1.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/src/HOL/Probability/Measure_Space.thy Mon Apr 23 12:14:35 2012 +0200
1.3 @@ -0,0 +1,1457 @@
1.4 +(* Title: HOL/Probability/Measure_Space.thy
1.5 + Author: Lawrence C Paulson
1.6 + Author: Johannes Hölzl, TU München
1.7 + Author: Armin Heller, TU München
1.8 +*)
1.9 +
1.10 +header {* Measure spaces and their properties *}
1.11 +
1.12 +theory Measure_Space
1.13 +imports
1.14 + Sigma_Algebra
1.15 + "~~/src/HOL/Multivariate_Analysis/Extended_Real_Limits"
1.16 +begin
1.17 +
1.18 +lemma suminf_eq_setsum:
1.19 + fixes f :: "nat \<Rightarrow> 'a::{comm_monoid_add, t2_space}"
1.20 + assumes "finite {i. f i \<noteq> 0}" (is "finite ?P")
1.21 + shows "(\<Sum>i. f i) = (\<Sum>i | f i \<noteq> 0. f i)"
1.22 +proof cases
1.23 + assume "?P \<noteq> {}"
1.24 + have [dest!]: "\<And>i. Suc (Max ?P) \<le> i \<Longrightarrow> f i = 0"
1.25 + using `finite ?P` `?P \<noteq> {}` by (auto simp: Suc_le_eq)
1.26 + have "(\<Sum>i. f i) = (\<Sum>i<Suc (Max ?P). f i)"
1.27 + by (rule suminf_finite) auto
1.28 + also have "\<dots> = (\<Sum>i | f i \<noteq> 0. f i)"
1.29 + using `finite ?P` `?P \<noteq> {}`
1.30 + by (intro setsum_mono_zero_right) (auto simp: less_Suc_eq_le)
1.31 + finally show ?thesis .
1.32 +qed simp
1.33 +
1.34 +lemma suminf_cmult_indicator:
1.35 + fixes f :: "nat \<Rightarrow> ereal"
1.36 + assumes "disjoint_family A" "x \<in> A i" "\<And>i. 0 \<le> f i"
1.37 + shows "(\<Sum>n. f n * indicator (A n) x) = f i"
1.38 +proof -
1.39 + have **: "\<And>n. f n * indicator (A n) x = (if n = i then f n else 0 :: ereal)"
1.40 + using `x \<in> A i` assms unfolding disjoint_family_on_def indicator_def by auto
1.41 + then have "\<And>n. (\<Sum>j<n. f j * indicator (A j) x) = (if i < n then f i else 0 :: ereal)"
1.42 + by (auto simp: setsum_cases)
1.43 + moreover have "(SUP n. if i < n then f i else 0) = (f i :: ereal)"
1.44 + proof (rule ereal_SUPI)
1.45 + fix y :: ereal assume "\<And>n. n \<in> UNIV \<Longrightarrow> (if i < n then f i else 0) \<le> y"
1.46 + from this[of "Suc i"] show "f i \<le> y" by auto
1.47 + qed (insert assms, simp)
1.48 + ultimately show ?thesis using assms
1.49 + by (subst suminf_ereal_eq_SUPR) (auto simp: indicator_def)
1.50 +qed
1.51 +
1.52 +lemma suminf_indicator:
1.53 + assumes "disjoint_family A"
1.54 + shows "(\<Sum>n. indicator (A n) x :: ereal) = indicator (\<Union>i. A i) x"
1.55 +proof cases
1.56 + assume *: "x \<in> (\<Union>i. A i)"
1.57 + then obtain i where "x \<in> A i" by auto
1.58 + from suminf_cmult_indicator[OF assms(1), OF `x \<in> A i`, of "\<lambda>k. 1"]
1.59 + show ?thesis using * by simp
1.60 +qed simp
1.61 +
1.62 +text {*
1.63 + The type for emeasure spaces is already defined in @{theory Sigma_Algebra}, as it is also used to
1.64 + represent sigma algebras (with an arbitrary emeasure).
1.65 +*}
1.66 +
1.67 +section "Extend binary sets"
1.68 +
1.69 +lemma LIMSEQ_binaryset:
1.70 + assumes f: "f {} = 0"
1.71 + shows "(\<lambda>n. \<Sum>i<n. f (binaryset A B i)) ----> f A + f B"
1.72 +proof -
1.73 + have "(\<lambda>n. \<Sum>i < Suc (Suc n). f (binaryset A B i)) = (\<lambda>n. f A + f B)"
1.74 + proof
1.75 + fix n
1.76 + show "(\<Sum>i < Suc (Suc n). f (binaryset A B i)) = f A + f B"
1.77 + by (induct n) (auto simp add: binaryset_def f)
1.78 + qed
1.79 + moreover
1.80 + have "... ----> f A + f B" by (rule tendsto_const)
1.81 + ultimately
1.82 + have "(\<lambda>n. \<Sum>i< Suc (Suc n). f (binaryset A B i)) ----> f A + f B"
1.83 + by metis
1.84 + hence "(\<lambda>n. \<Sum>i< n+2. f (binaryset A B i)) ----> f A + f B"
1.85 + by simp
1.86 + thus ?thesis by (rule LIMSEQ_offset [where k=2])
1.87 +qed
1.88 +
1.89 +lemma binaryset_sums:
1.90 + assumes f: "f {} = 0"
1.91 + shows "(\<lambda>n. f (binaryset A B n)) sums (f A + f B)"
1.92 + by (simp add: sums_def LIMSEQ_binaryset [where f=f, OF f] atLeast0LessThan)
1.93 +
1.94 +lemma suminf_binaryset_eq:
1.95 + fixes f :: "'a set \<Rightarrow> 'b::{comm_monoid_add, t2_space}"
1.96 + shows "f {} = 0 \<Longrightarrow> (\<Sum>n. f (binaryset A B n)) = f A + f B"
1.97 + by (metis binaryset_sums sums_unique)
1.98 +
1.99 +section {* Properties of a premeasure @{term \<mu>} *}
1.100 +
1.101 +text {*
1.102 + The definitions for @{const positive} and @{const countably_additive} should be here, by they are
1.103 + necessary to define @{typ "'a measure"} in @{theory Sigma_Algebra}.
1.104 +*}
1.105 +
1.106 +definition additive where
1.107 + "additive M \<mu> \<longleftrightarrow> (\<forall>x\<in>M. \<forall>y\<in>M. x \<inter> y = {} \<longrightarrow> \<mu> (x \<union> y) = \<mu> x + \<mu> y)"
1.108 +
1.109 +definition increasing where
1.110 + "increasing M \<mu> \<longleftrightarrow> (\<forall>x\<in>M. \<forall>y\<in>M. x \<subseteq> y \<longrightarrow> \<mu> x \<le> \<mu> y)"
1.111 +
1.112 +lemma positiveD_empty:
1.113 + "positive M f \<Longrightarrow> f {} = 0"
1.114 + by (auto simp add: positive_def)
1.115 +
1.116 +lemma additiveD:
1.117 + "additive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x \<in> M \<Longrightarrow> y \<in> M \<Longrightarrow> f (x \<union> y) = f x + f y"
1.118 + by (auto simp add: additive_def)
1.119 +
1.120 +lemma increasingD:
1.121 + "increasing M f \<Longrightarrow> x \<subseteq> y \<Longrightarrow> x\<in>M \<Longrightarrow> y\<in>M \<Longrightarrow> f x \<le> f y"
1.122 + by (auto simp add: increasing_def)
1.123 +
1.124 +lemma countably_additiveI:
1.125 + "(\<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))
1.126 + \<Longrightarrow> countably_additive M f"
1.127 + by (simp add: countably_additive_def)
1.128 +
1.129 +lemma (in ring_of_sets) disjointed_additive:
1.130 + assumes f: "positive M f" "additive M f" and A: "range A \<subseteq> M" "incseq A"
1.131 + shows "(\<Sum>i\<le>n. f (disjointed A i)) = f (A n)"
1.132 +proof (induct n)
1.133 + case (Suc n)
1.134 + then have "(\<Sum>i\<le>Suc n. f (disjointed A i)) = f (A n) + f (disjointed A (Suc n))"
1.135 + by simp
1.136 + also have "\<dots> = f (A n \<union> disjointed A (Suc n))"
1.137 + using A by (subst f(2)[THEN additiveD]) (auto simp: disjointed_incseq)
1.138 + also have "A n \<union> disjointed A (Suc n) = A (Suc n)"
1.139 + using `incseq A` by (auto dest: incseq_SucD simp: disjointed_incseq)
1.140 + finally show ?case .
