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src/HOL/Probability/Measure_Space.thy

author | wenzelm |

Thu, 18 Apr 2013 17:07:01 +0200 | |

changeset 51717 | 9e7d1c139569 |

parent 51351 | dd1dd470690b |

child 53374 | a14d2a854c02 |

permissions | -rw-r--r-- |

simplifier uses proper Proof.context instead of historic type simpset;

(* Title: HOL/Probability/Measure_Space.thy Author: Lawrence C Paulson Author: Johannes Hölzl, TU München Author: Armin Heller, TU München *) header {* Measure spaces and their properties *} theory Measure_Space imports Measurable "~~/src/HOL/Multivariate_Analysis/Extended_Real_Limits" begin lemma sums_def2: "f sums x \<longleftrightarrow> (\<lambda>n. (\<Sum>i\<le>n. f i)) ----> x" unfolding sums_def apply (subst LIMSEQ_Suc_iff[symmetric]) unfolding atLeastLessThanSuc_atLeastAtMost atLeast0AtMost .. subsection "Relate extended reals and the indicator function" lemma ereal_indicator: "ereal (indicator A x) = indicator A x" by (auto simp: indicator_def one_ereal_def) lemma ereal_indicator_pos[simp,intro]: "0 \<le> (indicator A x ::ereal)" unfolding indicator_def by auto lemma LIMSEQ_indicator_UN: "(\<lambda>k. indicator (\<Union> i<k. A i) x) ----> (indicator (\<Union>i. A i) x :: real)" proof cases assume "\<exists>i. x \<in> A i" then guess i .. note i = this then have *: "\<And>n. (indicator (\<Union> i<n + Suc i. A i) x :: real) = 1" "(indicator (\<Union> i. A i) x :: real) = 1" by (auto simp: indicator_def) show ?thesis apply (rule LIMSEQ_offset[of _ "Suc i"]) unfolding * by auto qed (auto simp: indicator_def) lemma indicator_add: "A \<inter> B = {} \<Longrightarrow> (indicator A x::_::monoid_add) + indicator B x = indicator (A \<union> B) x" unfolding indicator_def by auto lemma suminf_cmult_indicator: fixes f :: "nat \<Rightarrow> ereal" assumes "disjoint_family A" "x \<in> A i" "\<And>i. 0 \<le> f i" shows "(\<Sum>n. f n * indicator (A n) x) = f i" proof - have **: "\<And>n. f n * indicator (A n) x = (if n = i then f n else 0 :: ereal)" using `x \<in> A i` assms unfolding disjoint_family_on_def indicator_def by auto then have "\<And>n. (\<Sum>j<n. f j * indicator (A j) x) = (if i < n then f i else 0 :: ereal)" by (auto simp: setsum_cases) moreover have "(SUP n. if i < n then f i else 0) = (f i :: ereal)" proof (rule SUP_eqI) fix y :: ereal assume "\<And>n. n \<in> UNIV \<Longrightarrow> (if i < n then f i else 0) \<le> y" from this[of "Suc i"] show "f i \<le> y" by auto qed (insert assms, simp) ultimately show ?thesis using assms by (subst suminf_ereal_eq_SUPR) (auto simp: indicator_def) qed lemma suminf_indicator: assumes "disjoint_family A" shows "(\<Sum>n. indicator (A n) x :: ereal) = indicator (\<Union>i. A i) x" proof cases assume *: "x \<in> (\<Union>i. A i)" then obtain i where "x \<in> A i" by auto from suminf_cmult_indicator[OF assms(1), OF `x \<in> A i`, of "\<lambda>k. 1"] show ?thesis using * by simp qed simp text {* The type for emeasure spaces is already defined in @{theory Sigma_Algebra}, as it is also used to represent sigma algebras (with an arbitrary emeasure). *} section "Extend binary sets" lemma LIMSEQ_binaryset: assumes f: "f {} = 0" shows "(\<lambda>n. \<Sum>i<n. f (binaryset A B i)) ----> f A + f B" proof - have "(\<lambda>n. \<Sum>i < Suc (Suc n). f (binaryset A B i)) = (\<lambda>n. f A + f B)" proof fix n show "(\<Sum>i < Suc (Suc n). f (binaryset A B i)) = f A + f B" by (induct n) (auto simp add: binaryset_def f) qed moreover have "... ----> f A + f B" by (rule tendsto_const) ultimately have "(\<lambda>n. \<Sum>i< Suc (Suc n). f (binaryset A B i)) ----> f A + f B" by metis hence "(\<lambda>n. \<Sum>i< n+2. f (binaryset A B i)) ----> f A + f B" by simp thus ?thesis by (rule LIMSEQ_offset [where k=2]) qed lemma binaryset_sums: assumes f: "f {} = 0" shows "(\<lambda>n. f (binaryset A B n)) sums (f A + f B)" by (simp add: sums_def LIMSEQ_binaryset [where f=f, OF f] atLeast0LessThan) lemma suminf_binaryset_eq: fixes f :: "'a set \<Rightarrow> 'b::{comm_monoid_add, t2_space}" shows "f {} = 0 \<Longrightarrow> (\<Sum>n. f (binaryset A B n)) = f A + f B" by (metis binaryset_sums sums_unique) section {* Properties of a premeasure @{term \<mu>} *} text {* The definitions for @{const positive} and @{const countably_additive} should be here, by they are necessary to define @{typ "'a measure"} in @{theory Sigma_Algebra}. *} definition additive where "additive M \<mu> \<longleftrightarrow> (\<forall>x\<in>M. \<forall>y\<in>M. x \<inter> y = {} \<longrightarrow> \<mu> (x \<union> y) = \<mu> x + \<mu> y)" definition increasing where "increasing M \<mu> \<longleftrightarrow> (\<forall>x\<in>M. \<forall>y\<in>M. x \<subseteq> y \<longrightarrow> \<mu> x \<le> \<mu> y)" lemma positiveD1: "positive M f \<Longrightarrow> f {} = 0" by (auto simp: positive_def) lemma positiveD2: "positive M f \<Longrightarrow> A \<in> M \<Longrightarrow> 0 \<le> f A" by (auto simp: positive_def) lemma positiveD_empty: "positive M f \<Longrightarrow> f {} = 0" by (auto simp add: positive_def) lemma additiveD: "additive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x \<in> M \<Longrightarrow> y \<in> M \<Longrightarrow> f (x \<union> y) = f x + f y" by (auto simp add: additive_def) lemma increasingD: "increasing M f \<Longrightarrow> x \<subseteq> y \<Longrightarrow> x\<in>M \<Longrightarrow> y\<in>M \<Longrightarrow> f x \<le> f y" by (auto simp add: increasing_def) lemma countably_additiveI[case_names countably]: "(\<And>A. range A \<subseteq> M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i. A i) \<in> M \<Longrightarrow> (\<Sum>i. f (A i)) = f (\<Union>i. A i)) \<Longrightarrow> countably_additive M f" by (simp add: countably_additive_def) lemma (in ring_of_sets) disjointed_additive: assumes f: "positive M f" "additive M f" and A: "range A \<subseteq> M" "incseq A" shows "(\<Sum>i\<le>n. f (disjointed A i)) = f (A n)" proof (induct n) case (Suc n) then have "(\<Sum>i\<le>Suc n. f (disjointed A i)) = f (A n) + f (disjointed A (Suc n))" by simp also have "\<dots> = f (A n \<union> disjointed A (Suc n))" using A by (subst f(2)[THEN additiveD]) (auto simp: disjointed_incseq) also have "A n \<union> disjointed A (Suc n) = A (Suc n)" using `incseq A` by (auto dest: incseq_SucD simp: disjointed_incseq) finally show ?case . qed simp lemma (in ring_of_sets) additive_sum: fixes A:: "'i \<Rightarrow> 'a set" assumes f: "positive M f" and ad: "additive M f" and "finite S" and A: "A`S \<subseteq> M" and disj: "disjoint_family_on A S" shows "(\<Sum>i\<in>S. f (A i)) = f (\<Union>i\<in>S. A i)" using `finite S` disj A proof induct case empty show ?case using f by (simp add: positive_def) next case (insert s S) then have "A s \<inter> (\<Union>i\<in>S. A i) = {}" by (auto simp add: disjoint_family_on_def neq_iff) moreover have "A s \<in> M" using insert by blast moreover have "(\<Union>i\<in>S. A i) \<in> M" using insert `finite S` by auto moreover ultimately have "f (A s \<union> (\<Union>i\<in>S. A i)) = f (A s) + f(\<Union>i\<in>S. A i)" using ad UNION_in_sets A by (auto simp add: additive_def) with insert show ?case using ad disjoint_family_on_mono[of S "insert s S" A] by (auto simp add: additive_def subset_insertI) qed lemma (in ring_of_sets) additive_increasing: assumes posf: "positive M f" and addf: "additive M f" shows "increasing M f" proof (auto simp add: increasing_def) fix x y assume xy: "x \<in> M" "y \<in> M" "x \<subseteq> y" then have "y - x \<in> M" by auto then have "0 \<le> f (y-x)" using posf[unfolded positive_def] by auto then have "f x + 0 \<le> f x + f (y-x)" by (intro add_left_mono) auto also have "... = f (x \<union> (y-x))" using addf by (auto simp add: additive_def) (metis Diff_disjoint Un_Diff_cancel Diff xy(1,2)) also have "... = f y" by (metis Un_Diff_cancel Un_absorb1 xy(3)) finally show "f x \<le> f y" by simp qed lemma (in ring_of_sets) subadditive: assumes f: "positive M f" "additive M f" and A: "range A \<subseteq> M" and S: "finite S" shows "f (\<Union>i\<in>S. A i) \<le> (\<Sum>i\<in>S. f (A i))" using S proof (induct S) case empty thus ?case using f by (auto simp: positive_def) next case (insert x F) hence in_M: "A x \<in> M" "(\<Union> i\<in>F. A i) \<in> M" "(\<Union> i\<in>F. A i) - A x \<in> M" using A by force+ have subs: "(\<Union> i\<in>F. A i) - A x \<subseteq> (\<Union> i\<in>F. A i)" by auto have "(\<Union> i\<in>(insert x F). A i) = A x \<union> ((\<Union> i\<in>F. A i) - A x)" by auto hence "f (\<Union> i\<in>(insert x F). A i) = f (A x \<union> ((\<Union> i\<in>F. A i) - A x))" by simp also have "\<dots> = f (A x) + f ((\<Union> i\<in>F. A i) - A x)" using f(2) by (rule additiveD) (insert in_M, auto) also have "\<dots> \<le> f (A x) + f (\<Union> i\<in>F. A i)" using additive_increasing[OF f] in_M subs by (auto simp: increasing_def intro: add_left_mono) also have "\<dots> \<le> f (A x) + (\<Sum>i\<in>F. f (A i))" using insert by (auto intro: add_left_mono) finally show "f (\<Union> i\<in>(insert x F). A i) \<le> (\<Sum>i\<in>(insert x F). f (A i))" using insert by simp qed lemma (in ring_of_sets) countably_additive_additive: assumes posf: "positive M f" and ca: "countably_additive M f" shows "additive M f" proof (auto simp add: additive_def) fix x y assume x: "x \<in> M" and y: "y \<in> M" and "x \<inter> y = {}" hence "disjoint_family (binaryset x y)" by (auto simp add: disjoint_family_on_def binaryset_def) hence "range (binaryset x y) \<subseteq> M \<longrightarrow> (\<Union>i. binaryset x y i) \<in> M \<longrightarrow> f (\<Union>i. binaryset x y i) = (\<Sum> n. f (binaryset x y n))" using ca by (simp add: countably_additive_def) hence "{x,y,{}} \<subseteq> M \<longrightarrow> x \<union> y \<in> M \<longrightarrow> f (x \<union> y) = (\<Sum>n. f (binaryset x y n))" by (simp add: range_binaryset_eq UN_binaryset_eq) thus "f (x \<union> y) = f x + f y" using posf x y by (auto simp add: Un suminf_binaryset_eq positive_def) qed lemma (in algebra) increasing_additive_bound: fixes A:: "nat \<Rightarrow> 'a set" and f :: "'a set \<Rightarrow> ereal" assumes f: "positive M f" and ad: "additive M f" and inc: "increasing M f" and A: "range A \<subseteq> M" and disj: "disjoint_family A" shows "(\<Sum>i. f (A i)) \<le> f \<Omega>" proof (safe intro!: suminf_bound) fix N note disj_N = disjoint_family_on_mono[OF _ disj, of "{..<N}"] have "(\<Sum>i<N. f (A i)) = f (\<Union>i\<in>{..<N}. A i)" using A by (intro additive_sum [OF f ad _ _]) (auto simp: disj_N) also have "... \<le> f \<Omega>" using space_closed A by (intro increasingD[OF inc] finite_UN) auto finally show "(\<Sum>i<N. f (A i)) \<le> f \<Omega>" by simp qed (insert f A, auto simp: positive_def) lemma (in ring_of_sets) countably_additiveI_finite: assumes "finite \<Omega>" "positive M \<mu>" "additive M \<mu>" shows "countably_additive M \<mu>" proof (rule countably_additiveI) fix F :: "nat \<Rightarrow> 'a set" assume F: "range F \<subseteq> M" "(\<Union>i. F i) \<in> M" and disj: "disjoint_family F" have "\<forall>i\<in>{i. F i \<noteq> {}}. \<exists>x. x \<in> F i" by auto from bchoice[OF this] obtain f where f: "\<And>i. F i \<noteq> {} \<Longrightarrow> f i \<in> F i" by auto have inj_f: "inj_on f {i. F i \<noteq> {}}" proof (rule inj_onI, simp) fix i j a b assume *: "f i = f j" "F i \<noteq> {}" "F j \<noteq> {}" then have "f i \<in> F i" "f j \<in> F j" using f by force+ with disj * show "i = j" by (auto simp: disjoint_family_on_def) qed have "finite (\<Union>i. F i)" by (metis F(2) assms(1) infinite_super sets_into_space) have F_subset: "{i. \<mu> (F i) \<noteq> 0} \<subseteq> {i. F i \<noteq> {}}" by (auto simp: positiveD_empty[OF `positive M \<mu>`]) moreover have fin_not_empty: "finite {i. F i \<noteq> {}}" proof (rule finite_imageD) from f have "f`{i. F i \<noteq> {}} \<subseteq> (\<Union>i. F i)" by auto then show "finite (f`{i. F i \<noteq> {}})" by (rule finite_subset) fact qed fact ultimately have fin_not_0: "finite {i. \<mu> (F i) \<noteq> 0}" by (rule finite_subset) have disj_not_empty: "disjoint_family_on F {i. F i \<noteq> {}}" using disj by (auto simp: disjoint_family_on_def) from fin_not_0 have "(\<Sum>i. \<mu> (F i)) = (\<Sum>i | \<mu> (F i) \<noteq> 0. \<mu> (F i))" by (rule suminf_finite) auto also have "\<dots> = (\<Sum>i | F i \<noteq> {}. \<mu> (F i))" using fin_not_empty F_subset by (rule setsum_mono_zero_left) auto also have "\<dots> = \<mu> (\<Union>i\<in>{i. F i \<noteq> {}}. F i)" using `positive M \<mu>` `additive M \<mu>` fin_not_empty disj_not_empty F by (intro additive_sum) auto also have "\<dots> = \<mu> (\<Union>i. F i)" by (rule arg_cong[where f=\<mu>]) auto finally show "(\<Sum>i. \<mu> (F i)) = \<mu> (\<Union>i. F i)" . qed lemma (in ring_of_sets) countably_additive_iff_continuous_from_below: assumes f: "positive M f" "additive M f" shows "countably_additive M f \<longleftrightarrow> (\<forall>A. range A \<subseteq> M \<longrightarrow> incseq A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Union>i. A i))" unfolding countably_additive_def proof safe assume count_sum: "\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> UNION UNIV A \<in> M \<longrightarrow> (\<Sum>i. f (A i)) = f (UNION UNIV A)" fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "incseq A" "(\<Union>i. A i) \<in> M" then have dA: "range (disjointed A) \<subseteq> M" by (auto simp: range_disjointed_sets) with count_sum[THEN spec, of "disjointed A"] A(3) have f_UN: "(\<Sum>i. f (disjointed A i)) = f (\<Union>i. A i)" by (auto simp: UN_disjointed_eq disjoint_family_disjointed) moreover have "(\<lambda>n. (\<Sum>i=0..<n. f (disjointed A i))) ----> (\<Sum>i. f (disjointed A i))" using f(1)[unfolded positive_def] dA by (auto intro!: summable_sumr_LIMSEQ_suminf summable_ereal_pos) from LIMSEQ_Suc[OF this] have "(\<lambda>n. (\<Sum>i\<le>n. f (disjointed A i))) ----> (\<Sum>i. f (disjointed A i))" unfolding atLeastLessThanSuc_atLeastAtMost atLeast0AtMost . moreover have "\<And>n. (\<Sum>i\<le>n. f (disjointed A i)) = f (A n)" using disjointed_additive[OF f A(1,2)] . ultimately show "(\<lambda>i. f (A i)) ----> f (\<Union>i. A i)" by simp next assume cont: "\<forall>A. range A \<subseteq> M \<longrightarrow> incseq A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Union>i. A i)" fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "disjoint_family A" "(\<Union>i. A i) \<in> M" have *: "(\<Union>n. (\<Union>i\<le>n. A i)) = (\<Union>i. A i)" by auto have "(\<lambda>n. f (\<Union>i\<le>n. A i)) ----> f (\<Union>i. A i)" proof (unfold *[symmetric], intro cont[rule_format]) show "range (\<lambda>i. \<Union> i\<le>i. A i) \<subseteq> M" "(\<Union>i. \<Union> i\<le>i. A i) \<in> M" using A * by auto qed (force intro!: incseq_SucI) moreover have "\<And>n. f (\<Union>i\<le>n. A i) = (\<Sum>i\<le>n. f (A i))" using A by (intro additive_sum[OF f, of _ A, symmetric]) (auto intro: disjoint_family_on_mono[where B=UNIV]) ultimately have "(\<lambda>i. f (A i)) sums f (\<Union>i. A i)" unfolding sums_def2 by simp from sums_unique[OF this] show "(\<Sum>i. f (A i)) = f (\<Union>i. A i)" by simp qed lemma (in ring_of_sets) continuous_from_above_iff_empty_continuous: assumes f: "positive M f" "additive M f" shows "(\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) \<in> M \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Inter>i. A i)) \<longleftrightarrow> (\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> 0)" proof safe assume cont: "(\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) \<in> M \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Inter>i. A i))" fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "decseq A" "(\<Inter>i. A i) = {}" "\<forall>i. f (A i) \<noteq> \<infinity>" with cont[THEN spec, of A] show "(\<lambda>i. f (A i)) ----> 0" using `positive M f`[unfolded positive_def] by auto next assume cont: "\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> 0" fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "decseq A" "(\<Inter>i. A i) \<in> M" "\<forall>i. f (A i) \<noteq> \<infinity>" have f_mono: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<subseteq> b \<Longrightarrow> f a \<le> f b" using additive_increasing[OF f] unfolding increasing_def by simp have decseq_fA: "decseq (\<lambda>i. f (A i))" using A by (auto simp: decseq_def intro!: f_mono) have decseq: "decseq (\<lambda>i. A i - (\<Inter>i. A i))" using A by (auto simp: decseq_def) then have decseq_f: "decseq (\<lambda>i. f (A i - (\<Inter>i. A i)))" using A unfolding decseq_def by (auto intro!: f_mono Diff) have "f (\<Inter>x. A x) \<le> f (A 0)" using A by (auto intro!: f_mono) then have f_Int_fin: "f (\<Inter>x. A x) \<noteq> \<infinity>" using A by auto { fix i have "f (A i - (\<Inter>i. A i)) \<le> f (A i)" using A by (auto intro!: f_mono) then have "f (A i - (\<Inter>i. A i)) \<noteq> \<infinity>" using A by auto } note f_fin = this have "(\<lambda>i. f (A i - (\<Inter>i. A i))) ----> 0" proof (intro cont[rule_format, OF _ decseq _ f_fin]) show "range (\<lambda>i. A i - (\<Inter>i. A i)) \<subseteq> M" "(\<Inter>i. A i - (\<Inter>i. A i)) = {}" using A by auto qed from INF_Lim_ereal[OF decseq_f this] have "(INF n. f (A n - (\<Inter>i. A i))) = 0" . moreover have "(INF n. f (\<Inter>i. A i)) = f (\<Inter>i. A i)" by auto ultimately have "(INF n. f (A n - (\<Inter>i. A i)) + f (\<Inter>i. A i)) = 0 + f (\<Inter>i. A i)" using A(4) f_fin f_Int_fin by (subst INFI_ereal_add) (auto simp: decseq_f) moreover { fix n have "f (A n - (\<Inter>i. A i)) + f (\<Inter>i. A i) = f ((A n - (\<Inter>i. A i)) \<union> (\<Inter>i. A i))" using A by (subst f(2)[THEN additiveD]) auto also have "(A n - (\<Inter>i. A i)) \<union> (\<Inter>i. A i) = A n" by auto finally have "f (A n - (\<Inter>i. A i)) + f (\<Inter>i. A i) = f (A n)" . } ultimately have "(INF n. f (A n)) = f (\<Inter>i. A i)" by simp with LIMSEQ_INF[OF decseq_fA] show "(\<lambda>i. f (A i)) ----> f (\<Inter>i. A