {x\<in>space M. P j x}))"
hoelzl@56994:       proof (simp add: set_eq_iff, safe)
hoelzl@56994:         fix x assume neq: "(LEAST j. False) \<noteq> (LEAST j. P j x)"
hoelzl@56994:         have "\<exists>j. P j x"
hoelzl@56994:           by (rule ccontr) (insert neq, auto)
hoelzl@56994:         then show "P (LEAST j. P j x) x" by (rule LeastI_ex)
hoelzl@56994:       qed (auto dest: Least_le intro!: Least_equality)
hoelzl@56994:       with meas show ?thesis
hoelzl@56994:         by auto
hoelzl@56994:     qed }
hoelzl@56994:   then have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M) \<in> sets M"
hoelzl@56994:     by (intro sets.countable_UN) auto
hoelzl@56994:   moreover have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M) =
hoelzl@56994:     (\<lambda>x. LEAST j. P j x) -` A \<inter> space M" by auto
hoelzl@56994:   ultimately show ?thesis by auto
hoelzl@56994: qed
hoelzl@56994: 
hoelzl@56994: lemma measurable_strong:
hoelzl@56994:   fixes f :: "'a \<Rightarrow> 'b" and g :: "'b \<Rightarrow> 'c"
hoelzl@56994:   assumes f: "f \<in> measurable a b" and g: "g \<in> space b \<rightarrow> space c"
hoelzl@56994:       and t: "f ` (space a) \<subseteq> t"
hoelzl@56994:       and cb: "\<And>s. s \<in> sets c \<Longrightarrow> (g -` s) \<inter> t \<in> sets b"
hoelzl@56994:   shows "(g o f) \<in> measurable a c"
hoelzl@56994: proof -
hoelzl@56994:   have fab: "f \<in> (space a -> space b)"
hoelzl@56994:    and ba: "\<And>y. y \<in> sets b \<Longrightarrow> (f -` y) \<inter> (space a) \<in> sets a" using f
hoelzl@56994:      by (auto simp add: measurable_def)
hoelzl@56994:   have eq: "\<And>y. (g \<circ> f) -` y \<inter> space a = f -` (g -` y \<inter> t) \<inter> space a" using t
hoelzl@56994:     by force
hoelzl@56994:   show ?thesis
hoelzl@56994:     apply (auto simp add: measurable_def vimage_comp)
hoelzl@56994:     apply (metis funcset_mem fab g)
hoelzl@56994:     apply (subst eq)
hoelzl@56994:     apply (metis ba cb)
hoelzl@56994:     done
hoelzl@56994: qed
hoelzl@56994: 
hoelzl@57275: lemma measurable_discrete_difference:
hoelzl@57275:   assumes f: "f \<in> measurable M N"
hoelzl@57275:   assumes X: "countable X"
hoelzl@57275:   assumes sets: "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"
hoelzl@57275:   assumes space: "\<And>x. x \<in> X \<Longrightarrow> g x \<in> space N"
hoelzl@57275:   assumes eq: "\<And>x. x \<in> space M \<Longrightarrow> x \<notin> X \<Longrightarrow> f x = g x"
hoelzl@57275:   shows "g \<in> measurable M N"
hoelzl@57275:   unfolding measurable_def
hoelzl@57275: proof safe
hoelzl@57275:   fix x assume "x \<in> space M" then show "g x \<in> space N"
hoelzl@57275:     using measurable_space[OF f, of x] eq[of x] space[of x] by (cases "x \<in> X") auto
hoelzl@57275: next
hoelzl@57275:   fix S assume S: "S \<in> sets N"
hoelzl@57275:   have "g -` S \<inter> space M = (f -` S \<inter> space M) - (\<Union>x\<in>X. {x}) \<union> (\<Union>x\<in>{x\<in>X. g x \<in> S}. {x})"
hoelzl@57275:     using sets.sets_into_space[OF sets] eq by auto
hoelzl@57275:   also have "\<dots> \<in> sets M"
hoelzl@57275:     by (safe intro!: sets.Diff sets.Un measurable_sets[OF f] S sets.countable_UN' X countable_Collect sets)
hoelzl@57275:   finally show "g -` S \<inter> space M \<in> sets M" .