1.141 +qed simp
1.142 +
1.143 +lemma (in ring_of_sets) additive_sum:
1.144 + fixes A:: "'i \<Rightarrow> 'a set"
1.145 + assumes f: "positive M f" and ad: "additive M f" and "finite S"
1.146 + and A: "A`S \<subseteq> M"
1.147 + and disj: "disjoint_family_on A S"
1.148 + shows "(\<Sum>i\<in>S. f (A i)) = f (\<Union>i\<in>S. A i)"
1.149 +using `finite S` disj A proof induct
1.150 + case empty show ?case using f by (simp add: positive_def)
1.151 +next
1.152 + case (insert s S)
1.153 + then have "A s \<inter> (\<Union>i\<in>S. A i) = {}"
1.154 + by (auto simp add: disjoint_family_on_def neq_iff)
1.155 + moreover
1.156 + have "A s \<in> M" using insert by blast
1.157 + moreover have "(\<Union>i\<in>S. A i) \<in> M"
1.158 + using insert `finite S` by auto
1.159 + moreover
1.160 + ultimately have "f (A s \<union> (\<Union>i\<in>S. A i)) = f (A s) + f(\<Union>i\<in>S. A i)"
1.161 + using ad UNION_in_sets A by (auto simp add: additive_def)
1.162 + with insert show ?case using ad disjoint_family_on_mono[of S "insert s S" A]
1.163 + by (auto simp add: additive_def subset_insertI)
1.164 +qed
1.165 +
1.166 +lemma (in ring_of_sets) additive_increasing:
1.167 + assumes posf: "positive M f" and addf: "additive M f"
1.168 + shows "increasing M f"
1.169 +proof (auto simp add: increasing_def)
1.170 + fix x y
1.171 + assume xy: "x \<in> M" "y \<in> M" "x \<subseteq> y"
1.172 + then have "y - x \<in> M" by auto
1.173 + then have "0 \<le> f (y-x)" using posf[unfolded positive_def] by auto
1.174 + then have "f x + 0 \<le> f x + f (y-x)" by (intro add_left_mono) auto
1.175 + also have "... = f (x \<union> (y-x))" using addf
1.176 + by (auto simp add: additive_def) (metis Diff_disjoint Un_Diff_cancel Diff xy(1,2))
1.177 + also have "... = f y"
1.178 + by (metis Un_Diff_cancel Un_absorb1 xy(3))
1.179 + finally show "f x \<le> f y" by simp
1.180 +qed
1.181 +
1.182 +lemma (in ring_of_sets) countably_additive_additive:
1.183 + assumes posf: "positive M f" and ca: "countably_additive M f"
1.184 + shows "additive M f"
1.185 +proof (auto simp add: additive_def)
1.186 + fix x y
1.187 + assume x: "x \<in> M" and y: "y \<in> M" and "x \<inter> y = {}"
1.188 + hence "disjoint_family (binaryset x y)"
1.189 + by (auto simp add: disjoint_family_on_def binaryset_def)
1.190 + hence "range (binaryset x y) \<subseteq> M \<longrightarrow>
1.191 + (\<Union>i. binaryset x y i) \<in> M \<longrightarrow>
1.192 + f (\<Union>i. binaryset x y i) = (\<Sum> n. f (binaryset x y n))"
1.193 + using ca
1.194 + by (simp add: countably_additive_def)
1.195 + hence "{x,y,{}} \<subseteq> M \<longrightarrow> x \<union> y \<in> M \<longrightarrow>
1.196 + f (x \<union> y) = (\<Sum>n. f (binaryset x y n))"
1.197 + by (simp add: range_binaryset_eq UN_binaryset_eq)
1.198 + thus "f (x \<union> y) = f x + f y" using posf x y
1.199 + by (auto simp add: Un suminf_binaryset_eq positive_def)
1.200 +qed
1.201 +
1.202 +lemma (in algebra) increasing_additive_bound:
1.203 + fixes A:: "nat \<Rightarrow> 'a set" and f :: "'a set \<Rightarrow> ereal"
1.204 + assumes f: "positive M f" and ad: "additive M f"
1.205 + and inc: "increasing M f"
1.206 + and A: "range A \<subseteq> M"
1.207 + and disj: "disjoint_family A"
1.208 + shows "(\<Sum>i. f (A i)) \<le> f \<Omega>"
1.209 +proof (safe intro!: suminf_bound)
1.210 + fix N
1.211 + note disj_N = disjoint_family_on_mono[OF _ disj, of "{..<N}"]
1.212 + have "(\<Sum>i<N. f (A i)) = f (\<Union>i\<in>{..<N}. A i)"
1.213 + using A by (intro additive_sum [OF f ad _ _]) (auto simp: disj_N)
1.214 + also have "... \<le> f \<Omega>" using space_closed A
1.215 + by (intro increasingD[OF inc] finite_UN) auto
1.216 + finally show "(\<Sum>i<N. f (A i)) \<le> f \<Omega>" by simp
1.217 +qed (insert f A, auto simp: positive_def)
1.218 +
1.219 +lemma (in ring_of_sets) countably_additiveI_finite:
1.220 + assumes "finite \<Omega>" "positive M \<mu>" "additive M \<mu>"
1.221 + shows "countably_additive M \<mu>"
1.222 +proof (rule countably_additiveI)
1.223 + fix F :: "nat \<Rightarrow> 'a set" assume F: "range F \<subseteq> M" "(\<Union>i. F i) \<in> M" and disj: "disjoint_family F"
1.224 +
1.225 + have "\<forall>i\<in>{i. F i \<noteq> {}}. \<exists>x. x \<in> F i" by auto
1.226 + from bchoice[OF this] obtain f where f: "\<And>i. F i \<noteq> {} \<Longrightarrow> f i \<in> F i" by auto
1.227 +
1.228 + have inj_f: "inj_on f {i. F i \<noteq> {}}"
1.229 + proof (rule inj_onI, simp)
1.230 + fix i j a b assume *: "f i = f j" "F i \<noteq> {}" "F j \<noteq> {}"
1.231 + then have "f i \<in> F i" "f j \<in> F j" using f by force+
1.232 + with disj * show "i = j" by (auto simp: disjoint_family_on_def)
1.233 + qed
1.234 + have "finite (\<Union>i. F i)"
1.235 + by (metis F(2) assms(1) infinite_super sets_into_space)
1.236 +
1.237 + have F_subset: "{i. \<mu> (F i) \<noteq> 0} \<subseteq> {i. F i \<noteq> {}}"
1.238 + by (auto simp: positiveD_empty[OF `positive M \<mu>`])
1.239 + moreover have fin_not_empty: "finite {i. F i \<noteq> {}}"
1.240 + proof (rule finite_imageD)
1.241 + from f have "f`{i. F i \<noteq> {}} \<subseteq> (\<Union>i. F i)" by auto
1.242 + then show "finite (f`{i. F i \<noteq> {}})"
1.243 + by (rule finite_subset) fact
1.244 + qed fact
1.245 + ultimately have fin_not_0: "finite {i. \<mu> (F i) \<noteq> 0}"
1.246 + by (rule finite_subset)
1.247 +
1.248 + have disj_not_empty: "disjoint_family_on F {i. F i \<noteq> {}}"
1.249 + using disj by (auto simp: disjoint_family_on_def)
1.250 +
1.251 + from fin_not_0 have "(\<Sum>i. \<mu> (F i)) = (\<Sum>i | \<mu> (F i) \<noteq> 0. \<mu> (F i))"
1.252 + by (rule suminf_eq_setsum)
1.253 + also have "\<dots> = (\<Sum>i | F i \<noteq> {}. \<mu> (F i))"
1.254 + using fin_not_empty F_subset by (rule setsum_mono_zero_left) auto
1.255 + also have "\<dots> = \<mu> (\<Union>i\<in>{i. F i \<noteq> {}}. F i)"
1.256 + using `positive M \<mu>` `additive M \<mu>` fin_not_empty disj_not_empty F by (intro additive_sum) auto
1.257 + also have "\<dots> = \<mu> (\<Union>i. F i)"
1.258 + by (rule arg_cong[where f=\<mu>]) auto
1.259 + finally show "(\<Sum>i. \<mu> (F i)) = \<mu> (\<Union>i. F i)" .
1.260 +qed
1.261 +
1.262 +section {* Properties of @{const emeasure} *}
1.263 +
1.264 +lemma emeasure_positive: "positive (sets M) (emeasure M)"
1.265 + by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)
1.266 +
1.267 +lemma emeasure_empty[simp, intro]: "emeasure M {} = 0"
1.268 + using emeasure_positive[of M] by (simp add: positive_def)
1.269 +
1.270 +lemma emeasure_nonneg[intro!]: "0 \<le> emeasure M A"
1.271 + using emeasure_notin_sets[of A M] emeasure_positive[of M]
1.272 + by (cases "A \<in> sets M") (auto simp: positive_def)
1.273 +
1.274 +lemma emeasure_not_MInf[simp]: "emeasure M A \<noteq> - \<infinity>"
1.275 + using emeasure_nonneg[of M A] by auto
1.276 +
1.277 +lemma emeasure_countably_additive: "countably_additive (sets M) (emeasure M)"
1.278 + by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)
1.279 +
1.280 +lemma suminf_emeasure:
1.281 + "range A \<subseteq> sets M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Sum>i. emeasure M (A i)) = emeasure M (\<Union>i. A i)"
1.282 + using countable_UN[of A UNIV M] emeasure_countably_additive[of M]
1.283 + by (simp add: countably_additive_def)
1.284 +
1.285 +lemma emeasure_additive: "additive (sets M) (emeasure M)"
1.286 + by (metis countably_additive_additive emeasure_positive emeasure_countably_additive)
1.287 +
1.288 +lemma plus_emeasure:
1.289 + "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)"
1.290 + using additiveD[OF emeasure_additive] ..
1.291 +
1.292 +lemma setsum_emeasure:
1.293 + "F`I \<subseteq> sets M \<Longrightarrow> disjoint_family_on F I \<Longrightarrow> finite I \<Longrightarrow>
1.294 + (\<Sum>i\<in>I. emeasure M (F i)) = emeasure M (\<Union>i\<in>I. F i)"
1.295 + by (metis additive_sum emeasure_positive emeasure_additive)
1.296 +
1.297 +lemma emeasure_mono:
1.298 + "a \<subseteq> b \<Longrightarrow> b \<in> sets M \<Longrightarrow> emeasure M a \<le> emeasure M b"
1.299 + by (metis additive_increasing emeasure_additive emeasure_nonneg emeasure_notin_sets
1.300 + emeasure_positive increasingD)
1.301 +
1.302 +lemma emeasure_space:
1.303 + "emeasure M A \<le> emeasure M (space M)"
1.304 + by (metis emeasure_mono emeasure_nonneg emeasure_notin_sets sets_into_space top)
1.305 +
1.306 +lemma emeasure_compl:
1.307 + assumes s: "s \<in> sets M" and fin: "emeasure M s \<noteq> \<infinity>"
1.308 + shows "emeasure M (space M - s) = emeasure M (space M) - emeasure M s"
1.309 +proof -
1.310 + from s have "0 \<le> emeasure M s" by auto
1.311 + have "emeasure M (space M) = emeasure M (s \<union> (space M - s))" using s
1.312 + by (metis Un_Diff_cancel Un_absorb1 s sets_into_space)
1.313 + also have "... = emeasure M s + emeasure M (space M - s)"
1.314 + by (rule plus_emeasure[symmetric]) (auto simp add: s)
1.315 + finally have "emeasure M (space M) = emeasure M s + emeasure M (space M - s)" .
1.316 + then show ?thesis
1.317 + using fin `0 \<le> emeasure M s`
1.318 + unfolding ereal_eq_minus_iff by (auto simp: ac_simps)
1.319 +qed
1.320 +
1.321 +lemma emeasure_Diff:
1.322 + assumes finite: "emeasure M B \<noteq> \<infinity>"
1.323 + and measurable: "A \<in> sets M" "B \<in> sets M" "B \<subseteq> A"
1.324 + shows "emeasure M (A - B) = emeasure M A - emeasure M B"
1.325 +proof -
1.326 + have "0 \<le> emeasure M B" using assms by auto
1.327 + have "(A - B) \<union> B = A" using `B \<subseteq> A` by auto
1.328 + then have "emeasure M A = emeasure M ((A - B) \<union> B)" by simp
1.329 + also have "\<dots> = emeasure M (A - B) + emeasure M B"
1.330 + using measurable by (subst plus_emeasure[symmetric]) auto
1.331 + finally show "emeasure M (A - B) = emeasure M A - emeasure M B"
1.332 + unfolding ereal_eq_minus_iff
1.333 + using finite `0 \<le> emeasure M B` by auto
1.334 +qed
1.335 +
1.336 +lemma emeasure_countable_increasing:
1.337 + assumes A: "range A \<subseteq> sets M"
1.338 + and A0: "A 0 = {}"
1.339 + and ASuc: "\<And>n. A n \<subseteq> A (Suc n)"
1.340 + shows "(SUP n. emeasure M (A n)) = emeasure M (\<Union>i. A i)"
1.341 +proof -
1.342 + { fix n
1.343 + have "emeasure M (A n) = (\<Sum>i<n. emeasure M (A (Suc i) - A i))"
1.344 + proof (induct n)
1.345 + case 0 thus ?case by (auto simp add: A0)
1.346 + next
1.347 + case (Suc m)
1.348 + have "A (Suc m) = A m \<union> (A (Suc m) - A m)"
1.349 + by (metis ASuc Un_Diff_cancel Un_absorb1)
1.350 + hence "emeasure M (A (Suc m)) =
1.351 + emeasure M (A m) + emeasure M (A (Suc m) - A m)"
1.352 + by (subst plus_emeasure)
1.353 + (auto simp add: emeasure_additive range_subsetD [OF A])
1.354 + with Suc show ?case
1.355 + by simp
1.356 + qed }
1.357 + note Meq = this
1.358 + have Aeq: "(\<Union>i. A (Suc i) - A i) = (\<Union>i. A i)"
1.359 + proof (rule UN_finite2_eq [where k=1], simp)
1.360 + fix i
1.361 + show "(\<Union>i\<in>{0..<i}. A (Suc i) - A i) = (\<Union>i\<in>{0..<Suc i}. A i)"
1.362 + proof (induct i)
1.363 + case 0 thus ?case by (simp add: A0)
1.364 + next
1.365 + case (Suc i)
1.366 + thus ?case
1.367 + by (auto simp add: atLeastLessThanSuc intro: subsetD [OF ASuc])
1.368 + qed
1.369 + qed
1.370 + have A1: "\<And>i. A (Suc i) - A i \<in> sets M"
1.371 + by (metis A Diff range_subsetD)
1.372 + have A2: "(\<Union>i. A (Suc i) - A i) \<in> sets M"
1.373 + by (blast intro: range_subsetD [OF A])
1.374 + have "(SUP n. \<Sum>i<n. emeasure M (A (Suc i) - A i)) = (\<Sum>i. emeasure M (A (Suc i) - A i))"
1.375 + using A by (auto intro!: suminf_ereal_eq_SUPR[symmetric])
1.376 + also have "\<dots> = emeasure M (\<Union>i. A (Suc i) - A i)"
1.377 + by (rule suminf_emeasure)
1.378 + (auto simp add: disjoint_family_Suc ASuc A1 A2)
1.379 + also have "... = emeasure M (\<Union>i. A i)"
1.380 + by (simp add: Aeq)
1.381 + finally have "(SUP n. \<Sum>i<n. emeasure M (A (Suc i) - A i)) = emeasure M (\<Union>i. A i)" .