i)" by simp qed lemma (in ring_of_sets) empty_continuous_imp_continuous_from_below: assumes f: "positive M f" "additive M f" "\<forall>A\<in>M. f A \<noteq> \<infinity>" assumes cont: "\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<lambda>i. f (A i)) ----> 0" assumes A: "range A \<subseteq> M" "incseq A" "(\<Union>i. A i) \<in> M" shows "(\<lambda>i. f (A i)) ----> f (\<Union>i. A i)" proof - have "\<forall>A\<in>M. \<exists>x. f A = ereal x" proof fix A assume "A \<in> M" with f show "\<exists>x. f A = ereal x" unfolding positive_def by (cases "f A") auto qed from bchoice[OF this] guess f' .. note f' = this[rule_format] from A have "(\<lambda>i. f ((\<Union>i. A i) - A i)) ----> 0" by (intro cont[rule_format]) (auto simp: decseq_def incseq_def) moreover { fix i have "f ((\<Union>i. A i) - A i) + f (A i) = f ((\<Union>i. A i) - A i \<union> A i)" using A by (intro f(2)[THEN additiveD, symmetric]) auto also have "(\<Union>i. A i) - A i \<union> A i = (\<Union>i. A i)" by auto finally have "f' (\<Union>i. A i) - f' (A i) = f' ((\<Union>i. A i) - A i)" using A by (subst (asm) (1 2 3) f') auto then have "f ((\<Union>i. A i) - A i) = ereal (f' (\<Union>i. A i) - f' (A i))" using A f' by auto } ultimately have "(\<lambda>i. f' (\<Union>i. A i) - f' (A i)) ----> 0" by (simp add: zero_ereal_def) then have "(\<lambda>i. f' (A i)) ----> f' (\<Union>i. A i)" by (rule LIMSEQ_diff_approach_zero2[OF tendsto_const]) then show "(\<lambda>i. f (A i)) ----> f (\<Union>i. A i)" using A by (subst (1 2) f') auto qed lemma (in ring_of_sets) empty_continuous_imp_countably_additive: assumes f: "positive M f" "additive M f" and fin: "\<forall>A\<in>M. f A \<noteq> \<infinity>" assumes cont: "\<And>A. range A \<subseteq> M \<Longrightarrow> decseq A \<Longrightarrow> (\<Inter>i. A i) = {} \<Longrightarrow> (\<lambda>i. f (A i)) ----> 0" shows "countably_additive M f" using countably_additive_iff_continuous_from_below[OF f] using empty_continuous_imp_continuous_from_below[OF f fin] cont by blast section {* Properties of @{const emeasure} *} lemma emeasure_positive: "positive (sets M) (emeasure M)" by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def) lemma emeasure_empty[simp, intro]: "emeasure M {} = 0" using emeasure_positive[of M] by (simp add: positive_def) lemma emeasure_nonneg[intro!]: "0 \<le> emeasure M A" using emeasure_notin_sets[of A M] emeasure_positive[of M] by (cases "A \<in> sets M") (auto simp: positive_def) lemma emeasure_not_MInf[simp]: "emeasure M A \<noteq> - \<infinity>" using emeasure_nonneg[of M A] by auto lemma emeasure_le_0_iff: "emeasure M A \<le> 0 \<longleftrightarrow> emeasure M A = 0" using emeasure_nonneg[of M A] by auto lemma emeasure_less_0_iff: "emeasure M A < 0 \<longleftrightarrow> False" using emeasure_nonneg[of M A] by auto lemma emeasure_countably_additive: "countably_additive (sets M) (emeasure M)" by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def) lemma suminf_emeasure: "range A \<subseteq> sets M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Sum>i. emeasure M (A i)) = emeasure M (\<Union>i. A i)" using sets.countable_UN[of A UNIV M] emeasure_countably_additive[of M] by (simp add: countably_additive_def) lemma emeasure_additive: "additive (sets M) (emeasure M)" by (metis sets.countably_additive_additive emeasure_positive emeasure_countably_additive) lemma plus_emeasure: "a \<in> sets M \<Longrightarrow> b \<in> sets M \<Longrightarrow> a \<inter> b = {} \<Longrightarrow> emeasure M a + emeasure M b = emeasure M (a \<union> b)" using additiveD[OF emeasure_additive] .. lemma setsum_emeasure: "F`I \<subseteq> sets M \<Longrightarrow> disjoint_family_on F I \<Longrightarrow> finite I \<Longrightarrow> (\<Sum>i\<in>I. emeasure M (F i)) = emeasure M (\<Union>i\<in>I. F i)" by (metis sets.additive_sum emeasure_positive emeasure_additive) lemma emeasure_mono: "a \<subseteq> b \<Longrightarrow> b \<in> sets M \<Longrightarrow> emeasure M a \<le> emeasure M b" by (metis sets.additive_increasing emeasure_additive emeasure_nonneg emeasure_notin_sets emeasure_positive increasingD) lemma emeasure_space: "emeasure M A \<le> emeasure M (space M)" by (metis emeasure_mono emeasure_nonneg emeasure_notin_sets sets.sets_into_space sets.top) lemma emeasure_compl: assumes s: "s \<in> sets M" and fin: "emeasure M s \<noteq> \<infinity>" shows "emeasure M (space M - s) = emeasure M (space M) - emeasure M s" proof - from s have "0 \<le> emeasure M s" by auto have "emeasure M (space M) = emeasure M (s \<union> (space M - s))" using s by (metis Un_Diff_cancel Un_absorb1 s sets.sets_into_space) also have "... = emeasure M s + emeasure M (space M - s)" by (rule plus_emeasure[symmetric]) (auto simp add: s) finally have "emeasure M (space M) = emeasure M s + emeasure M (space M - s)" . then show ?thesis using fin `0 \<le> emeasure M s` unfolding ereal_eq_minus_iff by (auto simp: ac_simps) qed lemma emeasure_Diff: assumes finite: "emeasure M B \<noteq> \<infinity>" and [measurable]: "A \<in> sets M" "B \<in> sets M" and "B \<subseteq> A" shows "emeasure M (A - B) = emeasure M A - emeasure M B" proof - have "0 \<le> emeasure M B" using assms by auto have "(A - B) \<union> B = A" using `B \<subseteq> A` by auto then have "emeasure M A = emeasure M ((A - B) \<union> B)" by simp also have "\<dots> = emeasure M (A - B) + emeasure M B" by (subst plus_emeasure[symmetric]) auto finally show "emeasure M (A - B) = emeasure M A - emeasure M B" unfolding ereal_eq_minus_iff using finite `0 \<le> emeasure M B` by auto qed lemma Lim_emeasure_incseq: "range A \<subseteq> sets M \<Longrightarrow> incseq A \<Longrightarrow> (\<lambda>i. (emeasure M (A i))) ----> emeasure M (\<Union>i. A i)" using emeasure_countably_additive by (auto simp add: sets.countably_additive_iff_continuous_from_below emeasure_positive emeasure_additive) lemma incseq_emeasure: assumes "range B \<subseteq> sets M" "incseq B" shows "incseq (\<lambda>i. emeasure M (B i))" using assms by (auto simp: incseq_def intro!: emeasure_mono) lemma SUP_emeasure_incseq: assumes A: "range A \<subseteq> sets M" "incseq A" shows "(SUP n. emeasure M (A n)) = emeasure M (\<Union>i. A i)" using LIMSEQ_SUP[OF incseq_emeasure, OF A] Lim_emeasure_incseq[OF A] by (simp add: LIMSEQ_unique) lemma decseq_emeasure: assumes "range B \<subseteq> sets M" "decseq B" shows "decseq (\<lambda>i. emeasure M (B i))" using assms by (auto simp: decseq_def intro!: emeasure_mono) lemma INF_emeasure_decseq: assumes A: "range A \<subseteq> sets M" and "decseq A" and finite: "\<And>i. emeasure M (A i) \<noteq> \<infinity>" shows "(INF n. emeasure M (A n)) = emeasure M (\<Inter>i. A i)" proof - have le_MI: "emeasure M (\<Inter>i. A i) \<le> emeasure M (A 0)" using A by (auto intro!: emeasure_mono) hence *: "emeasure M (\<Inter>i. A i) \<noteq> \<infinity>" using finite[of 0] by auto have A0: "0 \<le> emeasure M (A 0)" using A by auto have "emeasure M (A 0) - (INF n. emeasure M (A n)) = emeasure M (A 0) + (SUP n. - emeasure M (A n))" by (simp add: ereal_SUPR_uminus minus_ereal_def) also have "\<dots> = (SUP n. emeasure M (A 0) - emeasure M (A n))" unfolding minus_ereal_def using A0 assms by (subst SUPR_ereal_add) (auto simp add: decseq_emeasure) also have "\<dots> = (SUP n. emeasure M (A 0 - A n))" using A finite `decseq A`[unfolded decseq_def] by (subst emeasure_Diff) auto also have "\<dots> = emeasure M (\<Union>i. A 0 - A i)" proof (rule SUP_emeasure_incseq) show "range (\<lambda>n. A 0 - A n) \<subseteq> sets M" using A by auto show "incseq (\<lambda>n. A 0 - A n)" using `decseq A` by (auto simp add: incseq_def decseq_def) qed also have "\<dots> = emeasure M (A 0) - emeasure M (\<Inter>i. A i)" using A finite * by (simp, subst emeasure_Diff) auto finally show ?thesis unfolding ereal_minus_eq_minus_iff using finite A0 by auto qed lemma Lim_emeasure_decseq: assumes A: "range A \<subseteq> sets M" "decseq A" and fin: "\<And>i. emeasure M (A i) \<noteq> \<infinity>" shows "(\<lambda>i. emeasure M (A i)) ----> emeasure M (\<Inter>i. A i)" using LIMSEQ_INF[OF decseq_emeasure, OF A] using INF_emeasure_decseq[OF A fin] by simp lemma emeasure_subadditive: assumes [measurable]: "A \<in> sets M" "B \<in> sets M" shows "emeasure M (A \<union> B) \<le> emeasure M A + emeasure M B" proof - from plus_emeasure[of A M "B - A"] have "emeasure M (A \<union> B) = emeasure M A + emeasure M (B - A)" by simp also have "\<dots> \<le> emeasure M A + emeasure M B" using assms by (auto intro!: add_left_mono emeasure_mono) finally show ?thesis . qed lemma emeasure_subadditive_finite: assumes "finite I" "A ` I \<subseteq> sets M" shows "emeasure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. emeasure M (A i))" using assms proof induct case (insert i I) then have "emeasure M (\<Union>i\<in>insert i I. A i) = emeasure M (A i \<union> (\<Union>i\<in>I. A i))" by simp also have "\<dots> \<le> emeasure M (A i) + emeasure M (\<Union>i\<in>I. A i)" using