hoelzl@57275: qed
hoelzl@57275: 
hoelzl@56994: lemma measurable_mono1:
hoelzl@56994:   "M' \<subseteq> Pow \<Omega> \<Longrightarrow> M \<subseteq> M' \<Longrightarrow>
hoelzl@56994:     measurable (measure_of \<Omega> M \<mu>) N \<subseteq> measurable (measure_of \<Omega> M' \<mu>') N"
hoelzl@56994:   using measure_of_subset[of M' \<Omega> M] by (auto simp add: measurable_def)
hoelzl@56994: 
hoelzl@56994: subsubsection {* Counting space *}
hoelzl@56994: 
hoelzl@56994: definition count_space :: "'a set \<Rightarrow> 'a measure" where
hoelzl@56994:   "count_space \<Omega> = measure_of \<Omega> (Pow \<Omega>) (\<lambda>A. if finite A then ereal (card A) else \<infinity>)"
hoelzl@56994: 
hoelzl@56994: lemma 
hoelzl@56994:   shows space_count_space[simp]: "space (count_space \<Omega>) = \<Omega>"
hoelzl@56994:     and sets_count_space[simp]: "sets (count_space \<Omega>) = Pow \<Omega>"
hoelzl@56994:   using sigma_sets_into_sp[of "Pow \<Omega>" \<Omega>]
hoelzl@56994:   by (auto simp: count_space_def)
hoelzl@56994: 
hoelzl@56994: lemma measurable_count_space_eq1[simp]:
hoelzl@56994:   "f \<in> measurable (count_space A) M \<longleftrightarrow> f \<in> A \<rightarrow> space M"
hoelzl@56994:  unfolding measurable_def by simp
hoelzl@56994: 
hoelzl@56994: lemma measurable_count_space_eq2:
hoelzl@56994:   assumes "finite A"
hoelzl@56994:   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))"
hoelzl@56994: proof -
hoelzl@56994:   { fix X assume "X \<subseteq> A" "f \<in> space M \<rightarrow> A"
hoelzl@56994:     with `finite A` have "f -` X \<inter> space M = (\<Union>a\<in>X. f -` {a} \<inter> space M)" "finite X"
hoelzl@56994:       by (auto dest: finite_subset)
hoelzl@56994:     moreover assume "\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M"
hoelzl@56994:     ultimately have "f -` X \<inter> space M \<in> sets M"
hoelzl@56994:       using `X \<subseteq> A` by (auto intro!: sets.finite_UN simp del: UN_simps) }
hoelzl@56994:   then show ?thesis
hoelzl@56994:     unfolding measurable_def by auto
hoelzl@56994: qed
hoelzl@56994: 
hoelzl@56994: lemma measurable_count_space_eq2_countable:
hoelzl@56994:   fixes f :: "'a => 'c::countable"
hoelzl@56994:   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))"
hoelzl@56994: proof -
hoelzl@56994:   { fix X assume "X \<subseteq> A" "f \<in> space M \<rightarrow> A"
hoelzl@56994:     assume *: "\<And>a. a\<in>A \<Longrightarrow> f -` {a} \<inter> space M \<in> sets M"
hoelzl@56994:     have "f -` X \<inter> space M = (\<Union>a\<in>X. f -` {a} \<inter> space M)"
hoelzl@56994:       by auto
hoelzl@56994:     also have "\<dots> \<in> sets M"
hoelzl@56994:       using * `X \<subseteq> A` by (intro sets.countable_UN) auto
hoelzl@56994:     finally have "f -` X \<inter> space M \<in> sets M" . }
hoelzl@56994:   then show ?thesis
hoelzl@56994:     unfolding measurable_def by auto
hoelzl@56994: qed
hoelzl@56994: 
hoelzl@56994: lemma measurable_compose_countable:
hoelzl@56994:   assumes f: "\<And>i::'i::countable. (\<lambda>x. f i x) \<in> measurable M N" and g: "g \<in> measurable M (count_space UNIV)"
hoelzl@56994:   shows "(\<lambda>x. f (g x) x) \<in> measurable M N"
hoelzl@56994:   unfolding measurable_def
hoelzl@56994: proof safe
hoelzl@56994:   fix x assume "x \<in> space M" then show "f (g x) x \<in> space N"
hoelzl@56994:     using f[THEN measurable_space] g[THEN measurable_space] by auto
hoelzl@56994: next
hoelzl@56994:   fix A assume A: "A \<in> sets N"
hoelzl@56994:   have "(\<lambda>x. f (g x) x) -` A \<inter> space M = (\<Union>i. (g -` {i} \<inter> space M) \<inter> (f i -` A \<inter> space M))"
hoelzl@56994:     by auto
hoelzl@56994:   also have "\<dots> \<in> sets M" using f[THEN measurable_sets, OF A] g[THEN measurable_sets]
hoelzl@56994:     by (auto intro!: sets.countable_UN measurable_sets)
hoelzl@56994:   finally show "(\<lambda>x. f (g x) x) -` A \<inter> space M \<in> sets M" .