1.382 + then show ?thesis by (auto simp add: Meq)
1.383 +qed
1.384 +
1.385 +lemma SUP_emeasure_incseq:
1.386 + assumes A: "range A \<subseteq> sets M" and "incseq A"
1.387 + shows "(SUP n. emeasure M (A n)) = emeasure M (\<Union>i. A i)"
1.388 +proof -
1.389 + have *: "(SUP n. emeasure M (nat_case {} A (Suc n))) = (SUP n. emeasure M (nat_case {} A n))"
1.390 + using A by (auto intro!: SUPR_eq exI split: nat.split)
1.391 + have ueq: "(\<Union>i. nat_case {} A i) = (\<Union>i. A i)"
1.392 + by (auto simp add: split: nat.splits)
1.393 + have meq: "\<And>n. emeasure M (A n) = (emeasure M \<circ> nat_case {} A) (Suc n)"
1.394 + by simp
1.395 + have "(SUP n. emeasure M (nat_case {} A n)) = emeasure M (\<Union>i. nat_case {} A i)"
1.396 + using range_subsetD[OF A] incseq_SucD[OF `incseq A`]
1.397 + by (force split: nat.splits intro!: emeasure_countable_increasing)
1.398 + also have "emeasure M (\<Union>i. nat_case {} A i) = emeasure M (\<Union>i. A i)"
1.399 + by (simp add: ueq)
1.400 + finally have "(SUP n. emeasure M (nat_case {} A n)) = emeasure M (\<Union>i. A i)" .
1.401 + thus ?thesis unfolding meq * comp_def .
1.402 +qed
1.403 +
1.404 +lemma incseq_emeasure:
1.405 + assumes "range B \<subseteq> sets M" "incseq B"
1.406 + shows "incseq (\<lambda>i. emeasure M (B i))"
1.407 + using assms by (auto simp: incseq_def intro!: emeasure_mono)
1.408 +
1.409 +lemma Lim_emeasure_incseq:
1.410 + assumes A: "range A \<subseteq> sets M" "incseq A"
1.411 + shows "(\<lambda>i. (emeasure M (A i))) ----> emeasure M (\<Union>i. A i)"
1.412 + using LIMSEQ_ereal_SUPR[OF incseq_emeasure, OF A]
1.413 + SUP_emeasure_incseq[OF A] by simp
1.414 +
1.415 +lemma decseq_emeasure:
1.416 + assumes "range B \<subseteq> sets M" "decseq B"
1.417 + shows "decseq (\<lambda>i. emeasure M (B i))"
1.418 + using assms by (auto simp: decseq_def intro!: emeasure_mono)
1.419 +
1.420 +lemma INF_emeasure_decseq:
1.421 + assumes A: "range A \<subseteq> sets M" and "decseq A"
1.422 + and finite: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
1.423 + shows "(INF n. emeasure M (A n)) = emeasure M (\<Inter>i. A i)"
1.424 +proof -
1.425 + have le_MI: "emeasure M (\<Inter>i. A i) \<le> emeasure M (A 0)"
1.426 + using A by (auto intro!: emeasure_mono)
1.427 + hence *: "emeasure M (\<Inter>i. A i) \<noteq> \<infinity>" using finite[of 0] by auto
1.428 +
1.429 + have A0: "0 \<le> emeasure M (A 0)" using A by auto
1.430 +
1.431 + have "emeasure M (A 0) - (INF n. emeasure M (A n)) = emeasure M (A 0) + (SUP n. - emeasure M (A n))"
1.432 + by (simp add: ereal_SUPR_uminus minus_ereal_def)
1.433 + also have "\<dots> = (SUP n. emeasure M (A 0) - emeasure M (A n))"
1.434 + unfolding minus_ereal_def using A0 assms
1.435 + by (subst SUPR_ereal_add) (auto simp add: decseq_emeasure)
1.436 + also have "\<dots> = (SUP n. emeasure M (A 0 - A n))"
1.437 + using A finite `decseq A`[unfolded decseq_def] by (subst emeasure_Diff) auto
1.438 + also have "\<dots> = emeasure M (\<Union>i. A 0 - A i)"
1.439 + proof (rule SUP_emeasure_incseq)
1.440 + show "range (\<lambda>n. A 0 - A n) \<subseteq> sets M"
1.441 + using A by auto
1.442 + show "incseq (\<lambda>n. A 0 - A n)"
1.443 + using `decseq A` by (auto simp add: incseq_def decseq_def)
1.444 + qed
1.445 + also have "\<dots> = emeasure M (A 0) - emeasure M (\<Inter>i. A i)"
1.446 + using A finite * by (simp, subst emeasure_Diff) auto
1.447 + finally show ?thesis
1.448 + unfolding ereal_minus_eq_minus_iff using finite A0 by auto
1.449 +qed
1.450 +
1.451 +lemma Lim_emeasure_decseq:
1.452 + assumes A: "range A \<subseteq> sets M" "decseq A" and fin: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
1.453 + shows "(\<lambda>i. emeasure M (A i)) ----> emeasure M (\<Inter>i. A i)"
1.454 + using LIMSEQ_ereal_INFI[OF decseq_emeasure, OF A]
1.455 + using INF_emeasure_decseq[OF A fin] by simp
1.456 +
1.457 +lemma emeasure_subadditive:
1.458 + assumes measurable: "A \<in> sets M" "B \<in> sets M"
1.459 + shows "emeasure M (A \<union> B) \<le> emeasure M A + emeasure M B"
1.460 +proof -
1.461 + from plus_emeasure[of A M "B - A"]
1.462 + have "emeasure M (A \<union> B) = emeasure M A + emeasure M (B - A)"
1.463 + using assms by (simp add: Diff)
1.464 + also have "\<dots> \<le> emeasure M A + emeasure M B"
1.465 + using assms by (auto intro!: add_left_mono emeasure_mono)
1.466 + finally show ?thesis .
1.467 +qed
1.468 +
1.469 +lemma emeasure_subadditive_finite:
1.470 + assumes "finite I" "A ` I \<subseteq> sets M"
1.471 + shows "emeasure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. emeasure M (A i))"
1.472 +using assms proof induct
1.473 + case (insert i I)
1.474 + then have "emeasure M (\<Union>i\<in>insert i I. A i) = emeasure M (A i \<union> (\<Union>i\<in>I. A i))"
1.475 + by simp
1.476 + also have "\<dots> \<le> emeasure M (A i) + emeasure M (\<Union>i\<in>I. A i)"
1.477 + using insert by (intro emeasure_subadditive finite_UN) auto
1.478 + also have "\<dots> \<le> emeasure M (A i) + (\<Sum>i\<in>I. emeasure M (A i))"
1.479 + using insert by (intro add_mono) auto
1.480 + also have "\<dots> = (\<Sum>i\<in>insert i I. emeasure M (A i))"
1.481 + using insert by auto
1.482 + finally show ?case .
1.483 +qed simp
1.484 +
1.485 +lemma emeasure_subadditive_countably:
1.486 + assumes "range f \<subseteq> sets M"
1.487 + shows "emeasure M (\<Union>i. f i) \<le> (\<Sum>i. emeasure M (f i))"
1.488 +proof -
1.489 + have "emeasure M (\<Union>i. f i) = emeasure M (\<Union>i. disjointed f i)"
1.490 + unfolding UN_disjointed_eq ..
1.491 + also have "\<dots> = (\<Sum>i. emeasure M (disjointed f i))"
1.492 + using range_disjointed_sets[OF assms] suminf_emeasure[of "disjointed f"]
1.493 + by (simp add: disjoint_family_disjointed comp_def)
1.494 + also have "\<dots> \<le> (\<Sum>i. emeasure M (f i))"
1.495 + using range_disjointed_sets[OF assms] assms
1.496 + by (auto intro!: suminf_le_pos emeasure_mono disjointed_subset)
1.497 + finally show ?thesis .
1.498 +qed
1.499 +
1.500 +lemma emeasure_insert:
1.501 + assumes sets: "{x} \<in> sets M" "A \<in> sets M" and "x \<notin> A"
1.502 + shows "emeasure M (insert x A) = emeasure M {x} + emeasure M A"
1.503 +proof -
1.504 + have "{x} \<inter> A = {}" using `x \<notin> A` by auto
1.505 + from plus_emeasure[OF sets this] show ?thesis by simp
1.506 +qed
1.507 +
1.508 +lemma emeasure_eq_setsum_singleton:
1.509 + assumes "finite S" "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
1.510 + shows "emeasure M S = (\<Sum>x\<in>S. emeasure M {x})"
1.511 + using setsum_emeasure[of "\<lambda>x. {x}" S M] assms
1.512 + by (auto simp: disjoint_family_on_def subset_eq)
1.513 +
1.514 +lemma setsum_emeasure_cover:
1.515 + assumes "finite S" and "A \<in> sets M" and br_in_M: "B ` S \<subseteq> sets M"
1.516 + assumes A: "A \<subseteq> (\<Union>i\<in>S. B i)"
1.517 + assumes disj: "disjoint_family_on B S"
1.518 + shows "emeasure M A = (\<Sum>i\<in>S. emeasure M (A \<inter> (B i)))"
1.519 +proof -
1.520 + have "(\<Sum>i\<in>S. emeasure M (A \<inter> (B i))) = emeasure M (\<Union>i\<in>S. A \<inter> (B i))"
1.521 + proof (rule setsum_emeasure)
1.522 + show "disjoint_family_on (\<lambda>i. A \<inter> B i) S"
1.523 + using `disjoint_family_on B S`
1.524 + unfolding disjoint_family_on_def by auto
1.525 + qed (insert assms, auto)
1.526 + also have "(\<Union>i\<in>S. A \<inter> (B i)) = A"
1.527 + using A by auto
1.528 + finally show ?thesis by simp
1.529 +qed
1.530 +
1.531 +lemma emeasure_eq_0:
1.532 + "N \<in> sets M \<Longrightarrow> emeasure M N = 0 \<Longrightarrow> K \<subseteq> N \<Longrightarrow> emeasure M K = 0"
1.533 + by (metis emeasure_mono emeasure_nonneg order_eq_iff)
1.534 +
1.535 +lemma emeasure_UN_eq_0:
1.536 + assumes "\<And>i::nat. emeasure M (N i) = 0" and "range N \<subseteq> sets M"
1.537 + shows "emeasure M (\<Union> i. N i) = 0"
1.538 +proof -
1.539 + have "0 \<le> emeasure M (\<Union> i. N i)" using assms by auto
1.540 + moreover have "emeasure M (\<Union> i. N i) \<le> 0"
1.541 + using emeasure_subadditive_countably[OF assms(2)] assms(1) by simp
1.542 + ultimately show ?thesis by simp
1.543 +qed
1.544 +
1.545 +lemma measure_eqI_finite:
1.546 + assumes [simp]: "sets M = Pow A" "sets N = Pow A" and "finite A"
1.547 + assumes eq: "\<And>a. a \<in> A \<Longrightarrow> emeasure M {a} = emeasure N {a}"
1.548 + shows "M = N"
1.549 +proof (rule measure_eqI)
1.550 + fix X assume "X \<in> sets M"
1.551 + then have X: "X \<subseteq> A" by auto
1.552 + then have "emeasure M X = (\<Sum>a\<in>X. emeasure M {a})"
1.553 + using `finite A` by (subst emeasure_eq_setsum_singleton) (auto dest: finite_subset)
1.554 + also have "\<dots> = (\<Sum>a\<in>X. emeasure N {a})"
1.555 + using X eq by (auto intro!: setsum_cong)
1.556 + also have "\<dots> = emeasure N X"
1.557 + using X `finite A` by (subst emeasure_eq_setsum_singleton) (auto dest: finite_subset)
1.558 + finally show "emeasure M X = emeasure N X" .