insert by (intro emeasure_subadditive) auto also have "\<dots> \<le> emeasure M (A i) + (\<Sum>i\<in>I. emeasure M (A i))" using insert by (intro add_mono) auto also have "\<dots> = (\<Sum>i\<in>insert i I. emeasure M (A i))" using insert by auto finally show ?case . qed simp lemma emeasure_subadditive_countably: assumes "range f \<subseteq> sets M" shows "emeasure M (\<Union>i. f i) \<le> (\<Sum>i. emeasure M (f i))" proof - have "emeasure M (\<Union>i. f i) = emeasure M (\<Union>i. disjointed f i)" unfolding UN_disjointed_eq .. also have "\<dots> = (\<Sum>i. emeasure M (disjointed f i))" using sets.range_disjointed_sets[OF assms] suminf_emeasure[of "disjointed f"] by (simp add: disjoint_family_disjointed comp_def) also have "\<dots> \<le> (\<Sum>i. emeasure M (f i))" using sets.range_disjointed_sets[OF assms] assms by (auto intro!: suminf_le_pos emeasure_mono disjointed_subset) finally show ?thesis . qed lemma emeasure_insert: assumes sets: "{x} \<in> sets M" "A \<in> sets M" and "x \<notin> A" shows "emeasure M (insert x A) = emeasure M {x} + emeasure M A" proof - have "{x} \<inter> A = {}" using `x \<notin> A` by auto from plus_emeasure[OF sets this] show ?thesis by simp qed lemma emeasure_eq_setsum_singleton: assumes "finite S" "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M" shows "emeasure M S = (\<Sum>x\<in>S. emeasure M {x})" using setsum_emeasure[of "\<lambda>x. {x}" S M] assms by (auto simp: disjoint_family_on_def subset_eq) lemma setsum_emeasure_cover: assumes "finite S" and "A \<in> sets M" and br_in_M: "B ` S \<subseteq> sets M" assumes A: "A \<subseteq> (\<Union>i\<in>S. B i)" assumes disj: "disjoint_family_on B S" shows "emeasure M A = (\<Sum>i\<in>S. emeasure M (A \<inter> (B i)))" proof - have "(\<Sum>i\<in>S. emeasure M (A \<inter> (B i))) = emeasure M (\<Union>i\<in>S. A \<inter> (B i))" proof (rule setsum_emeasure) show "disjoint_family_on (\<lambda>i. A \<inter> B i) S" using `disjoint_family_on B S` unfolding disjoint_family_on_def by auto qed (insert assms, auto) also have "(\<Union>i\<in>S. A \<inter> (B i)) = A" using A by auto finally show ?thesis by simp qed lemma emeasure_eq_0: "N \<in> sets M \<Longrightarrow> emeasure M N = 0 \<Longrightarrow> K \<subseteq> N \<Longrightarrow> emeasure M K = 0" by (metis emeasure_mono emeasure_nonneg order_eq_iff) lemma emeasure_UN_eq_0: assumes "\<And>i::nat. emeasure M (N i) = 0" and "range N \<subseteq> sets M" shows "emeasure M (\<Union> i. N i) = 0" proof - have "0 \<le> emeasure M (\<Union> i. N i)" using assms by auto moreover have "emeasure M (\<Union> i. N i) \<le> 0" using emeasure_subadditive_countably[OF assms(2)] assms(1) by simp ultimately show ?thesis by simp qed lemma measure_eqI_finite: assumes [simp]: "sets M = Pow A" "sets N = Pow A" and "finite A" assumes eq: "\<And>a. a \<in> A \<Longrightarrow> emeasure M {a} = emeasure N {a}" shows "M = N" proof (rule measure_eqI) fix X assume "X \<in> sets M" then have X: "X \<subseteq> A" by auto then have "emeasure M X = (\<Sum>a\<in>X. emeasure M {a})" using `finite A` by (subst emeasure_eq_setsum_singleton) (auto dest: finite_subset) also have "\<dots> = (\<Sum>a\<in>X. emeasure N {a})" using X eq by (auto intro!: setsum_cong) also have "\<dots> = emeasure N X" using X `finite A` by (subst emeasure_eq_setsum_singleton) (auto dest: finite_subset) finally show "emeasure M X = emeasure N X" . qed simp lemma measure_eqI_generator_eq: fixes M N :: "'a measure" and E :: "'a set set" and A :: "nat \<Rightarrow> 'a set" assumes "Int_stable E" "E \<subseteq> Pow \<Omega>" and eq: "\<And>X. X \<in> E \<Longrightarrow> emeasure M X = emeasure N X" and M: "sets M = sigma_sets \<Omega> E" and N: "sets N = sigma_sets \<Omega> E" and A: "range A \<subseteq> E" "(\<Union>i. A i) = \<Omega>" "\<And>i. emeasure M (A i) \<noteq> \<infinity>" shows "M = N" proof - let ?\<mu> = "emeasure M" and ?\<nu> = "emeasure N" interpret S: sigma_algebra \<Omega> "sigma_sets \<Omega> E" by (rule sigma_algebra_sigma_sets) fact have "space M = \<Omega>" using sets.top[of M] sets.space_closed[of M] S.top S.space_closed `sets M = sigma_sets \<Omega> E` by blast { fix F D assume "F \<in> E" and "?\<mu> F \<noteq> \<infinity>" then have [intro]: "F \<in> sigma_sets \<Omega> E" by auto have "?\<nu> F \<noteq> \<infinity>" using `?\<mu> F \<noteq> \<infinity>` `F \<in> E` eq by simp assume "D \<in> sets M" with `Int_stable E` `E \<subseteq> Pow \<Omega>` have "emeasure M (F \<inter> D) = emeasure N (F \<inter> D)" unfolding M proof (induct rule: sigma_sets_induct_disjoint) case (basic A) then have "F \<inter> A \<in> E" using `Int_stable E` `F \<in> E` by (auto simp: Int_stable_def) then show ?case using eq by auto next case empty then show ?case by simp next case (compl A) then have **: "F \<inter> (\<Omega> - A) = F - (F \<inter> A)" and [intro]: "F \<inter> A \<in> sigma_sets \<Omega> E" using `F \<in> E` S.sets_into_space by (auto simp: M) have "?\<nu> (F \<inter> A) \<le> ?\<nu> F" by (auto intro!: emeasure_mono simp: M N) then have "?\<nu> (F \<inter> A) \<noteq> \<infinity>" using `?\<nu> F \<noteq> \<infinity>` by auto have "?\<mu> (F \<inter> A) \<le> ?\<mu> F" by (auto intro!: emeasure_mono simp: M N) then have "?\<mu> (F \<inter> A) \<noteq> \<infinity>" using `?\<mu> F \<noteq> \<infinity>` by auto then have "?\<mu> (F \<inter> (\<Omega> - A)) = ?\<mu> F - ?\<mu> (F \<inter> A)" unfolding ** using `F \<inter> A \<in> sigma_sets \<Omega> E` by (auto intro!: emeasure_Diff simp: M N) also have "\<dots> = ?\<nu> F - ?\<nu> (F \<inter> A)" using eq `F \<in> E` compl by simp also have "\<dots> = ?\<nu> (F \<inter> (\<Omega> - A))" unfolding ** using `F \<inter> A \<in> sigma_sets \<Omega> E` `?\<nu> (F \<inter> A) \<noteq> \<infinity>` by (auto intro!: emeasure_Diff[symmetric] simp: M N) finally show ?case using `space M = \<Omega>` by auto next case (union A) then have "?\<mu> (\<Union>x. F \<inter> A x) = ?\<nu> (\<Union>x. F \<inter> A x)" by (subst (1 2) suminf_emeasure[symmetric]) (auto simp: disjoint_family_on_def subset_eq M N) with A show ?case by auto qed } note * = this show "M = N" proof (rule measure_eqI) show "sets M = sets N" using M N by simp have [simp, intro]: "\<And>i. A i \<in> sets M" using A(1) by (auto simp: subset_eq M) fix F assume "F \<in> sets M" let ?D = "disjointed (\<lambda>i. F \<inter> A i)" from `space M = \<Omega>` have F_eq: "F = (\<Union>i. ?D i)" using `F \<in> sets M`[THEN sets.sets_into_space] A(2)[symmetric] by (auto simp: UN_disjointed_eq) have [simp, intro]: "\<And>i. ?D i \<in> sets M" using sets.range_disjointed_sets[of "\<lambda>i. F \<inter> A i" M] `F \<in> sets M` by (auto simp: subset_eq) have "disjoint_family ?D" by (auto simp: disjoint_family_disjointed) moreover have "(\<Sum>i. emeasure M (?D i)) = (\<Sum>i. emeasure N (?D i))" proof (intro arg_cong[where f=suminf] ext) fix i have "A i \<inter> ?D i = ?D i" by (auto simp: disjointed_def) then show "emeasure M (?D i) = emeasure N (?D i)" using *[of "A i" "?D i", OF _ A(3)] A(1) by auto qed ultimately show "emeasure M F = emeasure N F" by (simp add: image_subset_iff `sets M = sets N`[symmetric] F_eq[symmetric] suminf_emeasure) qed qed lemma measure_of_of_measure: "measure_of (space M) (sets M) (emeasure M) = M" proof (intro measure_eqI emeasure_measure_of_sigma) show "sigma_algebra (space M) (sets M)" .. show "positive (sets M) (emeasure M)" by (simp add: positive_def emeasure_nonneg) show "countably_additive (sets M) (emeasure M)" by (simp add: emeasure_countably_additive) qed simp_all section "@{text \<mu>}-null sets" definition null_sets :: "'a measure \<Rightarrow> 'a set set" where "null_sets M = {N\<in>sets M. emeasure M N = 0}" lemma null_setsD1[dest]: "A \<in> null_sets M \<Longrightarrow> emeasure M A = 0" by (simp add: null_sets_def) lemma null_setsD2[dest]: "A \<in> null_sets M \<Longrightarrow> A \<in> sets M" unfolding null_sets_def by simp lemma null_setsI[intro]: "emeasure M A = 0 \<Longrightarrow> A \<in> sets M \<Longrightarrow> A \<in> null_sets M" unfolding null_sets_def by simp interpretation null_sets: ring_of_sets "space M" "null_sets M" for M proof (rule ring_of_setsI) show "null_sets M \<subseteq> Pow (space M)" using sets.sets_into_space by auto show "{} \<in> null_sets M" by auto fix A B assume sets: "A \<in> null_sets M" "B \<in> null_sets M" then have "A \<in> sets M" "B \<in> sets M" by auto moreover then have "emeasure M (A \<union> B) \<le> emeasure M A + emeasure M B" "emeasure M (A - B) \<le> emeasure M A" by (auto intro!: emeasure_subadditive emeasure_mono) moreover have "emeasure M B = 0" "emeasure M A = 0" using sets by auto ultimately show "A - B \<in> null_sets M" "A \<union> B \<in> null_sets M" by (auto intro!: antisym) qed lemma UN_from_nat: "(\<Union>i. N i) = (\<Union>i. N (Countable.from_nat i))" proof - have "(\<Union>i. N i) = (\<Union>i. (N \<circ> Countable.from_nat) i)" unfolding SUP_def image_compose unfolding