hoelzl@56994: qed
hoelzl@56994: 
hoelzl@56994: lemma measurable_count_space_const:
hoelzl@56994:   "(\<lambda>x. c) \<in> measurable M (count_space UNIV)"
hoelzl@56994:   by (simp add: measurable_const)
hoelzl@56994: 
hoelzl@56994: lemma measurable_count_space:
hoelzl@56994:   "f \<in> measurable (count_space A) (count_space UNIV)"
hoelzl@56994:   by simp
hoelzl@56994: 
hoelzl@56994: lemma measurable_compose_rev:
hoelzl@56994:   assumes f: "f \<in> measurable L N" and g: "g \<in> measurable M L"
hoelzl@56994:   shows "(\<lambda>x. f (g x)) \<in> measurable M N"
hoelzl@56994:   using measurable_compose[OF g f] .
hoelzl@56994: 
hoelzl@56994: lemma measurable_count_space_eq_countable:
hoelzl@56994:   assumes "countable A"
hoelzl@56994:   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))"
hoelzl@56994: proof -
hoelzl@56994:   { fix X assume "X \<subseteq> A" "f \<in> space M \<rightarrow> A"
hoelzl@56994:     with `countable A` have "f -` X \<inter> space M = (\<Union>a\<in>X. f -` {a} \<inter> space M)" "countable X"
hoelzl@56994:       by (auto dest: countable_subset)
hoelzl@56994:     moreover assume "\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M"
hoelzl@56994:     ultimately have "f -` X \<inter> space M \<in> sets M"
hoelzl@56994:       using `X \<subseteq> A` by (auto intro!: sets.countable_UN' simp del: UN_simps) }
hoelzl@56994:   then show ?thesis
hoelzl@56994:     unfolding measurable_def by auto
hoelzl@56994: qed
hoelzl@56994: 
hoelzl@56994: subsubsection {* Extend measure *}
hoelzl@56994: 
hoelzl@56994: definition "extend_measure \<Omega> I G \<mu> =
hoelzl@56994:   (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)
hoelzl@56994:       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>')
hoelzl@56994:       else measure_of \<Omega> (G`I) (\<lambda>_. 0))"
hoelzl@56994: 
hoelzl@56994: lemma space_extend_measure: "G ` I \<subseteq> Pow \<Omega> \<Longrightarrow> space (extend_measure \<Omega> I G \<mu>) = \<Omega>"
hoelzl@56994:   unfolding extend_measure_def by simp
hoelzl@56994: 
hoelzl@56994: lemma sets_extend_measure: "G ` I \<subseteq> Pow \<Omega> \<Longrightarrow> sets (extend_measure \<Omega> I G \<mu>) = sigma_sets \<Omega> (G`I)"
hoelzl@56994:   unfolding extend_measure_def by simp
hoelzl@56994: 
hoelzl@56994: lemma emeasure_extend_measure:
hoelzl@56994:   assumes M: "M = extend_measure \<Omega> I G \<mu>"
hoelzl@56994:     and eq: "\<And>i. i \<in> I \<Longrightarrow> \<mu>' (G i) = \<mu> i"
hoelzl@56994:     and ms: "G ` I \<subseteq> Pow \<Omega>" "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'"
hoelzl@56994:     and "i \<in> I"
hoelzl@56994:   shows "emeasure M (G i) = \<mu> i"
hoelzl@56994: proof cases
hoelzl@56994:   assume *: "(\<forall>i\<in>I. \<mu> i = 0)"
hoelzl@56994:   with M have M_eq: "M = measure_of \<Omega> (G`I) (\<lambda>_. 0)"
