--- a/src/HOL/Probability/Caratheodory.thy Thu Apr 14 12:17:44 2016 +0200
+++ b/src/HOL/Probability/Caratheodory.thy Thu Apr 14 15:48:11 2016 +0200
@@ -13,9 +13,8 @@
Originally from the Hurd/Coble measure theory development, translated by Lawrence Paulson.
\<close>
-lemma suminf_ereal_2dimen:
- fixes f:: "nat \<times> nat \<Rightarrow> ereal"
- assumes pos: "\<And>p. 0 \<le> f p"
+lemma suminf_ennreal_2dimen:
+ fixes f:: "nat \<times> nat \<Rightarrow> ennreal"
assumes "\<And>m. g m = (\<Sum>n. f (m,n))"
shows "(\<Sum>i. f (prod_decode i)) = suminf g"
proof -
@@ -23,46 +22,42 @@
using assms by (simp add: fun_eq_iff)
have reindex: "\<And>B. (\<Sum>x\<in>B. f (prod_decode x)) = setsum f (prod_decode ` B)"
by (simp add: setsum.reindex[OF inj_prod_decode] comp_def)
- { fix n
+ have "(SUP n. \<Sum>i<n. f (prod_decode i)) = (SUP p : UNIV \<times> UNIV. \<Sum>i<fst p. \<Sum>n<snd p. f (i, n))"
+ proof (intro SUP_eq; clarsimp simp: setsum.cartesian_product reindex)
+ fix n
let ?M = "\<lambda>f. Suc (Max (f ` prod_decode ` {..<n}))"
{ fix a b x assume "x < n" and [symmetric]: "(a, b) = prod_decode x"
then have "a < ?M fst" "b < ?M snd"
by (auto intro!: Max_ge le_imp_less_Suc image_eqI) }
then have "setsum f (prod_decode ` {..<n}) \<le> setsum f ({..<?M fst} \<times> {..<?M snd})"
- by (auto intro!: setsum_mono3 simp: pos)
- then have "\<exists>a b. setsum f (prod_decode ` {..<n}) \<le> setsum f ({..<a} \<times> {..<b})" by auto }
- moreover
- { fix a b
+ by (auto intro!: setsum_mono3)
+ then show "\<exists>a b. setsum f (prod_decode ` {..<n}) \<le> setsum f ({..<a} \<times> {..<b})" by auto
+ next
+ fix a b
let ?M = "prod_decode ` {..<Suc (Max (prod_encode ` ({..<a} \<times> {..<b})))}"
{ fix a' b' assume "a' < a" "b' < b" then have "(a', b') \<in> ?M"
by (auto intro!: Max_ge le_imp_less_Suc image_eqI[where x="prod_encode (a', b')"]) }
then have "setsum f ({..<a} \<times> {..<b}) \<le> setsum f ?M"
- by (auto intro!: setsum_mono3 simp: pos) }
- ultimately
- show ?thesis unfolding g_def using pos
- by (auto intro!: SUP_eq simp: setsum.cartesian_product reindex SUP_upper2
- suminf_ereal_eq_SUP SUP_pair
- SUP_ereal_setsum[symmetric] incseq_setsumI setsum_nonneg)
+ by (auto intro!: setsum_mono3)
+ then show "\<exists>n. setsum f ({..<a} \<times> {..<b}) \<le> setsum f (prod_decode ` {..<n})"
+ by auto
+ qed
+ also have "\<dots> = (SUP p. \<Sum>i<p. \<Sum>n. f (i, n))"
+ unfolding suminf_setsum[OF summableI, symmetric]
+ by (simp add: suminf_eq_SUP SUP_pair setsum.commute[of _ "{..< fst _}"])
+ finally show ?thesis unfolding g_def
+ by (simp add: suminf_eq_SUP)
qed
subsection \<open>Characterizations of Measures\<close>
-definition subadditive where
- "subadditive M f \<longleftrightarrow> (\<forall>x\<in>M. \<forall>y\<in>M. x \<inter> y = {} \<longrightarrow> f (x \<union> y) \<le> f x + f y)"
-
-definition countably_subadditive where
- "countably_subadditive M f \<longleftrightarrow>
