author | hoelzl |
Wed, 14 Sep 2011 10:08:52 -0400 | |
changeset 44928 | 7ef6505bde7f |
parent 44918 | 6a80fbc4e72c |
child 46731 | 5302e932d1e5 |
permissions | -rw-r--r-- |
42067 | 1 |
(* Title: HOL/Probability/Caratheodory.thy |
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Author: Lawrence C Paulson |
|
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Author: Johannes Hölzl, TU München |
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*) |
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||
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New theory Probability, which contains a development of measure theory
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header {*Caratheodory Extension Theorem*} |
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New theory Probability, which contains a development of measure theory
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|
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New theory Probability, which contains a development of measure theory
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theory Caratheodory |
44106 | 9 |
imports Sigma_Algebra "~~/src/HOL/Multivariate_Analysis/Extended_Real_Limits" |
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New theory Probability, which contains a development of measure theory
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begin |
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New theory Probability, which contains a development of measure theory
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|
42950
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move lemmas to Extended_Reals and Extended_Real_Limits
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lemma sums_def2: |
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move lemmas to Extended_Reals and Extended_Real_Limits
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"f sums x \<longleftrightarrow> (\<lambda>n. (\<Sum>i\<le>n. f i)) ----> x" |
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move lemmas to Extended_Reals and Extended_Real_Limits
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unfolding sums_def |
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move lemmas to Extended_Reals and Extended_Real_Limits
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parents:
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apply (subst LIMSEQ_Suc_iff[symmetric]) |
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move lemmas to Extended_Reals and Extended_Real_Limits
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parents:
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unfolding atLeastLessThanSuc_atLeastAtMost atLeast0AtMost .. |
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move lemmas to Extended_Reals and Extended_Real_Limits
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|
42067 | 18 |
text {* |
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Originally from the Hurd/Coble measure theory development, translated by Lawrence Paulson. |
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*} |
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||
43920 | 22 |
lemma suminf_ereal_2dimen: |
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fixes f:: "nat \<times> nat \<Rightarrow> ereal" |
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41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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assumes pos: "\<And>p. 0 \<le> f p" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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assumes "\<And>m. g m = (\<Sum>n. f (m,n))" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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shows "(\<Sum>i. f (prod_decode i)) = suminf g" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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proof - |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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have g_def: "g = (\<lambda>m. (\<Sum>n. f (m,n)))" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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using assms by (simp add: fun_eq_iff) |
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have reindex: "\<And>B. (\<Sum>x\<in>B. f (prod_decode x)) = setsum f (prod_decode ` B)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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by (simp add: setsum_reindex[OF inj_prod_decode] comp_def) |
cdf7693bbe08
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parents:
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{ fix n |
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let ?M = "\<lambda>f. Suc (Max (f ` prod_decode ` {..<n}))" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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{ fix a b x assume "x < n" and [symmetric]: "(a, b) = prod_decode x" |
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then have "a < ?M fst" "b < ?M snd" |
cdf7693bbe08
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by (auto intro!: Max_ge le_imp_less_Suc image_eqI) } |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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then have "setsum f (prod_decode ` {..<n}) \<le> setsum f ({..<?M fst} \<times> {..<?M snd})" |
cdf7693bbe08
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by (auto intro!: setsum_mono3 simp: pos) |
cdf7693bbe08
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then have "\<exists>a b. setsum f (prod_decode ` {..<n}) \<le> setsum f ({..<a} \<times> {..<b})" by auto } |
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moreover |
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{ fix a b |
cdf7693bbe08
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42 |
let ?M = "prod_decode ` {..<Suc (Max (prod_encode ` ({..<a} \<times> {..<b})))}" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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parents:
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{ fix a' b' assume "a' < a" "b' < b" then have "(a', b') \<in> ?M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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parents:
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by (auto intro!: Max_ge le_imp_less_Suc image_eqI[where x="prod_encode (a', b')"]) } |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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then have "setsum f ({..<a} \<times> {..<b}) \<le> setsum f ?M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
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by (auto intro!: setsum_mono3 simp: pos) } |
cdf7693bbe08
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ultimately |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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show ?thesis unfolding g_def using pos |
44928
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renamed Complete_Lattices lemmas, removed legacy names
hoelzl
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44918
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by (auto intro!: SUPR_eq simp: setsum_cartesian_product reindex SUP_upper2 |
43920 | 50 |
setsum_nonneg suminf_ereal_eq_SUPR SUPR_pair |
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SUPR_ereal_setsum[symmetric] incseq_setsumI setsum_nonneg) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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qed |
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53 |
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New theory Probability, which contains a development of measure theory
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subsection {* Measure Spaces *} |
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55 |
|
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the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
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record 'a measure_space = "'a algebra" + |
43920 | 57 |
measure :: "'a set \<Rightarrow> ereal" |
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58 |
|
43920 | 59 |
definition positive where "positive M f \<longleftrightarrow> f {} = (0::ereal) \<and> (\<forall>A\<in>sets M. 0 \<le> f A)" |
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60 |
|
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3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
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diff
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61 |
definition additive where "additive M f \<longleftrightarrow> |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
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(\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<inter> y = {} \<longrightarrow> f (x \<union> y) = f x + f y)" |
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New theory Probability, which contains a development of measure theory
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63 |
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43920 | 64 |
definition countably_additive :: "('a, 'b) algebra_scheme \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where |
41981
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"countably_additive M f \<longleftrightarrow> (\<forall>A. range A \<subseteq> sets M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> sets M \<longrightarrow> |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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66 |
(\<Sum>i. f (A i)) = f (\<Union>i. A i))" |
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New theory Probability, which contains a development of measure theory
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67 |
|
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3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
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68 |
definition increasing where "increasing M f \<longleftrightarrow> |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
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41023
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69 |
(\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<subseteq> y \<longrightarrow> f x \<le> f y)" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
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70 |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
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71 |
definition subadditive where "subadditive M f \<longleftrightarrow> |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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72 |
(\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<inter> y = {} \<longrightarrow> f (x \<union> y) \<le> f x + f y)" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
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parents:
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73 |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
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74 |
definition countably_subadditive where "countably_subadditive M f \<longleftrightarrow> |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
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|
75 |
(\<forall>A. range A \<subseteq> sets M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> sets M \<longrightarrow> |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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76 |
(f (\<Union>i. A i) \<le> (\<Sum>i. f (A i))))" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
77 |
|
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
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|
78 |