1.559 +qed simp
1.560 +
1.561 +lemma measure_eqI_generator_eq:
1.562 + fixes M N :: "'a measure" and E :: "'a set set" and A :: "nat \<Rightarrow> 'a set"
1.563 + assumes "Int_stable E" "E \<subseteq> Pow \<Omega>"
1.564 + and eq: "\<And>X. X \<in> E \<Longrightarrow> emeasure M X = emeasure N X"
1.565 + and M: "sets M = sigma_sets \<Omega> E"
1.566 + and N: "sets N = sigma_sets \<Omega> E"
1.567 + and A: "range A \<subseteq> E" "incseq A" "(\<Union>i. A i) = \<Omega>" "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
1.568 + shows "M = N"
1.569 +proof -
1.570 + let ?D = "\<lambda>F. {D. D \<in> sigma_sets \<Omega> E \<and> emeasure M (F \<inter> D) = emeasure N (F \<inter> D)}"
1.571 + interpret S: sigma_algebra \<Omega> "sigma_sets \<Omega> E" by (rule sigma_algebra_sigma_sets) fact
1.572 + { fix F assume "F \<in> E" and "emeasure M F \<noteq> \<infinity>"
1.573 + then have [intro]: "F \<in> sigma_sets \<Omega> E" by auto
1.574 + have "emeasure N F \<noteq> \<infinity>" using `emeasure M F \<noteq> \<infinity>` `F \<in> E` eq by simp
1.575 + interpret D: dynkin_system \<Omega> "?D F"
1.576 + proof (rule dynkin_systemI, simp_all)
1.577 + fix A assume "A \<in> sigma_sets \<Omega> E \<and> emeasure M (F \<inter> A) = emeasure N (F \<inter> A)"
1.578 + then show "A \<subseteq> \<Omega>" using S.sets_into_space by auto
1.579 + next
1.580 + have "F \<inter> \<Omega> = F" using `F \<in> E` `E \<subseteq> Pow \<Omega>` by auto
1.581 + then show "emeasure M (F \<inter> \<Omega>) = emeasure N (F \<inter> \<Omega>)"
1.582 + using `F \<in> E` eq by (auto intro: sigma_sets_top)
1.583 + next
1.584 + fix A assume *: "A \<in> sigma_sets \<Omega> E \<and> emeasure M (F \<inter> A) = emeasure N (F \<inter> A)"
1.585 + then have **: "F \<inter> (\<Omega> - A) = F - (F \<inter> A)"
1.586 + and [intro]: "F \<inter> A \<in> sigma_sets \<Omega> E"
1.587 + using `F \<in> E` S.sets_into_space by auto
1.588 + have "emeasure N (F \<inter> A) \<le> emeasure N F" by (auto intro!: emeasure_mono simp: M N)
1.589 + then have "emeasure N (F \<inter> A) \<noteq> \<infinity>" using `emeasure N F \<noteq> \<infinity>` by auto
1.590 + have "emeasure M (F \<inter> A) \<le> emeasure M F" by (auto intro!: emeasure_mono simp: M N)
1.591 + then have "emeasure M (F \<inter> A) \<noteq> \<infinity>" using `emeasure M F \<noteq> \<infinity>` by auto
1.592 + then have "emeasure M (F \<inter> (\<Omega> - A)) = emeasure M F - emeasure M (F \<inter> A)" unfolding **
1.593 + using `F \<inter> A \<in> sigma_sets \<Omega> E` by (auto intro!: emeasure_Diff simp: M N)
1.594 + also have "\<dots> = emeasure N F - emeasure N (F \<inter> A)" using eq `F \<in> E` * by simp
1.595 + also have "\<dots> = emeasure N (F \<inter> (\<Omega> - A))" unfolding **
1.596 + using `F \<inter> A \<in> sigma_sets \<Omega> E` `emeasure N (F \<inter> A) \<noteq> \<infinity>`
1.597 + by (auto intro!: emeasure_Diff[symmetric] simp: M N)
1.598 + finally show "\<Omega> - A \<in> sigma_sets \<Omega> E \<and> emeasure M (F \<inter> (\<Omega> - A)) = emeasure N (F \<inter> (\<Omega> - A))"
1.599 + using * by auto
1.600 + next
1.601 + fix A :: "nat \<Rightarrow> 'a set"
1.602 + assume "disjoint_family A" "range A \<subseteq> {X \<in> sigma_sets \<Omega> E. emeasure M (F \<inter> X) = emeasure N (F \<inter> X)}"
1.603 + then have A: "range (\<lambda>i. F \<inter> A i) \<subseteq> sigma_sets \<Omega> E" "F \<inter> (\<Union>x. A x) = (\<Union>x. F \<inter> A x)"
1.604 + "disjoint_family (\<lambda>i. F \<inter> A i)" "\<And>i. emeasure M (F \<inter> A i) = emeasure N (F \<inter> A i)" "range A \<subseteq> sigma_sets \<Omega> E"
1.605 + by (auto simp: disjoint_family_on_def subset_eq)
1.606 + then show "(\<Union>x. A x) \<in> sigma_sets \<Omega> E \<and> emeasure M (F \<inter> (\<Union>x. A x)) = emeasure N (F \<inter> (\<Union>x. A x))"
1.607 + by (auto simp: M N suminf_emeasure[symmetric] simp del: UN_simps)
1.608 + qed
1.609 + have *: "sigma_sets \<Omega> E = ?D F"
1.610 + using `F \<in> E` `Int_stable E`
1.611 + by (intro D.dynkin_lemma) (auto simp add: Int_stable_def eq)
1.612 + have "\<And>D. D \<in> sigma_sets \<Omega> E \<Longrightarrow> emeasure M (F \<inter> D) = emeasure N (F \<inter> D)"
1.613 + by (subst (asm) *) auto }
1.614 + note * = this
1.615 + show "M = N"
1.616 + proof (rule measure_eqI)
1.617 + show "sets M = sets N"
1.618 + using M N by simp
1.619 + fix X assume "X \<in> sets M"
1.620 + then have "X \<in> sigma_sets \<Omega> E"
1.621 + using M by simp
1.622 + let ?A = "\<lambda>i. A i \<inter> X"
1.623 + have "range ?A \<subseteq> sigma_sets \<Omega> E" "incseq ?A"
1.624 + using A(1,2) `X \<in> sigma_sets \<Omega> E` by (auto simp: incseq_def)
1.625 + moreover
1.626 + { fix i have "emeasure M (?A i) = emeasure N (?A i)"
1.627 + using *[of "A i" X] `X \<in> sigma_sets \<Omega> E` A finite by auto }
1.628 + ultimately show "emeasure M X = emeasure N X"
1.629 + using SUP_emeasure_incseq[of ?A M] SUP_emeasure_incseq[of ?A N] A(3) `X \<in> sigma_sets \<Omega> E`
1.630 + by (auto simp: M N SUP_emeasure_incseq)
1.631 + qed
1.632 +qed
1.633 +
1.634 +lemma measure_of_of_measure: "measure_of (space M) (sets M) (emeasure M) = M"
1.635 +proof (intro measure_eqI emeasure_measure_of_sigma)
1.636 + show "sigma_algebra (space M) (sets M)" ..
1.637 + show "positive (sets M) (emeasure M)"
1.638 + by (simp add: positive_def emeasure_nonneg)
1.639 + show "countably_additive (sets M) (emeasure M)"
1.640 + by (simp add: emeasure_countably_additive)
1.641 +qed simp_all
1.642 +
1.643 +section "@{text \<mu>}-null sets"
1.644 +
1.645 +definition null_sets :: "'a measure \<Rightarrow> 'a set set" where
1.646 + "null_sets M = {N\<in>sets M. emeasure M N = 0}"
1.647 +
1.648 +lemma null_setsD1[dest]: "A \<in> null_sets M \<Longrightarrow> emeasure M A = 0"
1.649 + by (simp add: null_sets_def)
1.650 +
1.651 +lemma null_setsD2[dest]: "A \<in> null_sets M \<Longrightarrow> A \<in> sets M"
1.652 + unfolding null_sets_def by simp
1.653 +
1.654 +lemma null_setsI[intro]: "emeasure M A = 0 \<Longrightarrow> A \<in> sets M \<Longrightarrow> A \<in> null_sets M"
1.655 + unfolding null_sets_def by simp
1.656 +
1.657 +interpretation null_sets: ring_of_sets "space M" "null_sets M" for M
1.658 +proof
1.659 + show "null_sets M \<subseteq> Pow (space M)"
1.660 + using sets_into_space by auto
1.661 + show "{} \<in> null_sets M"
1.662 + by auto
1.663 + fix A B assume sets: "A \<in> null_sets M" "B \<in> null_sets M"
1.664 + then have "A \<in> sets M" "B \<in> sets M"
1.665 + by auto
1.666 + moreover then have "emeasure M (A \<union> B) \<le> emeasure M A + emeasure M B"
1.667 + "emeasure M (A - B) \<le> emeasure M A"
1.668 + by (auto intro!: emeasure_subadditive emeasure_mono)
1.669 + moreover have "emeasure M B = 0" "emeasure M A = 0"
1.670 + using sets by auto
1.671 + ultimately show "A - B \<in> null_sets M" "A \<union> B \<in> null_sets M"
1.672 + by (auto intro!: antisym)
1.673 +qed
1.674 +
1.675 +lemma UN_from_nat: "(\<Union>i. N i) = (\<Union>i. N (Countable.from_nat i))"
1.676 +proof -
1.677 + have "(\<Union>i. N i) = (\<Union>i. (N \<circ> Countable.from_nat) i)"
1.678 + unfolding SUP_def image_compose
1.679 + unfolding surj_from_nat ..