hoelzl@56994:    by (simp add: extend_measure_def)
hoelzl@56994:   from measure_space_0[OF ms(1)] ms `i\<in>I`
hoelzl@56994:   have "emeasure M (G i) = 0"
hoelzl@56994:     by (intro emeasure_measure_of[OF M_eq]) (auto simp add: M measure_space_def sets_extend_measure)
hoelzl@56994:   with `i\<in>I` * show ?thesis
hoelzl@56994:     by simp
hoelzl@56994: next
hoelzl@56994:   def P \<equiv> "\<lambda>\<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G`I)) \<mu>'"
hoelzl@56994:   assume "\<not> (\<forall>i\<in>I. \<mu> i = 0)"
hoelzl@56994:   moreover
hoelzl@56994:   have "measure_space (space M) (sets M) \<mu>'"
hoelzl@56994:     using ms unfolding measure_space_def by auto default
hoelzl@56994:   with ms eq have "\<exists>\<mu>'. P \<mu>'"
hoelzl@56994:     unfolding P_def
hoelzl@56994:     by (intro exI[of _ \<mu>']) (auto simp add: M space_extend_measure sets_extend_measure)
hoelzl@56994:   ultimately have M_eq: "M = measure_of \<Omega> (G`I) (Eps P)"
hoelzl@56994:     by (simp add: M extend_measure_def P_def[symmetric])
hoelzl@56994: 
hoelzl@56994:   from `\<exists>\<mu>'. P \<mu>'` have P: "P (Eps P)" by (rule someI_ex)
hoelzl@56994:   show "emeasure M (G i) = \<mu> i"
hoelzl@56994:   proof (subst emeasure_measure_of[OF M_eq])
hoelzl@56994:     have sets_M: "sets M = sigma_sets \<Omega> (G`I)"
hoelzl@56994:       using M_eq ms by (auto simp: sets_extend_measure)
hoelzl@56994:     then show "G i \<in> sets M" using `i \<in> I` by auto
hoelzl@56994:     show "positive (sets M) (Eps P)" "countably_additive (sets M) (Eps P)" "Eps P (G i) = \<mu> i"
hoelzl@56994:       using P `i\<in>I` by (auto simp add: sets_M measure_space_def P_def)
hoelzl@56994:   qed fact
hoelzl@56994: qed
hoelzl@56994: 
hoelzl@56994: lemma emeasure_extend_measure_Pair:
hoelzl@56994:   assumes M: "M = extend_measure \<Omega> {(i, j). I i j} (\<lambda>(i, j). G i j) (\<lambda>(i, j). \<mu> i j)"
hoelzl@56994:     and eq: "\<And>i j. I i j \<Longrightarrow> \<mu>' (G i j) = \<mu> i j"
hoelzl@56994:     and ms: "\<And>i j. I i j \<Longrightarrow> G i j \<in> Pow \<Omega>" "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'"
hoelzl@56994:     and "I i j"
hoelzl@56994:   shows "emeasure M (G i j) = \<mu> i j"
hoelzl@56994:   using emeasure_extend_measure[OF M _ _ ms(2,3), of "(i,j)"] eq ms(1) `I i j`
hoelzl@56994:   by (auto simp: subset_eq)
hoelzl@56994: 
hoelzl@56994: subsubsection {* Sigma algebra generated by function preimages *}
hoelzl@56994: 
hoelzl@56994: definition
hoelzl@56994:   "vimage_algebra M S X = sigma S ((\<lambda>A. X -` A \<inter> S) ` sets M)"
hoelzl@56994: 
hoelzl@56994: lemma sigma_algebra_preimages:
hoelzl@56994:   fixes f :: "'x \<Rightarrow> 'a"
hoelzl@56994:   assumes "f \<in> S \<rightarrow> space M"
hoelzl@56994:   shows "sigma_algebra S ((\<lambda>A. f -` A \<inter> S) ` sets M)"