- (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow> (f (\<Union>i. A i) \<le> (\<Sum>i. f (A i))))"
-
definition outer_measure_space where
"outer_measure_space M f \<longleftrightarrow> positive M f \<and> increasing M f \<and> countably_subadditive M f"
-lemma subadditiveD: "subadditive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x \<in> M \<Longrightarrow> y \<in> M \<Longrightarrow> f (x \<union> y) \<le> f x + f y"
- by (auto simp add: subadditive_def)
-
subsubsection \<open>Lambda Systems\<close>
-definition lambda_system where
+definition lambda_system :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ennreal) \<Rightarrow> 'a set set"
+where
"lambda_system \<Omega> M f = {l \<in> M. \<forall>x \<in> M. f (l \<inter> x) + f ((\<Omega> - l) \<inter> x) = f x}"
lemma (in algebra) lambda_system_eq:
@@ -81,7 +76,7 @@
by (simp add: lambda_system_def)
lemma (in algebra) lambda_system_Compl:
- fixes f:: "'a set \<Rightarrow> ereal"
+ fixes f:: "'a set \<Rightarrow> ennreal"
assumes x: "x \<in> lambda_system \<Omega> M f"
shows "\<Omega> - x \<in> lambda_system \<Omega> M f"
proof -
@@ -94,7 +89,7 @@
qed
lemma (in algebra) lambda_system_Int:
- fixes f:: "'a set \<Rightarrow> ereal"
+ fixes f:: "'a set \<Rightarrow> ennreal"
assumes xl: "x \<in> lambda_system \<Omega> M f" and yl: "y \<in> lambda_system \<Omega> M f"
shows "x \<inter> y \<in> lambda_system \<Omega> M f"
proof -
@@ -128,7 +123,7 @@
qed
lemma (in algebra) lambda_system_Un:
- fixes f:: "'a set \<Rightarrow> ereal"
+ fixes f:: "'a set \<Rightarrow> ennreal"
assumes xl: "x \<in> lambda_system \<Omega> M f" and yl: "y \<in> lambda_system \<Omega> M f"
shows "x \<union> y \<in> lambda_system \<Omega> M f"
proof -
@@ -176,25 +171,6 @@
by (simp add: Un)
qed
-lemma (in ring_of_sets) countably_subadditive_subadditive:
- assumes f: "positive M f" and cs: "countably_subadditive M f"
- shows "subadditive M f"
-proof (auto simp add: subadditive_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) \<le> (\<Sum> n. f (binaryset x y n))"
- using cs by (auto simp add: countably_subadditive_def)
- hence "{x,y,{}} \<subseteq> M \<longrightarrow> x \<union> y \<in> M \<longrightarrow>
- f (x \<union> y) \<le> (\<Sum> n. f (binaryset x y n))"
- by (simp add: range_binaryset_eq UN_binaryset_eq)
- thus "f (x \<union> y) \<le> f x + f y" using f x y
- by (auto simp add: Un o_def suminf_binaryset_eq positive_def)
-qed
-
lemma lambda_system_increasing: "increasing M f \<Longrightarrow> increasing (lambda_system \<Omega> M f) f"
by (simp add: increasing_def lambda_system_def)
@@ -202,7 +178,7 @@
by (simp add: positive_def lambda_system_def)
lemma (in algebra) lambda_system_strong_sum:
- fixes A:: "nat \<Rightarrow> 'a set" and f :: "'a set \<Rightarrow> ereal"
+ fixes A:: "nat \<Rightarrow> 'a set" and f :: "'a set \<Rightarrow> ennreal"
assumes f: "positive M f" and a: "a \<in> M"
and A: "range A \<subseteq> lambda_system \<Omega> M f"