1.680 + then show ?thesis by simp
1.681 +qed
1.682 +
1.683 +lemma null_sets_UN[intro]:
1.684 + assumes "\<And>i::'i::countable. N i \<in> null_sets M"
1.685 + shows "(\<Union>i. N i) \<in> null_sets M"
1.686 +proof (intro conjI CollectI null_setsI)
1.687 + show "(\<Union>i. N i) \<in> sets M" using assms by auto
1.688 + have "0 \<le> emeasure M (\<Union>i. N i)" by (rule emeasure_nonneg)
1.689 + moreover have "emeasure M (\<Union>i. N i) \<le> (\<Sum>n. emeasure M (N (Countable.from_nat n)))"
1.690 + unfolding UN_from_nat[of N]
1.691 + using assms by (intro emeasure_subadditive_countably) auto
1.692 + ultimately show "emeasure M (\<Union>i. N i) = 0"
1.693 + using assms by (auto simp: null_setsD1)
1.694 +qed
1.695 +
1.696 +lemma null_set_Int1:
1.697 + assumes "B \<in> null_sets M" "A \<in> sets M" shows "A \<inter> B \<in> null_sets M"
1.698 +proof (intro CollectI conjI null_setsI)
1.699 + show "emeasure M (A \<inter> B) = 0" using assms
1.700 + by (intro emeasure_eq_0[of B _ "A \<inter> B"]) auto
1.701 +qed (insert assms, auto)
1.702 +
1.703 +lemma null_set_Int2:
1.704 + assumes "B \<in> null_sets M" "A \<in> sets M" shows "B \<inter> A \<in> null_sets M"
1.705 + using assms by (subst Int_commute) (rule null_set_Int1)
1.706 +
1.707 +lemma emeasure_Diff_null_set:
1.708 + assumes "B \<in> null_sets M" "A \<in> sets M"
1.709 + shows "emeasure M (A - B) = emeasure M A"
1.710 +proof -
1.711 + have *: "A - B = (A - (A \<inter> B))" by auto
1.712 + have "A \<inter> B \<in> null_sets M" using assms by (rule null_set_Int1)
1.713 + then show ?thesis
1.714 + unfolding * using assms
1.715 + by (subst emeasure_Diff) auto
1.716 +qed
1.717 +
1.718 +lemma null_set_Diff:
1.719 + assumes "B \<in> null_sets M" "A \<in> sets M" shows "B - A \<in> null_sets M"
1.720 +proof (intro CollectI conjI null_setsI)
1.721 + show "emeasure M (B - A) = 0" using assms by (intro emeasure_eq_0[of B _ "B - A"]) auto
1.722 +qed (insert assms, auto)
1.723 +
1.724 +lemma emeasure_Un_null_set:
1.725 + assumes "A \<in> sets M" "B \<in> null_sets M"
1.726 + shows "emeasure M (A \<union> B) = emeasure M A"
1.727 +proof -
1.728 + have *: "A \<union> B = A \<union> (B - A)" by auto
1.729 + have "B - A \<in> null_sets M" using assms(2,1) by (rule null_set_Diff)
1.730 + then show ?thesis
1.731 + unfolding * using assms
1.732 + by (subst plus_emeasure[symmetric]) auto
1.733 +qed
1.734 +
1.735 +section "Formalize almost everywhere"
1.736 +
1.737 +definition ae_filter :: "'a measure \<Rightarrow> 'a filter" where
1.738 + "ae_filter M = Abs_filter (\<lambda>P. \<exists>N\<in>null_sets M. {x \<in> space M. \<not> P x} \<subseteq> N)"
1.739 +
1.740 +abbreviation
1.741 + almost_everywhere :: "'a measure \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" where
1.742 + "almost_everywhere M P \<equiv> eventually P (ae_filter M)"
1.743 +
1.744 +syntax
1.745 + "_almost_everywhere" :: "pttrn \<Rightarrow> 'a \<Rightarrow> bool \<Rightarrow> bool" ("AE _ in _. _" [0,0,10] 10)
1.746 +
1.747 +translations
1.748 + "AE x in M. P" == "CONST almost_everywhere M (%x. P)"
1.749 +
1.750 +lemma eventually_ae_filter:
1.751 + fixes M P
1.752 + defines [simp]: "F \<equiv> \<lambda>P. \<exists>N\<in>null_sets M. {x \<in> space M. \<not> P x} \<subseteq> N"
1.753 + shows "eventually P (ae_filter M) \<longleftrightarrow> F P"
1.754 + unfolding ae_filter_def F_def[symmetric]
1.755 +proof (rule eventually_Abs_filter)
1.756 + show "is_filter F"
1.757 + proof
1.758 + fix P Q assume "F P" "F Q"
1.759 + then obtain N L where N: "N \<in> null_sets M" "{x \<in> space M. \<not> P x} \<subseteq> N"
1.760 + and L: "L \<in> null_sets M" "{x \<in> space M. \<not> Q x} \<subseteq> L"
1.761 + by auto
1.762 + 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
1.763 + then show "F (\<lambda>x. P x \<and> Q x)" by auto
1.764 + next
1.765 + fix P Q assume "F P"
1.766 + then obtain N where N: "N \<in> null_sets M" "{x \<in> space M. \<not> P x} \<subseteq> N" by auto
1.767 + moreover assume "\<forall>x. P x \<longrightarrow> Q x"
1.768 + ultimately have "N \<in> null_sets M" "{x \<in> space M. \<not> Q x} \<subseteq> N" by auto
1.769 + then show "F Q" by auto
1.770 + qed auto
1.771 +qed
1.772 +
1.773 +lemma AE_I':
1.774 + "N \<in> null_sets M \<Longrightarrow> {x\<in>space M. \<not> P x} \<subseteq> N \<Longrightarrow> (AE x in M. P x)"
1.775 + unfolding eventually_ae_filter by auto
1.776 +
1.777 +lemma AE_iff_null:
1.778 + assumes "{x\<in>space M. \<not> P x} \<in> sets M" (is "?P \<in> sets M")
1.779 + shows "(AE x in M. P x) \<longleftrightarrow> {x\<in>space M. \<not> P x} \<in> null_sets M"
1.780 +proof
1.781 + assume "AE x in M. P x" then obtain N where N: "N \<in> sets M" "?P \<subseteq> N" "emeasure M N = 0"
1.782 + unfolding eventually_ae_filter by auto
1.783 + have "0 \<le> emeasure M ?P" by auto
1.784 + moreover have "emeasure M ?P \<le> emeasure M N"
1.785 + using assms N(1,2) by (auto intro: emeasure_mono)
1.786 + ultimately have "emeasure M ?P = 0" unfolding `emeasure M N = 0` by auto
1.787 + then show "?P \<in> null_sets M" using assms by auto
1.788 +next
1.789 + assume "?P \<in> null_sets M" with assms show "AE x in M. P x" by (auto intro: AE_I')
1.790 +qed
1.791 +
1.792 +lemma AE_iff_null_sets:
1.793 + "N \<in> sets M \<Longrightarrow> N \<in> null_sets M \<longleftrightarrow> (AE x in M. x \<notin> N)"
1.794 + using Int_absorb1[OF sets_into_space, of N M]
1.795 + by (subst AE_iff_null) (auto simp: Int_def[symmetric])
1.796 +
1.797 +lemma AE_iff_measurable:
1.798 + "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"
1.799 + using AE_iff_null[of _ P] by auto
1.800 +
1.801 +lemma AE_E[consumes 1]:
1.802 + assumes "AE x in M. P x"
1.803 + obtains N where "{x \<in> space M. \<not> P x} \<subseteq> N" "emeasure M N = 0" "N \<in> sets M"
1.804 + using assms unfolding eventually_ae_filter by auto
1.805 +
1.806 +lemma AE_E2:
1.807 + assumes "AE x in M. P x" "{x\<in>space M. P x} \<in> sets M"
1.808 + shows "emeasure M {x\<in>space M. \<not> P x} = 0" (is "emeasure M ?P = 0")
1.809 +proof -
1.810 + have "{x\<in>space M. \<not> P x} = space M - {x\<in>space M. P x}" by auto
1.811 + with AE_iff_null[of M P] assms show ?thesis by auto
1.812 +qed
1.813 +
1.814 +lemma AE_I:
1.815 + assumes "{x \<in> space M. \<not> P x} \<subseteq> N" "emeasure M N = 0" "N \<in> sets M"
1.816 + shows "AE x in M. P x"
1.817 + using assms unfolding eventually_ae_filter by auto
1.818 +
1.819 +lemma AE_mp[elim!]:
1.820 + assumes AE_P: "AE x in M. P x" and AE_imp: "AE x in M. P x \<longrightarrow> Q x"
1.821 + shows "AE x in M. Q x"
1.822 +proof -
1.823 + from AE_P obtain A where P: "{x\<in>space M. \<not> P x} \<subseteq> A"
1.824 + and A: "A \<in> sets M" "emeasure M A = 0"
1.825 + by (auto elim!: AE_E)
1.826 +
1.827 + from AE_imp obtain B where imp: "{x\<in>space M. P x \<and> \<not> Q x} \<subseteq> B"
1.828 + and B: "B \<in> sets M" "emeasure M B = 0"
1.829 + by (auto elim!: AE_E)
1.830 +
1.831 + show ?thesis
1.832 + proof (intro AE_I)
1.833 + have "0 \<le> emeasure M (A \<union> B)" using A B by auto
1.834 + moreover have "emeasure M (A \<union> B) \<le> 0"
1.835 + using emeasure_subadditive[of A M B] A B by auto
1.836 + ultimately show "A \<union> B \<in> sets M" "emeasure M (A \<union> B) = 0" using A B by auto
1.837 + show "{x\<in>space M. \<not> Q x} \<subseteq> A \<union> B"
1.838 + using P imp by auto
1.839 + qed
1.840 +qed
1.841 +
1.842 +(* depricated replace by laws about eventually *)
1.843 +lemma
1.844 + 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"
1.845 + and AE_disjI1: "AE x in M. P x \<Longrightarrow> AE x in M. P x \<or> Q x"
1.846 + and AE_disjI2: "AE x in M. Q x \<Longrightarrow> AE x in M. P x \<or> Q x"
1.847 + 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"
1.848 + 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)"
1.849 + by auto
1.850 +
1.851 +lemma AE_impI:
1.852 + "(P \<Longrightarrow> AE x in M. Q x) \<Longrightarrow> AE x in M. P \<longrightarrow> Q x"
1.853 + by (cases P) auto
1.854 +
1.855 +lemma AE_measure:
1.856 + assumes AE: "AE x in M. P x" and sets: "{x\<in>space M. P x} \<in> sets M" (is "?P \<in> sets M")
1.857 + shows "emeasure M {x\<in>space M. P x} = emeasure M (space M)"
1.858 +proof -
1.859 + from AE_E[OF AE] guess N . note N = this
1.860 + with sets have "emeasure M (space M) \<le> emeasure M (?P \<union> N)"
1.861 + by (intro emeasure_mono) auto
1.862 + also have "\<dots> \<le> emeasure M ?P + emeasure M N"
1.863 + using sets N by (intro emeasure_subadditive) auto
1.864 + also have "\<dots> = emeasure M ?P" using N by simp
1.865 + finally show "emeasure M ?P = emeasure M (space M)"
1.866 + using emeasure_space[of M "?P"] by auto
1.867 +qed
1.868 +
1.869 +lemma AE_space: "AE x in M. x \<in> space M"
1.870 + by (rule AE_I[where N="{}"]) auto
1.871 +
1.872 +lemma AE_I2[simp, intro]:
1.873 + "(\<And>x. x \<in> space M \<Longrightarrow> P x) \<Longrightarrow> AE x in M. P x"
1.874 + using AE_space by force
1.875 +
1.876 +lemma AE_Ball_mp:
1.877 + "\<forall>x\<in>space M. P x \<Longrightarrow> AE x in M. P x \<longrightarrow> Q x \<Longrightarrow> AE x in M. Q x"
1.878 + by auto
1.879 +
1.880 +lemma AE_cong[cong]:
1.881 + "(\<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)"
1.882 + by auto
1.883 +
1.884 +lemma AE_all_countable:
1.885 + "(AE x in M. \<forall>i. P i x) \<longleftrightarrow> (\<forall>i::'i::countable. AE x in M. P i x)"
1.886 +proof
1.887 + assume "\<forall>i. AE x in M. P i x"
1.888 + from this[unfolded eventually_ae_filter Bex_def, THEN choice]
1.889 + 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
1.890 + have "{x\<in>space M. \<not> (\<forall>i. P i x)} \<subseteq> (\<Union>i. {x\<in>space M. \<not> P i x})" by auto
1.891 + also have "\<dots> \<subseteq> (\<Union>i. N i)" using N by auto
1.892 + finally have "{x\<in>space M. \<not> (\<forall>i. P i x)} \<subseteq> (\<Union>i. N i)" .
