src/HOL/Probability/Complete_Measure.thy
 author hoelzl Wed Dec 01 21:03:02 2010 +0100 (2010-12-01) changeset 40871 688f6ff859e1 parent 40859 de0b30e6c2d2 child 41023 9118eb4eb8dc permissions -rw-r--r--
Generalized simple_functionD and less_SUP_iff.
Moved theorems to appropriate places.
```     1 (*  Title:      Complete_Measure.thy
```
```     2     Author:     Robert Himmelmann, Johannes Hoelzl, TU Muenchen
```
```     3 *)
```
```     4 theory Complete_Measure
```
```     5 imports Product_Measure
```
```     6 begin
```
```     7
```
```     8 locale completeable_measure_space = measure_space
```
```     9
```
```    10 definition (in completeable_measure_space) completion :: "'a algebra" where
```
```    11   "completion = \<lparr> space = space M,
```
```    12     sets = { S \<union> N |S N N'. S \<in> sets M \<and> N' \<in> null_sets \<and> N \<subseteq> N' } \<rparr>"
```
```    13
```
```    14 lemma (in completeable_measure_space) space_completion[simp]:
```
```    15   "space completion = space M" unfolding completion_def by simp
```
```    16
```
```    17 lemma (in completeable_measure_space) sets_completionE:
```
```    18   assumes "A \<in> sets completion"
```
```    19   obtains S N N' where "A = S \<union> N" "N \<subseteq> N'" "N' \<in> null_sets" "S \<in> sets M"
```
```    20   using assms unfolding completion_def by auto
```
```    21
```
```    22 lemma (in completeable_measure_space) sets_completionI:
```
```    23   assumes "A = S \<union> N" "N \<subseteq> N'" "N' \<in> null_sets" "S \<in> sets M"
```
```    24   shows "A \<in> sets completion"
```
```    25   using assms unfolding completion_def by auto
```
```    26
```
```    27 lemma (in completeable_measure_space) sets_completionI_sets[intro]:
```
```    28   "A \<in> sets M \<Longrightarrow> A \<in> sets completion"
```
```    29   unfolding completion_def by force
```
```    30
```
```    31 lemma (in completeable_measure_space) null_sets_completion:
```
```    32   assumes "N' \<in> null_sets" "N \<subseteq> N'" shows "N \<in> sets completion"
```
```    33   apply(rule sets_completionI[of N "{}" N N'])
```
```    34   using assms by auto
```
```    35
```
```    36 sublocale completeable_measure_space \<subseteq> completion!: sigma_algebra completion
```
```    37 proof (unfold sigma_algebra_iff2, safe)
```
```    38   fix A x assume "A \<in> sets completion" "x \<in> A"
```
```    39   with sets_into_space show "x \<in> space completion"
```
```    40     by (auto elim!: sets_completionE)
```
```    41 next
```
```    42   fix A assume "A \<in> sets completion"
```
```    43   from this[THEN sets_completionE] guess S N N' . note A = this
```
```    44   let ?C = "space completion"
```
```    45   show "?C - A \<in> sets completion" using A
```
```    46     by (intro sets_completionI[of _ "(?C - S) \<inter> (?C - N')" "(?C - S) \<inter> N' \<inter> (?C - N)"])
```
```    47        auto
```
```    48 next
```
```    49   fix A ::"nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets completion"
```
```    50   then have "\<forall>n. \<exists>S N N'. A n = S \<union> N \<and> S \<in> sets M \<and> N' \<in> null_sets \<and> N \<subseteq> N'"
```
```    51     unfolding completion_def by (auto simp: image_subset_iff)
```
```    52   from choice[OF this] guess S ..
```
```    53   from choice[OF this] guess N ..
```
```    54   from choice[OF this] guess N' ..