hoelzl@56994:     (is "sigma_algebra _ (?F ` sets M)")
hoelzl@56994: proof (simp add: sigma_algebra_iff2, safe)
hoelzl@56994:   show "{} \<in> ?F ` sets M" by blast
hoelzl@56994: next
hoelzl@56994:   fix A assume "A \<in> sets M"
hoelzl@56994:   moreover have "S - ?F A = ?F (space M - A)"
hoelzl@56994:     using assms by auto
hoelzl@56994:   ultimately show "S - ?F A \<in> ?F ` sets M"
hoelzl@56994:     by blast
hoelzl@56994: next
hoelzl@56994:   fix A :: "nat \<Rightarrow> 'x set" assume *: "range A \<subseteq> ?F ` M"
hoelzl@56994:   have "\<forall>i. \<exists>b. b \<in> M \<and> A i = ?F b"
hoelzl@56994:   proof safe
hoelzl@56994:     fix i
hoelzl@56994:     have "A i \<in> ?F ` M" using * by auto
hoelzl@56994:     then show "\<exists>b. b \<in> M \<and> A i = ?F b" by auto
hoelzl@56994:   qed
hoelzl@56994:   from choice[OF this] obtain b where b: "range b \<subseteq> M" "\<And>i. A i = ?F (b i)"
hoelzl@56994:     by auto
hoelzl@56994:   then have "(\<Union>i. A i) = ?F (\<Union>i. b i)" by auto
hoelzl@56994:   then show "(\<Union>i. A i) \<in> ?F ` M" using b(1) by blast
hoelzl@56994: qed
hoelzl@56994: 
hoelzl@56994: lemma sets_vimage_algebra[simp]:
hoelzl@56994:   "f \<in> S \<rightarrow> space M \<Longrightarrow> sets (vimage_algebra M S f) = (\<lambda>A. f -` A \<inter> S) ` sets M"
hoelzl@56994:   using sigma_algebra.sets_measure_of_eq[OF sigma_algebra_preimages, of f S M]
hoelzl@56994:   by (simp add: vimage_algebra_def)
hoelzl@56994: 
hoelzl@56994: lemma space_vimage_algebra[simp]:
hoelzl@56994:   "f \<in> S \<rightarrow> space M \<Longrightarrow> space (vimage_algebra M S f) = S"
hoelzl@56994:   using sigma_algebra.space_measure_of_eq[OF sigma_algebra_preimages, of f S M]
hoelzl@56994:   by (simp add: vimage_algebra_def)
hoelzl@56994: 
hoelzl@56994: lemma in_vimage_algebra[simp]:
hoelzl@56994:   "f \<in> S \<rightarrow> space M \<Longrightarrow> A \<in> sets (vimage_algebra M S f) \<longleftrightarrow> (\<exists>B\<in>sets M. A = f -` B \<inter> S)"
hoelzl@56994:   by (simp add: image_iff)
hoelzl@56994: 
hoelzl@56994: lemma measurable_vimage_algebra:
hoelzl@56994:   fixes S :: "'c set" assumes "f \<in> S \<rightarrow> space M"
hoelzl@56994:   shows "f \<in> measurable (vimage_algebra M S f) M"
hoelzl@56994:   unfolding measurable_def using assms by force
hoelzl@56994: 
hoelzl@56994: lemma measurable_vimage:
hoelzl@56994:   fixes g :: "'a \<Rightarrow> 'c" and f :: "'d \<Rightarrow> 'a"
hoelzl@56994:   assumes "g \<in> measurable M M2" "f \<in> S \<rightarrow> space M"
hoelzl@56994:   shows "(\<lambda>x. g (f x)) \<in> measurable (vimage_algebra M S f) M2"
hoelzl@56994: proof -
hoelzl@56994:   note measurable_vimage_algebra[OF assms(2)]
hoelzl@56994:   from measurable_comp[OF this assms(1)]
hoelzl@56994:   show ?thesis by (simp add: comp_def)
hoelzl@56994: qed
hoelzl@56994: 