and disj: "disjoint_family A"
@@ -247,11 +223,10 @@
proof (rule antisym)
show "f (\<Union>i. A i) \<le> (\<Sum>i. f (A i))"
using csa[unfolded countably_subadditive_def] A'' disj U_in by auto
- have *: "\<And>i. 0 \<le> f (A i)" using pos A'' unfolding positive_def by auto
have dis: "\<And>N. disjoint_family_on A {..<N}" by (intro disjoint_family_on_mono[OF _ disj]) auto
show "(\<Sum>i. f (A i)) \<le> f (\<Union>i. A i)"
using ls.additive_sum [OF lambda_system_positive[OF pos] lambda_system_additive _ A' dis] A''
- by (intro suminf_bound[OF _ *]) (auto intro!: increasingD[OF inc] countable_UN)
+ by (intro suminf_le_const[OF summableI]) (auto intro!: increasingD[OF inc] countable_UN)
qed
have "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) = f a"
if a [iff]: "a \<in> M" for a
@@ -271,7 +246,7 @@
hence "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) \<le> (\<Sum>i. f (a \<inter> A i)) + f (a - (\<Union>i. A i))"
by (rule add_right_mono)
also have "\<dots> \<le> f a"
- proof (intro suminf_bound_add allI)
+ proof (intro ennreal_suminf_bound_add)
fix n
have UNION_in: "(\<Union>i\<in>{0..<n}. A i) \<in> M"
by (metis A'' UNION_in_sets)
@@ -285,14 +260,9 @@
by (blast intro: increasingD [OF inc] UNION_in U_in)
thus "(\<Sum>i<n. f (a \<inter> A i)) + f (a - (\<Union>i. A i)) \<le> f a"
by (simp add: lambda_system_strong_sum pos A disj eq_fa add_left_mono atLeast0LessThan[symmetric])
- next
- have "\<And>i. a \<inter> A i \<in> M" using A'' by auto
- then show "\<And>i. 0 \<le> f (a \<inter> A i)" using pos[unfolded positive_def] by auto
- have "\<And>i. a - (\<Union>i. A i) \<in> M" using A'' by auto
- then have "\<And>i. 0 \<le> f (a - (\<Union>i. A i))" using pos[unfolded positive_def] by auto
- then show "f (a - (\<Union>i. A i)) \<noteq> -\<infinity>" by auto
qed
- finally show "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) \<le> f a" .
+ finally show "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) \<le> f a"
+ by simp
next
have "f a \<le> f (a \<inter> (\<Union>i. A i) \<union> (a - (\<Union>i. A i)))"
by (blast intro: increasingD [OF inc] U_in)
@@ -327,7 +297,7 @@
using pos by (simp add: measure_space_def)
qed
-definition outer_measure :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> 'a set \<Rightarrow> ereal" where
+definition outer_measure :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ennreal) \<Rightarrow> 'a set \<Rightarrow> ennreal" where
"outer_measure M f X =
(INF A:{A. range A \<subseteq> M \<and> disjoint_family A \<and> X \<subseteq> (\<Union>i. A i)}. \<Sum>i. f (A i))"
@@ -346,7 +316,7 @@
by (intro ca[unfolded countably_additive_def, rule_format, symmetric])
(auto simp: Int_absorb1 disjoint_family_on_def)
also have "... \<le> (\<Sum>i. f (A i))"
- using A s by (intro suminf_le_pos increasingD[OF inc] positiveD2[OF posf]) auto
+ using A s by (auto intro!: suminf_le increasingD[OF inc])
finally show "f s \<le> (\<Sum>i. f (A i))" .