definition lambda_system where "lambda_system M f = {l \<in> sets M. |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
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79 |
\<forall>x \<in> sets M. f (l \<inter> x) + f ((space M - l) \<inter> x) = f x}" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
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80 |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
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81 |
definition outer_measure_space where "outer_measure_space M f \<longleftrightarrow> |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
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82 |
positive M f \<and> increasing M f \<and> countably_subadditive M f" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
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|
83 |
|
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
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|
84 |
definition measure_set where "measure_set M f X = {r. |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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85 |
\<exists>A. range A \<subseteq> sets M \<and> disjoint_family A \<and> X \<subseteq> (\<Union>i. A i) \<and> (\<Sum>i. f (A i)) = r}" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
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86 |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
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|
87 |
locale measure_space = sigma_algebra M for M :: "('a, 'b) measure_space_scheme" + |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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diff
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|
88 |
assumes measure_positive: "positive M (measure M)" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
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|
89 |
and ca: "countably_additive M (measure M)" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
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90 |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
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91 |
abbreviation (in measure_space) "\<mu> \<equiv> measure M" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
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92 |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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93 |
lemma (in measure_space) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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94 |
shows empty_measure[simp, intro]: "\<mu> {} = 0" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
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95 |
and positive_measure[simp, intro!]: "\<And>A. A \<in> sets M \<Longrightarrow> 0 \<le> \<mu> A" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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96 |
using measure_positive unfolding positive_def by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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97 |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
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98 |
lemma increasingD: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
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|
99 |
"increasing M f \<Longrightarrow> x \<subseteq> y \<Longrightarrow> x\<in>sets M \<Longrightarrow> y\<in>sets M \<Longrightarrow> f x \<le> f y" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
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100 |
by (auto simp add: increasing_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
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101 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
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102 |
lemma subadditiveD: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
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|
103 |
"subadditive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x \<in> sets M \<Longrightarrow> y \<in> sets M |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
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|
104 |
\<Longrightarrow> f (x \<union> y) \<le> f x + f y" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
105 |
by (auto simp add: subadditive_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
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|
106 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
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|
107 |
lemma additiveD: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
108 |
"additive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x \<in> sets M \<Longrightarrow> y \<in> sets M |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
109 |
\<Longrightarrow> f (x \<union> y) = f x + f y" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
110 |
by (auto simp add: additive_def) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
111 |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
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|
112 |
lemma countably_additiveI: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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|
113 |
assumes "\<And>A x. range A \<subseteq> sets M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i. A i) \<in> sets M |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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|
114 |
\<Longrightarrow> (\<Sum>i. f (A i)) = f (\<Union>i. A i)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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|
115 |
shows "countably_additive M f" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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|
116 |
using assms by (simp add: countably_additive_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
117 |
|
38656 | 118 |
section "Extend binary sets" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
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|
119 |
|
35582 | 120 |
lemma LIMSEQ_binaryset: |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
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|
121 |
assumes f: "f {} = 0" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
122 |
shows "(\<lambda>n. \<Sum>i<n. f (binaryset A B i)) ----> f A + f B" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
123 |
proof - |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
124 |
have "(\<lambda>n. \<Sum>i < Suc (Suc n). f (binaryset A B i)) = (\<lambda>n. f A + f B)" |
35582 | 125 |
proof |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
126 |
fix n |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
127 |
show "(\<Sum>i < Suc (Suc n). f (binaryset A B i)) = f A + f B" |
35582 | 128 |
by (induct n) (auto simp add: binaryset_def f) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
129 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
130 |
moreover |
44568
e6f291cb5810
discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents:
44106
diff
changeset
|
131 |
have "... ----> f A + f B" by (rule tendsto_const) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
132 |
ultimately |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
133 |
have "(\<lambda>n. \<Sum>i< Suc (Suc n). f (binaryset A B i)) ----> f A + f B" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
134 |
by metis |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
135 |
hence "(\<lambda>n. \<Sum>i< n+2. f (binaryset A B i)) ----> f A + f B" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
136 |
by simp |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
137 |
thus ?thesis by (rule LIMSEQ_offset [where k=2]) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
138 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
139 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
140 |
lemma binaryset_sums: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
141 |
assumes f: "f {} = 0" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
142 |
shows "(\<lambda>n. f (binaryset A B n)) sums (f A + f B)" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
143 |
by (simp add: sums_def LIMSEQ_binaryset [where f=f, OF f] atLeast0LessThan) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
144 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
145 |
lemma suminf_binaryset_eq: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
146 |
fixes f :: "'a set \<Rightarrow> 'b::{comm_monoid_add, t2_space}" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
147 |
shows "f {} = 0 \<Longrightarrow> (\<Sum>n. f (binaryset A B n)) = f A + f B" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
148 |
by (metis binaryset_sums sums_unique) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
149 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
150 |
subsection {* Lambda Systems *} |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
151 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
152 |
lemma (in algebra) lambda_system_eq: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
153 |
shows "lambda_system M f = {l \<in> sets M. |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
154 |
\<forall>x \<in> sets M. f (x \<inter> l) + f (x - l) = f x}" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
155 |
proof - |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
156 |
have [simp]: "!!l x. l \<in> sets M \<Longrightarrow> x \<in> sets M \<Longrightarrow> (space M - l) \<inter> x = x - l" |
37032 | 157 |
by (metis Int_Diff Int_absorb1 Int_commute sets_into_space) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
158 |
show ?thesis |
37032 | 159 |
by (auto simp add: lambda_system_def) (metis Int_commute)+ |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
160 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
161 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
162 |
lemma (in algebra) lambda_system_empty: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
163 |
"positive M f \<Longrightarrow> {} \<in> lambda_system M f" |
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
164 |
by (auto simp add: positive_def lambda_system_eq) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
165 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
166 |
lemma lambda_system_sets: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
167 |
"x \<in> lambda_system M f \<Longrightarrow> x \<in> sets M" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
168 |
by (simp add: lambda_system_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
169 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
170 |
lemma (in algebra) lambda_system_Compl: |
43920 | 171 |
fixes f:: "'a set \<Rightarrow> ereal" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
172 |
assumes x: "x \<in> lambda_system M f" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
173 |
shows "space M - x \<in> lambda_system M f" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
174 |
proof - |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
175 |
have "x \<subseteq> space M" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
176 |
by (metis sets_into_space lambda_system_sets x) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
177 |
hence "space M - (space M - x) = x" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
178 |
by (metis double_diff equalityE) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
179 |
with x show ?thesis |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
180 |
by (force simp add: lambda_system_def ac_simps) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
181 |
qed |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
182 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
183 |