hoelzl@56994: lemma sigma_sets_vimage:
hoelzl@56994:   assumes "f \<in> S' \<rightarrow> S" and "A \<subseteq> Pow S"
hoelzl@56994:   shows "sigma_sets S' ((\<lambda>X. f -` X \<inter> S') ` A) = (\<lambda>X. f -` X \<inter> S') ` sigma_sets S A"
hoelzl@56994: proof (intro set_eqI iffI)
hoelzl@56994:   let ?F = "\<lambda>X. f -` X \<inter> S'"
hoelzl@56994:   fix X assume "X \<in> sigma_sets S' (?F ` A)"
hoelzl@56994:   then show "X \<in> ?F ` sigma_sets S A"
hoelzl@56994:   proof induct
hoelzl@56994:     case (Basic X) then obtain X' where "X = ?F X'" "X' \<in> A"
hoelzl@56994:       by auto
hoelzl@56994:     then show ?case by auto
hoelzl@56994:   next
hoelzl@56994:     case Empty then show ?case
hoelzl@56994:       by (auto intro!: image_eqI[of _ _ "{}"] sigma_sets.Empty)
hoelzl@56994:   next
hoelzl@56994:     case (Compl X) then obtain X' where X: "X = ?F X'" and "X' \<in> sigma_sets S A"
hoelzl@56994:       by auto
hoelzl@56994:     then have "S - X' \<in> sigma_sets S A"
hoelzl@56994:       by (auto intro!: sigma_sets.Compl)
hoelzl@56994:     then show ?case
hoelzl@56994:       using X assms by (auto intro!: image_eqI[where x="S - X'"])
hoelzl@56994:   next
hoelzl@56994:     case (Union F)
hoelzl@56994:     then have "\<forall>i. \<exists>F'.  F' \<in> sigma_sets S A \<and> F i = f -` F' \<inter> S'"
hoelzl@56994:       by (auto simp: image_iff Bex_def)
hoelzl@56994:     from choice[OF this] obtain F' where
hoelzl@56994:       "\<And>i. F' i \<in> sigma_sets S A" and "\<And>i. F i = f -` F' i \<inter> S'"
hoelzl@56994:       by auto
hoelzl@56994:     then show ?case
hoelzl@56994:       by (auto intro!: sigma_sets.Union image_eqI[where x="\<Union>i. F' i"])
hoelzl@56994:   qed
hoelzl@56994: next
hoelzl@56994:   let ?F = "\<lambda>X. f -` X \<inter> S'"
hoelzl@56994:   fix X assume "X \<in> ?F ` sigma_sets S A"
hoelzl@56994:   then obtain X' where "X' \<in> sigma_sets S A" "X = ?F X'" by auto
hoelzl@56994:   then show "X \<in> sigma_sets S' (?F ` A)"
hoelzl@56994:   proof (induct arbitrary: X)
hoelzl@56994:     case Empty then show ?case by (auto intro: sigma_sets.Empty)
hoelzl@56994:   next
hoelzl@56994:     case (Compl X')
hoelzl@56994:     have "S' - (S' - X) \<in> sigma_sets S' (?F ` A)"
hoelzl@56994:       apply (rule sigma_sets.Compl)
hoelzl@56994:       using assms by (auto intro!: Compl.hyps simp: Compl.prems)
hoelzl@56994:     also have "S' - (S' - X) = X"
hoelzl@56994:       using assms Compl by auto
hoelzl@56994:     finally show ?case .
hoelzl@56994:   next
hoelzl@56994:     case (Union F)
hoelzl@56994:     have "(\<Union>i. f -` F i \<inter> S') \<in> sigma_sets S' (?F ` A)"
hoelzl@56994:       by (intro sigma_sets.Union Union.hyps) simp
hoelzl@56994:     also have "(\<Union>i. f -` F i \<inter> S') = X"
hoelzl@56994:       using assms Union by auto
hoelzl@56994:     finally show ?case .