1.893 + moreover from N have "(\<Union>i. N i) \<in> null_sets M"
1.894 + by (intro null_sets_UN) auto
1.895 + ultimately show "AE x in M. \<forall>i. P i x"
1.896 + unfolding eventually_ae_filter by auto
1.897 +qed auto
1.898 +
1.899 +lemma AE_finite_all:
1.900 + 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)"
1.901 + using f by induct auto
1.902 +
1.903 +lemma AE_finite_allI:
1.904 + assumes "finite S"
1.905 + 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"
1.906 + using AE_finite_all[OF `finite S`] by auto
1.907 +
1.908 +lemma emeasure_mono_AE:
1.909 + assumes imp: "AE x in M. x \<in> A \<longrightarrow> x \<in> B"
1.910 + and B: "B \<in> sets M"
1.911 + shows "emeasure M A \<le> emeasure M B"
1.912 +proof cases
1.913 + assume A: "A \<in> sets M"
1.914 + 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"
1.915 + by (auto simp: eventually_ae_filter)
1.916 + have "emeasure M A = emeasure M (A - N)"
1.917 + using N A by (subst emeasure_Diff_null_set) auto
1.918 + also have "emeasure M (A - N) \<le> emeasure M (B - N)"
1.919 + using N A B sets_into_space by (auto intro!: emeasure_mono)
1.920 + also have "emeasure M (B - N) = emeasure M B"
1.921 + using N B by (subst emeasure_Diff_null_set) auto
1.922 + finally show ?thesis .
1.923 +qed (simp add: emeasure_nonneg emeasure_notin_sets)
1.924 +
1.925 +lemma emeasure_eq_AE:
1.926 + assumes iff: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B"
1.927 + assumes A: "A \<in> sets M" and B: "B \<in> sets M"
1.928 + shows "emeasure M A = emeasure M B"
1.929 + using assms by (safe intro!: antisym emeasure_mono_AE) auto
1.930 +
1.931 +section {* @{text \<sigma>}-finite Measures *}
1.932 +
1.933 +locale sigma_finite_measure =
1.934 + fixes M :: "'a measure"
1.935 + assumes sigma_finite: "\<exists>A::nat \<Rightarrow> 'a set.
1.936 + range A \<subseteq> sets M \<and> (\<Union>i. A i) = space M \<and> (\<forall>i. emeasure M (A i) \<noteq> \<infinity>)"
1.937 +
1.938 +lemma (in sigma_finite_measure) sigma_finite_disjoint:
1.939 + obtains A :: "nat \<Rightarrow> 'a set"
1.940 + where "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>" "disjoint_family A"
1.941 +proof atomize_elim
1.942 + case goal1
1.943 + obtain A :: "nat \<Rightarrow> 'a set" where
1.944 + range: "range A \<subseteq> sets M" and
1.945 + space: "(\<Union>i. A i) = space M" and
1.946 + measure: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
1.947 + using sigma_finite by auto
1.948 + note range' = range_disjointed_sets[OF range] range
1.949 + { fix i
1.950 + have "emeasure M (disjointed A i) \<le> emeasure M (A i)"
1.951 + using range' disjointed_subset[of A i] by (auto intro!: emeasure_mono)
1.952 + then have "emeasure M (disjointed A i) \<noteq> \<infinity>"
1.953 + using measure[of i] by auto }
1.954 + with disjoint_family_disjointed UN_disjointed_eq[of A] space range'
1.955 + show ?case by (auto intro!: exI[of _ "disjointed A"])
1.956 +qed
1.957 +
1.958 +lemma (in sigma_finite_measure) sigma_finite_incseq:
1.959 + obtains A :: "nat \<Rightarrow> 'a set"
1.960 + where "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>" "incseq A"
1.961 +proof atomize_elim
1.962 + case goal1
1.963 + obtain F :: "nat \<Rightarrow> 'a set" where
1.964 + F: "range F \<subseteq> sets M" "(\<Union>i. F i) = space M" "\<And>i. emeasure M (F i) \<noteq> \<infinity>"
1.965 + using sigma_finite by auto
1.966 + then show ?case
1.967 + proof (intro exI[of _ "\<lambda>n. \<Union>i\<le>n. F i"] conjI allI)
1.968 + from F have "\<And>x. x \<in> space M \<Longrightarrow> \<exists>i. x \<in> F i" by auto
1.969 + then show "(\<Union>n. \<Union> i\<le>n. F i) = space M"
1.970 + using F by fastforce
1.971 + next
1.972 + fix n
1.973 + have "emeasure M (\<Union> i\<le>n. F i) \<le> (\<Sum>i\<le>n. emeasure M (F i))" using F
1.974 + by (auto intro!: emeasure_subadditive_finite)
1.975 + also have "\<dots> < \<infinity>"
1.976 + using F by (auto simp: setsum_Pinfty)
1.977 + finally show "emeasure M (\<Union> i\<le>n. F i) \<noteq> \<infinity>" by simp
1.978 + qed (force simp: incseq_def)+
1.979 +qed
1.980 +
1.981 +section {* Measure space induced by distribution of @{const measurable}-functions *}
1.982 +
1.983 +definition distr :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b measure" where
1.984 + "distr M N f = measure_of (space N) (sets N) (\<lambda>A. emeasure M (f -` A \<inter> space M))"
1.985 +
1.986 +lemma
1.987 + shows sets_distr[simp]: "sets (distr M N f) = sets N"
1.988 + and space_distr[simp]: "space (distr M N f) = space N"
1.989 + by (auto simp: distr_def)
1.990 +
1.991 +lemma
1.992 + shows measurable_distr_eq1[simp]: "measurable (distr Mf Nf f) Mf' = measurable Nf Mf'"
1.993 + and measurable_distr_eq2[simp]: "measurable Mg' (distr Mg Ng g) = measurable Mg' Ng"
1.994 + by (auto simp: measurable_def)
1.995 +
1.996 +lemma emeasure_distr:
1.997 + fixes f :: "'a \<Rightarrow> 'b"
1.998 + assumes f: "f \<in> measurable M N" and A: "A \<in> sets N"
1.999 + shows "emeasure (distr M N f) A = emeasure M (f -` A \<inter> space M)" (is "_ = ?\<mu> A")
1.1000 + unfolding distr_def
1.1001 +proof (rule emeasure_measure_of_sigma)
1.1002 + show "positive (sets N) ?\<mu>"
1.1003 + by (auto simp: positive_def)
1.1004 +
1.1005 + show "countably_additive (sets N) ?\<mu>"
1.1006 + proof (intro countably_additiveI)
1.1007 + fix A :: "nat \<Rightarrow> 'b set" assume "range A \<subseteq> sets N" "disjoint_family A"
1.1008 + then have A: "\<And>i. A i \<in> sets N" "(\<Union>i. A i) \<in> sets N" by auto
1.1009 + then have *: "range (\<lambda>i. f -` (A i) \<inter> space M) \<subseteq> sets M"
1.1010 + using f by (auto simp: measurable_def)
1.1011 + moreover have "(\<Union>i. f -` A i \<inter> space M) \<in> sets M"
1.1012 + using * by blast
1.1013 + moreover have **: "disjoint_family (\<lambda>i. f -` A i \<inter> space M)"
1.1014 + using `disjoint_family A` by (auto simp: disjoint_family_on_def)
1.1015 + ultimately show "(\<Sum>i. ?\<mu> (A i)) = ?\<mu> (\<Union>i. A i)"
1.1016 + using suminf_emeasure[OF _ **] A f
1.1017 + by (auto simp: comp_def vimage_UN)
1.1018 + qed
1.1019 + show "sigma_algebra (space N) (sets N)" ..
1.1020 +qed fact
1.1021 +
1.1022 +lemma AE_distrD:
1.1023 + assumes f: "f \<in> measurable M M'"
1.1024 + and AE: "AE x in distr M M' f. P x"
1.1025 + shows "AE x in M. P (f x)"
1.1026 +proof -
1.1027 + from AE[THEN AE_E] guess N .
1.1028 + with f show ?thesis
1.1029 + unfolding eventually_ae_filter
1.1030 + by (intro bexI[of _ "f -` N \<inter> space M"])
1.1031 + (auto simp: emeasure_distr measurable_def)
1.1032 +qed
1.1033 +
1.1034 +lemma null_sets_distr_iff:
1.1035 + "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"
1.1036 + by (auto simp add: null_sets_def emeasure_distr measurable_sets)
1.1037 +
1.1038 +lemma distr_distr:
1.1039 + assumes f: "g \<in> measurable N L" and g: "f \<in> measurable M N"
1.1040 + shows "distr (distr M N f) L g = distr M L (g \<circ> f)" (is "?L = ?R")
1.1041 + using measurable_comp[OF g f] f g
1.1042 + by (auto simp add: emeasure_distr measurable_sets measurable_space
1.1043 + intro!: arg_cong[where f="emeasure M"] measure_eqI)
1.1044 +
1.1045 +section {* Real measure values *}
1.1046 +
1.1047 +lemma measure_nonneg: "0 \<le> measure M A"
1.1048 + using emeasure_nonneg[of M A] unfolding measure_def by (auto intro: real_of_ereal_pos)
1.1049 +
1.1050 +lemma measure_empty[simp]: "measure M {} = 0"
1.1051 + unfolding measure_def by simp
1.1052 +
1.1053 +lemma emeasure_eq_ereal_measure:
1.1054 + "emeasure M A \<noteq> \<infinity> \<Longrightarrow> emeasure M A = ereal (measure M A)"
1.1055 + using emeasure_nonneg[of M A]
1.1056 + by (cases "emeasure M A") (auto simp: measure_def)
1.1057 +
1.1058 +lemma measure_Union:
1.1059 + assumes finite: "emeasure M A \<noteq> \<infinity>" "emeasure M B \<noteq> \<infinity>"
1.1060 + and measurable: "A \<in> sets M" "B \<in> sets M" "A \<inter> B = {}"
1.1061 + shows "measure M (A \<union> B) = measure M A + measure M B"
1.1062 + unfolding measure_def
1.1063 + using plus_emeasure[OF measurable, symmetric] finite
1.1064 + by (simp add: emeasure_eq_ereal_measure)
1.1065 +
1.1066 +lemma measure_finite_Union:
1.1067 + assumes measurable: "A`S \<subseteq> sets M" "disjoint_family_on A S" "finite S"
1.1068 + assumes finite: "\<And>i. i \<in> S \<Longrightarrow> emeasure M (A i) \<noteq> \<infinity>"
1.1069 + shows "measure M (\<Union>i\<in>S. A i) = (\<Sum>i\<in>S. measure M (A i))"
1.1070 + unfolding measure_def
1.1071 + using setsum_emeasure[OF measurable, symmetric] finite
1.1072 + by (simp add: emeasure_eq_ereal_measure)
1.1073 +
1.1074 +lemma measure_Diff:
1.1075 + assumes finite: "emeasure M A \<noteq> \<infinity>"
1.1076 + and measurable: "A \<in> sets M" "B \<in> sets M" "B \<subseteq> A"
1.1077 + shows "measure M (A - B) = measure M A - measure M B"
1.1078 +proof -
1.1079 + have "emeasure M (A - B) \<le> emeasure M A" "emeasure M B \<le> emeasure M A"
1.1080 + using measurable by (auto intro!: emeasure_mono)
1.1081 + hence "measure M ((A - B) \<union> B) = measure M (A - B) + measure M B"
1.1082 + using measurable finite by (rule_tac measure_Union) auto
1.1083 + thus ?thesis using `B \<subseteq> A` by (auto simp: Un_absorb2)
1.1084 +qed
1.1085 +
1.1086 +lemma measure_UNION:
1.1087 + assumes measurable: "range A \<subseteq> sets M" "disjoint_family A"
1.1088 + assumes finite: "emeasure M (\<Union>i. A i) \<noteq> \<infinity>"
1.1089 + shows "(\<lambda>i. measure M (A i)) sums (measure M (\<Union>i. A i))"
1.1090 +proof -
1.1091 + from summable_sums[OF summable_ereal_pos, of "\<lambda>i. emeasure M (A i)"]
1.1092 + suminf_emeasure[OF measurable] emeasure_nonneg[of M]
1.1093 + have "(\<lambda>i. emeasure M (A i)) sums (emeasure M (\<Union>i. A i))" by simp
1.1094 + moreover
1.1095 + { fix i
1.1096 + have "emeasure M (A i) \<le> emeasure M (\<Union>i. A i)"
1.1097 + using measurable by (auto intro!: emeasure_mono)
1.1098 + then have "emeasure M (A i) = ereal ((measure M (A i)))"
1.1099 + using finite by (intro emeasure_eq_ereal_measure) auto }
1.1100 + ultimately show ?thesis using finite
1.1101 + unfolding sums_ereal[symmetric] by (simp add: emeasure_eq_ereal_measure)
1.1102 +qed
1.1103 +
1.1104 +lemma measure_subadditive:
1.1105 + assumes measurable: "A \<in> sets M" "B \<in> sets M"
1.1106 + and fin: "emeasure M A \<noteq> \<infinity>" "emeasure M B \<noteq> \<infinity>"
1.1107 + shows "(measure M (A \<union> B)) \<le> (measure M A) + (measure M B)"
1.1108 +proof -
1.1109 + have "emeasure M (A \<union> B) \<noteq> \<infinity>"
1.1110 + using emeasure_subadditive[OF measurable] fin by auto
1.1111 + then show "(measure M (A \<union> B)) \<le> (measure M A) + (measure M B)"
1.1112 + using emeasure_subadditive[OF measurable] fin
1.1113 + by (auto simp: emeasure_eq_ereal_measure)
1.1114 +qed
1.1115 +
1.1116 +lemma measure_subadditive_finite:
1.1117 + assumes A: "finite I" "A`I \<subseteq> sets M" and fin: "\<And>i. i \<in> I \<Longrightarrow> emeasure M (A i) \<noteq> \<infinity>"
1.1118 + shows "measure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. measure M (A i))"
1.1119 +proof -
1.1120 + { have "emeasure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. emeasure M (A i))"
1.1121 + using emeasure_subadditive_finite[OF A] .