next
have "(\<Sum>i. f (if i = 0 then s else {})) \<le> f s"
@@ -356,20 +326,14 @@
(auto simp: disjoint_family_on_def)
qed
-lemma outer_measure_nonneg: "positive M f \<Longrightarrow> 0 \<le> outer_measure M f X"
- by (auto intro!: INF_greatest suminf_0_le intro: positiveD2 simp: outer_measure_def)
-
lemma outer_measure_empty:
- assumes posf: "positive M f" and "{} \<in> M"
- shows "outer_measure M f {} = 0"
-proof (rule antisym)
- show "outer_measure M f {} \<le> 0"
- using assms by (auto intro!: INF_lower2[of "\<lambda>_. {}"] simp: outer_measure_def disjoint_family_on_def positive_def)
-qed (intro outer_measure_nonneg posf)
+ "positive M f \<Longrightarrow> {} \<in> M \<Longrightarrow> outer_measure M f {} = 0"
+ unfolding outer_measure_def
+ by (intro antisym INF_lower2[of "\<lambda>_. {}"]) (auto simp: disjoint_family_on_def positive_def)
lemma (in ring_of_sets) positive_outer_measure:
assumes "positive M f" shows "positive (Pow \<Omega>) (outer_measure M f)"
- unfolding positive_def by (auto simp: assms outer_measure_empty outer_measure_nonneg)
+ unfolding positive_def by (auto simp: assms outer_measure_empty)
lemma (in ring_of_sets) increasing_outer_measure: "increasing (Pow \<Omega>) (outer_measure M f)"
by (force simp: increasing_def outer_measure_def intro!: INF_greatest intro: INF_lower)
@@ -383,22 +347,13 @@
by (auto intro!: A range_disjointed_sets)
have "\<forall>n. f (disjointed A n) \<le> f (A n)"
by (metis increasingD [OF inc] UNIV_I dA image_subset_iff disjointed_subset A)
- moreover have "\<forall>i. 0 \<le> f (disjointed A i)"
- using pos dA unfolding positive_def by auto
- ultimately show "(\<Sum>i. f (disjointed A i)) \<le> (\<Sum>i. f (A i))"
- by (blast intro!: suminf_le_pos)
+ then show "(\<Sum>i. f (disjointed A i)) \<le> (\<Sum>i. f (A i))"
+ by (blast intro!: suminf_le)
qed (auto simp: X UN_disjointed_eq disjoint_family_disjointed)
lemma (in ring_of_sets) outer_measure_close:
- assumes posf: "positive M f" and "0 < e" and "outer_measure M f X \<noteq> \<infinity>"
- shows "\<exists>A. range A \<subseteq> M \<and> disjoint_family A \<and> X \<subseteq> (\<Union>i. A i) \<and> (\<Sum>i. f (A i)) \<le> outer_measure M f X + e"
-proof -
- from \<open>outer_measure M f X \<noteq> \<infinity>\<close> have fin: "\<bar>outer_measure M f X\<bar> \<noteq> \<infinity>"
- using outer_measure_nonneg[OF posf, of X] by auto
- show ?thesis
- using Inf_ereal_close [OF fin [unfolded outer_measure_def], OF \<open>0 < e\<close>]
- by (auto intro: less_imp_le simp add: outer_measure_def)
-qed
+ "outer_measure M f X < e \<Longrightarrow> \<exists>A. range A \<subseteq> M \<and> disjoint_family A \<and> X \<subseteq> (\<Union>i. A i) \<and> (\<Sum>i. f (A i)) < e"
+ unfolding outer_measure_def INF_less_iff by auto
lemma (in ring_of_sets) countably_subadditive_outer_measure:
assumes posf: "positive M f" and inc: "increasing M f"
@@ -406,16 +361,16 @@
proof (simp add: countably_subadditive_def, safe)
fix A :: "nat \<Rightarrow> _" assume A: "range A \<subseteq> Pow (\<Omega>)" and sb: "(\<Union>i. A i) \<subseteq> \<Omega>"
let ?O = "outer_measure M f"
-