hoelzl@56994:   qed auto
hoelzl@56994: qed
hoelzl@56994: 
hoelzl@56994: subsubsection {* Restricted Space Sigma Algebra *}
hoelzl@56994: 
hoelzl@57025: definition restrict_space where
hoelzl@57025:   "restrict_space M \<Omega> = measure_of (\<Omega> \<inter> space M) ((op \<inter> \<Omega>) ` sets M) (emeasure M)"
hoelzl@56994: 
hoelzl@57025: lemma space_restrict_space: "space (restrict_space M \<Omega>) = \<Omega> \<inter> space M"
hoelzl@57025:   using sets.sets_into_space unfolding restrict_space_def by (subst space_measure_of) auto
hoelzl@57025: 
hoelzl@57025: lemma space_restrict_space2: "\<Omega> \<in> sets M \<Longrightarrow> space (restrict_space M \<Omega>) = \<Omega>"
hoelzl@57025:   by (simp add: space_restrict_space sets.sets_into_space)
hoelzl@56994: 
hoelzl@57025: lemma sets_restrict_space: "sets (restrict_space M \<Omega>) = (op \<inter> \<Omega>) ` sets M"
hoelzl@57025: proof -
hoelzl@57025:   have "sigma_sets (\<Omega> \<inter> space M) ((\<lambda>X. X \<inter> (\<Omega> \<inter> space M)) ` sets M) =
hoelzl@57025:     (\<lambda>X. X \<inter> (\<Omega> \<inter> space M)) ` sets M"
hoelzl@57025:     using sigma_sets_vimage[of "\<lambda>x. x" "\<Omega> \<inter> space M" "space M" "sets M"] sets.space_closed[of M]
hoelzl@57025:     by (simp add: sets.sigma_sets_eq)
hoelzl@57025:   moreover have "(\<lambda>X. X \<inter> (\<Omega> \<inter> space M)) ` sets M = (op \<inter> \<Omega>) ` sets M"
hoelzl@57025:     using sets.sets_into_space by (intro image_cong) auto
hoelzl@57025:   ultimately show ?thesis
hoelzl@57025:     using sets.sets_into_space[of _ M] unfolding restrict_space_def
hoelzl@57025:     by (subst sets_measure_of) fastforce+
hoelzl@57025: qed
hoelzl@56994: 
hoelzl@56994: lemma sets_restrict_space_iff:
hoelzl@57025:   "\<Omega> \<inter> space M \<in> sets M \<Longrightarrow> A \<in> sets (restrict_space M \<Omega>) \<longleftrightarrow> (A \<subseteq> \<Omega> \<and> A \<in> sets M)"
hoelzl@57025: proof (subst sets_restrict_space, safe)
hoelzl@57025:   fix A assume "\<Omega> \<inter> space M \<in> sets M" and A: "A \<in> sets M"
hoelzl@57025:   then have "(\<Omega> \<inter> space M) \<inter> A \<in> sets M"
hoelzl@57025:     by rule
hoelzl@57025:   also have "(\<Omega> \<inter> space M) \<inter> A = \<Omega> \<inter> A"
hoelzl@57025:     using sets.sets_into_space[OF A] by auto
hoelzl@57025:   finally show "\<Omega> \<inter> A \<in> sets M"
hoelzl@57025:     by auto
hoelzl@57025: qed auto
hoelzl@56994: 
hoelzl@56994: lemma measurable_restrict_space1:
hoelzl@57025:   assumes \<Omega>: "\<Omega> \<inter> space M \<in> sets M" and f: "f \<in> measurable M N"
hoelzl@57025:   shows "f \<in> measurable (restrict_space M \<Omega>) N"
hoelzl@56994:   unfolding measurable_def
hoelzl@56994: proof (intro CollectI conjI ballI)
hoelzl@56994:   show sp: "f \<in> space (restrict_space M \<Omega>) \<rightarrow> space N"
hoelzl@56994:     using measurable_space[OF f] sets.sets_into_space[OF \<Omega>] by (auto simp: space_restrict_space)
hoelzl@56994: 
hoelzl@56994:   fix A assume "A \<in> sets N"
hoelzl@57025:   have "f -` A \<inter> space (restrict_space M \<Omega>) = (f -` A \<inter> space M) \<inter> (\<Omega> \<inter> space M)"
hoelzl@56994:     using sets.sets_into_space[OF \<Omega>] by (auto simp: space_restrict_space)
hoelzl@56994:   also have "\<dots> \<in> sets (restrict_space M \<Omega>)"
hoelzl@56994:     unfolding sets_restrict_space_iff[OF \<Omega>]
hoelzl@56994:     using measurable_sets[OF f `A \<in> sets N`] \<Omega> by blast
hoelzl@56994:   finally show "f -` A \<inter> space (restrict_space M \<Omega>) \<in> sets (restrict_space M \<Omega>)" .