```
```    55   then show "UNION UNIV A \<in> sets completion"
```
```    56     using null_sets_UN[of N']
```
```    57     by (intro sets_completionI[of _ "UNION UNIV S" "UNION UNIV N" "UNION UNIV N'"])
```
```    58        auto
```
```    59 qed auto
```
```    60
```
```    61 definition (in completeable_measure_space)
```
```    62   "split_completion A p = (\<exists>N'. A = fst p \<union> snd p \<and> fst p \<inter> snd p = {} \<and>
```
```    63     fst p \<in> sets M \<and> snd p \<subseteq> N' \<and> N' \<in> null_sets)"
```
```    64
```
```    65 definition (in completeable_measure_space)
```
```    66   "main_part A = fst (Eps (split_completion A))"
```
```    67
```
```    68 definition (in completeable_measure_space)
```
```    69   "null_part A = snd (Eps (split_completion A))"
```
```    70
```
```    71 lemma (in completeable_measure_space) split_completion:
```
```    72   assumes "A \<in> sets completion"
```
```    73   shows "split_completion A (main_part A, null_part A)"
```
```    74   unfolding main_part_def null_part_def
```
```    75 proof (rule someI2_ex)
```
```    76   from assms[THEN sets_completionE] guess S N N' . note A = this
```
```    77   let ?P = "(S, N - S)"
```
```    78   show "\<exists>p. split_completion A p"
```
```    79     unfolding split_completion_def using A
```
```    80   proof (intro exI conjI)
```
```    81     show "A = fst ?P \<union> snd ?P" using A by auto
```
```    82     show "snd ?P \<subseteq> N'" using A by auto
```
```    83   qed auto
```
```    84 qed auto
```
```    85
```
```    86 lemma (in completeable_measure_space)
```
```    87   assumes "S \<in> sets completion"
```
```    88   shows main_part_sets[intro, simp]: "main_part S \<in> sets M"
```
```    89     and main_part_null_part_Un[simp]: "main_part S \<union> null_part S = S"
```
```    90     and main_part_null_part_Int[simp]: "main_part S \<inter> null_part S = {}"
```
```    91   using split_completion[OF assms] by (auto simp: split_completion_def)
```
```    92
```
```    93 lemma (in completeable_measure_space) null_part:
```
```    94   assumes "S \<in> sets completion" shows "\<exists>N. N\<in>null_sets \<and> null_part S \<subseteq> N"
```
```    95   using split_completion[OF assms] by (auto simp: split_completion_def)
```
```    96
```
```    97 lemma (in completeable_measure_space) null_part_sets[intro, simp]:
```
```    98   assumes "S \<in> sets M" shows "null_part S \<in> sets M" "\<mu> (null_part S) = 0"
```
```    99 proof -
```
```   100   have S: "S \<in> sets completion" using assms by auto
```
```   101   have "S - main_part S \<in> sets M" using assms by auto
```
```   102   moreover
```
```   103   from main_part_null_part_Un[OF S] main_part_null_part_Int[OF S]
```
```   104   have "S - main_part S = null_part S" by auto
```
```   105   ultimately show sets: "null_part S \<in> sets M" by auto
```
```   106   from null_part[OF S] guess N ..
```
```   107   with measure_eq_0[of N "null_part S"] sets
```
```   108   show "\<mu> (null_part S) = 0" by auto
```
```   109 qed
```
```   110
```
```   111 definition (in completeable_measure_space) "\<mu>' A = \<mu> (main_part A)"
```
```   112
```
```   113 lemma (in completeable_measure_space) \<mu>'_set[simp]:
```
```   114   assumes "S \<in> sets M" shows "\<mu>' S = \<mu> S"
```
```   115 proof -
```
```   116   have S: "S \<in> sets completion" using assms by auto
```
```   117   then have "\<mu> S = \<mu> (main_part S \<union> null_part S)" by simp
```
```   118   also have "\<dots> = \<mu> (main_part S)"
```
```   119     using S assms measure_additive[of "main_part S" "null_part S"]
```
```   120     by (auto simp: measure_additive)
```
```   121   finally show ?thesis unfolding \<mu>'_def by simp
```
```   122 qed
```
```   123
```
```   124 lemma (in completeable_measure_space) sets_completionI_sub:
```
```   125   assumes N: "N' \<in> null_sets" "N \<subseteq> N'"
```
```   126   shows "N \<in> sets completion"
```
```   127   using assms by (intro sets_completionI[of _ "{}" N N']) auto
```
```   128
```
```   129 lemma (in completeable_measure_space) \<mu>_main_part_UN:
```
```   130   fixes S :: "nat \<Rightarrow> 'a set"
```
```   131   assumes "range S \<subseteq> sets completion"
```
```   132   shows "\<mu>' (\<Union>i. (S i)) = \<mu> (\<Union>i. main_part (S i))"
```
```   133 proof -
```
```   134   have S: "\<And>i. S i \<in> sets completion" using assms by auto
```
```   135   then have UN: "(\<Union>i. S i) \<in> sets completion" by auto
```
```   136   have "\<forall>i. \<exists>N. N \<in> null_sets \<and> null_part (S i) \<subseteq> N"
```
```   137     using null_part[OF S] by auto
```
```   138   from choice[OF this] guess N .. note N = this
```
```   139   then have UN_N: "(\<Union>i. N i) \<in> null_sets" by (intro null_sets_UN) auto
```
```   140   have "(\<Union>i. S i) \<in> sets completion" using S by auto
```
```   141   from null_part[OF this] guess N' .. note N' = this
```
```   142   let ?N = "(\<Union>i. N i) \<union> N'"
```
```   143   have null_set: "?N \<in> null_sets" using N' UN_N by (intro null_sets_Un) auto
```
```   144   have "main_part (\<Union>i. S i) \<union> ?N = (main_part (\<Union>i. S i) \<union> null_part (\<Union>i. S i)) \<union> ?N"
```
```   145     using N' by auto
```
```   146   also have "\<dots> = (\<Union>i. main_part (S i) \<union> null_part (S i)) \<union> ?N"
```
```   147     unfolding main_part_null_part_Un[OF S] main_part_null_part_Un[OF UN] by auto
```
```   148   also have "\<dots> = (\<Union>i. main_part (S i)) \<union> ?N"
```
```   149     using N by auto
```
```   150   finally have *: "main_part (\<Union>i. S i) \<union> ?N = (\<Union>i. main_part (S i)) \<union> ?N" .