lemma (in algebra) lambda_system_Int: |
43920 | 184 |
fixes f:: "'a set \<Rightarrow> ereal" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
185 |
assumes xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
186 |
shows "x \<inter> y \<in> lambda_system M f" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
187 |
proof - |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
188 |
from xl yl show ?thesis |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
189 |
proof (auto simp add: positive_def lambda_system_eq Int) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
190 |
fix u |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
191 |
assume x: "x \<in> sets M" and y: "y \<in> sets M" and u: "u \<in> sets M" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
192 |
and fx: "\<forall>z\<in>sets M. f (z \<inter> x) + f (z - x) = f z" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
193 |
and fy: "\<forall>z\<in>sets M. f (z \<inter> y) + f (z - y) = f z" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
194 |
have "u - x \<inter> y \<in> sets M" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
195 |
by (metis Diff Diff_Int Un u x y) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
196 |
moreover |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
197 |
have "(u - (x \<inter> y)) \<inter> y = u \<inter> y - x" by blast |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
198 |
moreover |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
199 |
have "u - x \<inter> y - y = u - y" by blast |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
200 |
ultimately |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
201 |
have ey: "f (u - x \<inter> y) = f (u \<inter> y - x) + f (u - y)" using fy |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
202 |
by force |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
203 |
have "f (u \<inter> (x \<inter> y)) + f (u - x \<inter> y) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
204 |
= (f (u \<inter> (x \<inter> y)) + f (u \<inter> y - x)) + f (u - y)" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
205 |
by (simp add: ey ac_simps) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
206 |
also have "... = (f ((u \<inter> y) \<inter> x) + f (u \<inter> y - x)) + f (u - y)" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
207 |
by (simp add: Int_ac) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
208 |
also have "... = f (u \<inter> y) + f (u - y)" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
209 |
using fx [THEN bspec, of "u \<inter> y"] Int y u |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
210 |
by force |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
211 |
also have "... = f u" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
212 |
by (metis fy u) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
213 |
finally show "f (u \<inter> (x \<inter> y)) + f (u - x \<inter> y) = f u" . |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
214 |
qed |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
215 |
qed |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
216 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
217 |
lemma (in algebra) lambda_system_Un: |
43920 | 218 |
fixes f:: "'a set \<Rightarrow> ereal" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
219 |
assumes xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
220 |
shows "x \<union> y \<in> lambda_system M f" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
221 |
proof - |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
222 |
have "(space M - x) \<inter> (space M - y) \<in> sets M" |
38656 | 223 |
by (metis Diff_Un Un compl_sets lambda_system_sets xl yl) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
224 |
moreover |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
225 |
have "x \<union> y = space M - ((space M - x) \<inter> (space M - y))" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
226 |
by auto (metis subsetD lambda_system_sets sets_into_space xl yl)+ |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
227 |
ultimately show ?thesis |
38656 | 228 |
by (metis lambda_system_Compl lambda_system_Int xl yl) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
229 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
230 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
231 |
lemma (in algebra) lambda_system_algebra: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
232 |
"positive M f \<Longrightarrow> algebra (M\<lparr>sets := lambda_system M f\<rparr>)" |
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41981
diff
changeset
|
233 |
apply (auto simp add: algebra_iff_Un) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
234 |
apply (metis lambda_system_sets set_mp sets_into_space) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
235 |
apply (metis lambda_system_empty) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
236 |
apply (metis lambda_system_Compl) |
38656 | 237 |
apply (metis lambda_system_Un) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
238 |
done |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
239 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
240 |
lemma (in algebra) lambda_system_strong_additive: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
241 |
assumes z: "z \<in> sets M" and disj: "x \<inter> y = {}" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
242 |
and xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
243 |
shows "f (z \<inter> (x \<union> y)) = f (z \<inter> x) + f (z \<inter> y)" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
244 |
proof - |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
245 |
have "z \<inter> x = (z \<inter> (x \<union> y)) \<inter> x" using disj by blast |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
246 |
moreover |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
247 |
have "z \<inter> y = (z \<inter> (x \<union> y)) - x" using disj by blast |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
248 |
moreover |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
249 |
have "(z \<inter> (x \<union> y)) \<in> sets M" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
250 |
by (metis Int Un lambda_system_sets xl yl z) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
251 |
ultimately show ?thesis using xl yl |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
252 |
by (simp add: lambda_system_eq) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
253 |
qed |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
254 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
255 |
lemma (in algebra) lambda_system_additive: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
256 |
"additive (M (|sets := lambda_system M f|)) f" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
257 |
proof (auto simp add: additive_def) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
258 |
fix x and y |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
259 |
assume disj: "x \<inter> y = {}" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
260 |
and xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
261 |
hence "x \<in> sets M" "y \<in> sets M" by (blast intro: lambda_system_sets)+ |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
262 |
thus "f (x \<union> y) = f x + f y" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
263 |
using lambda_system_strong_additive [OF top disj xl yl] |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
264 |
by (simp add: Un) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
265 |
qed |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
266 |
|
42145 | 267 |
lemma (in ring_of_sets) disjointed_additive: |
268 |
assumes f: "positive M f" "additive M f" and A: "range A \<subseteq> sets M" "incseq A" |
|
269 |
shows "(\<Sum>i\<le>n. f (disjointed A i)) = f (A n)" |
|
270 |
proof (induct n) |
|
271 |
case (Suc n) |
|
272 |
then have "(\<Sum>i\<le>Suc n. f (disjointed A i)) = f (A n) + f (disjointed A (Suc n))" |
|
273 |
by simp |
|
274 |
also have "\<dots> = f (A n \<union> disjointed A (Suc n))" |
|
275 |
using A by (subst f(2)[THEN additiveD]) (auto simp: disjointed_incseq) |
|
276 |
also have "A n \<union> disjointed A (Suc n) = A (Suc n)" |
|
277 |
using `incseq A` by (auto dest: incseq_SucD simp: disjointed_incseq) |
|
278 |
finally show ?case . |
|
279 |
qed simp |
|
280 |
||
281 |
lemma (in ring_of_sets) countably_subadditive_subadditive: |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
282 |
assumes f: "positive M f" and cs: "countably_subadditive M f" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
283 |
shows "subadditive M f" |
35582 | 284 |
proof (auto simp add: subadditive_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
285 |
fix x y |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
286 |
assume x: "x \<in> sets M" and y: "y \<in> sets M" and "x \<inter> y = {}" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
287 |
hence "disjoint_family (binaryset x y)" |
35582 | 288 |
by (auto simp add: disjoint_family_on_def binaryset_def) |
289 |
hence "range (binaryset x y) \<subseteq> sets M \<longrightarrow> |
|
290 |
(\<Union>i. binaryset x y i) \<in> sets M \<longrightarrow> |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
291 |
f (\<Union>i. binaryset x y i) \<le> (\<Sum> n. f (binaryset x y n))" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
292 |
using cs by (auto simp add: countably_subadditive_def) |
35582 | 293 |
hence "{x,y,{}} \<subseteq> sets M \<longrightarrow> x \<union> y \<in> sets M \<longrightarrow> |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
294 |
f (x \<union> y) \<le> (\<Sum> n. f (binaryset x y n))" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
295 |
by (simp add: range_binaryset_eq UN_binaryset_eq) |
38656 | 296 |
thus "f (x \<union> y) \<le> f x + f y" using f x y |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
297 |
by (auto simp add: Un o_def suminf_binaryset_eq positive_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
298 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
299 |
|
42145 | 300 |
lemma (in ring_of_sets) additive_sum: |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
301 |
fixes A:: "nat \<Rightarrow> 'a set" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
302 |
assumes f: "positive M f" and ad: "additive M f" and "finite S" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
303 |
and A: "range A \<subseteq> sets M" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
304 |
and disj: "disjoint_family_on A S" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
305 |
shows "(\<Sum>i\<in>S. f (A i)) = f (\<Union>i\<in>S. A i)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
306 |
using `finite S` disj proof induct |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
307 |
case empty show ?case using f by (simp add: positive_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
308 |
next |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
309 |
case (insert s S) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
310 |
then have "A s \<inter> (\<Union>i\<in>S. A i) = {}" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
311 |
by (auto simp add: disjoint_family_on_def neq_iff) |
38656 | 312 |
moreover |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
313 |
have "A s \<in> sets M" using A by blast |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
314 |
moreover have "(\<Union>i\<in>S. A i) \<in> sets M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
315 |
using A `finite S` by auto |
38656 | 316 |
moreover |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
317 |
ultimately have "f (A s \<union> (\<Union>i\<in>S. A i)) = f (A s) + f(\<Union>i\<in>S. A i)" |
38656 | 318 |