hoelzl@56994: qed
hoelzl@56994: 
hoelzl@56994: lemma measurable_restrict_space2:
hoelzl@57137:   "\<Omega> \<inter> space N \<in> sets N \<Longrightarrow> f \<in> space M \<rightarrow> \<Omega> \<Longrightarrow> f \<in> measurable M N \<Longrightarrow>
hoelzl@57025:     f \<in> measurable M (restrict_space N \<Omega>)"
hoelzl@57137:   by (simp add: measurable_def space_restrict_space sets_restrict_space_iff Pi_Int[symmetric])
hoelzl@56994: 
hoelzl@57138: lemma measurable_restrict_space_iff:
hoelzl@57138:   assumes \<Omega>[simp, intro]: "\<Omega> \<inter> space M \<in> sets M" "c \<in> space N"
hoelzl@57138:   shows "f \<in> measurable (restrict_space M \<Omega>) N \<longleftrightarrow>
hoelzl@57138:     (\<lambda>x. if x \<in> \<Omega> then f x else c) \<in> measurable M N" (is "f \<in> measurable ?R N \<longleftrightarrow> ?f \<in> measurable M N")
hoelzl@57138:   unfolding measurable_def
hoelzl@57138: proof safe
hoelzl@57138:   fix x assume "f \<in> space ?R \<rightarrow> space N" "x \<in> space M" then show "?f x \<in> space N"
hoelzl@57138:     using `c\<in>space N` by (auto simp: space_restrict_space)
hoelzl@57138: next
hoelzl@57138:   fix x assume "?f \<in> space M \<rightarrow> space N" "x \<in> space ?R" then show "f x \<in> space N"
hoelzl@57138:     using `c\<in>space N` by (auto simp: space_restrict_space Pi_iff)
hoelzl@57138: next
hoelzl@57138:   fix X assume X: "X \<in> sets N"
hoelzl@57138:   assume *[THEN bspec]: "\<forall>y\<in>sets N. f -` y \<inter> space ?R \<in> sets ?R"
hoelzl@57138:   have "?f -` X \<inter> space M = (f -` X \<inter> (\<Omega> \<inter> space M)) \<union> (if c \<in> X then (space M - (\<Omega> \<inter> space M)) else {})"
hoelzl@57138:     by (auto split: split_if_asm)
hoelzl@57138:   also have "\<dots> \<in> sets M"
hoelzl@57138:     using *[OF X] by (auto simp add: space_restrict_space sets_restrict_space_iff)
hoelzl@57138:   finally show "?f -` X \<inter> space M \<in> sets M" .
hoelzl@57138: next
hoelzl@57138:   assume *[THEN bspec]: "\<forall>y\<in>sets N. ?f -` y \<inter> space M \<in> sets M"
hoelzl@57138:   fix X :: "'b set" assume X: "X \<in> sets N"
hoelzl@57138:   have "f -` X \<inter> (\<Omega> \<inter> space M) = (?f -` X \<inter> space M) \<inter> (\<Omega> \<inter> space M)"
hoelzl@57138:     by (auto simp: space_restrict_space)
hoelzl@57138:   also have "\<dots> \<in> sets M"
hoelzl@57138:     using *[OF X] by auto
hoelzl@57138:   finally show "f -` X \<inter> space ?R \<in> sets ?R"
hoelzl@57138:     by (auto simp add: sets_restrict_space_iff space_restrict_space)
hoelzl@57138: qed
hoelzl@57138: 
paulson@33271: end
hoelzl@57025: 