surj_from_nat .. then show ?thesis by simp qed lemma null_sets_UN[intro]: assumes "\<And>i::'i::countable. N i \<in> null_sets M" shows "(\<Union>i. N i) \<in> null_sets M" proof (intro conjI CollectI null_setsI) show "(\<Union>i. N i) \<in> sets M" using assms by auto have "0 \<le> emeasure M (\<Union>i. N i)" by (rule emeasure_nonneg) moreover have "emeasure M (\<Union>i. N i) \<le> (\<Sum>n. emeasure M (N (Countable.from_nat n)))" unfolding UN_from_nat[of N] using assms by (intro emeasure_subadditive_countably) auto ultimately show "emeasure M (\<Union>i. N i) = 0" using assms by (auto simp: null_setsD1) qed lemma null_set_Int1: assumes "B \<in> null_sets M" "A \<in> sets M" shows "A \<inter> B \<in> null_sets M" proof (intro CollectI conjI null_setsI) show "emeasure M (A \<inter> B) = 0" using assms by (intro emeasure_eq_0[of B _ "A \<inter> B"]) auto qed (insert assms, auto) lemma null_set_Int2: assumes "B \<in> null_sets M" "A \<in> sets M" shows "B \<inter> A \<in> null_sets M" using assms by (subst Int_commute) (rule null_set_Int1) lemma emeasure_Diff_null_set: assumes "B \<in> null_sets M" "A \<in> sets M" shows "emeasure M (A - B) = emeasure M A" proof - have *: "A - B = (A - (A \<inter> B))" by auto have "A \<inter> B \<in> null_sets M" using assms by (rule null_set_Int1) then show ?thesis unfolding * using assms by (subst emeasure_Diff) auto qed lemma null_set_Diff: assumes "B \<in> null_sets M" "A \<in> sets M" shows "B - A \<in> null_sets M" proof (intro CollectI conjI null_setsI) show "emeasure M (B - A) = 0" using assms by (intro emeasure_eq_0[of B _ "B - A"]) auto qed (insert assms, auto) lemma emeasure_Un_null_set: assumes "A \<in> sets M" "B \<in> null_sets M" shows "emeasure M (A \<union> B) = emeasure M A" proof - have *: "A \<union> B = A \<union> (B - A)" by auto have "B - A \<in> null_sets M" using assms(2,1) by (rule null_set_Diff) then show ?thesis unfolding * using assms by (subst plus_emeasure[symmetric]) auto qed section "Formalize almost everywhere" definition ae_filter :: "'a measure \<Rightarrow> 'a filter" where "ae_filter M = Abs_filter (\<lambda>P. \<exists>N\<in>null_sets M. {x \<in> space M. \<not> P x} \<subseteq> N)" abbreviation almost_everywhere :: "'a measure \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" where "almost_everywhere M P \<equiv> eventually P (ae_filter M)" syntax "_almost_everywhere" :: "pttrn \<Rightarrow> 'a \<Rightarrow> bool \<Rightarrow> bool" ("AE _ in _. _" [0,0,10] 10) translations "AE x in M. P" == "CONST almost_everywhere M (%x. P)" lemma eventually_ae_filter: fixes M P defines [simp]: "F \<equiv> \<lambda>P. \<exists>N\<in>null_sets M. {x \<in> space M. \<not> P x} \<subseteq> N" shows "eventually P (ae_filter M) \<longleftrightarrow> F P" unfolding ae_filter_def F_def[symmetric] proof (rule eventually_Abs_filter) show "is_filter F" proof fix P Q assume "F P" "F Q" then obtain N L where N: "N \<in> null_sets M" "{x \<in> space M. \<not> P x} \<subseteq> N" and L: "L \<in> null_sets M" "{x \<in> space M. \<not> Q x} \<subseteq> L" by auto then have "L \<union> N \<in> null_sets M" "{x \<in> space M. \<not> (P x \<and> Q x)} \<subseteq> L \<union> N" by auto then show "F (\<lambda>x. P x \<and> Q x)" by auto next fix P Q assume "F P" then obtain N where N: "N \<in> null_sets M" "{x \<in> space M. \<not> P x} \<subseteq> N" by auto moreover assume "\<forall>x. P x \<longrightarrow> Q x" ultimately have "N \<in> null_sets M" "{x \<in> space M. \<not> Q x} \<subseteq> N" by auto then show "F Q" by auto qed auto qed lemma AE_I': "N \<in> null_sets M \<Longrightarrow> {x\<in>space M. \<not> P x} \<subseteq> N \<Longrightarrow> (AE x in M. P x)" unfolding eventually_ae_filter by auto lemma AE_iff_null: assumes "{x\<in>space M. \<not> P x} \<in> sets M" (is "?P \<in> sets M") shows "(AE x in M. P x) \<longleftrightarrow> {x\<in>space M. \<not> P x} \<in> null_sets M" proof assume "AE x in M. P x" then obtain N where N: "N \<in> sets M" "?P \<subseteq> N" "emeasure M N = 0" unfolding eventually_ae_filter by auto have "0 \<le> emeasure M ?P" by auto moreover have "emeasure M ?P \<le> emeasure M N" using assms N(1,2) by (auto intro: emeasure_mono) ultimately have "emeasure M ?P = 0" unfolding `emeasure M N = 0` by auto then show "?P \<in> null_sets M" using assms by auto next assume "?P \<in> null_sets M" with assms show "AE x in M. P x" by (auto intro: AE_I') qed lemma AE_iff_null_sets: "N \<in> sets M \<Longrightarrow> N \<in> null_sets M \<longleftrightarrow> (AE x in M. x \<notin> N)" using Int_absorb1[OF sets.sets_into_space, of N M] by (subst AE_iff_null) (auto simp: Int_def[symmetric]) lemma AE_not_in: "N \<in> null_sets M \<Longrightarrow> AE x in M. x \<notin> N" by (metis AE_iff_null_sets null_setsD2) lemma AE_iff_measurable: "N \<in> sets M \<Longrightarrow> {x\<in>space M. \<not> P x} = N \<Longrightarrow> (AE x in M. P x) \<longleftrightarrow> emeasure M N = 0" using AE_iff_null[of _ P] by auto lemma AE_E[consumes 1]: assumes "AE x in M. P x" obtains N where "{x \<in> space M. \<not> P x} \<subseteq> N" "emeasure M N = 0" "N \<in> sets M" using assms unfolding eventually_ae_filter by auto lemma AE_E2: assumes "AE x in M. P x" "{x\<in>space M. P x} \<in> sets M" shows "emeasure M {x\<in>space M. \<not> P x} = 0" (is "emeasure M ?P = 0") proof - have "{x\<in>space M. \<not> P x} = space M - {x\<in>space M. P x}" by auto with AE_iff_null[of M P] assms show ?thesis by auto qed lemma AE_I: assumes "{x \<in> space M. \<not> P x} \<subseteq> N" "emeasure M N = 0" "N \<in> sets M" shows "AE x in M. P x" using assms unfolding eventually_ae_filter by auto lemma AE_mp[elim!]: assumes AE_P: "AE x in M. P x" and AE_imp: "AE x in M. P x \<longrightarrow> Q x" shows "AE x in M. Q x" proof - from AE_P obtain A where P: "{x\<in>space M. \<not> P x} \<subseteq> A" and A: "A \<in> sets M" "emeasure M A = 0" by (auto elim!: AE_E) from AE_imp obtain B where imp: "{x\<in>space M. P x \<and> \<not> Q x} \<subseteq> B" and B: "B \<in> sets M" "emeasure M B = 0" by (auto elim!: AE_E) show ?thesis proof (intro AE_I) have "0 \<le> emeasure M (A \<union> B)" using A B by auto moreover have "emeasure M (A \<union> B) \<le> 0" using emeasure_subadditive[of A M B] A B by auto ultimately show "A \<union> B \<in> sets M" "emeasure M (A \<union> B) = 0" using A B by auto show "{x\<in>space M. \<not> Q x} \<subseteq> A \<union> B" using P imp by auto qed qed (* depricated replace by laws about eventually *) lemma shows AE_iffI: "AE x in M. P x \<Longrightarrow> AE x in M. P x \<longleftrightarrow> Q x \<Longrightarrow> AE x in M. Q x" and AE_disjI1: "AE x in M. P x \<Longrightarrow> AE x in M. P x \<or> Q x" and AE_disjI2: "AE x in M. Q x \<Longrightarrow> AE x in M. P x \<or> Q x" and AE_conjI: "AE x in M. P x \<Longrightarrow> AE x in M. Q x \<Longrightarrow> AE x in M. P x \<and> Q x" and AE_conj_iff[simp]: "(AE x in M. P x \<and> Q x) \<longleftrightarrow> (AE x in M. P x) \<and> (AE x in M. Q x)" by auto lemma AE_impI: "(P \<Longrightarrow> AE x in M. Q x) \<Longrightarrow> AE x in M. P \<longrightarrow> Q x" by (cases P) auto lemma AE_measure: assumes AE: "AE x in M. P x" and sets: "{x\<in>space M. P x} \<in> sets M" (is "?P \<in> sets M") shows "emeasure M {x\<in>space M. P x} = emeasure M (space M)" proof - from AE_E[OF AE] guess N . note N = this with sets have "emeasure M (space M) \<le> emeasure M (?P \<union> N)" by (intro emeasure_mono) auto also have "\<dots> \<le> emeasure M ?P + emeasure M N" using sets N by (intro emeasure_subadditive) auto also have "\<dots> = emeasure M ?P" using N by simp finally show "emeasure M ?P = emeasure M (space M)" using emeasure_space[of M "?P"] by auto qed lemma AE_space: "AE x in M. x \<in> space M" by (rule AE_I[where N="{}"]) auto lemma AE_I2[simp, intro]: "(\<And>x. x \<in> space M \<Longrightarrow> P x) \<Longrightarrow> AE x in M. P x" using AE_space by force lemma AE_Ball_mp: "\<forall>x\<in>space M. P x \<Longrightarrow> AE x in M. P x \<longrightarrow> Q x \<Longrightarrow> AE x in M. Q x" by auto lemma AE_cong[cong]: "(\<And>x. x \<in> space M \<Longrightarrow> P x \<longleftrightarrow> Q x) \<Longrightarrow> (AE x in M. P x) \<longleftrightarrow> (AE x in M. Q x)" by auto lemma AE_all_countable: "(AE x in M. \<forall>i. P i x) \<longleftrightarrow> (\<forall>i::'i::countable. AE x in M. P i x)" proof assume "\<forall>i. AE x in M. P i x" from this[unfolded eventually_ae_filter Bex_def, THEN choice] obtain N where N: "\<And>i. N i \<in> null_sets M" "\<And>i. {x\<in>space M. \<not> P i x} \<subseteq> N i" by auto have "{x\<in>space M. \<not> (\<forall>i. P i x)} \<subseteq> (\<Union>i. {x\<in>space M. \<not> P i x})" by auto also have "\<dots> \<subseteq> (\<Union>i. N i)" using N by auto finally have "{x\<in>space M. \<not> (\<forall>i. P i x)} \<subseteq> (\<Union>i. N i)" . moreover from N have "(\<Union>i. N i) \<in> null_sets M" by (intro