1.1122 + also have "\<dots> < \<infinity>"
1.1123 + using fin by (simp add: setsum_Pinfty)
1.1124 + finally have "emeasure M (\<Union>i\<in>I. A i) \<noteq> \<infinity>" by simp }
1.1125 + then show ?thesis
1.1126 + using emeasure_subadditive_finite[OF A] fin
1.1127 + unfolding measure_def by (simp add: emeasure_eq_ereal_measure suminf_ereal measure_nonneg)
1.1128 +qed
1.1129 +
1.1130 +lemma measure_subadditive_countably:
1.1131 + assumes A: "range A \<subseteq> sets M" and fin: "(\<Sum>i. emeasure M (A i)) \<noteq> \<infinity>"
1.1132 + shows "measure M (\<Union>i. A i) \<le> (\<Sum>i. measure M (A i))"
1.1133 +proof -
1.1134 + from emeasure_nonneg fin have "\<And>i. emeasure M (A i) \<noteq> \<infinity>" by (rule suminf_PInfty)
1.1135 + moreover
1.1136 + { have "emeasure M (\<Union>i. A i) \<le> (\<Sum>i. emeasure M (A i))"
1.1137 + using emeasure_subadditive_countably[OF A] .
1.1138 + also have "\<dots> < \<infinity>"
1.1139 + using fin by simp
1.1140 + finally have "emeasure M (\<Union>i. A i) \<noteq> \<infinity>" by simp }
1.1141 + ultimately show ?thesis
1.1142 + using emeasure_subadditive_countably[OF A] fin
1.1143 + unfolding measure_def by (simp add: emeasure_eq_ereal_measure suminf_ereal measure_nonneg)
1.1144 +qed
1.1145 +
1.1146 +lemma measure_eq_setsum_singleton:
1.1147 + assumes S: "finite S" "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
1.1148 + and fin: "\<And>x. x \<in> S \<Longrightarrow> emeasure M {x} \<noteq> \<infinity>"
1.1149 + shows "(measure M S) = (\<Sum>x\<in>S. (measure M {x}))"
1.1150 + unfolding measure_def
1.1151 + using emeasure_eq_setsum_singleton[OF S] fin
1.1152 + by simp (simp add: emeasure_eq_ereal_measure)
1.1153 +
1.1154 +lemma Lim_measure_incseq:
1.1155 + assumes A: "range A \<subseteq> sets M" "incseq A" and fin: "emeasure M (\<Union>i. A i) \<noteq> \<infinity>"
1.1156 + shows "(\<lambda>i. (measure M (A i))) ----> (measure M (\<Union>i. A i))"
1.1157 +proof -
1.1158 + have "ereal ((measure M (\<Union>i. A i))) = emeasure M (\<Union>i. A i)"
1.1159 + using fin by (auto simp: emeasure_eq_ereal_measure)
1.1160 + then show ?thesis
1.1161 + using Lim_emeasure_incseq[OF A]
1.1162 + unfolding measure_def
1.1163 + by (intro lim_real_of_ereal) simp
1.1164 +qed
1.1165 +
1.1166 +lemma Lim_measure_decseq:
1.1167 + assumes A: "range A \<subseteq> sets M" "decseq A" and fin: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
1.1168 + shows "(\<lambda>n. measure M (A n)) ----> measure M (\<Inter>i. A i)"
1.1169 +proof -
1.1170 + have "emeasure M (\<Inter>i. A i) \<le> emeasure M (A 0)"
1.1171 + using A by (auto intro!: emeasure_mono)
1.1172 + also have "\<dots> < \<infinity>"
1.1173 + using fin[of 0] by auto
1.1174 + finally have "ereal ((measure M (\<Inter>i. A i))) = emeasure M (\<Inter>i. A i)"
1.1175 + by (auto simp: emeasure_eq_ereal_measure)
1.1176 + then show ?thesis
1.1177 + unfolding measure_def
1.1178 + using Lim_emeasure_decseq[OF A fin]
1.1179 + by (intro lim_real_of_ereal) simp
1.1180 +qed
1.1181 +
1.1182 +section {* Measure spaces with @{term "emeasure M (space M) < \<infinity>"} *}
1.1183 +
1.1184 +locale finite_measure = sigma_finite_measure M for M +
1.1185 + assumes finite_emeasure_space: "emeasure M (space M) \<noteq> \<infinity>"
1.1186 +
1.1187 +lemma finite_measureI[Pure.intro!]:
1.1188 + assumes *: "emeasure M (space M) \<noteq> \<infinity>"
1.1189 + shows "finite_measure M"
1.1190 +proof
1.1191 + show "\<exists>A. range A \<subseteq> sets M \<and> (\<Union>i. A i) = space M \<and> (\<forall>i. emeasure M (A i) \<noteq> \<infinity>)"
1.1192 + using * by (auto intro!: exI[of _ "\<lambda>_. space M"])
1.1193 +qed fact
1.1194 +
1.1195 +lemma (in finite_measure) emeasure_finite[simp, intro]: "emeasure M A \<noteq> \<infinity>"
1.1196 + using finite_emeasure_space emeasure_space[of M A] by auto
1.1197 +
1.1198 +lemma (in finite_measure) emeasure_eq_measure: "emeasure M A = ereal (measure M A)"
1.1199 + unfolding measure_def by (simp add: emeasure_eq_ereal_measure)
1.1200 +
1.1201 +lemma (in finite_measure) emeasure_real: "\<exists>r. 0 \<le> r \<and> emeasure M A = ereal r"
1.1202 + using emeasure_finite[of A] emeasure_nonneg[of M A] by (cases "emeasure M A") auto
1.1203 +
1.1204 +lemma (in finite_measure) bounded_measure: "measure M A \<le> measure M (space M)"
1.1205 + using emeasure_space[of M A] emeasure_real[of A] emeasure_real[of "space M"] by (auto simp: measure_def)
1.1206 +
1.1207 +lemma (in finite_measure) finite_measure_Diff:
1.1208 + assumes sets: "A \<in> sets M" "B \<in> sets M" and "B \<subseteq> A"
1.1209 + shows "measure M (A - B) = measure M A - measure M B"
1.1210 + using measure_Diff[OF _ assms] by simp
1.1211 +
1.1212 +lemma (in finite_measure) finite_measure_Union:
1.1213 + assumes sets: "A \<in> sets M" "B \<in> sets M" and "A \<inter> B = {}"
1.1214 + shows "measure M (A \<union> B) = measure M A + measure M B"
1.1215 + using measure_Union[OF _ _ assms] by simp
1.1216 +
1.1217 +lemma (in finite_measure) finite_measure_finite_Union:
1.1218 + assumes measurable: "A`S \<subseteq> sets M" "disjoint_family_on A S" "finite S"
1.1219 + shows "measure M (\<Union>i\<in>S. A i) = (\<Sum>i\<in>S. measure M (A i))"
1.1220 + using measure_finite_Union[OF assms] by simp
1.1221 +
1.1222 +lemma (in finite_measure) finite_measure_UNION:
1.1223 + assumes A: "range A \<subseteq> sets M" "disjoint_family A"
1.1224 + shows "(\<lambda>i. measure M (A i)) sums (measure M (\<Union>i. A i))"
1.1225 + using measure_UNION[OF A] by simp
1.1226 +
1.1227 +lemma (in finite_measure) finite_measure_mono:
1.1228 + assumes "A \<subseteq> B" "B \<in> sets M" shows "measure M A \<le> measure M B"
1.1229 + using emeasure_mono[OF assms] emeasure_real[of A] emeasure_real[of B] by (auto simp: measure_def)
1.1230 +
1.1231 +lemma (in finite_measure) finite_measure_subadditive:
1.1232 + assumes m: "A \<in> sets M" "B \<in> sets M"
1.1233 + shows "measure M (A \<union> B) \<le> measure M A + measure M B"
1.1234 + using measure_subadditive[OF m] by simp
1.1235 +
1.1236 +lemma (in finite_measure) finite_measure_subadditive_finite:
1.1237 + 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))"
1.1238 + using measure_subadditive_finite[OF assms] by simp
1.1239 +
1.1240 +lemma (in finite_measure) finite_measure_subadditive_countably:
1.1241 + assumes A: "range A \<subseteq> sets M" and sum: "summable (\<lambda>i. measure M (A i))"
1.1242 + shows "measure M (\<Union>i. A i) \<le> (\<Sum>i. measure M (A i))"
1.1243 +proof -
1.1244 + from `summable (\<lambda>i. measure M (A i))`
1.1245 + have "(\<lambda>i. ereal (measure M (A i))) sums ereal (\<Sum>i. measure M (A i))"
1.1246 + by (simp add: sums_ereal) (rule summable_sums)
1.1247 + from sums_unique[OF this, symmetric]
1.1248 + measure_subadditive_countably[OF A]
1.1249 + show ?thesis by (simp add: emeasure_eq_measure)
1.1250 +qed
1.1251 +
1.1252 +lemma (in finite_measure) finite_measure_eq_setsum_singleton:
1.1253 + assumes "finite S" and *: "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
1.1254 + shows "measure M S = (\<Sum>x\<in>S. measure M {x})"
1.1255 + using measure_eq_setsum_singleton[OF assms] by simp
1.1256 +
1.1257 +lemma (in finite_measure) finite_Lim_measure_incseq:
1.1258 + assumes A: "range A \<subseteq> sets M" "incseq A"
1.1259 + shows "(\<lambda>i. measure M (A i)) ----> measure M (\<Union>i. A i)"
1.1260 + using Lim_measure_incseq[OF A] by simp
1.1261 +
1.1262 +lemma (in finite_measure) finite_Lim_measure_decseq:
1.1263 + assumes A: "range A \<subseteq> sets M" "decseq A"
1.1264 + shows "(\<lambda>n. measure M (A n)) ----> measure M (\<Inter>i. A i)"
1.1265 + using Lim_measure_decseq[OF A] by simp
1.1266 +
1.1267 +lemma (in finite_measure) finite_measure_compl:
1.1268 + assumes S: "S \<in> sets M"
1.1269 + shows "measure M (space M - S) = measure M (space M) - measure M S"
1.1270 + using measure_Diff[OF _ top S sets_into_space] S by simp
1.1271 +
1.1272 +lemma (in finite_measure) finite_measure_mono_AE:
1.1273 + assumes imp: "AE x in M. x \<in> A \<longrightarrow> x \<in> B" and B: "B \<in> sets M"
1.1274 + shows "measure M A \<le> measure M B"
1.1275 + using assms emeasure_mono_AE[OF imp B]
1.1276 + by (simp add: emeasure_eq_measure)
1.1277 +
1.1278 +lemma (in finite_measure) finite_measure_eq_AE:
1.1279 + assumes iff: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B"
1.1280 + assumes A: "A \<in> sets M" and B: "B \<in> sets M"