using ad UNION_in_sets A by (auto simp add: additive_def) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
319 |
with insert show ?case using ad disjoint_family_on_mono[of S "insert s S" A] |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
320 |
by (auto simp add: additive_def subset_insertI) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
321 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
322 |
|
38656 | 323 |
lemma (in algebra) increasing_additive_bound: |
43920 | 324 |
fixes A:: "nat \<Rightarrow> 'a set" and f :: "'a set \<Rightarrow> ereal" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
325 |
assumes f: "positive M f" and ad: "additive M f" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
326 |
and inc: "increasing M f" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
327 |
and A: "range A \<subseteq> sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
328 |
and disj: "disjoint_family A" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
329 |
shows "(\<Sum>i. f (A i)) \<le> f (space M)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
330 |
proof (safe intro!: suminf_bound) |
38656 | 331 |
fix N |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
332 |
note disj_N = disjoint_family_on_mono[OF _ disj, of "{..<N}"] |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
333 |
have "(\<Sum>i<N. f (A i)) = f (\<Union>i\<in>{..<N}. A i)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
334 |
by (rule additive_sum [OF f ad _ A]) (auto simp: disj_N) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
335 |
also have "... \<le> f (space M)" using space_closed A |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
336 |
by (intro increasingD[OF inc] finite_UN) auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
337 |
finally show "(\<Sum>i<N. f (A i)) \<le> f (space M)" by simp |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
338 |
qed (insert f A, auto simp: positive_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
339 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
340 |
lemma lambda_system_increasing: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
341 |
"increasing M f \<Longrightarrow> increasing (M (|sets := lambda_system M f|)) f" |
38656 | 342 |
by (simp add: increasing_def lambda_system_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
343 |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
344 |
lemma lambda_system_positive: |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
345 |
"positive M f \<Longrightarrow> positive (M (|sets := lambda_system M f|)) f" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
346 |
by (simp add: positive_def lambda_system_def) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
347 |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
348 |
lemma (in algebra) lambda_system_strong_sum: |
43920 | 349 |
fixes A:: "nat \<Rightarrow> 'a set" and f :: "'a set \<Rightarrow> ereal" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
350 |
assumes f: "positive M f" and a: "a \<in> sets M" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
351 |
and A: "range A \<subseteq> lambda_system M f" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
352 |
and disj: "disjoint_family A" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
353 |
shows "(\<Sum>i = 0..<n. f (a \<inter>A i)) = f (a \<inter> (\<Union>i\<in>{0..<n}. A i))" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
354 |
proof (induct n) |
38656 | 355 |
case 0 show ?case using f by (simp add: positive_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
356 |
next |
38656 | 357 |
case (Suc n) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
358 |
have 2: "A n \<inter> UNION {0..<n} A = {}" using disj |
38656 | 359 |
by (force simp add: disjoint_family_on_def neq_iff) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
360 |
have 3: "A n \<in> lambda_system M f" using A |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
361 |
by blast |
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41981
diff
changeset
|
362 |
interpret l: algebra "M\<lparr>sets := lambda_system M f\<rparr>" |
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41981
diff
changeset
|
363 |
using f by (rule lambda_system_algebra) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
364 |
have 4: "UNION {0..<n} A \<in> lambda_system M f" |
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41981
diff
changeset
|
365 |
using A l.UNION_in_sets by simp |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
366 |
from Suc.hyps show ?case |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
367 |
by (simp add: atLeastLessThanSuc lambda_system_strong_additive [OF a 2 3 4]) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
368 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
369 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
370 |
lemma (in sigma_algebra) lambda_system_caratheodory: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
371 |
assumes oms: "outer_measure_space M f" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
372 |
and A: "range A \<subseteq> lambda_system M f" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
373 |
and disj: "disjoint_family A" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
374 |
shows "(\<Union>i. A i) \<in> lambda_system M f \<and> (\<Sum>i. f (A i)) = f (\<Union>i. A i)" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
375 |
proof - |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
376 |
have pos: "positive M f" and inc: "increasing M f" |
38656 | 377 |
and csa: "countably_subadditive M f" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
378 |
by (metis oms outer_measure_space_def)+ |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
379 |
have sa: "subadditive M f" |
38656 | 380 |
by (metis countably_subadditive_subadditive csa pos) |
381 |
have A': "range A \<subseteq> sets (M(|sets := lambda_system M f|))" using A |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
382 |
by simp |
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41981
diff
changeset
|
383 |
interpret ls: algebra "M\<lparr>sets := lambda_system M f\<rparr>" |
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41981
diff
changeset
|
384 |
using pos by (rule lambda_system_algebra) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
385 |
have A'': "range A \<subseteq> sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
386 |
by (metis A image_subset_iff lambda_system_sets) |
38656 | 387 |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
388 |
have U_in: "(\<Union>i. A i) \<in> sets M" |
37032 | 389 |
by (metis A'' countable_UN) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
390 |
have U_eq: "f (\<Union>i. A i) = (\<Sum>i. f (A i))" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
391 |
proof (rule antisym) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
392 |
show "f (\<Union>i. A i) \<le> (\<Sum>i. f (A i))" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
393 |
using csa[unfolded countably_subadditive_def] A'' disj U_in by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
394 |
have *: "\<And>i. 0 \<le> f (A i)" using pos A'' unfolding positive_def by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
395 |
have dis: "\<And>N. disjoint_family_on A {..<N}" by (intro disjoint_family_on_mono[OF _ disj]) auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
396 |
show "(\<Sum>i. f (A i)) \<le> f (\<Union>i. A i)" |
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41981
diff
changeset
|
397 |
using ls.additive_sum [OF lambda_system_positive[OF pos] lambda_system_additive _ A' dis] |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
398 |
using A'' |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
399 |
by (intro suminf_bound[OF _ *]) (auto intro!: increasingD[OF inc] allI countable_UN) |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
400 |
qed |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
401 |
{ |
38656 | 402 |
fix a |
403 |
assume a [iff]: "a \<in> sets M" |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
404 |
have "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) = f a" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
405 |
proof - |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
406 |
show ?thesis |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
407 |
proof (rule antisym) |
33536 | 408 |
have "range (\<lambda>i. a \<inter> A i) \<subseteq> sets M" using A'' |
409 |
by blast |
|
38656 | 410 |
moreover |
33536 | 411 |
have "disjoint_family (\<lambda>i. a \<inter> A i)" using disj |
38656 | 412 |
by (auto simp add: disjoint_family_on_def) |
413 |
moreover |
|
33536 | 414 |
have "a \<inter> (\<Union>i. A i) \<in> sets M" |
415 |
by (metis Int U_in a) |
|
38656 | 416 |
ultimately |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
417 |
have "f (a \<inter> (\<Union>i. A i)) \<le> (\<Sum>i. f (a \<inter> A i))" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
418 |
using csa[unfolded countably_subadditive_def, rule_format, of "(\<lambda>i. a \<inter> A i)"] |
38656 | 419 |
by (simp add: o_def) |
420 |
hence "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) \<le> |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
421 |
(\<Sum>i. f (a \<inter> A i)) + f (a - (\<Union>i. A i))" |
38656 | 422 |
by (rule add_right_mono) |
423 |
moreover |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
424 |
have "(\<Sum>i. f (a \<inter> A i)) + f (a - (\<Union>i. A i)) \<le> f a" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
425 |
proof (intro suminf_bound_add allI) |
33536 | 426 |
fix n |
427 |
have UNION_in: "(\<Union>i\<in>{0..<n}. A i) \<in> sets M" |
|
38656 | 428 |
by (metis A'' UNION_in_sets) |
33536 | 429 |
have le_fa: "f (UNION {0..<n} A \<inter> a) \<le> f a" using A'' |
37032 | 430 |
by (blast intro: increasingD [OF inc] A'' UNION_in_sets) |
33536 | 431 |
have ls: "(\<Union>i\<in>{0..<n}. A i) \<in> lambda_system M f" |
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41981
diff
changeset
|
432 |
using ls.UNION_in_sets by (simp add: A) |
38656 | 433 |
hence eq_fa: "f a = f (a \<inter> (\<Union>i\<in>{0..<n}. A i)) + f (a - (\<Union>i\<in>{0..<n}. A i))" |
37032 | 434 |
by (simp add: lambda_system_eq UNION_in) |
33536 | 435 |
have "f (a - (\<Union>i. A i)) \<le> f (a - (\<Union>i\<in>{0..<n}. A i))" |
44106 | 436 |
by (blast intro: increasingD [OF inc] UNION_in U_in) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
437 |
thus "(\<Sum>i<n. f (a \<inter> A i)) + f (a - (\<Union>i. A i)) \<le> f a" |
38656 | 438 |
by (simp add: lambda_system_strong_sum pos A disj eq_fa add_left_mono atLeast0LessThan[symmetric]) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
439 |
next |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
440 |
have "\<And>i. a \<inter> A i \<in> sets M" using A'' by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
441 |
then show "\<And>i. 0 \<le> f (a \<inter> A i)" using pos[unfolded positive_def] by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
442 |
have "\<And>i. a - (\<Union>i. A i) \<in> sets M" using A'' by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
443 |
then have "\<And>i. 0 \<le> f (a - (\<Union>i. A i))" using pos[unfolded positive_def] by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
444 |
then show "f (a - (\<Union>i. A i)) \<noteq> -\<infinity>" by auto |
33536 | 445 |
qed |
38656 | 446 |
ultimately show "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) \<le> f a" |
447 |
by (rule order_trans) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
448 |
next |
38656 | 449 |
have "f a \<le> f (a \<inter> (\<Union>i. A i) \<union> (a - (\<Union>i. A i)))" |
37032 | 450 |
by (blast intro: increasingD [OF inc] U_in) |
33536 | 451 |
also have "... \<le> f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i))" |