null_sets_UN) auto ultimately show "AE x in M. \<forall>i. P i x" unfolding eventually_ae_filter by auto qed auto lemma AE_finite_all: assumes f: "finite S" shows "(AE x in M. \<forall>i\<in>S. P i x) \<longleftrightarrow> (\<forall>i\<in>S. AE x in M. P i x)" using f by induct auto lemma AE_finite_allI: assumes "finite S" shows "(\<And>s. s \<in> S \<Longrightarrow> AE x in M. Q s x) \<Longrightarrow> AE x in M. \<forall>s\<in>S. Q s x" using AE_finite_all[OF `finite S`] by auto lemma emeasure_mono_AE: assumes imp: "AE x in M. x \<in> A \<longrightarrow> x \<in> B" and B: "B \<in> sets M" shows "emeasure M A \<le> emeasure M B" proof cases assume A: "A \<in> sets M" from imp obtain N where N: "{x\<in>space M. \<not> (x \<in> A \<longrightarrow> x \<in> B)} \<subseteq> N" "N \<in> null_sets M" by (auto simp: eventually_ae_filter) have "emeasure M A = emeasure M (A - N)" using N A by (subst emeasure_Diff_null_set) auto also have "emeasure M (A - N) \<le> emeasure M (B - N)" using N A B sets.sets_into_space by (auto intro!: emeasure_mono) also have "emeasure M (B - N) = emeasure M B" using N B by (subst emeasure_Diff_null_set) auto finally show ?thesis . qed (simp add: emeasure_nonneg emeasure_notin_sets) lemma emeasure_eq_AE: assumes iff: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B" assumes A: "A \<in> sets M" and B: "B \<in> sets M" shows "emeasure M A = emeasure M B" using assms by (safe intro!: antisym emeasure_mono_AE) auto section {* @{text \<sigma>}-finite Measures *} locale sigma_finite_measure = fixes M :: "'a measure" assumes sigma_finite: "\<exists>A::nat \<Rightarrow> 'a set. range A \<subseteq> sets M \<and> (\<Union>i. A i) = space M \<and> (\<forall>i. emeasure M (A i) \<noteq> \<infinity>)" lemma (in sigma_finite_measure) sigma_finite_disjoint: obtains A :: "nat \<Rightarrow> 'a set" where "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>" "disjoint_family A" proof atomize_elim case goal1 obtain A :: "nat \<Rightarrow> 'a set" where range: "range A \<subseteq> sets M" and space: "(\<Union>i. A i) = space M" and measure: "\<And>i. emeasure M (A i) \<noteq> \<infinity>" using sigma_finite by auto note range' = sets.range_disjointed_sets[OF range] range { fix i have "emeasure M (disjointed A i) \<le> emeasure M (A i)" using range' disjointed_subset[of A i] by (auto intro!: emeasure_mono) then have "emeasure M (disjointed A i) \<noteq> \<infinity>" using measure[of i] by auto } with disjoint_family_disjointed UN_disjointed_eq[of A] space range' show ?case by (auto intro!: exI[of _ "disjointed A"]) qed lemma (in sigma_finite_measure) sigma_finite_incseq: obtains A :: "nat \<Rightarrow> 'a set" where "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>" "incseq A" proof atomize_elim case goal1 obtain F :: "nat \<Rightarrow> 'a set" where F: "range F \<subseteq> sets M" "(\<Union>i. F i) = space M" "\<And>i. emeasure M (F i) \<noteq> \<infinity>" using sigma_finite by auto then show ?case proof (intro exI[of _ "\<lambda>n. \<Union>i\<le>n. F i"] conjI allI) from F have "\<And>x. x \<in> space M \<Longrightarrow> \<exists>i. x \<in> F i" by auto then show "(\<Union>n. \<Union> i\<le>n. F i) = space M" using F by fastforce next fix n have "emeasure M (\<Union> i\<le>n. F i) \<le> (\<Sum>i\<le>n. emeasure M (F i))" using F by (auto intro!: emeasure_subadditive_finite) also have "\<dots> < \<infinity>" using F by (auto simp: setsum_Pinfty) finally show "emeasure M (\<Union> i\<le>n. F i) \<noteq> \<infinity>" by simp qed (force simp: incseq_def)+ qed section {* Measure space induced by distribution of @{const measurable}-functions *} definition distr :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b measure" where "distr M N f = measure_of (space N) (sets N) (\<lambda>A. emeasure M (f -` A \<inter> space M))" lemma shows sets_distr[simp]: "sets (distr M N f) = sets N" and space_distr[simp]: "space (distr M N f) = space N" by (auto simp: distr_def) lemma shows measurable_distr_eq1[simp]: "measurable (distr Mf Nf f) Mf' = measurable Nf Mf'" and measurable_distr_eq2[simp]: "measurable Mg' (distr Mg Ng g) = measurable Mg' Ng" by (auto simp: measurable_def) lemma emeasure_distr: fixes f :: "'a \<Rightarrow> 'b" assumes f: "f \<in> measurable M N" and A: "A \<in> sets N" shows "emeasure (distr M N f) A = emeasure M (f -` A \<inter> space M)" (is "_ = ?\<mu> A") unfolding distr_def proof (rule emeasure_measure_of_sigma) show "positive (sets N) ?\<mu>" by (auto simp: positive_def) show "countably_additive (sets N) ?\<mu>" proof (intro countably_additiveI) fix A :: "nat \<Rightarrow> 'b set" assume "range A \<subseteq> sets N" "disjoint_family A" then have A: "\<And>i. A i \<in> sets N" "(\<Union>i. A i) \<in> sets N" by auto then have *: "range (\<lambda>i. f -` (A i) \<inter> space M) \<subseteq> sets M" using f by (auto simp: measurable_def) moreover have "(\<Union>i. f -` A i \<inter> space M) \<in> sets M" using * by blast moreover have **: "disjoint_family (\<lambda>i. f -` A i \<inter> space M)" using `disjoint_family A` by (auto simp: disjoint_family_on_def) ultimately show "(\<Sum>i. ?\<mu> (A i)) = ?\<mu> (\<Union>i. A i)" using suminf_emeasure[OF _ **] A f by (auto simp: comp_def vimage_UN) qed show "sigma_algebra (space N) (sets N)" .. qed fact lemma distr_id[simp]: "distr N N (\<lambda>x. x) = N" by (rule measure_eqI) (auto simp: emeasure_distr) lemma measure_distr: "f \<in> measurable M N \<Longrightarrow> S \<in> sets N \<Longrightarrow> measure (distr M N f) S = measure M (f -` S \<inter> space M)" by (simp add: emeasure_distr measure_def) lemma AE_distrD: assumes f: "f \<in> measurable M M'" and AE: "AE x in distr M M' f. P x" shows "AE x in M. P (f x)" proof - from AE[THEN AE_E] guess N . with f show ?thesis unfolding eventually_ae_filter by (intro bexI[of _ "f -` N \<inter> space M"]) (auto simp: emeasure_distr measurable_def) qed lemma AE_distr_iff: assumes f[measurable]: "f \<in> measurable M N" and P[measurable]: "{x \<in> space N. P x} \<in> sets N" shows "(AE x in distr M N f. P x) \<longleftrightarrow> (AE x in M. P (f x))" proof (subst (1 2) AE_iff_measurable[OF _ refl]) have "f -` {x\<in>space N. \<not> P x} \<inter> space M = {x \<in> space M. \<not> P (f x)}" using f[THEN measurable_space] by auto then show "(emeasure (distr M N f) {x \<in> space (distr M N f). \<not> P x} = 0) = (emeasure M {x \<in> space M. \<not> P (f x)} = 0)" by (simp add: emeasure_distr) qed auto lemma null_sets_distr_iff: "f \<in> measurable M N \<Longrightarrow> A \<in> null_sets (distr M N f) \<longleftrightarrow> f -` A \<inter> space M \<in> null_sets M \<and> A \<in> sets N" by (auto simp add: null_sets_def emeasure_distr) lemma distr_distr: "g \<in> measurable N L \<Longrightarrow> f \<in> measurable M N \<Longrightarrow> distr (distr M N f) L g = distr M L (g \<circ> f)" by (auto simp add: emeasure_distr measurable_space intro!: arg_cong[where f="emeasure M"] measure_eqI) section {* Real measure values *} lemma measure_nonneg: "0 \<le> measure M A" using emeasure_nonneg[of M A] unfolding measure_def by (auto intro: real_of_ereal_pos) lemma measure_empty[simp]: "measure M {} = 0" unfolding measure_def by simp lemma emeasure_eq_ereal_measure: "emeasure M A \<noteq> \<infinity> \<Longrightarrow> emeasure M A = ereal (measure M A)" using emeasure_nonneg[of M A] by (cases "emeasure M A") (auto simp: measure_def) lemma measure_Union: assumes finite: "emeasure M A \<noteq> \<infinity>" "emeasure M B \<noteq> \<infinity>" and measurable: "A \<in> sets M" "B \<in> sets M" "A \<inter> B = {}" shows "measure M (A \<union> B) = measure M A + measure M B" unfolding measure_def using plus_emeasure[OF measurable, symmetric] finite by (simp add: emeasure_eq_ereal_measure) lemma measure_finite_Union: assumes measurable: "A`S \<subseteq> sets M" "disjoint_family_on A S" "finite S" assumes finite: "\<And>i. i \<in> S \<Longrightarrow> emeasure M (A i) \<noteq> \<infinity>" shows "measure M (\<Union>i\<in>S. A i) = (\<Sum>i\<in>S. measure M (A i))" unfolding measure_def using setsum_emeasure[OF measurable, symmetric] finite by (simp add: emeasure_eq_ereal_measure) lemma measure_Diff: assumes finite: "emeasure M A \<noteq> \<infinity>" and measurable: "A \<in> sets M" "B \<in> sets M" "B \<subseteq> A" shows "measure M (A - B) = measure M A - measure M B" proof - have "emeasure M (A - B) \<le> emeasure M A" "emeasure M B \<le> emeasure M A" using measurable by (auto intro!: emeasure_mono) hence "measure M ((A - B) \<union> B) = measure M (A - B) + measure M B" using measurable finite by (rule_tac measure_Union) auto thus ?thesis using `B \<subseteq> A` by (auto simp: Un_absorb2) qed lemma measure_UNION: assumes measurable: "range A \<subseteq> sets M" "disjoint_family A" assumes finite: "emeasure M (\<Union>i. A i) \<noteq> \<infinity>" shows "(\<lambda>i. measure M (A i)) sums (measure M (\<Union>i. A i))" proof - from summable_sums[OF summable_ereal_pos, of "\<lambda>i. emeasure M (A i)"] suminf_emeasure[OF measurable] emeasure_nonneg[of M] have "(\<lambda>i. emeasure M (A i)) sums (emeasure M (\<Union>i. A i))" by simp moreover { fix i have "emeasure M (A i) \<le> emeasure M (\<Union>i. A i)" using measurable by (auto intro!: emeasure_mono) then have "emeasure M (A i) = ereal ((measure M (A i)))" using finite by (intro emeasure_eq_ereal_measure) auto } ultimately show ?thesis using finite unfolding sums_ereal[symmetric] by (simp add: emeasure_eq_ereal_measure) qed lemma measure_subadditive: assumes measurable: "A \<in> sets M" "B \<in> sets M" and fin: "emeasure M A \<noteq> \<infinity>" "emeasure M B \<noteq> \<infinity>" shows "(measure M (A \<union> B)) \<le> (measure M A) + (measure M B)" proof - have "emeasure M (A \<union> B) \<noteq> \<infinity>" using emeasure_subadditive[OF measurable] fin by auto then show "(measure M (A \<union> B)) \<le> (measure M A) + (measure M B)" using emeasure_subadditive[OF measurable] fin by (auto simp: emeasure_eq_ereal_measure) qed lemma measure_subadditive_finite: assumes A: "finite I" "A`I \<subseteq> sets M" and fin: "\<And>i. i \<in> I \<Longrightarrow> emeasure M (A i) \<noteq> \<infinity>" shows "measure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. measure M (A i))" proof - { have "emeasure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. emeasure M (A i))" using emeasure_subadditive_finite[OF A] . also have "\<dots> < \<infinity>" using fin by (simp add: setsum_Pinfty) finally have "emeasure M (\<Union>i\<in>I. A i) \<noteq> \<infinity>" by simp } then show ?thesis using emeasure_subadditive_finite[OF A] fin unfolding measure_def by (simp add: emeasure_eq_ereal_measure suminf_ereal measure_nonneg) qed lemma measure_subadditive_countably: assumes A: "range A \<subseteq> sets M" and fin: "(\<Sum>i. emeasure M (A i)) \<noteq> \<infinity>" shows "measure M (\<Union>i. A i) \<le> (\<Sum>i. measure M (A i))" proof - from emeasure_nonneg fin have "\<And>i. emeasure M (A i) \<noteq> \<infinity>" by (rule suminf_PInfty) moreover { have "emeasure M (\<Union>i. A i) \<le> (\<Sum>i. emeasure M (A i))" using emeasure_subadditive_countably[OF A] . also have "\<dots> < \<infinity>" using fin by simp finally have "emeasure M (\<Union>i. A i) \<noteq> \<infinity>" by simp } ultimately show ?thesis using emeasure_subadditive_countably[OF A] fin unfolding measure_def by (simp add: emeasure_eq_ereal_measure suminf_ereal measure_nonneg) qed lemma measure_eq_setsum_singleton: assumes S: "finite S" "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M" and fin: "\<And>x. x \<in> S \<Longrightarrow> emeasure M {x} \<noteq> \<infinity>" shows "(measure M S) = (\<Sum>x\<in>S. (measure M {x}))" unfolding measure_def using emeasure_eq_setsum_singleton[OF S] fin by simp (simp add: emeasure_eq_ereal_measure) lemma Lim_measure_incseq: assumes A: "range A \<subseteq> sets M" "incseq A" and fin: "emeasure M (\<Union>i. A i) \<noteq> \<infinity>" shows "(\<lambda>i. (measure M (A i))) ----> (measure M (\<Union>i. A i))" proof - have "ereal ((measure M (\<Union>i. A i))) = emeasure M (\<Union>i. A i)" using fin by (auto simp: emeasure_eq_ereal_measure) then show ?thesis using Lim_emeasure_incseq[OF A] unfolding measure_def by (intro lim_real_of_ereal) simp qed lemma Lim_measure_decseq: assumes A: "range A \<subseteq> sets M" "decseq A" and fin: "\<And>i. emeasure M (A i) \<noteq> \<infinity>" shows "(\<lambda>n. measure M (A n)) ----> measure M (\<Inter>i. A i)" proof - have "emeasure M (\<Inter>i. A i) \<le> emeasure M (A 0)" using A by (auto intro!: emeasure_mono) also have "\<dots> < \<infinity>" using fin[of 0] by auto finally have "ereal ((measure M (\<Inter>i. A i))) = emeasure M (\<Inter>i. A i)" by (auto simp: emeasure_eq_ereal_measure) then show ?thesis unfolding measure_def using Lim_emeasure_decseq[OF A fin] by (intro lim_real_of_ereal) simp qed section {* Measure spaces with @{term "emeasure M (space M) < \<infinity>"} *} locale finite_measure = sigma_finite_measure M for M + assumes finite_emeasure_space: "emeasure M (space M) \<noteq> \<infinity>" lemma finite_measureI[Pure.intro!]: assumes *: "emeasure M (space M) \<noteq> \<infinity>" shows "finite_measure M" proof show "\<exists>A. range A \<subseteq> sets M \<and> (\<Union>i. A i) = space M \<and> (\<forall>i. emeasure M (A i) \<noteq> \<infinity>)" using * by (auto intro!: exI[of _ "\<lambda>_. space M"]) qed fact lemma (in finite_measure) emeasure_finite[simp, intro]: "emeasure M A \<noteq> \<infinity>" using finite_emeasure_space emeasure_space[of M A] by auto lemma (in finite_measure) emeasure_eq_measure: "emeasure M A = ereal (measure M A)" unfolding measure_def by (simp add: emeasure_eq_ereal_measure) lemma (in finite_measure) emeasure_real: "\<exists>r. 0 \<le> r \<and> emeasure M A = ereal r" using emeasure_finite[of A] emeasure_nonneg[of M A] by (cases "emeasure M A") auto lemma (in finite_measure) bounded_measure: "measure M A \<le> measure M (space M)" using emeasure_space[of M A] emeasure_real[of A] emeasure_real[of "space M"] by (auto simp: measure_def) lemma (in finite_measure) finite_measure_Diff: assumes sets: "A \<in> sets M" "B \<in> sets M" and "B \<subseteq> A" shows "measure M (A - B) = measure M A - measure M B" using measure_Diff[OF _ assms] by simp lemma (in finite_measure) finite_measure_Union: assumes sets: "A \<in> sets M" "B \<in> sets M" and "A \<inter> B = {}" shows "measure M (A \<union> B) = measure M A + measure M B" using measure_Union[OF _ _ assms] by simp lemma (in finite_measure) finite_measure_finite_Union: assumes measurable: "A`S \<subseteq> sets M" "disjoint_family_on A S" "finite S" shows "measure M (\<Union>i\<in>S. A i) = (\<Sum>i\<in>S. measure M (A i))" using measure_finite_Union[OF assms] by simp lemma (in finite_measure) finite_measure_UNION: assumes A: "range A \<subseteq> sets M" "disjoint_family A" shows "(\<lambda>i. measure M (A i)) sums (measure M (\<Union>i. A i))" using measure_UNION[OF A] by simp lemma (in finite_measure) finite_measure_mono: assumes "A \<subseteq> B" "B \<in> sets M" shows "measure M A \<le> measure M B" using emeasure_mono[OF assms] emeasure_real[of A] emeasure_real[of B] by (auto simp: measure_def) lemma (in finite_measure) finite_measure_subadditive: assumes m: "A \<in> sets M" "B \<in> sets M" shows "measure M (A \<union> B) \<le> measure M A + measure M B" using measure_subadditive[OF m] by simp lemma (in finite_measure) finite_measure_subadditive_finite: assumes "finite I" "A`I \<subseteq> sets M" shows "measure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. measure M (A i))" using measure_subadditive_finite[OF assms] by simp lemma (in finite_measure) finite_measure_subadditive_countably: assumes A: "range A \<subseteq> sets M" and sum: "summable (\<lambda>i. measure M (A i))" shows "measure M (\<Union>i. A i) \<le> (\<Sum>i. measure M (A i))" proof - from `summable (\<lambda>i. measure M (A i))` have "(\<lambda>i. ereal (measure M (A i))) sums ereal (\<Sum>i. measure M (A i))" by (simp add: sums_ereal) (rule summable_sums) from sums_unique[OF this, symmetric] measure_subadditive_countably[OF A] show ?thesis by (simp add: emeasure_eq_measure) qed lemma (in finite_measure) finite_measure_eq_setsum_singleton: assumes "finite S" and *: "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M" shows "measure M S = (\<Sum>x\<in>S. measure M {x})" using measure_eq_setsum_singleton[OF assms] by simp lemma (in finite_measure) finite_Lim_measure_incseq: assumes A: "range A \<subseteq> sets M" "incseq A" shows "(\<lambda>i. measure M (A i)) ----> measure M (\<Union>i. A i)" using Lim_measure_incseq[OF A] by simp lemma (in finite_measure) finite_Lim_measure_decseq: assumes A: "range A \<subseteq> sets M" "decseq A" shows "(\<lambda>n. measure M (A n)) ----> measure M (\<Inter>i. A i)" using Lim_measure_decseq[OF A] by simp lemma (in finite_measure) finite_measure_compl: assumes S: "S \<in> sets M" shows "measure M (space M - S) = measure M (space M) - measure M S" using measure_Diff[OF _ sets.top S sets.sets_into_space] S by simp lemma (in finite_measure) finite_measure_mono_AE: assumes imp: "AE x in M. x \<in> A \<longrightarrow> x \<in> B" and B: "B \<in> sets M" shows "measure M A \<le> measure M B" using assms emeasure_mono_AE[OF imp B] by (simp add: emeasure_eq_measure) lemma (in finite_measure) finite_measure_eq_AE: assumes iff: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B" assumes A: "A \<in> sets M" and B: "B \<in> sets M" shows "measure M A = measure M B" using assms emeasure_eq_AE[OF assms] by (simp add: emeasure_eq_measure) lemma (in finite_measure) measure_increasing: "increasing M (measure M)" by (auto intro!: finite_measure_mono simp: increasing_def) lemma (in finite_measure) measure_zero_union: assumes "s \<in> sets M" "t \<in> sets M" "measure M t = 0" shows "measure M (s \<union> t) = measure M s" using