```
```   151   have "\<mu> (main_part (\<Union>i. S i)) = \<mu> (main_part (\<Union>i. S i) \<union> ?N)"
```
```   152     using null_set UN by (intro measure_Un_null_set[symmetric]) auto
```
```   153   also have "\<dots> = \<mu> ((\<Union>i. main_part (S i)) \<union> ?N)"
```
```   154     unfolding * ..
```
```   155   also have "\<dots> = \<mu> (\<Union>i. main_part (S i))"
```
```   156     using null_set S by (intro measure_Un_null_set) auto
```
```   157   finally show ?thesis unfolding \<mu>'_def .
```
```   158 qed
```
```   159
```
```   160 lemma (in completeable_measure_space) \<mu>_main_part_Un:
```
```   161   assumes S: "S \<in> sets completion" and T: "T \<in> sets completion"
```
```   162   shows "\<mu>' (S \<union> T) = \<mu> (main_part S \<union> main_part T)"
```
```   163 proof -
```
```   164   have UN: "(\<Union>i. binary (main_part S) (main_part T) i) = (\<Union>i. main_part (binary S T i))"
```
```   165     unfolding binary_def by (auto split: split_if_asm)
```
```   166   show ?thesis
```
```   167     using \<mu>_main_part_UN[of "binary S T"] assms
```
```   168     unfolding range_binary_eq Un_range_binary UN by auto
```
```   169 qed
```
```   170
```
```   171 sublocale completeable_measure_space \<subseteq> completion!: measure_space completion \<mu>'
```
```   172 proof
```
```   173   show "\<mu>' {} = 0" by auto
```
```   174 next
```
```   175   show "countably_additive completion \<mu>'"
```
```   176   proof (unfold countably_additive_def, intro allI conjI impI)
```
```   177     fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets completion" "disjoint_family A"
```
```   178     have "disjoint_family (\<lambda>i. main_part (A i))"
```
```   179     proof (intro disjoint_family_on_bisimulation[OF A(2)])
```
```   180       fix n m assume "A n \<inter> A m = {}"
```
```   181       then have "(main_part (A n) \<union> null_part (A n)) \<inter> (main_part (A m) \<union> null_part (A m)) = {}"
```
```   182         using A by (subst (1 2) main_part_null_part_Un) auto
```
```   183       then show "main_part (A n) \<inter> main_part (A m) = {}" by auto
```
```   184     qed
```
```   185     then have "(\<Sum>\<^isub>\<infinity>n. \<mu>' (A n)) = \<mu> (\<Union>i. main_part (A i))"
```
```   186       unfolding \<mu>'_def using A by (intro measure_countably_additive) auto
```
```   187     then show "(\<Sum>\<^isub>\<infinity>n. \<mu>' (A n)) = \<mu>' (UNION UNIV A)"
```
```   188       unfolding \<mu>_main_part_UN[OF A(1)] .