37032 | 452 |
by (blast intro: subadditiveD [OF sa] U_in) |
33536 | 453 |
finally show "f a \<le> f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i))" . |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
454 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
455 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
456 |
} |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
457 |
thus ?thesis |
38656 | 458 |
by (simp add: lambda_system_eq sums_iff U_eq U_in) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
459 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
460 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
461 |
lemma (in sigma_algebra) caratheodory_lemma: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
462 |
assumes oms: "outer_measure_space M f" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
463 |
shows "measure_space \<lparr> space = space M, sets = lambda_system M f, measure = f \<rparr>" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
464 |
(is "measure_space ?M") |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
465 |
proof - |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
466 |
have pos: "positive M f" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
467 |
by (metis oms outer_measure_space_def) |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
468 |
have alg: "algebra ?M" |
38656 | 469 |
using lambda_system_algebra [of f, OF pos] |
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41981
diff
changeset
|
470 |
by (simp add: algebra_iff_Un) |
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41981
diff
changeset
|
471 |
then |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
472 |
have "sigma_algebra ?M" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
473 |
using lambda_system_caratheodory [OF oms] |
38656 | 474 |
by (simp add: sigma_algebra_disjoint_iff) |
475 |
moreover |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
476 |
have "measure_space_axioms ?M" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
477 |
using pos lambda_system_caratheodory [OF oms] |
38656 | 478 |
by (simp add: measure_space_axioms_def positive_def lambda_system_sets |
479 |
countably_additive_def o_def) |
|
480 |
ultimately |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
481 |
show ?thesis |
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41981
diff
changeset
|
482 |
by (simp add: measure_space_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
483 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
484 |
|
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
485 |
lemma (in ring_of_sets) additive_increasing: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
486 |
assumes posf: "positive M f" and addf: "additive M f" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
487 |
shows "increasing M f" |
38656 | 488 |
proof (auto simp add: increasing_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
489 |
fix x y |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
490 |
assume xy: "x \<in> sets M" "y \<in> sets M" "x \<subseteq> y" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
491 |
then have "y - x \<in> sets M" by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
492 |
then have "0 \<le> f (y-x)" using posf[unfolded positive_def] by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
493 |
then have "f x + 0 \<le> f x + f (y-x)" by (intro add_left_mono) auto |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
494 |
also have "... = f (x \<union> (y-x))" using addf |
37032 | 495 |
by (auto simp add: additive_def) (metis Diff_disjoint Un_Diff_cancel Diff xy(1,2)) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
496 |
also have "... = f y" |
37032 | 497 |
by (metis Un_Diff_cancel Un_absorb1 xy(3)) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
498 |
finally show "f x \<le> f y" by simp |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
499 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
500 |
|
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
501 |
lemma (in ring_of_sets) countably_additive_additive: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
502 |
assumes posf: "positive M f" and ca: "countably_additive M f" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
503 |
shows "additive M f" |
38656 | 504 |
proof (auto simp add: additive_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
505 |
fix x y |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
506 |
assume x: "x \<in> sets M" and y: "y \<in> sets M" and "x \<inter> y = {}" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
507 |
hence "disjoint_family (binaryset x y)" |
38656 | 508 |
by (auto simp add: disjoint_family_on_def binaryset_def) |
509 |
hence "range (binaryset x y) \<subseteq> sets M \<longrightarrow> |
|
510 |
(\<Union>i. binaryset x y i) \<in> sets M \<longrightarrow> |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
511 |
f (\<Union>i. binaryset x y i) = (\<Sum> n. f (binaryset x y n))" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
512 |
using ca |
38656 | 513 |
by (simp add: countably_additive_def) |
514 |
hence "{x,y,{}} \<subseteq> sets M \<longrightarrow> x \<union> y \<in> sets M \<longrightarrow> |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
515 |
f (x \<union> y) = (\<Sum>n. f (binaryset x y n))" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
516 |
by (simp add: range_binaryset_eq UN_binaryset_eq) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
517 |
thus "f (x \<union> y) = f x + f y" using posf x y |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
518 |
by (auto simp add: Un suminf_binaryset_eq positive_def) |
38656 | 519 |
qed |
520 |
||
39096 | 521 |
lemma inf_measure_nonempty: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
522 |
assumes f: "positive M f" and b: "b \<in> sets M" and a: "a \<subseteq> b" "{} \<in> sets M" |
39096 | 523 |
shows "f b \<in> measure_set M f a" |
524 |
proof - |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
525 |
let ?A = "\<lambda>i::nat. (if i = 0 then b else {})" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
526 |
have "(\<Sum>i. f (?A i)) = (\<Sum>i<1::nat. f (?A i))" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
527 |
by (rule suminf_finite) (simp add: f[unfolded positive_def]) |
39096 | 528 |
also have "... = f b" |
529 |
by simp |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
530 |
finally show ?thesis using assms |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
531 |
by (auto intro!: exI [of _ ?A] |
39096 | 532 |
simp: measure_set_def disjoint_family_on_def split_if_mem2 comp_def) |
533 |
qed |
|
534 |
||
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
535 |
lemma (in ring_of_sets) inf_measure_agrees: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
536 |
assumes posf: "positive M f" and ca: "countably_additive M f" |
38656 | 537 |
and s: "s \<in> sets M" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
538 |
shows "Inf (measure_set M f s) = f s" |
43920 | 539 |
unfolding Inf_ereal_def |
38656 | 540 |
proof (safe intro!: Greatest_equality) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
541 |
fix z |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
542 |
assume z: "z \<in> measure_set M f s" |
38656 | 543 |
from this obtain A where |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
544 |
A: "range A \<subseteq> sets M" and disj: "disjoint_family A" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
545 |
and "s \<subseteq> (\<Union>x. A x)" and si: "(\<Sum>i. f (A i)) = z" |
38656 | 546 |
by (auto simp add: measure_set_def comp_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
547 |
hence seq: "s = (\<Union>i. A i \<inter> s)" by blast |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
548 |
have inc: "increasing M f" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
549 |
by (metis additive_increasing ca countably_additive_additive posf) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
550 |
have sums: "(\<Sum>i. f (A i \<inter> s)) = f (\<Union>i. A i \<inter> s)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
551 |
proof (rule ca[unfolded countably_additive_def, rule_format]) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
552 |
show "range (\<lambda>n. A n \<inter> s) \<subseteq> sets M" using A s |
33536 | 553 |
by blast |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
554 |
show "disjoint_family (\<lambda>n. A n \<inter> s)" using disj |
35582 | 555 |
by (auto simp add: disjoint_family_on_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
556 |
show "(\<Union>i. A i \<inter> s) \<in> sets M" using A s |
33536 | 557 |
by (metis UN_extend_simps(4) s seq) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
558 |
qed |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
559 |
hence "f s = (\<Sum>i. f (A i \<inter> s))" |
37032 | 560 |
using seq [symmetric] by (simp add: sums_iff) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
561 |
also have "... \<le> (\<Sum>i. f (A i))" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
562 |
proof (rule suminf_le_pos) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
563 |
fix n show "f (A n \<inter> s) \<le> f (A n)" using A s |
38656 | 564 |
by (force intro: increasingD [OF inc]) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
565 |
fix N have "A N \<inter> s \<in> sets M" using A s by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
566 |
then show "0 \<le> f (A N \<inter> s)" using posf unfolding positive_def by auto |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
567 |
qed |
38656 | 568 |
also have "... = z" by (rule si) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
569 |
finally show "f s \<le> z" . |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
570 |
next |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
571 |
fix y |
38656 | 572 |
assume y: "\<forall>u \<in> measure_set M f s. y \<le> u" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
573 |
thus "y \<le> f s" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
574 |
by (blast intro: inf_measure_nonempty [of _ f, OF posf s subset_refl]) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
575 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
576 |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
577 |
lemma measure_set_pos: |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
578 |
assumes posf: "positive M f" "r \<in> measure_set M f X" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
579 |
shows "0 \<le> r" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
580 |
proof - |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
581 |
obtain A where "range A \<subseteq> sets M" and r: "r = (\<Sum>i. f (A i))" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
582 |
using `r \<in> measure_set M f X` unfolding measure_set_def by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
583 |
then show "0 \<le> r" using posf unfolding r positive_def |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
584 |
by (intro suminf_0_le) auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
585 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
586 |
|
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
587 |
lemma inf_measure_pos: |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
588 |
assumes posf: "positive M f" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
589 |
shows "0 \<le> Inf (measure_set M f X)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
590 |
proof (rule complete_lattice_class.Inf_greatest) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
591 |
fix r assume "r \<in> measure_set M f X" with posf show "0 \<le> r" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
592 |
by (rule measure_set_pos) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