1.1281 + shows "measure M A = measure M B"
1.1282 + using assms emeasure_eq_AE[OF assms] by (simp add: emeasure_eq_measure)
1.1283 +
1.1284 +section {* Counting space *}
1.1285 +
1.1286 +definition count_space :: "'a set \<Rightarrow> 'a measure" where
1.1287 + "count_space \<Omega> = measure_of \<Omega> (Pow \<Omega>) (\<lambda>A. if finite A then ereal (card A) else \<infinity>)"
1.1288 +
1.1289 +lemma
1.1290 + shows space_count_space[simp]: "space (count_space \<Omega>) = \<Omega>"
1.1291 + and sets_count_space[simp]: "sets (count_space \<Omega>) = Pow \<Omega>"
1.1292 + using sigma_sets_into_sp[of "Pow \<Omega>" \<Omega>]
1.1293 + by (auto simp: count_space_def)
1.1294 +
1.1295 +lemma measurable_count_space_eq1[simp]:
1.1296 + "f \<in> measurable (count_space A) M \<longleftrightarrow> f \<in> A \<rightarrow> space M"
1.1297 + unfolding measurable_def by simp
1.1298 +
1.1299 +lemma measurable_count_space_eq2[simp]:
1.1300 + assumes "finite A"
1.1301 + 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))"
1.1302 +proof -
1.1303 + { fix X assume "X \<subseteq> A" "f \<in> space M \<rightarrow> A"
1.1304 + with `finite A` have "f -` X \<inter> space M = (\<Union>a\<in>X. f -` {a} \<inter> space M)" "finite X"
1.1305 + by (auto dest: finite_subset)
1.1306 + moreover assume "\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M"
1.1307 + ultimately have "f -` X \<inter> space M \<in> sets M"
1.1308 + using `X \<subseteq> A` by (auto intro!: finite_UN simp del: UN_simps) }
1.1309 + then show ?thesis
1.1310 + unfolding measurable_def by auto
1.1311 +qed
1.1312 +
1.1313 +lemma emeasure_count_space:
1.1314 + assumes "X \<subseteq> A" shows "emeasure (count_space A) X = (if finite X then ereal (card X) else \<infinity>)"
1.1315 + (is "_ = ?M X")
1.1316 + unfolding count_space_def
1.1317 +proof (rule emeasure_measure_of_sigma)
1.1318 + show "sigma_algebra A (Pow A)" by (rule sigma_algebra_Pow)
1.1319 +
1.1320 + show "positive (Pow A) ?M"
1.1321 + by (auto simp: positive_def)
1.1322 +
1.1323 + show "countably_additive (Pow A) ?M"
1.1324 + proof (unfold countably_additive_def, safe)
1.1325 + fix F :: "nat \<Rightarrow> 'a set" assume disj: "disjoint_family F"
1.1326 + show "(\<Sum>i. ?M (F i)) = ?M (\<Union>i. F i)"
1.1327 + proof cases
1.1328 + assume "\<forall>i. finite (F i)"
1.1329 + then have finite_F: "\<And>i. finite (F i)" by auto
1.1330 + have "\<forall>i\<in>{i. F i \<noteq> {}}. \<exists>x. x \<in> F i" by auto
1.1331 + from bchoice[OF this] obtain f where f: "\<And>i. F i \<noteq> {} \<Longrightarrow> f i \<in> F i" by auto
1.1332 +
1.1333 + have inj_f: "inj_on f {i. F i \<noteq> {}}"
1.1334 + proof (rule inj_onI, simp)
1.1335 + fix i j a b assume *: "f i = f j" "F i \<noteq> {}" "F j \<noteq> {}"
1.1336 + then have "f i \<in> F i" "f j \<in> F j" using f by force+
1.1337 + with disj * show "i = j" by (auto simp: disjoint_family_on_def)
1.1338 + qed
1.1339 + have fin_eq: "finite (\<Union>i. F i) \<longleftrightarrow> finite {i. F i \<noteq> {}}"
1.1340 + proof
1.1341 + assume "finite (\<Union>i. F i)"
1.1342 + show "finite {i. F i \<noteq> {}}"
1.1343 + proof (rule finite_imageD)
1.1344 + from f have "f`{i. F i \<noteq> {}} \<subseteq> (\<Union>i. F i)" by auto
1.1345 + then show "finite (f`{i. F i \<noteq> {}})"
1.1346 + by (rule finite_subset) fact
1.1347 + qed fact
1.1348 + next
1.1349 + assume "finite {i. F i \<noteq> {}}"
1.1350 + with finite_F have "finite (\<Union>i\<in>{i. F i \<noteq> {}}. F i)"
1.1351 + by auto
1.1352 + also have "(\<Union>i\<in>{i. F i \<noteq> {}}. F i) = (\<Union>i. F i)"
1.1353 + by auto
1.1354 + finally show "finite (\<Union>i. F i)" .
1.1355 + qed
1.1356 +
1.1357 + show ?thesis
1.1358 + proof cases
1.1359 + assume *: "finite (\<Union>i. F i)"
1.1360 + with finite_F have "finite {i. ?M (F i) \<noteq> 0} "
1.1361 + by (simp add: fin_eq)
1.1362 + then have "(\<Sum>i. ?M (F i)) = (\<Sum>i | ?M (F i) \<noteq> 0. ?M (F i))"
1.1363 + by (rule suminf_eq_setsum)
1.1364 + also have "\<dots> = ereal (\<Sum>i | F i \<noteq> {}. card (F i))"
1.1365 + using finite_F by simp
1.1366 + also have "\<dots> = ereal (card (\<Union>i \<in> {i. F i \<noteq> {}}. F i))"
1.1367 + using * finite_F disj
1.1368 + by (subst card_UN_disjoint) (auto simp: disjoint_family_on_def fin_eq)
1.1369 + also have "\<dots> = ?M (\<Union>i. F i)"
1.1370 + using * by (auto intro!: arg_cong[where f=card])
1.1371 + finally show ?thesis .
1.1372 + next
1.1373 + assume inf: "infinite (\<Union>i. F i)"
1.1374 + { fix i
1.1375 + have "\<exists>N. i \<le> (\<Sum>i<N. card (F i))"
1.1376 + proof (induct i)
1.1377 + case (Suc j)
1.1378 + from Suc obtain N where N: "j \<le> (\<Sum>i<N. card (F i))" by auto
1.1379 + have "infinite ({i. F i \<noteq> {}} - {..< N})"
1.1380 + using inf by (auto simp: fin_eq)
1.1381 + then have "{i. F i \<noteq> {}} - {..< N} \<noteq> {}"
1.1382 + by (metis finite.emptyI)
1.1383 + then obtain i where i: "F i \<noteq> {}" "N \<le> i"
1.1384 + by (auto simp: not_less[symmetric])
1.1385 +
1.1386 + note N
1.1387 + also have "(\<Sum>i<N. card (F i)) \<le> (\<Sum>i<i. card (F i))"
1.1388 + by (rule setsum_mono2) (auto simp: i)
1.1389 + also have "\<dots> < (\<Sum>i<i. card (F i)) + card (F i)"
1.1390 + using finite_F `F i \<noteq> {}` by (simp add: card_gt_0_iff)
1.1391 + finally have "j < (\<Sum>i<Suc i. card (F i))"
1.1392 + by simp
1.1393 + then show ?case unfolding Suc_le_eq by blast
1.1394 + qed simp }
1.1395 + with finite_F inf show ?thesis
1.1396 + by (auto simp del: real_of_nat_setsum intro!: SUP_PInfty
1.1397 + simp add: suminf_ereal_eq_SUPR real_of_nat_setsum[symmetric])
1.1398 + qed
1.1399 + next
1.1400 + assume "\<not> (\<forall>i. finite (F i))"
1.1401 + then obtain j where j: "infinite (F j)" by auto
1.1402 + then have "infinite (\<Union>i. F i)"
1.1403 + using finite_subset[of "F j" "\<Union>i. F i"] by auto
1.1404 + moreover have "\<And>i. 0 \<le> ?M (F i)" by auto
1.1405 + ultimately show ?thesis
1.1406 + using suminf_PInfty[of "\<lambda>i. ?M (F i)" j] j by auto
1.1407 + qed
1.1408 + qed
1.1409 + show "X \<in> Pow A" using `X \<subseteq> A` by simp
1.1410 +qed
1.1411 +
1.1412 +lemma emeasure_count_space_finite[simp]:
1.1413 + "X \<subseteq> A \<Longrightarrow> finite X \<Longrightarrow> emeasure (count_space A) X = ereal (card X)"
1.1414 + using emeasure_count_space[of X A] by simp
1.1415 +
1.1416 +lemma emeasure_count_space_infinite[simp]:
1.1417 + "X \<subseteq> A \<Longrightarrow> infinite X \<Longrightarrow> emeasure (count_space A) X = \<infinity>"
1.1418 + using emeasure_count_space[of X A] by simp
1.1419 +
1.1420 +lemma emeasure_count_space_eq_0:
1.1421 + "emeasure (count_space A) X = 0 \<longleftrightarrow> (X \<subseteq> A \<longrightarrow> X = {})"
1.1422 +proof cases
1.1423 + assume X: "X \<subseteq> A"
1.1424 + then show ?thesis
1.1425 + proof (intro iffI impI)
1.1426 + assume "emeasure (count_space A) X = 0"
1.1427 + with X show "X = {}"
1.1428 + by (subst (asm) emeasure_count_space) (auto split: split_if_asm)
1.1429 + qed simp
1.1430 +qed (simp add: emeasure_notin_sets)
1.1431 +
1.1432 +lemma null_sets_count_space: "null_sets (count_space A) = { {} }"
1.1433 + unfolding null_sets_def by (auto simp add: emeasure_count_space_eq_0)
1.1434 +
1.1435 +lemma AE_count_space: "(AE x in count_space A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x)"
1.1436 + unfolding eventually_ae_filter by (auto simp add: null_sets_count_space)
1.1437 +
1.1438 +lemma sigma_finite_measure_count_space:
1.1439 + fixes A :: "'a::countable set"
1.1440 + shows "sigma_finite_measure (count_space A)"
1.1441 +proof
1.1442 + show "\<exists>F::nat \<Rightarrow> 'a set. range F \<subseteq> sets (count_space A) \<and> (\<Union>i. F i) = space (count_space A) \<and>
1.1443 + (\<forall>i. emeasure (count_space A) (F i) \<noteq> \<infinity>)"
1.1444 + using surj_from_nat by (intro exI[of _ "\<lambda>i. {from_nat i} \<inter> A"]) (auto simp del: surj_from_nat)
1.1445 +qed
1.1446 +
1.1447 +lemma finite_measure_count_space:
1.1448 + assumes [simp]: "finite A"
1.1449 + shows "finite_measure (count_space A)"
1.1450 + by rule simp
1.1451 +
1.1452 +lemma sigma_finite_measure_count_space_finite:
1.1453 + assumes A: "finite A" shows "sigma_finite_measure (count_space A)"
1.1454 +proof -
1.1455 + interpret finite_measure "count_space A" using A by (rule finite_measure_count_space)
1.1456 + show "sigma_finite_measure (count_space A)" ..
1.1457 +qed
1.1458 +
1.1459 +end
1.1460 +