assms proof - have "measure M (s \<union> t) \<le> measure M s" using finite_measure_subadditive[of s t] assms by auto moreover have "measure M (s \<union> t) \<ge> measure M s" using assms by (blast intro: finite_measure_mono) ultimately show ?thesis by simp qed lemma (in finite_measure) measure_eq_compl: assumes "s \<in> sets M" "t \<in> sets M" assumes "measure M (space M - s) = measure M (space M - t)" shows "measure M s = measure M t" using assms finite_measure_compl by auto lemma (in finite_measure) measure_eq_bigunion_image: assumes "range f \<subseteq> sets M" "range g \<subseteq> sets M" assumes "disjoint_family f" "disjoint_family g" assumes "\<And> n :: nat. measure M (f n) = measure M (g n)" shows "measure M (\<Union> i. f i) = measure M (\<Union> i. g i)" using assms proof - have a: "(\<lambda> i. measure M (f i)) sums (measure M (\<Union> i. f i))" by (rule finite_measure_UNION[OF assms(1,3)]) have b: "(\<lambda> i. measure M (g i)) sums (measure M (\<Union> i. g i))" by (rule finite_measure_UNION[OF assms(2,4)]) show ?thesis using sums_unique[OF b] sums_unique[OF a] assms by simp qed lemma (in finite_measure) measure_countably_zero: assumes "range c \<subseteq> sets M" assumes "\<And> i. measure M (c i) = 0" shows "measure M (\<Union> i :: nat. c i) = 0" proof (rule antisym) show "measure M (\<Union> i :: nat. c i) \<le> 0" using finite_measure_subadditive_countably[OF assms(1)] by (simp add: assms(2)) qed (simp add: measure_nonneg) lemma (in finite_measure) measure_space_inter: assumes events:"s \<in> sets M" "t \<in> sets M" assumes "measure M t = measure M (space M)" shows "measure M (s \<inter> t) = measure M s" proof - have "measure M ((space M - s) \<union> (space M - t)) = measure M (space M - s)" using events assms finite_measure_compl[of "t"] by (auto intro!: measure_zero_union) also have "(space M - s) \<union> (space M - t) = space M - (s \<inter> t)" by blast finally show "measure M (s \<inter> t) = measure M s" using events by (auto intro!: measure_eq_compl[of "s \<inter> t" s]) qed lemma (in finite_measure) measure_equiprobable_finite_unions: assumes s: "finite s" "\<And>x. x \<in> s \<Longrightarrow> {x} \<in> sets M" assumes "\<And> x y. \<lbrakk>x \<in> s; y \<in> s\<rbrakk> \<Longrightarrow> measure M {x} = measure M {y}" shows "measure M s = real (card s) * measure M {SOME x. x \<in> s}" proof cases assume "s \<noteq> {}" then have "\<exists> x. x \<in> s" by blast from someI_ex[OF this] assms have prob_some: "\<And> x. x \<in> s \<Longrightarrow> measure M {x} = measure M {SOME y. y \<in> s}" by blast have "measure M s = (\<Sum> x \<in> s. measure M {x})" using finite_measure_eq_setsum_singleton[OF s] by simp also have "\<dots> = (\<Sum> x \<in> s. measure M {SOME y. y \<in> s})" using prob_some by auto also have "\<dots> = real (card s) * measure M {(SOME x. x \<in> s)}" using setsum_constant assms by (simp add: real_eq_of_nat) finally show ?thesis by simp qed simp lemma (in finite_measure) measure_real_sum_image_fn: assumes "e \<in> sets M" assumes "\<And> x. x \<in> s \<Longrightarrow> e \<inter> f x \<in> sets M" assumes "finite s" assumes disjoint: "\<And> x y. \<lbrakk>x \<in> s ; y \<in> s ; x \<noteq> y\<rbrakk> \<Longrightarrow> f x \<inter> f y = {}" assumes upper: "space M \<subseteq> (\<Union> i \<in> s. f i)" shows "measure M e = (\<Sum> x \<in> s. measure M (e \<inter> f x))" proof - have e: "e = (\<Union> i \<in> s. e \<inter> f i)" using `e \<in> sets M` sets.sets_into_space upper by blast hence "measure M e = measure M (\<Union> i \<in> s. e \<inter> f i)" by simp also have "\<dots> = (\<Sum> x \<in> s. measure M (e \<inter> f x))" proof (rule finite_measure_finite_Union) show "finite s" by fact show "(\<lambda>i. e \<inter> f i)`s \<subseteq> sets M" using assms(2) by auto show "disjoint_family_on (\<lambda>i. e \<inter> f i) s" using disjoint by (auto simp: disjoint_family_on_def) qed finally show ?thesis . qed lemma (in finite_measure) measure_exclude: assumes "A \<in> sets M" "B \<in> sets M" assumes "measure M A = measure M (space M)" "A \<inter> B = {}" shows "measure M B = 0" using measure_space_inter[of B A] assms by (auto simp: ac_simps) section {* Counting space *} lemma strict_monoI_Suc: assumes ord [simp]: "(\<And>n. f n < f (Suc n))" shows "strict_mono f" unfolding strict_mono_def proof safe fix n m :: nat assume "n < m" then show "f n < f m" by (induct m) (auto simp: less_Suc_eq intro: less_trans ord) qed lemma emeasure_count_space: assumes "X \<subseteq> A" shows "emeasure (count_space A) X = (if finite X then ereal (card X) else \<infinity>)" (is "_ = ?M X") unfolding count_space_def proof (rule emeasure_measure_of_sigma) show "X \<in> Pow A" using `X \<subseteq> A` by auto show "sigma_algebra A (Pow A)" by (rule sigma_algebra_Pow) show positive: "positive (Pow A) ?M" by (auto simp: positive_def) have additive: "additive (Pow A) ?M" by (auto simp: card_Un_disjoint additive_def) interpret ring_of_sets A "Pow A" by (rule ring_of_setsI) auto show "countably_additive (Pow A) ?M" unfolding countably_additive_iff_continuous_from_below[OF positive additive] proof safe fix F :: "nat \<Rightarrow> 'a set" assume "incseq F" show "(\<lambda>i. ?M (F i)) ----> ?M (\<Union>i. F i)" proof cases assume "\<exists>i. \<forall>j\<ge>i. F i = F j" then guess i .. note i = this { fix j from i `incseq F` have "F j \<subseteq> F i" by (cases "i \<le> j") (auto simp: incseq_def) } then have eq: "(\<Union>i. F i) = F i" by auto with i show ?thesis by (auto intro!: Lim_eventually eventually_sequentiallyI[where c=i]) next assume "\<not> (\<exists>i. \<forall>j\<ge>i. F i = F j)" then obtain f where "\<And>i. i \<le> f i" "\<And>i. F i \<noteq> F (f i)" by metis moreover then have "\<And>i. F i \<subseteq> F (f i)" using `incseq F` by (auto simp: incseq_def) ultimately have *: "\<And>i. F i \<subset> F (f i)" by auto have "incseq (\<lambda>i. ?M (F i))" using `incseq F` unfolding incseq_def by (auto simp: card_mono dest: finite_subset) then have "(\<lambda>i. ?M (F i)) ----> (SUP n. ?M (F n))" by (rule LIMSEQ_SUP) moreover have "(SUP n. ?M (F n)) = \<infinity>" proof (rule SUP_PInfty) fix n :: nat show "\<exists>k::nat\<in>UNIV. ereal n \<le> ?M (F k)" proof (induct n) case (Suc n) then guess k .. note k = this moreover have "finite (F k) \<Longrightarrow> finite (F (f k)) \<Longrightarrow> card (F k) < card (F (f k))" using `F k \<subset> F (f k)` by (simp add: psubset_card_mono) moreover have "finite (F (f k)) \<Longrightarrow> finite (F k)" using `k \<le> f k` `incseq F` by (auto simp: incseq_def dest: finite_subset) ultimately show ?case by (auto intro!: exI[of _ "f k"]) qed auto qed moreover have "inj (\<lambda>n. F ((f ^^ n) 0))" by (intro strict_mono_imp_inj_on strict_monoI_Suc) (simp add: *) then have 1: "infinite (range (\<lambda>i. F ((f ^^ i) 0)))" by (rule range_inj_infinite) have "infinite (Pow (\<Union>i. F i))" by (rule infinite_super[OF _ 1]) auto then have "infinite (\<Union>i. F i)" by auto ultimately show ?thesis by auto qed qed qed lemma emeasure_count_space_finite[simp]: "X \<subseteq> A \<Longrightarrow> finite X \<Longrightarrow> emeasure (count_space A) X = ereal (card X)" using emeasure_count_space[of X A] by simp lemma emeasure_count_space_infinite[simp]: "X \<subseteq> A \<Longrightarrow> infinite X \<Longrightarrow> emeasure (count_space A) X = \<infinity>" using emeasure_count_space[of X A] by simp lemma emeasure_count_space_eq_0: "emeasure (count_space A) X = 0 \<longleftrightarrow> (X \<subseteq> A \<longrightarrow> X = {})" proof cases assume X: "X \<subseteq> A" then show ?thesis proof (intro iffI impI) assume "emeasure (count_space A) X = 0" with X show "X = {}" by (subst (asm) emeasure_count_space) (auto split: split_if_asm) qed simp qed (simp add: emeasure_notin_sets) lemma null_sets_count_space: "null_sets (count_space A) = { {} }" unfolding null_sets_def by (auto simp add: emeasure_count_space_eq_0) lemma AE_count_space: "(AE x in count_space A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x)" unfolding eventually_ae_filter by (auto simp add: null_sets_count_space) lemma sigma_finite_measure_count_space: fixes A :: "'a::countable set" shows "sigma_finite_measure (count_space A)" proof show "\<exists>F::nat \<Rightarrow> 'a set. range F \<subseteq> sets (count_space A) \<and> (\<Union>i. F i) = space (count_space A) \<and> (\<forall>i. emeasure (count_space A) (F i) \<noteq> \<infinity>)" using surj_from_nat by (intro exI[of _ "\<lambda>i. {from_nat i} \<inter> A"]) (auto simp del: surj_from_nat) qed lemma finite_measure_count_space: assumes [simp]: "finite A" shows "finite_measure (count_space A)" by rule simp lemma sigma_finite_measure_count_space_finite: assumes A: "finite A" shows "sigma_finite_measure (count_space A)" proof - interpret finite_measure "count_space A" using A by (rule finite_measure_count_space) show "sigma_finite_measure (count_space A)" .. qed end