593 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
594 |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
595 |
lemma inf_measure_empty: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
596 |
assumes posf: "positive M f" and "{} \<in> sets M" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
597 |
shows "Inf (measure_set M f {}) = 0" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
598 |
proof (rule antisym) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
599 |
show "Inf (measure_set M f {}) \<le> 0" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
600 |
by (metis complete_lattice_class.Inf_lower `{} \<in> sets M` |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
601 |
inf_measure_nonempty[OF posf] subset_refl posf[unfolded positive_def]) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
602 |
qed (rule inf_measure_pos[OF posf]) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
603 |
|
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
604 |
lemma (in ring_of_sets) inf_measure_positive: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
605 |
assumes p: "positive M f" and "{} \<in> sets M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
606 |
shows "positive M (\<lambda>x. Inf (measure_set M f x))" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
607 |
proof (unfold positive_def, intro conjI ballI) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
608 |
show "Inf (measure_set M f {}) = 0" using inf_measure_empty[OF assms] by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
609 |
fix A assume "A \<in> sets M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
610 |
qed (rule inf_measure_pos[OF p]) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
611 |
|
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
612 |
lemma (in ring_of_sets) inf_measure_increasing: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
613 |
assumes posf: "positive M f" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
614 |
shows "increasing \<lparr> space = space M, sets = Pow (space M) \<rparr> |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
615 |
(\<lambda>x. Inf (measure_set M f x))" |
44918 | 616 |
apply (clarsimp simp add: increasing_def) |
38656 | 617 |
apply (rule complete_lattice_class.Inf_greatest) |
618 |
apply (rule complete_lattice_class.Inf_lower) |
|
37032 | 619 |
apply (clarsimp simp add: measure_set_def, rule_tac x=A in exI, blast) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
620 |
done |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
621 |
|
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
622 |
lemma (in ring_of_sets) inf_measure_le: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
623 |
assumes posf: "positive M f" and inc: "increasing M f" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
624 |
and x: "x \<in> {r . \<exists>A. range A \<subseteq> sets M \<and> s \<subseteq> (\<Union>i. A i) \<and> (\<Sum>i. f (A i)) = r}" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
625 |
shows "Inf (measure_set M f s) \<le> x" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
626 |
proof - |
38656 | 627 |
obtain A where A: "range A \<subseteq> sets M" and ss: "s \<subseteq> (\<Union>i. A i)" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
628 |
and xeq: "(\<Sum>i. f (A i)) = x" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
629 |
using x by auto |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
630 |
have dA: "range (disjointed A) \<subseteq> sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
631 |
by (metis A range_disjointed_sets) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
632 |
have "\<forall>n. f (disjointed A n) \<le> f (A n)" |
38656 | 633 |
by (metis increasingD [OF inc] UNIV_I dA image_subset_iff disjointed_subset A comp_def) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
634 |
moreover have "\<forall>i. 0 \<le> f (disjointed A i)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
635 |
using posf dA unfolding positive_def by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
636 |
ultimately have sda: "(\<Sum>i. f (disjointed A i)) \<le> (\<Sum>i. f (A i))" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
637 |
by (blast intro!: suminf_le_pos) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
638 |
hence ley: "(\<Sum>i. f (disjointed A i)) \<le> x" |
38656 | 639 |
by (metis xeq) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
640 |
hence y: "(\<Sum>i. f (disjointed A i)) \<in> measure_set M f s" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
641 |
apply (auto simp add: measure_set_def) |
38656 | 642 |
apply (rule_tac x="disjointed A" in exI) |
643 |
apply (simp add: disjoint_family_disjointed UN_disjointed_eq ss dA comp_def) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
644 |
done |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
645 |
show ?thesis |
38656 | 646 |
by (blast intro: y order_trans [OF _ ley] posf complete_lattice_class.Inf_lower) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
647 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
648 |
|
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
649 |
lemma (in ring_of_sets) inf_measure_close: |
43920 | 650 |
fixes e :: ereal |
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
651 |
assumes posf: "positive M f" and e: "0 < e" and ss: "s \<subseteq> (space M)" and "Inf (measure_set M f s) \<noteq> \<infinity>" |
38656 | 652 |
shows "\<exists>A. range A \<subseteq> sets M \<and> disjoint_family A \<and> s \<subseteq> (\<Union>i. A i) \<and> |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
653 |
(\<Sum>i. f (A i)) \<le> Inf (measure_set M f s) + e" |
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
654 |
proof - |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
655 |
from `Inf (measure_set M f s) \<noteq> \<infinity>` have fin: "\<bar>Inf (measure_set M f s)\<bar> \<noteq> \<infinity>" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
656 |
using inf_measure_pos[OF posf, of s] by auto |
38656 | 657 |
obtain l where "l \<in> measure_set M f s" "l \<le> Inf (measure_set M f s) + e" |
43920 | 658 |
using Inf_ereal_close[OF fin e] by auto |
38656 | 659 |
thus ?thesis |
660 |
by (auto intro!: exI[of _ l] simp: measure_set_def comp_def) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
661 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
662 |
|
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
663 |
lemma (in ring_of_sets) inf_measure_countably_subadditive: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
664 |
assumes posf: "positive M f" and inc: "increasing M f" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
665 |
shows "countably_subadditive (| space = space M, sets = Pow (space M) |) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
666 |
(\<lambda>x. Inf (measure_set M f x))" |
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
667 |
proof (simp add: countably_subadditive_def, safe) |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
668 |
fix A :: "nat \<Rightarrow> 'a set" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
669 |
let "?outer B" = "Inf (measure_set M f B)" |
38656 | 670 |
assume A: "range A \<subseteq> Pow (space M)" |
671 |
and disj: "disjoint_family A" |
|
672 |
and sb: "(\<Union>i. A i) \<subseteq> space M" |
|
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
673 |
|
43920 | 674 |
{ fix e :: ereal assume e: "0 < e" and "\<forall>i. ?outer (A i) \<noteq> \<infinity>" |
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
675 |
hence "\<exists>BB. \<forall>n. range (BB n) \<subseteq> sets M \<and> disjoint_family (BB n) \<and> |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
676 |
A n \<subseteq> (\<Union>i. BB n i) \<and> (\<Sum>i. f (BB n i)) \<le> ?outer (A n) + e * (1/2)^(Suc n)" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
677 |
apply (safe intro!: choice inf_measure_close [of f, OF posf]) |
43920 | 678 |
using e sb by (auto simp: ereal_zero_less_0_iff one_ereal_def) |
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
679 |
then obtain BB |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
680 |
where BB: "\<And>n. (range (BB n) \<subseteq> sets M)" |
38656 | 681 |
and disjBB: "\<And>n. disjoint_family (BB n)" |
682 |
and sbBB: "\<And>n. A n \<subseteq> (\<Union>i. BB n i)" |
|
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
683 |
and BBle: "\<And>n. (\<Sum>i. f (BB n i)) \<le> ?outer (A n) + e * (1/2)^(Suc n)" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
684 |
by auto blast |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
685 |
have sll: "(\<Sum>n. \<Sum>i. (f (BB n i))) \<le> (\<Sum>n. ?outer (A n)) + e" |
38656 | 686 |
proof - |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
687 |
have sum_eq_1: "(\<Sum>n. e*(1/2) ^ Suc n) = e" |
43920 | 688 |
using suminf_half_series_ereal e |
689 |
by (simp add: ereal_zero_le_0_iff zero_le_divide_ereal suminf_cmult_ereal) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
690 |
have "\<And>n i. 0 \<le> f (BB n i)" using posf[unfolded positive_def] BB by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
691 |
then have "\<And>n. 0 \<le> (\<Sum>i. f (BB n i))" by (rule suminf_0_le) |
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
692 |
then have "(\<Sum>n. \<Sum>i. (f (BB n i))) \<le> (\<Sum>n. ?outer (A n) + e*(1/2) ^ Suc n)" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
693 |
by (rule suminf_le_pos[OF BBle]) |
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
694 |
also have "... = (\<Sum>n. ?outer (A n)) + e" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
695 |
using sum_eq_1 inf_measure_pos[OF posf] e |
43920 | 696 |
by (subst suminf_add_ereal) (auto simp add: ereal_zero_le_0_iff) |
38656 | 697 |
finally show ?thesis . |
698 |
qed |
|
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
699 |
def C \<equiv> "(split BB) o prod_decode" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
700 |
have C: "!!n. C n \<in> sets M" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
701 |
apply (rule_tac p="prod_decode n" in PairE) |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
702 |
apply (simp add: C_def) |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
703 |
apply (metis BB subsetD rangeI) |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
704 |
done |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
705 |
have sbC: "(\<Union>i. A i) \<subseteq> (\<Union>i. C i)" |
38656 | 706 |
proof (auto simp add: C_def) |
707 |
fix x i |
|
708 |
assume x: "x \<in> A i" |
|
709 |
with sbBB [of i] obtain j where "x \<in> BB i j" |
|
710 |
by blast |
|
711 |
thus "\<exists>i. x \<in> split BB (prod_decode i)" |
|
712 |
by (metis prod_encode_inverse prod.cases) |
|
713 |
qed |
|
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
714 |
have "(f \<circ> C) = (f \<circ> (\<lambda>(x, y). BB x y)) \<circ> prod_decode" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
715 |
by (rule ext) (auto simp add: C_def) |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
716 |
moreover have "suminf ... = (\<Sum>n. \<Sum>i. f (BB n i))" using BBle |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
717 |
using BB posf[unfolded positive_def] |
43920 | 718 |
by (force intro!: suminf_ereal_2dimen simp: o_def) |
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
719 |
ultimately have Csums: "(\<Sum>i. f (C i)) = (\<Sum>n. \<Sum>i. f (BB n i))" by (simp add: o_def) |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
720 |
have "?outer (\<Union>i. A i) \<le> (\<Sum>n. \<Sum>i. f (BB n i))" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
721 |
apply (rule inf_measure_le [OF posf(1) inc], auto) |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
722 |