```
```   189   qed
```
```   190 qed
```
```   191
```
```   192 lemma (in completeable_measure_space) completion_ex_simple_function:
```
```   193   assumes f: "completion.simple_function f"
```
```   194   shows "\<exists>f'. simple_function f' \<and> (AE x. f x = f' x)"
```
```   195 proof -
```
```   196   let "?F x" = "f -` {x} \<inter> space M"
```
```   197   have F: "\<And>x. ?F x \<in> sets completion" and fin: "finite (f`space M)"
```
```   198     using completion.simple_functionD[OF f]
```
```   199       completion.simple_functionD[OF f] by simp_all
```
```   200   have "\<forall>x. \<exists>N. N \<in> null_sets \<and> null_part (?F x) \<subseteq> N"
```
```   201     using F null_part by auto
```
```   202   from choice[OF this] obtain N where
```
```   203     N: "\<And>x. null_part (?F x) \<subseteq> N x" "\<And>x. N x \<in> null_sets" by auto
```
```   204   let ?N = "\<Union>x\<in>f`space M. N x" let "?f' x" = "if x \<in> ?N then undefined else f x"
```
```   205   have sets: "?N \<in> null_sets" using N fin by (intro null_sets_finite_UN) auto
```
```   206   show ?thesis unfolding simple_function_def
```
```   207   proof (safe intro!: exI[of _ ?f'])
```
```   208     have "?f' ` space M \<subseteq> f`space M \<union> {undefined}" by auto
```
```   209     from finite_subset[OF this] completion.simple_functionD(1)[OF f]
```
```   210     show "finite (?f' ` space M)" by auto
```
```   211   next
```
```   212     fix x assume "x \<in> space M"
```
```   213     have "?f' -` {?f' x} \<inter> space M =
```
```   214       (if x \<in> ?N then ?F undefined \<union> ?N
```
```   215        else if f x = undefined then ?F (f x) \<union> ?N
```
```   216        else ?F (f x) - ?N)"
```
```   217       using N(2) sets_into_space by (auto split: split_if_asm)
```
```   218     moreover { fix y have "?F y \<union> ?N \<in> sets M"
```
```   219       proof cases
```
```   220         assume y: "y \<in> f`space M"
```
```   221         have "?F y \<union> ?N = (main_part (?F y) \<union> null_part (?F y)) \<union> ?N"
```
```   222           using main_part_null_part_Un[OF F] by auto
```
```   223         also have "\<dots> = main_part (?F y) \<union> ?N"
```
```   224           using y N by auto
```
```   225         finally show ?thesis
```
```   226           using F sets by auto
```
```   227       next
```
```   228         assume "y \<notin> f`space M" then have "?F y = {}" by auto
```
```   229         then show ?thesis using sets by auto
```
```   230       qed }
```
```   231     moreover {
```
```   232       have "?F (f x) - ?N = main_part (?F (f x)) \<union> null_part (?F (f x)) - ?N"
```
```   233         using main_part_null_part_Un[OF F] by auto
```
```   234       also have "\<dots> = main_part (?F (f x)) - ?N"
```
```   235         using N `x \<in> space M` by auto
```
```   236       finally have "?F (f x) - ?N \<in> sets M"
```
```   237         using F sets by auto }
```
```   238     ultimately show "?f' -` {?f' x} \<inter> space M \<in> sets M" by auto
```
```   239   next
```
```   240     show "AE x. f x = ?f' x"
```
```   241       by (rule AE_I', rule sets) auto
```
```   242   qed
```
```   243 qed
```
```   244
```
```   245 lemma (in completeable_measure_space) completion_ex_borel_measurable:
```
```   246   fixes g :: "'a \<Rightarrow> pinfreal"
```
```   247   assumes g: "g \<in> borel_measurable completion"
```
```   248   shows "\<exists>g'\<in>borel_measurable M. (AE x. g x = g' x)"
```
```   249 proof -
```
```   250   from g[THEN completion.borel_measurable_implies_simple_function_sequence]
```
```   251   obtain f where "\<And>i. completion.simple_function (f i)" "f \<up> g" by auto
```
```   252   then have "\<forall>i. \<exists>f'. simple_function f' \<and> (AE x. f i x = f' x)"
```
```   253     using completion_ex_simple_function by auto
```
```   254   from this[THEN choice] obtain f' where
```
```   255     sf: "\<And>i. simple_function (f' i)" and
```
```   256     AE: "\<forall>i. AE x. f i x = f' i x" by auto
```
```   257   show ?thesis
```
```   258   proof (intro bexI)
```
```   259     from AE[unfolded all_AE_countable]
```
```   260     show "AE x. g x = (SUP i. f' i) x" (is "AE x. g x = ?f x")
```
```   261     proof (rule AE_mp, safe intro!: AE_cong)
```
```   262       fix x assume eq: "\<forall>i. f i x = f' i x"
```
```   263       have "g x = (SUP i. f i x)"
```
```   264         using `f \<up> g` unfolding isoton_def SUPR_fun_expand by auto
```
```   265       then show "g x = ?f x"
```
```   266         using eq unfolding SUPR_fun_expand by auto
```
```   267     qed
```
```   268     show "?f \<in> borel_measurable M"
```
```   269       using sf by (auto intro!: borel_measurable_SUP
```
```   270         intro: borel_measurable_simple_function)
```
```   271   qed
```
```   272 qed
```
```   273
```
```   274 end
```