- { fix e :: ereal assume e: "0 < e" and "\<forall>i. ?O (A i) \<noteq> \<infinity>"
- hence "\<exists>B. \<forall>n. range (B n) \<subseteq> M \<and> disjoint_family (B n) \<and> A n \<subseteq> (\<Union>i. B n i) \<and>
- (\<Sum>i. f (B n i)) \<le> ?O (A n) + e * (1/2)^(Suc n)"
- using e sb by (auto intro!: choice outer_measure_close [of f, OF posf] simp: ereal_zero_less_0_iff one_ereal_def)
- then obtain B
+ show "?O (\<Union>i. A i) \<le> (\<Sum>n. ?O (A n))"
+ proof (rule ennreal_le_epsilon)
+ fix b and e :: real assume "0 < e" "(\<Sum>n. outer_measure M f (A n)) < top"
+ then have *: "\<And>n. outer_measure M f (A n) < outer_measure M f (A n) + e * (1/2)^Suc n"
+ by (auto simp add: less_top dest!: ennreal_suminf_lessD)
+ obtain B
where B: "\<And>n. range (B n) \<subseteq> M"
and sbB: "\<And>n. A n \<subseteq> (\<Union>i. B n i)"
and Ble: "\<And>n. (\<Sum>i. f (B n i)) \<le> ?O (A n) + e * (1/2)^(Suc n)"
- by auto blast
+ by (metis less_imp_le outer_measure_close[OF *])
def C \<equiv> "case_prod B o prod_decode"
from B have B_in_M: "\<And>i j. B i j \<in> M"
@@ -425,26 +380,22 @@
have A_C: "(\<Union>i. A i) \<subseteq> (\<Union>i. C i)"
using sbB by (auto simp add: C_def subset_eq) (metis prod.case prod_encode_inverse)
- have "?O (\<Union>i. A i) \<le> ?O (\<Union>i. C i)"
+ have "?O (\<Union>i. A i) \<le> ?O (\<Union>i. C i)"
using A_C A C by (intro increasing_outer_measure[THEN increasingD]) (auto dest!: sets_into_space)
also have "\<dots> \<le> (\<Sum>i. f (C i))"
using C by (intro outer_measure_le[OF posf inc]) auto
also have "\<dots> = (\<Sum>n. \<Sum>i. f (B n i))"
- using B_in_M unfolding C_def comp_def by (intro suminf_ereal_2dimen positiveD2[OF posf]) auto
- also have "\<dots> \<le> (\<Sum>n. ?O (A n) + e*(1/2) ^ Suc n)"
- using B_in_M by (intro suminf_le_pos[OF Ble] suminf_0_le posf[THEN positiveD2]) auto
- also have "... = (\<Sum>n. ?O (A n)) + (\<Sum>n. e*(1/2) ^ Suc n)"
- using e by (subst suminf_add_ereal) (auto simp add: ereal_zero_le_0_iff outer_measure_nonneg posf)
- also have "(\<Sum>n. e*(1/2) ^ Suc n) = e"
- using suminf_half_series_ereal e by (simp add: ereal_zero_le_0_iff suminf_cmult_ereal)
- finally have "?O (\<Union>i. A i) \<le> (\<Sum>n. ?O (A n)) + e" . }
- note * = this
-
- show "?O (\<Union>i. A i) \<le> (\<Sum>n. ?O (A n))"
- proof cases
- assume "\<forall>i. ?O (A i) \<noteq> \<infinity>" with * show ?thesis
- by (intro ereal_le_epsilon) auto
- qed (metis suminf_PInfty[OF outer_measure_nonneg, OF posf] ereal_less_eq(1))
+ using B_in_M unfolding C_def comp_def by (intro suminf_ennreal_2dimen) auto
+ also have "\<dots> \<le> (\<Sum>n. ?O (A n) + e * (1/2) ^ Suc n)"
+ using B_in_M by (intro suminf_le suminf_nonneg allI Ble) auto
+ also have "... = (\<Sum>n. ?O (A n)) + (\<Sum>n. ennreal e * ennreal ((1/2) ^ Suc n))"
+ using \<open>0 < e\<close> by (subst suminf_add[symmetric])
+ (auto simp del: ennreal_suminf_cmult simp add: ennreal_mult[symmetric])
+ also have "\<dots> = (\<Sum>n. ?O (A n)) + e"
+ unfolding ennreal_suminf_cmult
+ by (subst suminf_ennreal_eq[OF zero_le_power power_half_series]) auto
+ finally show "?O (\<Union>i. A i) \<le> (\<Sum>n. ?O (A n)) + e" .