apply (rule_tac x="C" in exI) |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
723 |
apply (auto simp add: C sbC Csums) |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
724 |
done |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
725 |
also have "... \<le> (\<Sum>n. ?outer (A n)) + e" using sll |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
726 |
by blast |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
727 |
finally have "?outer (\<Union>i. A i) \<le> (\<Sum>n. ?outer (A n)) + e" . } |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
728 |
note for_finite_Inf = this |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
729 |
|
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
730 |
show "?outer (\<Union>i. A i) \<le> (\<Sum>n. ?outer (A n))" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
731 |
proof cases |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
732 |
assume "\<forall>i. ?outer (A i) \<noteq> \<infinity>" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
733 |
with for_finite_Inf show ?thesis |
43920 | 734 |
by (intro ereal_le_epsilon) auto |
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
735 |
next |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
736 |
assume "\<not> (\<forall>i. ?outer (A i) \<noteq> \<infinity>)" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
737 |
then have "\<exists>i. ?outer (A i) = \<infinity>" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
738 |
by auto |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
739 |
then have "(\<Sum>n. ?outer (A n)) = \<infinity>" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
740 |
using suminf_PInfty[OF inf_measure_pos, OF posf] |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
741 |
by metis |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
742 |
then show ?thesis by simp |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
743 |
qed |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
744 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
745 |
|
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
746 |
lemma (in ring_of_sets) inf_measure_outer: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
747 |
"\<lbrakk> positive M f ; increasing M f \<rbrakk> |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
748 |
\<Longrightarrow> outer_measure_space \<lparr> space = space M, sets = Pow (space M) \<rparr> |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
749 |
(\<lambda>x. Inf (measure_set M f x))" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
750 |
using inf_measure_pos[of M f] |
38656 | 751 |
by (simp add: outer_measure_space_def inf_measure_empty |
752 |
inf_measure_increasing inf_measure_countably_subadditive positive_def) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
753 |
|
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
754 |
lemma (in ring_of_sets) algebra_subset_lambda_system: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
755 |
assumes posf: "positive M f" and inc: "increasing M f" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
756 |
and add: "additive M f" |
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
757 |
shows "sets M \<subseteq> lambda_system \<lparr> space = space M, sets = Pow (space M) \<rparr> |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
758 |
(\<lambda>x. Inf (measure_set M f x))" |
38656 | 759 |
proof (auto dest: sets_into_space |
760 |
simp add: algebra.lambda_system_eq [OF algebra_Pow]) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
761 |
fix x s |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
762 |
assume x: "x \<in> sets M" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
763 |
and s: "s \<subseteq> space M" |
38656 | 764 |
have [simp]: "!!x. x \<in> sets M \<Longrightarrow> s \<inter> (space M - x) = s-x" using s |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
765 |
by blast |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
766 |
have "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x)) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
767 |
\<le> Inf (measure_set M f s)" |
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
768 |
proof cases |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
769 |
assume "Inf (measure_set M f s) = \<infinity>" then show ?thesis by simp |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
770 |
next |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
771 |
assume fin: "Inf (measure_set M f s) \<noteq> \<infinity>" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
772 |
then have "measure_set M f s \<noteq> {}" |
43920 | 773 |
by (auto simp: top_ereal_def) |
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
774 |
show ?thesis |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
775 |
proof (rule complete_lattice_class.Inf_greatest) |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
776 |
fix r assume "r \<in> measure_set M f s" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
777 |
then obtain A where A: "disjoint_family A" "range A \<subseteq> sets M" "s \<subseteq> (\<Union>i. A i)" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
778 |
and r: "r = (\<Sum>i. f (A i))" unfolding measure_set_def by auto |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
779 |
have "Inf (measure_set M f (s \<inter> x)) \<le> (\<Sum>i. f (A i \<inter> x))" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
780 |
unfolding measure_set_def |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
781 |
proof (safe intro!: complete_lattice_class.Inf_lower exI[of _ "\<lambda>i. A i \<inter> x"]) |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
782 |
from A(1) show "disjoint_family (\<lambda>i. A i \<inter> x)" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
783 |
by (rule disjoint_family_on_bisimulation) auto |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
784 |
qed (insert x A, auto) |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
785 |
moreover |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
786 |
have "Inf (measure_set M f (s - x)) \<le> (\<Sum>i. f (A i - x))" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
787 |
unfolding measure_set_def |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
788 |
proof (safe intro!: complete_lattice_class.Inf_lower exI[of _ "\<lambda>i. A i - x"]) |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
789 |
from A(1) show "disjoint_family (\<lambda>i. A i - x)" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
790 |
by (rule disjoint_family_on_bisimulation) auto |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
791 |
qed (insert x A, auto) |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
792 |
ultimately have "Inf (measure_set M f (s \<inter> x)) + Inf (measure_set M f (s - x)) \<le> |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
793 |
(\<Sum>i. f (A i \<inter> x)) + (\<Sum>i. f (A i - x))" by (rule add_mono) |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
794 |
also have "\<dots> = (\<Sum>i. f (A i \<inter> x) + f (A i - x))" |
43920 | 795 |
using A(2) x posf by (subst suminf_add_ereal) (auto simp: positive_def) |
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
796 |
also have "\<dots> = (\<Sum>i. f (A i))" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
797 |
using A x |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
798 |
by (subst add[THEN additiveD, symmetric]) |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
799 |
(auto intro!: arg_cong[where f=suminf] arg_cong[where f=f]) |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
800 |
finally show "Inf (measure_set M f (s \<inter> x)) + Inf (measure_set M f (s - x)) \<le> r" |
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
801 |
using r by simp |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
802 |
qed |
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
803 |
qed |
38656 | 804 |
moreover |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
805 |
have "Inf (measure_set M f s) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
806 |
\<le> Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))" |
42145 | 807 |
proof - |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
808 |
have "Inf (measure_set M f s) = Inf (measure_set M f ((s\<inter>x) \<union> (s-x)))" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
809 |
by (metis Un_Diff_Int Un_commute) |
38656 | 810 |
also have "... \<le> Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))" |
811 |
apply (rule subadditiveD) |
|
42145 | 812 |
apply (rule ring_of_sets.countably_subadditive_subadditive [OF ring_of_sets_Pow]) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41689
diff
changeset
|
813 |
apply (simp add: positive_def inf_measure_empty[OF posf] inf_measure_pos[OF posf]) |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
814 |
apply (rule inf_measure_countably_subadditive) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
815 |
using s by (auto intro!: posf inc) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
816 |
finally show ?thesis . |
42145 | 817 |
qed |
38656 | 818 |
ultimately |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
819 |
show "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x)) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
820 |
= Inf (measure_set M f s)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
821 |
by (rule order_antisym) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
822 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
823 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
824 |
lemma measure_down: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
825 |
"measure_space N \<Longrightarrow> sigma_algebra M \<Longrightarrow> sets M \<subseteq> sets N \<Longrightarrow> measure N = measure M \<Longrightarrow> measure_space M" |
38656 | 826 |
by (simp add: measure_space_def measure_space_axioms_def positive_def |
827 |
countably_additive_def) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
828 |
blast |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
829 |
|
42066
6db76c88907a
generalized Caratheodory from algebra to ring_of_sets
hoelzl
parents:
42065
diff
changeset
|
830 |
theorem (in ring_of_sets) caratheodory: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
831 |
assumes posf: "positive M f" and ca: "countably_additive M f" |
43920 | 832 |
shows "\<exists>\<mu> :: 'a set \<Rightarrow> ereal. (\<forall>s \<in> sets M. \<mu> s = f s) \<and> |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
833 |
measure_space \<lparr> space = space M, sets = sets (sigma M), measure = \<mu> \<rparr>" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
834 |
proof - |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
835 |
have inc: "increasing M f" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
836 |
by (metis additive_increasing ca countably_additive_additive posf) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
837 |
let ?infm = "(\<lambda>x. Inf (measure_set M f x))" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
838 |
def ls \<equiv> "lambda_system (|space = space M, sets = Pow (space M)|) ?infm" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
839 |
have mls: "measure_space \<lparr>space = space M, sets = ls, measure = ?infm\<rparr>" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
840 |
using sigma_algebra.caratheodory_lemma |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
841 |
[OF sigma_algebra_Pow inf_measure_outer [OF posf inc]] |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
842 |
by (simp add: ls_def) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
843 |
hence sls: "sigma_algebra (|space = space M, sets = ls, measure = ?infm|)" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
844 |
by (simp add: measure_space_def) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
845 |
have "sets M \<subseteq> ls" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
846 |
by (simp add: ls_def) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
847 |
(metis ca posf inc countably_additive_additive algebra_subset_lambda_system) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
848 |
hence sgs_sb: "sigma_sets (space M) (sets M) \<subseteq> ls" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
849 |
using sigma_algebra.sigma_sets_subset [OF sls, of "sets M"] |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
850 |
by simp |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
851 |
have "measure_space \<lparr> space = space M, sets = sets (sigma M), measure = ?infm \<rparr>" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
852 |
unfolding sigma_def |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
853 |
by (rule measure_down [OF mls], rule sigma_algebra_sigma_sets) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
854 |
(simp_all add: sgs_sb space_closed) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
855 |
thus ?thesis using inf_measure_agrees [OF posf ca] |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
856 |
by (intro exI[of _ ?infm]) auto |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41023
diff
changeset
|
857 |
qed |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
858 |
|
42145 | 859 |
subsubsection {*Alternative instances of caratheodory*} |
860 |
||
861 |
lemma (in ring_of_sets) countably_additive_iff_continuous_from_below: |
|
862 |
assumes f: "positive M f" "additive M f" |
|
863 |
shows "countably_additive M f \<longleftrightarrow> |
|
864 |
(\<forall>A. range A \<subseteq> sets M \<longrightarrow> incseq A \<longrightarrow> (\<Union>i. A i) \<in> sets M \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Union>i. A i))" |
|
865 |
unfolding countably_additive_def |
|
866 |
proof safe |
|
867 |
assume count_sum: "\<forall>A. range A \<subseteq> sets M \<longrightarrow> disjoint_family A \<longrightarrow> UNION UNIV A \<in> sets M \<longrightarrow> (\<Sum>i. f (A i)) = f (UNION UNIV A)" |
|
868 |
fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets M" "incseq A" "(\<Union>i. A i) \<in> sets M" |
|
869 |
then have dA: "range (disjointed A) \<subseteq> sets M" by (auto simp: range_disjointed_sets) |
|
870 |
with count_sum[THEN spec, of "disjointed A"] A(3) |
|
871 |
have f_UN: "(\<Sum>i. f (disjointed A i)) = f (\<Union>i. A i)" |
|
872 |
by (auto simp: UN_disjointed_eq disjoint_family_disjointed) |
|
873 |
moreover have "(\<lambda>n. (\<Sum>i=0..<n. f (disjointed A i))) ----> (\<Sum>i. f (disjointed A i))" |
|
874 |
using f(1)[unfolded positive_def] dA |
|
43920 | 875 |
by (auto intro!: summable_sumr_LIMSEQ_suminf summable_ereal_pos) |
42145 | 876 |
from LIMSEQ_Suc[OF this] |
877 |
have "(\<lambda>n. (\<Sum>i\<le>n. f (disjointed A i))) ----> (\<Sum>i. f (disjointed A i))" |
|
878 |
unfolding atLeastLessThanSuc_atLeastAtMost atLeast0AtMost . |
|
879 |
moreover have "\<And>n. (\<Sum>i\<le>n. f (disjointed A i)) = f (A n)" |
|
880 |
using disjointed_additive[OF f A(1,2)] . |
|
881 |
ultimately show "(\<lambda>i. f (A i)) ----> f (\<Union>i. A i)" by simp |
|
882 |
next |
|
883 |
assume cont: "\<forall>A. range A \<subseteq> sets M \<longrightarrow> incseq A \<longrightarrow> (\<Union>i. A i) \<in> sets M \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Union>i. A i)" |
|
884 |
fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets M" "disjoint_family A" "(\<Union>i. A i) \<in> sets M" |
|
885 |
have *: "(\<Union>n. (\<Union>i\<le>n. A i)) = (\<Union>i. A i)" by auto |
|
886 |
have "(\<lambda>n. f (\<Union>i\<le>n. A i)) ----> f (\<Union>i. A i)" |
|
887 |
proof (unfold *[symmetric], intro cont[rule_format]) |
|
888 |
show "range (\<lambda>i. \<Union> i\<le>i. A i) \<subseteq> sets M" "(\<Union>i. \<Union> i\<le>i. A i) \<in> sets M" |
|
889 |
using A * by auto |
|
890 |
qed (force intro!: incseq_SucI) |
|
891 |
moreover have "\<And>n. f (\<Union>i\<le>n. A i) = (\<Sum>i\<le>n. f (A i))" |
|
892 |
using A |
|
893 |
by (intro additive_sum[OF f, of _ A, symmetric]) |
|
894 |
(auto intro: disjoint_family_on_mono[where B=UNIV]) |
|
895 |
ultimately |
|
896 |
have "(\<lambda>i. f (A i)) sums f (\<Union>i. A i)" |
|
897 |
unfolding sums_def2 by simp |
|
898 |
from sums_unique[OF this] |
|
899 |
show "(\<Sum>i. f (A i)) = f (\<Union>i. A i)" by simp |
|
900 |
qed |
|
901 |
||
902 |
lemma (in ring_of_sets) continuous_from_above_iff_empty_continuous: |
|
903 |
assumes f: "positive M f" "additive M f" |
|
904 |
shows "(\<forall>A. range A \<subseteq> sets M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) \<in> sets M \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Inter>i. A i)) |
|
905 |
\<longleftrightarrow> (\<forall>A. range A \<subseteq> sets M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> 0)" |
|
906 |
proof safe |
|
907 |
assume cont: "(\<forall>A. range A \<subseteq> sets M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) \<in> sets M \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> f (\<Inter>i. A i))" |
|
908 |
fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets M" "decseq A" "(\<Inter>i. A i) = {}" "\<forall>i. f (A i) \<noteq> \<infinity>" |
|
909 |
with cont[THEN spec, of A] show "(\<lambda>i. f (A i)) ----> 0" |
|
910 |
using `positive M f`[unfolded positive_def] by auto |
|
911 |
next |
|
912 |
assume cont: "\<forall>A. range A \<subseteq> sets M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) ----> 0" |
|
913 |
fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets M" "decseq A" "(\<Inter>i. A i) \<in> sets M" "\<forall>i. f (A i) \<noteq> \<infinity>" |
|
914 |
||
915 |
have f_mono: "\<And>a b. a \<in> sets M \<Longrightarrow> b \<in> sets M \<Longrightarrow> a \<subseteq> b \<Longrightarrow> f a \<le> f b" |
|
916 |
using additive_increasing[OF f] unfolding increasing_def by simp |
|
917 |
||
918 |
have decseq_fA: "decseq (\<lambda>i. f (A i))" |
|
919 |
using A by (auto simp: decseq_def intro!: f_mono) |
|
920 |
have decseq: "decseq (\<lambda>i. A i - (\<Inter>i. A i))" |
|
921 |
using A by (auto simp: decseq_def) |
|
922 |
then have decseq_f: "decseq (\<lambda>i. f (A i - (\<Inter>i. A i)))" |
|
923 |
using A unfolding decseq_def by (auto intro!: f_mono Diff) |
|
924 |
have "f (\<Inter>x. A x) \<le> f (A 0)" |
|
925 |
using A by (auto intro!: f_mono) |
|
926 |
then have f_Int_fin: "f (\<Inter>x. A x) \<noteq> \<infinity>" |
|
927 |
using A by auto |
|
928 |
{ fix i |
|
929 |
have "f (A i - (\<Inter>i. A i)) \<le> f (A i)" using A by (auto intro!: f_mono) |
|
930 |
then have "f (A i - (\<Inter>i. A i)) \<noteq> \<infinity>" |
|
931 |
using A by auto } |
|
932 |
note f_fin = this |
|
933 |
have "(\<lambda>i. f (A i - (\<Inter>i. A i))) ----> 0" |
|
934 |
proof (intro cont[rule_format, OF _ decseq _ f_fin]) |
|
935 |
show "range (\<lambda>i. A i - (\<Inter>i. A i)) \<subseteq> sets M" "(\<Inter>i. A i - (\<Inter>i. A i)) = {}" |
|
936 |
using A by auto |
|
937 |
qed |
|
43920 | 938 |
from INF_Lim_ereal[OF decseq_f this] |
42145 | 939 |
have "(INF n. f (A n - (\<Inter>i. A i))) = 0" . |
940 |
moreover have "(INF n. f (\<Inter>i. A i)) = f (\<Inter>i. A i)" |
|
941 |
by auto |
|
942 |
ultimately have "(INF n. f (A n - (\<Inter>i. A i)) + f (\<Inter>i. A i)) = 0 + f (\<Inter>i. A i)" |
|
943 |
using A(4) f_fin f_Int_fin |
|
43920 | 944 |
by (subst INFI_ereal_add) (auto simp: decseq_f) |
42145 | 945 |
moreover { |
946 |
fix n |
|
947 |
have "f (A n - (\<Inter>i. A i)) + f (\<Inter>i. A i) = f ((A n - (\<Inter>i. A i)) \<union> (\<Inter>i. A i))" |
|
948 |
using A by (subst f(2)[THEN additiveD]) auto |
|
949 |
also have "(A n - (\<Inter>i. A i)) \<union> (\<Inter>i. A i) = A n" |
|
950 |
by auto |
|
951 |
finally have "f (A n - (\<Inter>i. A i)) + f (\<Inter>i. A i) = f (A n)" . } |
|
952 |
ultimately have "(INF n. f (A n)) = f (\<Inter>i. A i)" |
|
953 |
by simp |
|
43920 | 954 |
with LIMSEQ_ereal_INFI[OF decseq_fA] |
42145 | 955 |
show "(\<lambda>i. f (A i)) ----> f (\<Inter>i. A i)" by simp |
956 |
qed |
|
957 |
||
958 |
lemma positiveD1: "positive M f \<Longrightarrow> f {} = 0" by (auto simp: positive_def) |
|
959 |
lemma positiveD2: "positive M f \<Longrightarrow> A \<in> sets M \<Longrightarrow> 0 \<le> f A" by (auto simp: positive_def) |
|
960 |
||
961 |
lemma (in ring_of_sets) empty_continuous_imp_continuous_from_below: |
|
962 |
assumes f: "positive M f" "additive M f" "\<forall>A\<in>sets M. f A \<noteq> \<infinity>" |
|
963 |
assumes cont: "\<forall>A. range A \<subseteq> sets M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<lambda>i. f (A i)) ----> 0" |
|
964 |
assumes A: "range A \<subseteq> sets M" "incseq A" "(\<Union>i. A i) \<in> sets M" |
|
965 |
shows "(\<lambda>i. f (A i)) ----> f (\<Union>i. A i)" |
|
966 |
proof - |
|
43920 | 967 |
have "\<forall>A\<in>sets M. \<exists>x. f A = ereal x" |
42145 | 968 |
proof |
43920 | 969 |
fix A assume "A \<in> sets M" with f show "\<exists>x. f A = ereal x" |
42145 | 970 |
unfolding positive_def by (cases "f A") auto |
971 |
qed |
|
972 |
from bchoice[OF this] guess f' .. note f' = this[rule_format] |
|
973 |
from A have "(\<lambda>i. f ((\<Union>i. A i) - A i)) ----> 0" |
|
974 |
by (intro cont[rule_format]) (auto simp: decseq_def incseq_def) |
|
975 |
moreover |
|
976 |
{ fix i |
|
977 |
have "f ((\<Union>i. A i) - A i) + f (A i) = f ((\<Union>i. A i) - A i \<union> A i)" |
|
978 |
using A by (intro f(2)[THEN additiveD, symmetric]) auto |
|
979 |
also have "(\<Union>i. A i) - A i \<union> A i = (\<Union>i. A i)" |
|
980 |
by auto |
|
981 |
finally have "f' (\<Union>i. A i) - f' (A i) = f' ((\<Union>i. A i) - A i)" |
|
982 |
using A by (subst (asm) (1 2 3) f') auto |
|
43920 | 983 |
then have "f ((\<Union>i. A i) - A i) = ereal (f' (\<Union>i. A i) - f' (A i))" |
42145 | 984 |
using A f' by auto } |
985 |
ultimately have "(\<lambda>i. f' (\<Union>i. A i) - f' (A i)) ----> 0" |
|
43920 | 986 |
by (simp add: zero_ereal_def) |
42145 | 987 |
then have "(\<lambda>i. f' (A i)) ----> f' (\<Union>i. A i)" |
44568
e6f291cb5810
discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents:
44106
diff
changeset
|
988 |
by (rule LIMSEQ_diff_approach_zero2[OF tendsto_const]) |
42145 | 989 |
then show "(\<lambda>i. f (A i)) ----> f (\<Union>i. A i)" |
990 |
using A by (subst (1 2) f') auto |
|
991 |
qed |
|
992 |
||
993 |
lemma (in ring_of_sets) empty_continuous_imp_countably_additive: |
|
994 |
assumes f: "positive M f" "additive M f" and fin: "\<forall>A\<in>sets M. f A \<noteq> \<infinity>" |
|
995 |
assumes cont: "\<And>A. range A \<subseteq> sets M \<Longrightarrow> decseq A \<Longrightarrow> (\<Inter>i. A i) = {} \<Longrightarrow> (\<lambda>i. f (A i)) ----> 0" |
|
996 |
shows "countably_additive M f" |
|
997 |
using countably_additive_iff_continuous_from_below[OF f] |
|
998 |
using empty_continuous_imp_continuous_from_below[OF f fin] cont |
|
999 |
by blast |
|
1000 |
||
1001 |
lemma (in ring_of_sets) caratheodory_empty_continuous: |
|
1002 |
assumes f: "positive M f" "additive M f" and fin: "\<And>A. A \<in> sets M \<Longrightarrow> f A \<noteq> \<infinity>" |
|
1003 |
assumes cont: "\<And>A. range A \<subseteq> sets M \<Longrightarrow> decseq A \<Longrightarrow> (\<Inter>i. A i) = {} \<Longrightarrow> (\<lambda>i. f (A i)) ----> 0" |
|
43920 | 1004 |
shows "\<exists>\<mu> :: 'a set \<Rightarrow> ereal. (\<forall>s \<in> sets M. \<mu> s = f s) \<and> |
42145 | 1005 |
measure_space \<lparr> space = space M, sets = sets (sigma M), measure = \<mu> \<rparr>" |
1006 |
proof (intro caratheodory empty_continuous_imp_countably_additive f) |
|
1007 |
show "\<forall>A\<in>sets M. f A \<noteq> \<infinity>" using fin by auto |
|
1008 |
qed (rule cont) |
|
1009 |
||
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
1010 |
end |