+ qed
qed
lemma (in ring_of_sets) outer_measure_space_outer_measure:
@@ -481,7 +432,7 @@
ultimately have "outer_measure M f (s \<inter> x) + outer_measure M f (s - x) \<le>
(\<Sum>i. f (A i \<inter> x)) + (\<Sum>i. f (A i - x))" by (rule add_mono)
also have "\<dots> = (\<Sum>i. f (A i \<inter> x) + f (A i - x))"
- using A(2) x posf by (subst suminf_add_ereal) (auto simp: positive_def)
+ using A(2) x posf by (subst suminf_add) (auto simp: positive_def)
also have "\<dots> = (\<Sum>i. f (A i))"
using A x
by (subst add[THEN additiveD, symmetric])
@@ -496,7 +447,7 @@
also have "... \<le> outer_measure M f (s \<inter> x) + outer_measure M f (s - x)"
apply (rule subadditiveD)
apply (rule ring_of_sets.countably_subadditive_subadditive [OF ring_of_sets_Pow])
- apply (simp add: positive_def outer_measure_empty[OF posf] outer_measure_nonneg[OF posf])
+ apply (simp add: positive_def outer_measure_empty[OF posf])
apply (rule countably_subadditive_outer_measure)
using s by (auto intro!: posf inc)
finally show ?thesis .
@@ -513,7 +464,7 @@
theorem (in ring_of_sets) caratheodory':
assumes posf: "positive M f" and ca: "countably_additive M f"
- shows "\<exists>\<mu> :: 'a set \<Rightarrow> ereal. (\<forall>s \<in> M. \<mu> s = f s) \<and> measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"
+ shows "\<exists>\<mu> :: 'a set \<Rightarrow> ennreal. (\<forall>s \<in> M. \<mu> s = f s) \<and> measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"
proof -
have inc: "increasing M f"
by (metis additive_increasing ca countably_additive_additive posf)
@@ -541,14 +492,14 @@
lemma (in ring_of_sets) caratheodory_empty_continuous:
assumes f: "positive M f" "additive M f" and fin: "\<And>A. A \<in> M \<Longrightarrow> 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)) \<longlonglongrightarrow> 0"
- shows "\<exists>\<mu> :: 'a set \<Rightarrow> ereal. (\<forall>s \<in> M. \<mu> s = f s) \<and> measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"
+ shows "\<exists>\<mu> :: 'a set \<Rightarrow> ennreal. (\<forall>s \<in> M. \<mu> s = f s) \<and> measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"
proof (intro caratheodory' empty_continuous_imp_countably_additive f)
show "\<forall>A\<in>M. f A \<noteq> \<infinity>" using fin by auto
qed (rule cont)
subsection \<open>Volumes\<close>
-definition volume :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
+definition volume :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ennreal) \<Rightarrow> bool" where
"volume M f \<longleftrightarrow>
(f {} = 0) \<and> (\<forall>a\<in>M. 0 \<le> f a) \<and>
(\<forall>C\<subseteq>M. disjoint C \<longrightarrow> finite C \<longrightarrow> \<Union>C \<in> M \<longrightarrow> f (\<Union>C) = (\<Sum>c\<in>C. f c))"
@@ -569,7 +520,7 @@
by (auto simp: volume_def)
lemma volume_finite_additive:
- assumes "volume M f"
+ assumes "volume M f"
assumes A: "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M" "disjoint_family_on A I" "finite I" "UNION I A \<in> M"
shows "f (UNION I A) = (\<Sum>i\<in>I. f (A i))"
proof -
@@ -590,7 +541,7 @@
qed
lemma (in ring_of_sets) volume_additiveI:
- assumes pos: "\<And>a. a \<in> M \<Longrightarrow> 0 \<le> \<mu> a"
+ assumes pos: "\<And>a. a \<in> M \<Longrightarrow> 0 \<le> \<mu> a"
assumes [simp]: "\<mu> {} = 0"
assumes add: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<inter> b = {} \<Longrightarrow> \<mu> (a \<union> b) = \<mu> a + \<mu> b"
shows "volume M \<mu>"
@@ -614,7 +565,7 @@
have "\<forall>a\<in>?R. \<exists>m. \<exists>C\<subseteq>M. a = \<Union>C \<and> finite C \<and> disjoint C \<and> m = (\<Sum>c\<in>C. \<mu> c)"
by (auto simp: generated_ring_def)
from bchoice[OF this] guess \<mu>' .. note \<mu>'_spec = this
-
+
{ fix C assume C: "C \<subseteq> M" "finite C" "disjoint C"
fix D assume D: "D \<subseteq> M" "finite D" "disjoint D"
assume "\<Union>C = \<Union>D"
@@ -688,7 +639,7 @@
theorem (in semiring_of_sets) caratheodory:
assumes pos: "positive M \<mu>" and ca: "countably_additive M \<mu>"
- shows "\<exists>\<mu>' :: 'a set \<Rightarrow> ereal. (\<forall>s \<in> M. \<mu>' s = \<mu> s) \<and> measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>'"
+ shows "\<exists>\<mu>' :: 'a set \<Rightarrow> ennreal. (\<forall>s \<in> M. \<mu>' s = \<mu> s) \<and> measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>'"
proof -
have "volume M \<mu>"
proof (rule volumeI)
@@ -787,7 +738,7 @@
then have "\<Union>range f = A i"
using f C Ai unfolding bij_betw_def
by (auto simp add: f_def cong del: strong_SUP_cong)
- moreover
+ moreover
{ have "(\<Sum>j. \<mu>_r (f j)) = (\<Sum>j. if j \<in> {..< card C} then \<mu>_r (f j) else 0)"
using volume_empty[OF V(1)] by (auto intro!: arg_cong[where f=suminf] simp: f_def)
also have "\<dots> = (\<Sum>j<card C. \<mu>_r (f j))"
@@ -827,7 +778,7 @@
qed
qed
from f have "(\<Sum>n. \<mu>_r (A n)) = (\<Sum>n. \<mu>_r (case_prod f (prod_decode n)))"
- by (intro suminf_ereal_2dimen[symmetric] positiveD2[OF pos] generated_ringI_Basic)
+ by (intro suminf_ennreal_2dimen[symmetric] generated_ringI_Basic)
(auto split: prod.split)
also have "\<dots> = (\<Sum>n. \<mu> (case_prod f (prod_decode n)))"
using f V(2) by (auto intro!: arg_cong[where f=suminf] split: prod.split)
@@ -847,8 +798,7 @@
intro: generated_ringI_Basic)
also have "\<dots> = (\<Sum>c\<in>C'. \<Sum>n. \<mu>_r (A' n \<inter> c))"
using C' A'
- by (intro suminf_setsum_ereal positiveD2[OF pos] G.Int G.finite_Union)
- (auto intro: generated_ringI_Basic)
+ by (intro suminf_setsum G.Int G.finite_Union) (auto intro: generated_ringI_Basic)
also have "\<dots> = (\<Sum>c\<in>C'. \<mu>_r c)"
using eq V C' by (auto intro!: setsum.cong)
also have "\<dots> = \<mu>_r (\<Union>C')"
@@ -909,7 +859,7 @@
using \<mu>' by (simp_all add: M sets_extend_measure measure_space_def)
qed fact
qed
-
+
lemma extend_measure_caratheodory_pair:
fixes G :: "'i \<Rightarrow> 'j \<Rightarrow> 'a set"
assumes M: "M = extend_measure \<Omega> {(a, b). P a b} (\<lambda>(a, b). G a b) (\<lambda>(a, b). \<mu> a b)"